hcju/isabelleps huggingface dataset:
log2_le: fixes x :: "nat" assumes "0 < x" shows "2 ^ log2 x \<le> x"
assms: shows "nat_to_bv (bv_to_nat w) = norm_unsigned w"
union_inter: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a measure" assumes "finite A"   and "finite B" shows "sup (sup_measure.F g (A \<union> B)) (sup_measure.F g (A \<inter> B)) = sup (sup_measure.F g A) (sup_measure.F g B)"
absc_valid_cfg_eq: shows "absc ` (cfg_on_div s \<inter> {cfg. alternating cfg}) = R_G_cfg_on_div s' \<inter> {cfg. alternating cfg}"
blinding_respects_hashes_def: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "blinding_respects_hashes h bo \<equiv> bo \<le> vimage2p h h (=)"
cut: fixes H :: "fm set"   and A :: "fm"   and H' :: "fm set"   and B :: "fm" assumes "H \<turnstile> A"   and "insert A H' \<turnstile> B" shows "H \<union> H' \<turnstile> B"
lipschitz_on_def: fixes C :: "real"   and U :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "C-lipschitz_on U f = (0 \<le> C \<and> (\<forall>x\<in>U. \<forall>y\<in>U. dist (f x) (f y) \<le> C * dist x y))"
BEGIN_before: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "BEGIN i \<lless> BEGIN j = ((i, j) \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>)"
U_def: fixes s :: "'s"   and t :: "'s" shows "U s t = T.prob s {\<omega> \<in> space (T s). ev (HLD {t}) \<omega>}"
absorb2: fixes b :: "'a upred"   and a :: "'a upred" assumes "a \<sqsubseteq> b" shows "(a \<and> b) = b"
heap_read_total: fixes h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and T :: "ty" assumes "hconf h"   and "P,h \<turnstile> a@al : T" shows "\<exists>v. heap_read h a al v \<and> P,h \<turnstile> v :\<le> T"
atom_map_bot_pres: shows "atom_map \<bottom> = {}"
id_closes: fixes \<Gamma> :: "(name \<times> ty) list" shows "Lam_ml.id \<Gamma> closes \<Gamma>"
mor_def: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "('f, 'g, 'h) F1 \<Rightarrow> 'g"   and s2' :: "('f, 'g, 'h) F2 \<Rightarrow> 'h"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h" shows "mor B1 B2 s1 s2 B1' B2' s1' s2' f g = (((\<forall>a\<in>B1. f a \<in> B1') \<and> (\<forall>a\<in>B2. g a \<in> B2')) \<and> (\<forall>z\<in>F1in UNIV B1 B2. f (s1 z) = s1' (F1map id f g z)) \<and> (\<forall>z\<in>F2in UNIV B1 B2. g (s2 z) = s2' (F2map id f g z)))"
outside_convex: fixes S :: "'a set" assumes "convex S" shows "outside S = - S"
step: fixes N :: "'s stream measure"   and t :: "'s" assumes "\<exists>s. N = stream_space (count_space UNIV) \<and> t = s" shows "Measurable.pred N (A t)"
assms_3: fixes x :: "'var" assumes "x \<in> S" shows "\<Gamma> (Var x) = Var a"
length_filter_mset_sorted_nth: fixes xs :: "'a list"   and n :: "nat" assumes "distinct xs"   and "n < length xs"   and "sorted xs" shows "{#x \<in># mset xs. x \<le> xs ! n#} = mset (take (n + 1) xs)"
not_eq_zero: shows "vec_to_iarray A !! ([mod_type_class.to_nat i..<IArray.length (vec_to_iarray A)] ! ia) \<noteq> (0::'a)"
sym_simps_2: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "local.dom \
[a, b] = a \<otimes> b"
case_prodE2: fixes Q :: "'g \<Rightarrow> bool"   and P :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and z :: "'h \<times> 'i" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'h" and y :: "'i" where "z = (x, y)" and "Q (P x y)"
mset_left_cancel_elem: fixes a :: "'f"   and b :: "'f"   and A :: "'f multiset" assumes "a \<in># {#b#} + A"   and "a \<noteq> b" shows "a \<in># A"
fold_deriv_FAnd: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and xs :: "'x list"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" shows "fold (deriv d0) xs (FAnd \<phi> \<psi>) = FAnd (fold (deriv d0) xs \<phi>) (fold (deriv d0) xs \<psi>)"
prefixes_conv_suffixes: fixes xs :: "'b list" shows "prefixes xs = map rev (suffixes (rev xs))"
lspace_iff': fixes p :: "ennreal"   and S :: "'a set" assumes "p < \<infinity>"   and "0 < p"   and "S \<in> sets lebesgue" shows "lspace (lebesgue_on S) p = {f \<in> borel_measurable (lebesgue_on S). (\<lambda>x. norm (f x) powr enn2real p) integrable_on S}"
r_result1_bivalent': fixes i :: "nat"   and x :: "nat"   and v :: "nat"   and t :: "nat" assumes "eval r_phi [i, x] \<down>= v" shows "eval r_result1 [t, i, x] \<down>= Suc v \<or> eval r_result1 [t, i, x] \<down>= 0"
simps_5: fixes e1 :: "expr"   and e2 :: "expr" shows "radicals (Multiplication e1 e2) = radicals e1 \<union> radicals e2"
ring_zero: fixes R :: "('e, 'f) Ring_scheme" assumes "Ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
guard_elims_2: fixes b :: "bool" assumes "guard b = None" obtains "\<not> b"
min_rvar_incdec_eq_None': fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and eq :: "nat \<times> linear_poly"   and "is" :: "'i list"   and I :: "'i set"   and v :: "nat \<Rightarrow> 'a" assumes "dir = Positive \<or> dir = Negative"   and "min_rvar_incdec_eq dir s eq = Inl is"   and "I = set is"   and "(I, v) \<Turnstile>\<^sub>i\<^sub>b \<B>\<I> s" shows "lt dir rhs eq \<lbrace> v \<rbrace> rhs eq \<lbrace> \<langle>\<V> s\<rangle> \<rbrace> \<or> rhs eq \<lbrace> v \<rbrace> = rhs eq \<lbrace> \<langle>\<V> s\<rangle> \<rbrace>"
collapse_RBT_set_def: fixes r :: "'b set_rbt"   and M :: "'b set" shows "collapse_RBT_set r M = set (RBT_Set2.keys r) \<union> M"
fp_cop_F_closed_inv': fixes x :: "'x"   and ds :: "'d set"   and c :: "'x" assumes "x \<in> fp_cop_F ds"   and "c \<in> above (Pd (Xd x)) x" shows "c \<in> fp_cop_F ds"
pred_def: fixes i :: "'a" shows "pred i \<equiv> SOME j. j \<in> Field r \<and> local.aboveS j \<noteq> {} \<and> succ j = i"
insertion_altdef: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly" shows "insertion f p = (\<Sum>m. MPoly_Type.coeff p m * (\<Prod>i. f i ^ lookup m i))"
a2: shows "I \<subseteq> {sa. (\<exists>i\<in>I. i \<rightarrow>\<^sub>i* sa) \<and> sa \<in> EF (- s)}"
tiny_dghm_is_tm_dghm: shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a" assumes "bounded_quasi_semilattice f top bot normalize" shows "bounded_quasi_semilattice_set f top bot normalize"
CallObj: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and M :: "char list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s,b\<rangle> \<rightarrow> \<langle>e'\<bullet>M(es),s',b'\<rangle>"
xivp2: shows "X \<in> Sols (\<lambda>t. f) (\<lambda>s. Collect ((\<le>) 0)) UNIV 0 s"
locate_locate_pred_unique: fixes al :: "'a list"   and f :: "'a \<Rightarrow> nat"   and i :: "nat"   and n_j :: "nat \<times> nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "locate_pred f al i n_j" shows "n_j = locate f al i"
override_on_bot_1: fixes m :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "\<bottom> ++\<^bsub>S\<^esub> m = m f|` S"
lookup_pair_update_by_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k1 :: "'a"   and v :: "'c"   and k2 :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (update_by_pair x (k1, v) xs) k2 = (if k1 = k2 then v else lookup_pair x xs k2)"
Step: fixes x :: "name"   and xs :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "\<lparr>\<nu>*(x # xs)\<rparr>B = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xs\<rparr>B)"
mpoly_to_nested_poly_eq_zero_iff: fixes mp :: "'b mpoly"   and v :: "nat" shows "(mpoly_to_nested_poly mp v = 0) = (mp = 0)"
list_augment_def: fixes xs :: "'a list"   and k :: "nat"   and v :: "'a" shows "list_augment xs k v = (list_pad_out xs k)[k := v]"
index_map_mat_2: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c mat" shows "dim_row (map_mat f A) = dim_row A"
guar_def: fixes X :: "'b program set"   and Y :: "'b program set" shows "X guarantees Y \<equiv> {F. \<forall>G. F ok G \<longrightarrow> F \<squnion> G \<in> X \<longrightarrow> F \<squnion> G \<in> Y}"
triangulize_closed: fixes A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "local.triangulize A = (r', A')"   and "A \<in> carrier_mat m n" shows "A' \<in> carrier_mat m n"
distance_direct: fixes s :: "'v"   and u :: "'v" shows "\<delta> s u \<le> w (s, u)"
orbit_def: fixes x :: "Vertex" shows "tetrahedral.orbit x = {y. \<exists>g\<in>carrier tetrahedral_group. y = apply_rotation g x}"
ap_some_option: fixes f :: "('b \<Rightarrow> 'a) option"   and x :: "'b" shows "Applicative.ap f (pure x) = map_option (\<lambda>g. g x) f"
mk_eqcl_snd_mon: fixes zs :: "nat list"   and i :: "nat"   and j1 :: "nat"   and j2 :: "nat"   and xs :: "nat option list"   and T :: "bool list list" assumes "\<And>j1 j2. \<lbrakk>j1 < j2; j2 < length zs\<rbrakk> \<Longrightarrow> zs ! j1 < zs ! j2"   and "\<And>x. x \<in> set zs \<Longrightarrow> x < i"   and "j1 < j2"   and "j2 < length (snd (mk_eqcl xs zs i T))" shows "snd (mk_eqcl xs zs i T) ! j1 < snd (mk_eqcl xs zs i T) ! j2"
atMost_shift: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {..n} = g 0 + (\<Sum>i<n. g (Suc i))"
add_mset_not_empty: fixes a :: "'d"   and A :: "'d multiset" shows "add_mset a A \<noteq> {#}"
rank_take_cols_leq: fixes R :: "'b mat"   and n :: "nat"   and nc :: "nat"   and ls :: "nat list" assumes "R \<in> carrier_mat n nc" shows "vec_space.rank n (take_cols R ls) \<le> vec_space.rank n R"
find_closest_bf_dist: fixes ps :: "(int \<times> int) list"   and p :: "int \<times> int" shows "\<forall>q\<in>set ps. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case find_closest_bf p ps of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case q of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"
in_Fvars_psubstD: fixes y :: "'var"   and \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "y \<in> Fvars (psubst \<phi> txs)"   and "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "y \<in> Fvars \<phi> - snd ` set txs \<union> \<Union> {if x \<in> Fvars \<phi> then FvarsT t else {} |t x. (t, x) \<in> set txs}"
wfzf_is_Elem_of: shows "wfzf is_Elem_of"
less_list_of: fixes K :: "nat set"   and j :: "nat"   and i :: "nat"   and l :: "nat" assumes "K \<in> \<K> j i"   and "\<forall>j\<in>K. j < l"   and "i \<le> j"   and "j \<le> l" shows "BB j i K < list_of (b l (j, i))"
int_pow_of_inv: fixes a :: "'a"   and n :: "int" assumes "a \<in> Units G" shows "inv a [^] n = inv (a [^] n)"
inject_1: fixes x11 :: "nat"   and x12 :: "tf"   and x13 :: "t list"   and x14 :: "b_e list"   and y11 :: "nat"   and y12 :: "tf"   and y13 :: "t list"   and y14 :: "b_e list" shows "(Func_native x11 x12 x13 x14 = Func_native y11 y12 y13 y14) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13 \<and> x14 = y14)"
span_def: fixes S :: "'a Matrix.vec set" shows "NC.span S = {NC.lincomb a A |a A. finite A \<and> A \<subseteq> S}"
winning_region_extends_VVp: fixes v :: "'a"   and p :: "Player"   and w :: "'a" assumes "v \<in> G'.VV p"   and "v \<rightarrow>\<^bsub>G'\<^esub> w"   and "w \<in> G'.winning_region p" shows "v \<in> G'.winning_region p"
simplices_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "order.greater_eq Y (f \<turnstile> X)"
simps_8: fixes \<phi> :: "'a fm" shows "depth (AllQ \<phi>) = depth \<phi> + 1"
assms_3: shows "LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3"
is_span: shows "C.span g.cod.leg0 g.cod.leg1"
renaming_cl_def: fixes C :: "'b eclause"   and D :: "'b eclause" shows "renaming_cl C D = (\<exists>\<eta>. renaming \<eta> (vars_of_cl (cl_ecl C)) \<and> D = subst_ecl C \<eta>)"
hsub: fixes s :: "'s" shows "nearly_healthy (wlp (p (f s)))"
le_pp: fixes x :: "'a"   and y :: "'a" assumes "rep_nat_pp x \<le> rep_nat_pp y" shows "x \<le> y"
c_snd_of_c_pair: fixes x :: "nat"   and y :: "nat" shows "c_snd (c_pair x y) = y"
utp_theory_rel_axioms: shows "utp_theory_rel \<H>"
simps_2: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and p :: "'d"   and Tl :: "'c trm list" shows "substA \<pi> (Pr p Tl) = Pr p (map (subst \<pi>) Tl)"
prems: shows "set ((ps, a) # branch) \<subseteq> set branch'"
cases: fixes x :: "'a tree \<times> nat \<times> nat" obtains t :: "'a tree" and n :: "nat" and dl :: "nat" where "x = (t, n, dl)"
inf_truncate_append: fixes z :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "inf_truncate z i @ [z (Suc i)] = inf_truncate z (Suc i)"
enum_less: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "enum i < enum j"
assms_1: shows "h \<turnstile> remove_child parent child \<rightarrow>\<^sub>h h'"
quasinorm_intersection_1: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm"   and f :: "'b" shows "eNorm (N1 \<inter>\<^sub>N N2) f = eNorm N1 f + eNorm N2 f"
my_locale1_def: fixes A :: "bool" shows "my_locale1 A \<equiv> A"
mark_out_def: fixes n :: "nat"   and bs :: "bool list" shows "mark_out n bs = map (\<lambda>(q, b). b \<and> \<not> Suc n dvd Suc (Suc q)) (enumerate n bs)"
Erdos_Rado_sunflower: fixes F :: "'b set set"   and k :: "nat"   and r :: "nat" assumes "\<forall>A\<in>F. finite A \<and> card A \<le> k"   and "(r - 1) ^ k * fact k < card F" shows "\<exists>S\<subseteq>F. sunflower S \<and> card S = r"
that: fixes G :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "G \<subseteq> dgrad_p_set d m"   and "finite G"   and "is_reduced_GB G"   and "pmdl G = pmdl F" shows "thesis"
setdist_ltE: fixes S :: "'c set"   and T :: "'c set"   and b :: "real" assumes "setdist S T < b"   and "S \<noteq> {}"   and "T \<noteq> {}" obtains x :: "'c" and y :: "'c" where "x \<in> S" and "y \<in> T" and "dist x y < b"
assms_1: fixes f :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F" shows "homogeneous f"
simps: fixes a1 :: "ord0"   and a2 :: "ord0" shows "((a1, a2) \<in> ord0_leq) = (\<exists>x y. a1 = x \<and> a2 = y \<and> (\<forall>a. (a, x) \<in> ord0_prec\<^sup>+ \<longrightarrow> (\<exists>b. (b, y) \<in> ord0_prec\<^sup>+ \<and> (a, b) \<in> ord0_leq)))"
pdom_res_alt_def: fixes A :: "'a set"   and f :: "('a, 'b) pfun" shows "A \<lhd>\<^sub>p f = f \<circ>\<^sub>p pId_on A"
time_delayed_sub: fixes sub :: "'b run"   and r :: "'b run"   and a :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and ms :: "clock"   and b :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> a time-delayed by \<delta>\<tau> on ms implies b \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> a time-delayed by \<delta>\<tau> on ms implies b \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
quasi_isometry_on_euclidean: fixes lambda :: "real"   and C :: "real"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry f" shows "DIM('b) \<le> DIM('c)"
HMA_V_def: shows "HMA_V = (\<lambda>v w. v = from_hma\<^sub>v w)"
divide_divide_eq_left': fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (divide a b) c = divide a (times c b)"
rdomp_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> assn"   and a :: "'e" shows "rdomp R a \<equiv> \<exists>h c. h \<Turnstile> R a c"
Tag_Aenc: fixes X :: "tag"   and X' :: "msg"   and Y' :: "msg" shows "Tag X \<noteq> Aenc X' Y'"
wf_dom_w: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w" shows "Sorted_Less.sorted w"
assms_5: shows "f (real_of_float u) \<le> real_of_float (ub prec u)"
cSup_eq_non_empty: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and a :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x a"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> less_eq x y) \<Longrightarrow> less_eq a y" shows "Sup X = a"
associated_eqI: fixes a :: "'a"   and b :: "'a" assumes "a dvd b"   and "b dvd a"   and "normalize a = a"   and "normalize b = b" shows "a = b"
map_upd_nonempty: fixes t :: "'a \<Rightarrow> 'b option"   and k :: "'a"   and x :: "'b" shows "t(k \<mapsto> x) \<noteq> Map.empty"
hyps: fixes b :: "(symbol \<times> character list) list" assumes "is_prefix a b" shows "is_prefix (chars a) (chars b)"
assert_bound_nolhs_unsat: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "index_valid as s"   and "\<U> (assert_bound a s)" shows "minimal_unsat_state_core (assert_bound a s)"
R_D_1: fixes Aa :: "'a stream measure"   and Ba :: "'a stream measure" assumes "R Aa Ba" shows "prob_space Aa"
rev_is_Nil_conv: fixes xs :: "'b list" shows "(rev xs = []) = (xs = [])"
k_invariant_def: fixes A :: "('ma, 'mt) AlgState" shows "k_invariant A \<equiv> (\<forall>ec ec'. k_isNode ec \<and> k_isNode ec' \<and> simAbs ec' = simAbs ec \<longrightarrow> lookup aOps (aActs A) ec = lookup aOps (aActs A) ec') \<and> (\<forall>ec ec' obs. k_isNode ec \<and> k_isNode ec' \<and> simAbs ec' = simAbs ec \<longrightarrow> lookup tOps (aTrans A) (ec, obs) = lookup tOps (aTrans A) (ec', obs)) \<and> (\<forall>ec. k_isNode ec \<and> k_memb ec A \<longrightarrow> (\<exists>acts. lookup aOps (aActs A) ec = Some acts \<and> set acts = set (simAction a ec))) \<and> (\<forall>ec obs. k_isNode ec \<and> k_memb ec A \<and> obs \<in> simObs a ` set (k_succs ec) \<longrightarrow> (\<exists>ec'. lookup tOps (aTrans A) (ec, obs) = Some ec' \<and> simAbs ec' \<in> simAbs ` set (k_succs ec) \<and> simObs a ec' = obs))"
well_analyzed_singleton_4: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "well_analyzed [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t]"
distinct_concat: fixes xs :: "'a list list" assumes "distinct xs"   and "\<And>ys. ys \<in> set xs \<Longrightarrow> distinct ys"   and "\<And>ys zs. \<lbrakk>ys \<in> set xs; zs \<in> set xs; ys \<noteq> zs\<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}" shows "distinct (concat xs)"
set_maps: fixes f :: "'c \<Rightarrow> 'b list"   and a :: "'c list" shows "set (List.maps f a) = (\<Union>a\<in>set a. set (f a))"
that_1: fixes b :: "ennreal" assumes "0 \<le> a"   and "a = enn2ereal b" shows "thesis"
diff_fun_space_times: fixes f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "f \<in> dest.diff_fun_space"   and "g \<in> dest.diff_fun_space" shows "f * g \<in> dest.diff_fun_space"
independent_insert: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "vector_space scale" shows "(\<not> module.dependent scale (insert a S)) = (if a \<in> S then \<not> module.dependent scale S else \<not> module.dependent scale S \<and> a \<notin> module.span scale S)"
af_interchange: fixes f :: "('e \<Rightarrow> 'd) af"   and x :: "'e" shows "f \<diamondop> pure x = pure (\<lambda>g. g x) \<diamondop> f"
diff_conv_add_uminus: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and uminus :: "'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g" assumes "class.group_add minus plus zero uminus" shows "minus a b = plus a (uminus b)"
horner_schema': fixes a :: "nat \<Rightarrow> real"   and x :: "real"   and n :: "nat" shows "a 0 - x * (\<Sum>i = 0..<n. (- 1) ^ i * a (Suc i) * x ^ i) = (\<Sum>i = 0..<Suc n. (- 1) ^ i * a i * x ^ i)"
type_definition_nat_term_order: shows "type_definition nat_term_compare Abs_nat_term_order {cmp. comparator cmp \<and> nat_term_comp cmp}"
csk1: shows "cs\<^bsup>\<rho>\<^esup> k = cs\<^bsup>\<pi>\<^esup> k"
Var\<^sub>0_def: fixes n :: "'a" shows "Var\<^sub>0 n \<equiv> Poly_Mapping.single (Poly_Mapping.single n 1) (1::'b)"
expand_set: fixes g :: "'b \<Rightarrow> 'a" shows "Sum_any g = sum g {a. g a \<noteq> (0::'a)}"
fundchamber_vertex_stabilizer2: fixes s :: "'a permutation"   and w :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "w \<in> \<langle>(\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) - {s}\<rangle>" shows "w \<rightarrow> fundantivertex s = fundantivertex s"
lu_props_6: fixes x :: "real" assumes "u \<le> x" shows "sgn (poly p x) = sgn (poly p u)"
subset_iprev_closed: fixes n :: "nat"   and B :: "nat set"   and A :: "nat set" assumes "n \<in> B"   and "A \<subseteq> B" shows "iprev n A \<in> B"
no_grey_refs_not_grey_reachableD: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state"   and x :: "'d" assumes "no_grey_refs s" shows "\<not> grey_reachable x s"
sup_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "sup_prod x y = (fst x \<squnion> fst y, snd x \<squnion> snd y)"
idr: shows "(card {x. poly p x = 0} = degree p) = distinct as"
\<phi>_def: fixes i :: "nat" shows "\<phi> i = (g ^^ i) (Suc n)"
Inf_nres_def: fixes X :: "'z nres set" shows "Inf X \<equiv> if \<exists>x. RES x \<in> X then RES (\<Inter> {x. RES x \<in> X}) else FAILi"
extremum_uniqueI: fixes a :: "'b" assumes "a \<le> \<bottom>" shows "a = \<bottom>"
assms_6: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> T" shows "orthogonal a b"
measurable_restrict: fixes I :: "'c set"   and X :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and N :: "'d measure"   and M :: "'c \<Rightarrow> 'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> N \<rightarrow>\<^sub>M M i" shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
traces_prefixclosed_def: fixes ES :: "'b ES_rec" shows "traces_prefixclosed ES \<equiv> prefixclosed Tr\<^bsub>ES\<^esub>"
alphaInput: fixes c :: "name"   and I :: "('a, 'b, 'c) input"   and x :: "name" assumes "c \<sharp> I" shows "\<nu>xI = \<nu>c([(x, c)] \<bullet> I)"
inject_20: fixes x201 :: "char list"   and x202 :: "'a exp"   and x203 :: "char list list"   and x204 :: "'a exp"   and y201 :: "char list"   and y202 :: "'a exp"   and y203 :: "char list list"   and y204 :: "'a exp" shows "(RI (x201,x202) ; x203 \<leftarrow> x204 = RI (y201,y202) ; y203 \<leftarrow> y204) = (x201 = y201 \<and> x202 = y202 \<and> x203 = y203 \<and> x204 = y204)"
H_1: shows "Cauchy x \<and> Cauchy y \<and> (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0"
help_InitConf: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'g, 'h) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates A = {}" shows "{p. fst p \<noteq> InitState SA \<and> snd p \<noteq> InitState SA \<and> p \<in> insert (InitState SA) (HAInitStates A) \<times> insert (InitState SA) (HAInitStates A) \<and> (p \<in> {S} \<times> States SA \<or> p \<in> ChiRel A)} = HAInitStates A \<times> HAInitStates A \<inter> ChiRel A"
hom_add: fixes c :: "'a"   and x :: "'a"   and y :: "'a" shows "c * (x + y) = c * x + c * y"
bidirected_digraph_def: fixes G :: "('e, 'f) pre_digraph"   and arev :: "'f \<Rightarrow> 'f" shows "bidirected_digraph G arev \<equiv> wf_digraph G \<and> bidirected_digraph_axioms G arev"