hcju/isabelleps huggingface dataset:
differentiable_on_open_Union: fixes S :: "'b set set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>s. s \<in> S \<Longrightarrow> f differentiable_on s"   and "\<And>s. s \<in> S \<Longrightarrow> open s" shows "f differentiable_on \<Union> S"
subg_lcs: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "H \<subseteq> K"   and "x \<in> K" shows "x \<diamondsuit>\<^bsub>\<natural>K\<^esub> H = x \<diamondsuit> H"
asymp_equiv_powr_real: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and y :: "real" assumes "f \<sim>[F] g"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x" shows "(\<lambda>x. f x powr y) \<sim>[F] (\<lambda>x. g x powr y)"
match_2: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a" assumes "\<nu> \<star> \<mu> \<star> \<tau> \<noteq> null"   and "\<mu> \<star> \<tau> \<noteq> null" shows "\<nu> \<star> \<mu> \<noteq> null"
kuratowski_planar_def: fixes G :: "('b, 'c) pre_digraph" shows "kuratowski_planar G \<equiv> \<nexists>H. subgraph H G \<and> (\<exists>K rev_K rev_H. subdivision (K, rev_K) (H, rev_H) \<and> (K\<^bsub>3,3\<^esub> K \<or> K\<^bsub>5\<^esub> K))"
bal_list_split: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ (a, b) # rs) t)" shows "bal_up\<^sub>i (Up\<^sub>i (Node ls a) b (Node rs t))"
IH_2: fixes k :: "('fun, 'var) Term.term" assumes "k \<in> set K"   and "M \<subseteq> M'" shows "intruder_deduct M' k"
fx_def: shows "fx = fcompose x1 [:a, b:] [:1, 1:]"
periodic_simps_8: fixes nc :: "nat"   and x :: "real" shows "pbernpoly nc (x - 1) = pbernpoly nc x"
is_generic_def: fixes p :: "'a point"   and q :: "'a point" shows "is_generic p q = (p \<noteq> Infinity \<and> q \<noteq> Infinity \<and> p \<noteq> q \<and> p \<noteq> opp q)"
ncols_diagonal_to_Smith_aux: fixes A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "ncols (diagonal_to_Smith_aux A xs bezout) = ncols A"
bind_lunit: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a option" shows "Some x \<bind> f = f x"
le_FD_adm: fixes u :: "'a \<Rightarrow> 'b process"   and v :: "'a \<Rightarrow> 'b process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<sqsubseteq>\<^sub>F\<^sub>D v x)"
disj_conj_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<^bold>\<sqinter> (y \<^bold>\<squnion> z) = x \<^bold>\<sqinter> y \<^bold>\<squnion> x \<^bold>\<sqinter> z"
insertTauAssertion: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" shows "insertAssertion (extractFrame (\<tau>.P)) \<Psi> \<simeq>\<^sub>F \<langle>\<epsilon>, \<Psi>\<rangle>"
rel_interior_aff_dim: fixes S :: "'a set" assumes "convex S" shows "aff_dim (rel_interior S) = aff_dim S"
g: shows "(\<lambda>u'. - (w \<bullet>c u') / (u' \<bullet>c u') \<cdot>\<^sub>v u') \<in> set us \<rightarrow> carrier_vec n"
degree_slist_eq_zeroD: fixes xs :: "(nat, 'd) slist" assumes "degree_slist xs = 0" shows "degree (Pdevs xs) = 0"
bisim1ALengthNull: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list" shows "P,a\<bullet>length,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, length (compE2 a), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
K2_acc: fixes x :: "'s"   and y :: "'s option" shows "((Some x, y) \<in> K2.acc) = (\<exists>z. y = Some z \<and> (x, z) \<in> acc)"
intros_6: fixes s :: "'d"   and t :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> Normal t"
prod_coprime_right: fixes A :: "'b set"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>i. i \<in> A \<Longrightarrow> coprime a (f i)" shows "coprime a (prod f A)"
IH: fixes y :: "nat"   and ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" assumes "y < a"   and "enat y < llength E"   and "action_obs E y = NormalAction (ReadMem ad al v)" shows "\<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T"
dom_comp: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "D.dom (g \<cdot>\<^sub>D f) = D.dom f"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'a list" shows "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)"
finite_abs_int_segment: fixes a :: "'a" shows "finite {k \<in> \<int>. \<bar>k\<bar> \<le> a}"
rcs_subset_elem: fixes H :: "'a set"   and a :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> H \<bullet> a" shows "x \<in> carrier G"
cis_minus_pi_half: shows "cis (- (pi / 2)) = - \<i>"
simps_2: fixes i :: "nat" shows "local.goodstein (Suc i) = decode (i + 3) 0 (encode (i + 2) 0 (local.goodstein i)) - 1"
invpst_combine: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "invpst l"   and "invpst r" shows "invpst (combine l r)"
g: shows "g \<in> L \<rightarrow>\<^sub>M subprob_algebra M"
rep_eq: fixes x :: "'b \<Longrightarrow> 'a"   and xa :: "'a upred" shows "\<lbrakk>\<exists> x \<bullet> xa\<rbrakk>\<^sub>e = (\<lambda>b. \<exists>v. \<lbrakk>xa\<rbrakk>\<^sub>e (put\<^bsub>x\<^esub> b v))"
deriv_def: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and ABI :: "((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)" shows "deriv G C m l ABI = (let (A, B, I) = ABI in G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I)"
sdom_neq: fixes n2 :: "nat"   and n1 :: "nat" assumes "reachable n2"   and "strict_dominate n1 n2" shows "n1 \<noteq> n2"
similar_mat_witD_4: fixes n :: "nat"   and A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "A \<in> carrier_mat n n"
imonad1: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'a dres" shows "dRETURN u \<bind> f = f u"
has_visible_field: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F,b:T in D" shows "P \<turnstile> C has F,b:T in D"
match_final: fixes i :: "'index"   and s1 :: "'state1"   and s2 :: "'state2" assumes "match i s1 s2"   and "final2 s2" shows "final1 s1"
map_relation_def: fixes f :: "'b \<Rightarrow> 'c"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "map_relation f R = (\<lambda>x y. R (f x) (f y))"
derf: fixes x :: "'n" assumes "x \<in> S" shows "(f has_derivative f' x) (at x)"
tauEq: fixes P :: "pi"   and P' :: "pi" shows "P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P' = P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'"
not_ss: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>" shows "\<not> {(x, ix), (y, iy)} \<subseteq> b \<times> I"
big_1_mult': fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "h \<in> O[F](\<lambda>_. 1::'b)"   and "f \<in> O[F](g)" shows "(\<lambda>x. f x * h x) \<in> O[F](g)"
scaleRat_right_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "rat"   and x :: "'a"   and y :: "'a" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat a (plus x y) = plus (scaleRat a x) (scaleRat a y)"
rtrancl_beta_sred: fixes r :: "lam"   and r' :: "lam" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* r'" shows "r \<rightarrow>\<^sub>s r'"
left_unit: fixes a :: "'a" assumes "a \<in> M" shows "\<one> \<cdot> a = a"
commutative: fixes x :: "'b"   and y :: "'b" assumes "x \<in> B"   and "y \<in> B" shows "addB x y = addB y x"
rel_fundef_arities: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and gd1 :: "'a fundef"   and gd2 :: "'b fundef" assumes "rel_fundef r gd1 gd2" shows "arity gd1 = arity gd2"
cap: shows "\<forall>e. (0::'capacity) \<le> f e \<and> f e \<le> c e"
labelAtIn_def: fixes v :: "'form \<times> nat list"   and p :: "('form, 'var) antecedent" shows "labelAtIn v p = subst (inst v) (freshen (vidx v) (labelsIn (nodeOf v) p))"
set_intros_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 \<in> set (x21 # x22)"
not_LCons_le_LNil: fixes a :: "'a"   and l :: "'a llist" shows "\<not> a ## l \<le> LNil"
IF: fixes \<sigma> :: "'\<sigma>" assumes "I {} \<sigma>" shows "P \<sigma>"
vcomp_assoc: fixes r :: "V"   and s :: "V"   and t :: "V" shows "(r \<circ>\<^sub>\<circ> s) \<circ>\<^sub>\<circ> t = r \<circ>\<^sub>\<circ> s \<circ>\<^sub>\<circ> t"
chars_append: fixes a :: "(symbol \<times> character list) list"   and b :: "(symbol \<times> character list) list" shows "chars (a @ b) = chars a @ chars b"
swaprows_carrier: fixes k :: "nat"   and l :: "nat"   and A :: "'c mat"   and n :: "nat"   and nc :: "nat" shows "(swaprows k l A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
simps_4: fixes x41 :: "typ"   and x42 :: "Core.term"   and y41 :: "typ"   and y42 :: "Core.term" shows "(Abs x41 x42 = Abs y41 y42) = (x41 = y41 \<and> x42 = y42)"
cBallI: fixes A :: "'f cset"   and P :: "'f \<Rightarrow> bool" assumes "\<And>x. cin x A \<Longrightarrow> P x" shows "cBall A P"
perm_xss: shows "perm_restrict f (\<Union> (sset xss)) permutes \<Union> (sset xss)"
Q'_init_eq: shows "Q' initQ init\<pi> r = (\<lambda>u. if (u, r) \<in> edges rg then enat (w {u, r}) else \<infinity>)"
S\<^sub>0_in_hom: shows "Maps.in_hom S\<^sub>0 (Maps.MkIde (src (tab\<^sub>0 s))) (Maps.MkIde (src s))"
Kruskal_interface_axioms: shows "Kruskal_interface E V vertices joins forest connected"
gtarget_alt_def: fixes r :: "(nat \<times> 'a) list"   and v :: "nat \<times> 'a" shows "gtarget r v = last (v # r)"
F2_def: shows "F2 = Subx.Fenv.from_list xs"
run_state_bind: fixes x :: "('b, 'c) state"   and f :: "'c \<Rightarrow> ('b, 'a) state"   and s :: "'b" shows "run_state (x \<bind> f) s = (case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
starlike_convex_tweak_boundary_points: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "S \<noteq> {}"   and "rel_interior S \<subseteq> T"   and "T \<subseteq> closure S" shows "starlike T"
Cons_1: fixes a :: "'a list" assumes "x \<notin> set a" shows "list_diff a b' = list_diff a (removeAll x b')"
vars_minus_Var_subset: fixes p' :: "'a mpoly"   and x :: "nat"   and \<V> :: "nat set" assumes "vars (p' - Var x) \<subseteq> \<V>" shows "vars p' \<subseteq> insert x \<V>"
read_only_def: fixes \<S> :: "nat \<Rightarrow> bool option" shows "read_only \<S> \<equiv> {a. \<S> a = Some False}"
Suc_7: shows "x \<in> rvars (\<T> (l ! j))"
Un_absorb1: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A \<union> B = B"
transpose_matrix: fixes A :: "'b matrix"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (transpose_matrix A) j i = Rep_matrix A i j"
C_def: shows "C = A1 * Q1 @\<^sub>r P1 * A2 * Q1"
set_spmf_bind_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "set_spmf (p \<bind> f) = set_pmf p \<bind> set_spmf \<circ> f"
update_trace_def: fixes k :: "'a"   and v :: "'c" shows "update_trace k v = State (\<lambda>(log, m). case run_state (update k v) m of (x, m) \<Rightarrow> ((), (''Stored'', k) # log, m))"
map_of_filter_in: fixes xs :: "('b \<times> 'a) list"   and k :: "'b"   and z :: "'a"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "table_of xs k = Some z"   and "P k z" shows "table_of (filter (\<lambda>(x, y). P x y) xs) k = Some z"
oldest_token_always_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "\<exists>i\<le>x. semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) n = Some i"
roots_mem_iff_is_root: fixes p :: "'a list list"   and x :: "'a list" assumes "p \<in> carrier (poly_ring (poly_ring R))" shows "(x \<in># UP.roots p) = UP.is_root p x"
relates_def: fixes \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a stream" shows "relates \<phi> s = \<phi> (shd s) (shd (stl s))"
a_def: shows "a = exp (fds_nth f (Suc 0))"
cong: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l"   and x :: "'k"   and y :: "'k" assumes "f = g"   and "x = y" shows "f x = g y"
finite_ReZ_segments_subpath: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and u :: "real"   and v :: "real" assumes "finite_ReZ_segments g z"   and "0 \<le> u"   and "u \<le> v"   and "v \<le> 1" shows "finite_ReZ_segments (subpath u v g) z"
LTLGlobal: shows "closed (\<G>\<^sub>F\<^sub>G \<phi> w) w"
nth_root_nat_aux2_1: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "finite {m. m ^ k \<le> n}"
simps_1: fixes b :: "bool"   and m :: "'b \<Rightarrow> 'b trie option" shows "Trie_Fun.insert [] (Nd b m) = Nd True m"
acc_bound: shows "\<Union> (range acc) \<subseteq> {0..<num_acc}"
not_eq_extremum: fixes a :: "'b" shows "(a \<noteq> top) = (a < top)"
pred_map: fixes Q :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d list" shows "list_all Q (map f x) = list_all (Q \<circ> f) x"
liftn_0: fixes t :: "dB"   and k :: "nat" shows "liftn 0 t k = t"
compose_Id: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "g \<in> A \<rightarrow> B"   and "g \<in> extensional A" shows "compose A g (\<lambda>x\<in>A. x) = g"
suminf_nonneg: assumes "summable f"   and "\<And>n. (0::'a) \<le> f n" shows "(0::'a) \<le> suminf f"
gcd_dvd1: fixes a :: "'c"   and b :: "'c" shows "normalization_euclidean_semiring_class.gcd a b dvd a"
basis_exists: fixes V :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "V \<subseteq> cotangent_space p" obtains B :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set" where "B \<subseteq> cotangent_space p" and "B \<subseteq> V" and "independent B" and "V \<subseteq> span B" and "card B = cotangent_space.dim p V"
quasi_isometry_between_def: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c" shows "lambda C -quasi_isometry_between X Y f = (lambda C -quasi_isometry_on X f \<and> f ` X \<subseteq> Y \<and> (\<forall>y\<in>Y. \<exists>x\<in>X. dist (f x) y \<le> C))"
pwd_spy_iv_def: fixes n :: "nat" shows "pwd_spy_iv n \<equiv> let (s, S, A, B, C, D) = pwd_spy_con n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, Token n (Auth_PriK n) B C SK) s, S, A, B, C, D)"
evaluate_total: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" shows "\<exists>s' r. BigStep.evaluate True env s e (s', r)"
left_lang_def: fixes q :: "hf" shows "left_lang q \<equiv> {u. nextl (dfa.init M) u = q}"
size1_size: fixes t :: "'c tree" shows "size1 t = size t + 1"
fminus_fsubset: fixes A :: "'c fset"   and B :: "'c fset" shows "A |-| B |\<subseteq>| A"
Cauchy_theorem_convex: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and K :: "complex set"   and g :: "real \<Rightarrow> complex" assumes "continuous_on S f"   and "convex S"   and "finite K"   and "\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x"   and "valid_path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g" shows "(f has_contour_integral 0) g"
trans_def: fixes r :: "('k \<times> 'k) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
\<theta>_pos: fixes x :: "real" assumes "2 \<le> x" shows "0 < \<theta> x"
cases: fixes x :: "'d ifex" obtains "x = Trueif"   | "x = Falseif"   | v :: "'d" and t :: "'d ifex" and e :: "'d ifex" where "x = IF v t e"
fst_add: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "fst (x + y) = fst x + fst y"
member_le_sum_list: fixes x :: "'b multiset"   and xs :: "'b multiset list" assumes "x \<in> set xs" shows "x \<subseteq># monoid_add.sum_list (+) {#} xs"
assms: shows "heuristicPicker n L F = Some (var, step)"
cr_uint64_def: shows "cr_uint64 \<equiv> \<lambda>x y. x = Rep_uint64 y"
is_generalisation_of_subsumes_original: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat"   and c :: "nat \<Rightarrow>f value option" assumes "is_generalisation_of t' t i r"   and "c $ r = None" shows "subsumes t' c t"
simps_1: fixes k :: "nat"   and \<tau> :: "ty" shows "subst k \<tau> TNat = TNat"
that: fixes u :: "'a list"   and a :: "'a" assumes "w = u @- sconst a" shows "thesis"
all_total_def: fixes F :: "'d program" shows "all_total F \<equiv> \<forall>act\<in>Acts F. Domain act = UNIV"
primitive_extractor_correct_1: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr"   and \<gamma> :: "('b \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and a :: "action"   and p :: "'d" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "(matches \<gamma> (alist_and (NegPos_map C as)) a p \<and> matches \<gamma> ms a p) = matches \<gamma> m a p"
hyps_free_vertices_distinct': fixes v :: "'vertex"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "terminal_path v v' pth" shows "distinct (v # map fst (map snd pth))"
Union_insert: fixes a :: "'c set"   and B :: "'c set set" shows "\<Union> (insert a B) = a \<union> \<Union> B"
dist_nz: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open" shows "(x \<noteq> y) = (0 < dist x y)"
CondTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and n :: "nat"   and t :: "('d, 'f) xstate"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
Entry_target: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and a :: "'f" assumes "CFG sourcenode targetnode valid_edge Entry"   and "valid_edge a"   and "targetnode a = Entry" shows "False"
horizontal_chop_leq1: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<longrightarrow> u \<le> v"
delete_index_altdef: fixes n :: "nat"   and xs :: "'b list" shows "delete_index n xs = take n xs @ drop (Suc n) xs"
seq_def: fixes P :: "'b process"   and Q :: "'b process" shows "P `;` Q \<equiv> Abs_process ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)"
token_succeeds_def: fixes x :: "nat" shows "token_succeeds x = (\<exists>n. token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
funion_finsert_right: fixes A :: "'a fset"   and a :: "'a"   and B :: "'a fset" shows "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
InvariantConflictClauseCharacterizationAfterNotifyWatches: fixes state :: "State"   and M :: "(Literal \<times> bool) list"   and literal :: "Literal"   and decision :: "bool"   and Wl :: "nat list"   and newWl :: "nat list" assumes "getM state = M @ [(opposite literal, decision)]"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"   and "uniq Wl" shows "let state' = notifyWatches_loop literal Wl newWl state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "hom.in_hom f a' a"   and "hom'.iso (trg f \<star> f \<star> e)" shows "hom.iso f"
g_def: shows "g = (\<lambda>w. if w = z then c else f w * (w - z))"
strip_eq_While: fixes c :: "'c acom"   and b :: "bexp"   and c1 :: "com" shows "(strip c = WHILE b DO c1) = (\<exists>I d1 P. c = {I} WHILE b DO d1 {P} \<and> strip d1 = c1)"
dimXAB: shows "\<forall>n. B - X n \<in> carrier_mat m m \<and> B - A \<in> carrier_mat m m"
mergesort_by_rel_merge_simps_2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "mergesort_by_rel_merge R xs [] = xs"
new: shows "new \<in> DeclConcepts.imethds G I sig"
bexI': fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
chine_composite_in_hom: shows "\<guillemotleft>gf.chine : g.leg0 \<down>\<down> f.leg1 \<rightarrow>\<^sub>C g.cod.leg0 \<down>\<down> f.cod.leg1\<guillemotright>"
times_sat_def: fixes x :: "'b sat"   and y :: "'b sat" shows "x * y = Abs_sat' (nat_of x * nat_of y)"
vf: shows "verticesFrom f v = v # tl (verticesFrom f v)"
fst_splitE: fixes s' :: "'b \<times> 'c"   and x' :: "'b" assumes "fst s' = x'" obtains x :: "'b" and s :: "'c" where "s' = (x, s)" and "x = x'"
left_neutral: fixes a :: "'a upred" shows "(true \<and> a) = a"
RSpan_contains_RSpans_Cons_left: fixes ms :: "'m list"   and m :: "'m" assumes "set ms \<subseteq> M" shows "RSpan [m] \<subseteq> RSpan (m # ms)"
Px: shows "distributed M S X (\<lambda>x. ennreal (Px x))"