hcju/isabelleps huggingface dataset:
bdd_below_def: fixes A :: "'a multiset set" shows "subset_mset.bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<subseteq># x)"
card_roots_unity: fixes n :: "nat" assumes "1 \<le> n" shows "card {z. z ^ n = (1::'b)} \<le> n"
aezfun_decomp_aezdeltafun: fixes c :: "('b, 'c) aezfun" shows "\<exists>ras. set (map snd ras) = supp (aezfun c) \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
privateKey_neq_symKey: fixes K :: "nat"   and b :: "bool"   and A :: "agent" assumes "K \<in> symKeys" shows "invKey (publicKey b A) \<noteq> K"
distinct_17: fixes x3 :: "bool"   and x5 :: "'a" shows "Bool x3 \<noteq> Addr x5"
integrable_in_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "subalgebra M F"   and "f \<in> borel_measurable F"   and "integrable M f" shows "integrable (restr_to_subalg M F) f"
dvd_add_times_triv_right_iff: fixes 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b (times c a)) = dvd.dvd times a b"
res_sub_eq: fixes x :: "int"   and y :: "int" shows "x mod m \<ominus> y mod m = (x mod m - y mod m) mod m"
coplanar_5_bis: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and \<alpha> :: "Points"   and \<gamma> :: "Points" assumes "rk {A, B, C} = 3"   and "rk {B, C, \<alpha>} = 2"   and "rk {A, B, \<gamma>} = 2" shows "rk {A, B, C, \<alpha>, \<gamma>} = 3"
K_cfg_same_loc_iff: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "(\<forall>cfg'\<in>set_pmf (K_cfg cfg). fst (state cfg') = fst (state cfg)) = (\<forall>cfg'\<in>set_pmf (K_cfg (absc cfg)). fst (state cfg') = fst (state (absc cfg)))"
gen_dfs_post_imp_eq: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and V0 :: "'a set"   and u0 :: "'a"   and V :: "'a set" assumes "gen_dfs_pre E U {} V0 u0"   and "gen_dfs_post E U {} V0 u0 V False" shows "V = V0 \<union> E\<^sup>* `` {u0}"
step_2: assumes "0 < n" shows "harm n < ln (real (2 * n + 1))"
lmult_0: fixes R :: "('c, 'd) ring_scheme"   and M :: "('c, 'e, 'f) module_scheme"   and m :: "'e" assumes "module R M"   and "m \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> m = \<zero>\<^bsub>M\<^esub>"
lim_1_over_Ln: shows "(\<lambda>n. 1 / Ln (of_nat n)) \<longlonglongrightarrow> 0"
pos_img: shows "\<forall>p\<in>path_image g. 0 \<le> Re p"
match_rews_4: fixes a :: "'r"   and tree1 :: "'r tree"   and tree2 :: "'r tree"   and k :: "'s match" assumes "a \<noteq> \<bottom>" shows "match_Null\<cdot>(Node\<cdot>a\<cdot>tree1\<cdot>tree2)\<cdot>k = Fixrec.fail"
assms_5: shows "has_least_prefixpoint (h \<circ> g)"
ivl_cong: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "sum g {a..<b} = sum h {c..<d}"
assms: fixes x :: "'a" shows "return (f x) \<le> F x"
e_lam_intro: fixes v :: "val"   and f :: "(val \<times> val) fset"   and e :: "exp"   and x :: "nat"   and \<rho> :: "(nat \<times> val) list" assumes "v = VFun f"   and "\<forall>v1 v2. (v1, v2) \<in> fset f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)" shows "v \<in> E (ELam x e) \<rho>"
LL_def: shows "LL = ((1, 1), 0, 1)"
impl_OrdSum_def: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "a l\<rightarrow> b \<equiv> if second a = (1::'c) \<and> second b = (1::'c) then Abs_OrdSum (first a l\<rightarrow> first b, 1::'c) else if first a = (1::'b) \<and> first b = (1::'b) then Abs_OrdSum (1::'b, second a l\<rightarrow> second b) else if first a = (1::'b) \<and> second b = (1::'c) then b else 1"
finite_minimal_antichain: fixes A :: "'b set" assumes "finite A" shows "finite (minimal_antichain A)"
et_1: fixes v :: "'a"   and e :: "'a"   and t :: "'a" assumes "v \<sqinter> top = v"   and "e \<le> v \<sqinter> - id v"   and "t \<le> v \<sqinter> id v" shows "e \<sqinter> t = bot"
simps_1: fixes v :: "nat" shows "find_lucky_eq v [] = None"
pivot_fun_mono_strict: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i1 :: "nat"   and i2 :: "nat" assumes "pivot_fun A f nc"   and "i1 < i2"   and "i2 < dim_row A"   and "f i1 < nc" shows "f i1 < f i2"
A_2: fixes z :: "complex" assumes "\<bar>Re z\<bar> < 1" shows "isCont h2' z"
c_cmod_id: fixes a :: "complex" assumes "a \<in> \<real>"   and "0 \<le> Re a" shows "complex_of_real (cmod a) = a"
count_roots_between_def: fixes p :: "real poly"   and a :: "real"   and b :: "real" shows "count_roots_between p a b = (if a \<le> b \<and> p \<noteq> 0 then let ps = sturm_squarefree p in sign_changes ps a - sign_changes ps b else 0)"
forest_axioms_def: fixes G :: "('c, 'd) graph" shows "forest_axioms G \<equiv> \<forall>(a, w, b)\<in>edges G. \<not> nodes_connected (delete_edge a w b G) a b"
update_simps_5: fixes up :: "(nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and s :: "('a, 'b) state" shows "\<U> (\<B>\<^sub>i\<^sub>u_update up s) = \<U> s"
S'_def: shows "S' \<equiv> {x. \<exists>u. u \<in> {0..1} \<and> (u, x) \<in> {0..1} \<times> T - V}"
bij_if_span_eq_span_bases: fixes B :: "'b set"   and C :: "'b set" assumes "B \<subseteq> U\<^sub>V\<^sub>S"   and "C \<subseteq> U\<^sub>V\<^sub>S"   and "\<not> local.dependent B"   and "\<not> local.dependent C"   and "local.span B = local.span C" shows "\<exists>x. bij_betw x B C \<and> (\<forall>a\<in>U\<^sub>V\<^sub>S. x a \<in> U\<^sub>V\<^sub>S)"
invertible_inverse_closed: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "inverse u \<in> G"
wf_tuple_restrict: fixes n :: "nat"   and B :: "nat set"   and v :: "'c option list"   and A :: "nat set"   and C :: "nat set" assumes "wf_tuple n B v"   and "A \<inter> B = C" shows "wf_tuple n C (restrict A v)"
add_right_isotone: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.aggregation_order plus bot less_eq less" shows "x \<noteq> bot \<and> less_eq (plus x bot) (plus y bot) \<longrightarrow> less_eq (plus x z) (plus y z)"
cancel: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<otimes> c = b \<otimes> c"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a = b"
unit_simps_5: fixes a :: "'a" assumes "C.obj a" shows "D.cod (G.unit a) = G a"
G_No_Op: fixes T :: "nat \<Rightarrow> nat set option"   and i :: "nat"   and p :: "local_expr"   and \<gamma> :: "group_state"   and a :: "abs_level" assumes "\<forall>j\<in>the (T i). \<not> eval_bool p (the (\<gamma> \<^sub>t\<^sub>s j))" shows "step_g a i T (\<gamma>, Barrier, p) (Some \<gamma>)"
rep_eq: fixes x :: "'a mod_ring"   and xa :: "'a mod_ring" shows "Rep_mod_ring (x + xa) = (Rep_mod_ring x + Rep_mod_ring xa) mod int CARD('a)"
isin_set: fixes t :: "'b trie"   and xs :: "'b list" shows "isin t xs = (xs \<in> Trie_Fun.set t)"
dims_matricize_1: fixes rmodes :: "nat set"   and T :: "'b tensor" shows "dim_row (matricize rmodes T) = prod_list (nths (dims T) rmodes)"
mnext: fixes I :: "\<I>"   and xs :: "'a option list set list"   and ts :: "nat list"   and ys :: "'a option list set list"   and xs' :: "'a option list set list"   and ts' :: "nat list"   and P :: "nat \<Rightarrow> 'a option list set \<Rightarrow> bool"   and i :: "nat"   and j' :: "nat"   and \<sigma> :: "'b trace"   and j :: "nat" assumes "mprev_next I xs ts = (ys, xs', ts')"   and "list_all2 P [Suc i..<j'] xs"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [i..<j] ts"   and "Suc i \<le> j'"   and "i < j" shows "list_all2 (\<lambda>i X. if mem (\<tau> \<sigma> (Suc i) - \<tau> \<sigma> i) I then P (Suc i) X else X = empty_table) [i..<min (j' - 1) (j - 1)] ys \<and> list_all2 P [Suc (min (j' - 1) (j - 1))..<j'] xs' \<and> list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (j' - 1) (j - 1)..<j] ts'"
set_rel_br: fixes a :: "'e \<Rightarrow> 'f"   and I :: "'e \<Rightarrow> bool" shows "\<langle>br a I\<rangle>set_rel = br ((`) a) (\<lambda>X. Ball X I)"
substs_preserve_occurs_in: fixes s :: "'b trm"   and t :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "occurs_in s t" shows "occurs_in (s \<lhd> \<sigma>) (t \<lhd> \<sigma>)"
ic: shows "inverse (lead_coeff p) \<noteq> (0::'a)"
Melem_subset_eq_sum_list: fixes x :: "'b multiset"   and T :: "'b multiset list" assumes "x \<in># mset T" shows "x \<subseteq># sum_list T"
D_LUB: fixes S :: "nat \<Rightarrow> 'b process" assumes "chain S" shows "D (lim_proc (range S)) = \<Inter> (D ` range S)"
aI_simps: fixes i :: "int" shows "aI (intgV i) = i"
sum_subtractf: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. f x - g x) = sum f A - sum g A"
onl_invariant_sterms_TT: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and A :: "('e \<times> ('b, 'c, 'a, 'd) seqp, 'f) automaton"   and P :: "'e \<times> 'd \<Rightarrow> bool"   and \<xi> :: "'e"   and p :: "('b, 'c, 'a, 'd) seqp"   and I :: "'f \<Rightarrow> bool"   and p' :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d" assumes "wellformed \<Gamma>"   and "A \<TTurnstile> onl \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P (\<xi>, l)"
abs_conv_abscissa_const: fixes c :: "'a" shows "abs_conv_abscissa (fds_const c) = - \<infinity>"
assms_2: shows "token_run x n \<notin> F - {q\<^sub>0}"
legendre_aux_altdef1: fixes x :: "real"   and p :: "nat" shows "legendre_aux x p = (if prime p \<and> 1 \<le> x then \<Sum>m\<in>{0<..nat \<lfloor>log (real p) x\<rfloor>}. nat \<lfloor>x / real (p ^ m)\<rfloor> else 0)"
foldl_spmf_pair_left: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'a spmf"   and l :: "'a"   and r :: "'b"   and es :: "'c list" shows "foldl_spmf (\<lambda>(l, r) e. map_spmf (\<lambda>l'. (l', r)) (f l e)) (return_spmf (l, r)) es = map_spmf (\<lambda>l'. (l', r)) (foldl_spmf f (return_spmf l) es)"
sum_list_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero \<equiv> monoid_list.F plus zero"
osum_congR: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and t :: "('f \<times> 'f) set" assumes "r =o s"   and "Well_order t" shows "t +o r =o t +o s"
Diff_infinite_finite: fixes T :: "'q set"   and S :: "'q set" assumes "finite T"   and "infinite S" shows "infinite (S - T)"
taut_def: shows "taut \<equiv> map_fun Rep_uexpr id All"
induct: fixes P :: "(int \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> value option \<Rightarrow> value option \<Rightarrow> bool"   and a0 :: "int \<Rightarrow> int \<Rightarrow> int"   and a1 :: "value option"   and a2 :: "value option" assumes "\<And>f x y. P f (Some (Num x)) (Some (Num y))"   and "\<And>uu_ uw_. P uu_ None uw_"   and "\<And>uu_ va uw_. P uu_ (Some (Str va)) uw_"   and "\<And>uu_ uv_. P uu_ uv_ None"   and "\<And>uu_ uv_ va. P uu_ uv_ (Some (Str va))" shows "P a0 a1 a2"
Predict_def: fixes k :: "nat"   and I :: "item set" shows "Predict k I = I \<union> {uu_. \<exists>r x. uu_ = init_item r k \<and> r \<in> \<RR> \<and> x \<in> bin I k \<and> next_symbol x = Some (fst r)}"
bij_betw_multiset_number_partition_with_atmost_size: fixes n :: "nat"   and k :: "nat" shows "bij_betw count {N. number_partition n N \<and> size N \<le> k} {p. p partitions n \<and> sum p {..n} \<le> k}"
assms_2: shows "\<forall>\<^sub>F x in F. f x = g x"
inj_on_arev: fixes S :: "'b set" shows "inj_on rev_G S"
A: fixes H :: "msg set" shows "H \<subseteq> range Key \<union> range Pri_AgrK - U \<longrightarrow> analz (H \<union> A \<union> spies evsFR5) = H \<union> analz (A \<union> spies evsFR5)"
Rep_filter: fixes x :: "'b Filters.filter" shows "Filters.filter.Rep_filter x \<in> filters"
list_split_two_elems: fixes cs :: "'c list"   and x :: "'c"   and y :: "'c" assumes "distinct cs"   and "x \<in> set cs"   and "y \<in> set cs"   and "x \<noteq> y" shows "\<exists>pre mid suf. cs = pre @ x # mid @ y # suf \<or> cs = pre @ y # mid @ x # suf"
prod_uminus: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Prod>x\<in>A. - f x) = (- (1::'c)) ^ card A * prod f A"
lazy_sample_vars: fixes V :: "'v set"   and e :: "'v exp"   and f :: "int \<Rightarrow> 'm" assumes "finite V" shows "sample_vars X V (bind (lazy X e) f) = bind (lazy X e) (\<lambda>i. sample_vars X V (f i))"
ta_in_repc_NodesSucnb: shows "set_of ta \<subseteq> repc ` Nodes (Suc nb) levellista"
frameChainAlpha: fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and F :: "'d frame" assumes "(p \<bullet> xvec) \<sharp>* F"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" shows "\<lparr>\<nu>*xvec\<rparr>F = \<lparr>\<nu>*p \<bullet> xvec\<rparr>p \<bullet> F"
i_set_cont_IN: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<in> i_set_cont"
wlp_and_term_imp_wp: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "wlp \<pi> c Q s \<and> \<pi>: (c, s) \<Rightarrow> t" shows "wp \<pi> c Q s"
sorted_wrt_linorder_index_le_imp: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "linorder_on A R"   and "set xs \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "index xs x \<le> index xs y" shows "(x, y) \<in> R"
simps: fixes R :: "('d \<times> 'c) set"   and a :: "'d" shows "R ,, a = the_elem (R `` {a})"
g_pos: fixes h :: "nat \<Rightarrow> real" assumes "g \<in> \<Omega>(h)"   and "\<forall>\<^sub>F x in sequentially. 0 < h x" shows "\<forall>\<^sub>F x in sequentially. 0 < g x"
coplanar_def: fixes S :: "'a set" shows "coplanar S \<equiv> \<exists>u v w. S \<subseteq> affine hull {u, v, w}"
imageI: fixes x :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "x \<in> A" shows "f x \<in> f ` A"
simps_6: fixes v :: "char list"   and va :: "aexp"   and vb :: "'b"   and vc :: "'b" shows "(v ::= va {vb} \<sqsubseteq> SKIP {vc}) = False"
hash_blindable_simps_2: fixes h :: "'d \<Rightarrow> 'c"   and y :: "'c blindable\<^sub>h" shows "hash_blindable h (Blinded y) = y"
commute: fixes a :: "'f multiset"   and b :: "'f multiset" shows "a \<inter># b = b \<inter># a"
F1bd'_Cinfinite: fixes r2 :: "('d \<times> 'd) set" shows "Cinfinite (bd_F1 +c r2)"
valid_SDG_node_parent_Entry: fixes n :: "'node SDG_node" assumes "valid_SDG_node n"   and "parent_node n = (_Entry_)" shows "n = CFG_node (_Entry_)"
special_points_on_x_axis'_3: shows "on_circline x_axis \<infinity>\<^sub>h"
simps_5: fixes va :: "dag"   and vb :: "ref"   and vc :: "dag"   and v :: "ref" shows "ordered (Node (Node va vb vc) v Tip) = (\<lambda>var. var (root (Node va vb vc)) < var v \<and> ordered (Node va vb vc) var)"
exec_move_SeqI1: 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 e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e;; e') h s ta h' s'"
Or_Some: fixes e2 :: "'a option"   and e1 :: "'a option"   and x :: "'a" shows "(e1 OR e2 = Some x) = (e1 = Some x \<or> e1 = None \<and> e2 = Some x)"
BNTr2: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x" shows "{a} \<in> WWa D f a"
lasso_accepted: fixes G :: "('e, 'f) gb_graph_rec_scheme"   and L :: "'e lasso" assumes "gb_graph G"   and "gb_graph.is_lasso G L" shows "gb_graph.is_acc_run G (run_of_lasso L)"
abs_eq: fixes y :: "'a" shows "force (Abs_lazy y) = y"
typing_elims_1: fixes e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and T :: "LambdaType.type" assumes "e \<turnstile> Var i : T" obtains "e i = T"
Basic: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> \<Rightarrow> Normal (f s)"
dens'_3: shows "(\<lambda>(\<rho>, x). f \<rho> x) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t)"
simps_64: fixes x7 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type" shows "Bag x7 \<noteq> Set x5"
Max_wrt_among_map_relation_bij: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "Order_Predicates.preorder_on carrier le"   and "bij f" shows "f ` Max_wrt_among le A = Max_wrt_among (map_relation (inv f) le) (f ` A)"
tiny_dg_Dom_in_Vset: shows "\<CC>\<lparr>Dom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
map_def: fixes \<mu> :: "'a" shows "local.map \<mu> = (if S.arr \<mu> then \<mu> else B.null)"
homogeneous_monomial: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "homogeneous (monomial c t)"
simps_16: fixes a :: "'a"   and b :: "'a" shows "local.le_rexp (Atom a) (Atom b) = (a \<le> b)"
in_small_setE: fixes a :: "V"   and A :: "V set" assumes "a \<in>\<^sub>\<circ> ZFC_in_HOL.set A"   and "small A" obtains "a \<in> A"
id_Zp_is_Zp_continuous: shows "is_Zp_continuous ring_id"
is_top_sorted_cons: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and l :: "'b list" shows "is_top_sorted R (x # l) = ({x} \<times> set l \<inter> (R\<^sup>*)\<inverse> = {} \<and> is_top_sorted R l)"
regular_closed_p: fixes uminus :: "'b \<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 x :: "'b" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus x = uminus (uminus (uminus x))"
wpt_is_pre: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com" assumes "finite (support Q)" shows "\<turnstile>\<^sub>t {wp\<^sub>t c Q} c {Q}"
classes_eq: fixes A :: "'a set" assumes "A \<subseteq> carrier S" shows "A {.=} closure_of A"
prop_restrict: fixes x :: "'c"   and Z :: "'c set"   and X :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> Z"   and "Z \<subseteq> {x \<in> X. P x}" shows "P x"
notin: shows "a \<notin> (\<Union>(uu_, uu_, uu_, sb, uu_, uu_, uu_)\<in>set ls. outstanding_refs (Not \<circ> is_volatile) sb)"
Dirichlet_L_principal: fixes s :: "complex" shows "Dirichlet_L n \<chi>\<^sub>0 s = (\<Prod>x\<in>{p. prime p \<and> p dvd n}. 1 - 1 / of_nat x powr s) * zeta s"
iT_defs_5: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<equiv> {x. x mod m = r mod m \<and> r \<le> x \<and> x \<le> r + m * c}"
ex_pos_a: shows "\<exists>a\<in>set as. 0 < a"
F_def: fixes P :: "'b process" shows "F P = FAILURES (Rep_process P)"
strip_all_vars_bind_fv: fixes v :: "variable \<times> typ"   and lev :: "nat"   and t :: "Core.term" shows "strip_all_vars (bind_fv2 v lev t) = strip_all_vars t"
defined_args_valid: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> Bag('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<delta> x->includes\<^sub>B\<^sub>a\<^sub>g(y)) = (\<tau> \<Turnstile> \<delta> x \<and> \<tau> \<Turnstile> \<upsilon> y)"
Domain_Union: fixes S :: "('e \<times> 'f) set set" shows "Domain (\<Union> S) = \<Union> (Domain ` S)"
dim_eq_full: fixes S :: "'a set" shows "(dim S = DIM('a)) = (span S = UNIV)"
diff_add: fixes M :: "'b multiset"   and N :: "'b multiset"   and Q :: "'b multiset" shows "M - (N + Q) = M - N - Q"
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A" shows "sum g (insert x A) = g x + sum g (A - {x})"
range_reduce_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "range_reduce_aform_err p X = range_reducer p (lower (ivl_of_aform_err p X)) \<bind> (\<lambda>r. Some (add_aform' p X (ivl_err (real_interval r))))"
subst_typ_stepwise: fixes instT :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" assumes "distinct (map fst instT)"   and "\<And>x. x \<in> tvsT_Set (snd ` set instT) \<Longrightarrow> x \<notin> fst ` set instT" shows "subst_typ instT T = fold (\<lambda>single. subst_typ [single]) instT T"
arctan_series: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "arctan x = (\<Sum>k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
simps_2: fixes var :: "'c"   and t :: "('a, 'c) trm" shows "SIGP (var := t) = {Inl x |x. x \<in> SIGT t}"
isomorphic_ide_rep: fixes f :: "'a" assumes "ide f" shows "f \<cong> rep f"
uminus_poly_mapping_def: shows "uminus \<equiv> map_fun lookup Abs_poly_mapping uminus"
abs_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(abs_nat m = abs_nat n) = (m = n)"
diff_right_commute: fixes M :: "'c multiset"   and N :: "'c multiset"   and Q :: "'c multiset" shows "M - N - Q = M - Q - N"
at_within_domain: fixes x :: "'a"   and c :: "('a, 'b) chart" assumes "x \<in> domain c" shows "at x within domain c = at x"
BinOpRedsVal: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop"   and v :: "val" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2\<rangle>"
cases: fixes p :: "nat"   and n :: "nat"   and a :: "(nat \<Rightarrow> nat) set" assumes "ksimplex p n a" obtains base :: "nat \<Rightarrow> nat" and upd :: "nat \<Rightarrow> nat" and s :: "(nat \<Rightarrow> nat) set" where "a = s" and "kuhn_simplex p n base upd s"
Word_compare: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme" assumes "Worder D"   and "Worder E" shows "(\<exists>a\<in>carrier D. ord_equiv (Iod D (segment D a)) E) \<or> ord_equiv D E \<or> (\<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b)))"
sig_inv_set_closed_sig_red_single: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_red :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "p \<in> sig_inv_set' (length fs)"   and "f \<in> sig_inv_set' (length fs)"   and "sig_red_single sing_red top_tail p q f t" shows "q \<in> sig_inv_set' (length fs)"