hcju/isabelleps huggingface dataset:
systemIN_noLOC: fixes x :: "chanID"   and i :: "AbstrLevelsID" assumes "systemIN x i" shows "\<not> systemLOC x i"
stopping_time_def: fixes F :: "'b \<Rightarrow> 'c measure"   and T :: "'c \<Rightarrow> 'b" shows "stopping_time F T = (\<forall>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t))"
homeomorphic_map_closure_of: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "homeomorphic_map X Y f"   and "S \<subseteq> topspace X" shows "Y closure_of f ` S = f ` (X closure_of S)"
ipd_not_self: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "x \<noteq> IFC_def.return prob" shows "x \<noteq> IFC_def.ipd prob x"
order_refl: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.preorder less_eq less" shows "less_eq x x"
intro: fixes M1 :: "'a measure"   and M2 :: "'b measure" assumes "sigma_finite_measure M1"   and "sigma_finite_measure M2" shows "pair_sigma_finite M1 M2"
update_def: fixes f :: "('v \<Rightarrow> int option) \<Rightarrow> 'v \<Rightarrow> int option"   and m :: "'m" shows "update f m = get (\<lambda>s. put (f s) m)"
cmp_components_are_iso: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.iso (S.cmp\<^sub>U\<^sub>P (f, g))"
rnn: shows "real_nonneg_mat (of_real_hom.mat_hom A)"
has_sgnx_derivative_at_right: fixes g :: "real \<Rightarrow> real"   and c :: "real"   and x :: "real" assumes "(g has_real_derivative c) (at x)"   and "g x = 0"   and "c \<noteq> 0" shows "(g has_sgnx sgn c) (at_right x)"
l_null: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
node_castTE: fixes i :: "nat"   and s :: "'c"   and R :: "nat set"   and RM :: "nat set"   and m :: "'d"   and s' :: "'c"   and R' :: "nat set"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i s R, RM:*cast(m), NodeS i s' R') \<in> node_sos S" obtains "RM = R" and "R' = R" and "(s, broadcast m, s') \<in> S"   | D :: "nat set" where "RM = R \<inter> D" and "R' = R" and "(s, groupcast D m, s') \<in> S"   | d :: "nat" where "RM = {d}" and "R' = R" and "(s, unicast d m, s') \<in> S" and "d \<in> R"
ks1: shows "\<forall>p. \<exists>ks. p = prob ks"
update_convs_2: fixes incoming' :: "nat set \<Rightarrow> nat set"   and name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "incoming_update incoming' \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = \<lparr>name = name, incoming = incoming' incoming, new = new, old = old, next = next, \<dots> = more\<rparr>"
simps_3: fixes r :: "'form" shows "inPorts' (Conclusion r) = [plain_ant r]"
unsigned_ucast_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "'d word" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.unsigned one plus zero (UCAST('d \<rightarrow> 'c) w) = take_bit LENGTH('c) (semiring_1.unsigned one plus zero w)"
m1_secrecyI: fixes x :: "'a m1_state_scheme" assumes "knC (runs x) \<subseteq> azC (runs x) \<union> Domain (m1_state.leak x) \<times> UNIV" shows "x \<in> m1_secrecy"
assms_2: shows "\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>a. F a \<bullet> i) j"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "P [] []"   and "\<And>v va. P [] (v # va)"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
conjugate_scalar_prod: fixes v :: "'c vec"   and w :: "'c vec" assumes "dim_vec v = dim_vec w" shows "conjugate (v \<bullet> w) = inner_prod w (conjugate v)"
idmap_def: fixes A :: "'c set" shows "idmap A = (\<lambda>x\<in>A. x)"
preserves_modes_mem_mem\<^sub>A_simp: fixes mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "\<forall>x\<^sub>A. mem\<^sub>A x\<^sub>A = mem\<^sub>C (var\<^sub>C_of x\<^sub>A)" shows "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"
af\<^sub>G_F: fixes \<phi> :: "'a ltl"   and w :: "'a set list" shows "af\<^sub>G (F \<phi>) w = Or (F \<phi> # map (af\<^sub>G \<phi>) (suffixes w))"
finite_comm_group_adjoin_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "finite_comm_group_adjoin G H a \<equiv> finite_comm_group G \<and> subgroup H G \<and> finite_comm_group_adjoin_axioms G H a"
simps_3: fixes v :: "(int \<times> int) set set"   and va :: "(int \<times> int) set set"   and xs :: "((int \<times> int) set set, int) Combinators list" shows "wellformed_policy2 (DenyAllFromTo v va # xs) = ((\<forall>c a b. c = DenyAllFromTo a b \<and> c \<in> set xs \<longrightarrow> dom (C (DenyAllFromTo v va)) \<inter> dom (C c) = {}) \<and> wellformed_policy2 xs)"
B: shows "\<forall>x\<in>A. \<exists>y\<in>B. x \<le> y"
psimps: fixes n :: "nat" assumes "f91_dom n" shows "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
af\<^sub>G_simps_2: fixes w :: "'a set list" shows "af\<^sub>G false w = false"
one_subgroup_generated: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" shows "\<one>\<^bsub>subgroup_generated G S\<^esub> = \<one>\<^bsub>G\<^esub>"
FW_zone_equiv_spec: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" shows "[M]\<^bsub>v,n\<^esub> = [FW M n]\<^bsub>v,n\<^esub>"
spec_is_locally_ringed_space: shows "locally_ringed_space Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
elims: fixes x :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and y :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" assumes "translate_ars x = y" obtains ars :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" where "x = ars" and "y = map_of (map (apsnd map_of) ars)"
assms_2: shows "A = P\<^sup>-\<^sup>1 * sq_mtx_diag f * P"
bdd_lookup_binop: fixes bs :: "bool list"   and l :: "'a bdd"   and r :: "'b bdd"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bddh (length bs) l"   and "bddh (length bs) r" shows "bdd_lookup (bdd_binop f l r) bs = f (bdd_lookup l bs) (bdd_lookup r bs)"
allUses_narrows: fixes n :: "'node" assumes "n \<in> set (\<alpha>n g)" shows "step.allUses g n \<subseteq> substNext g ` allUses g n"
flatten_single: fixes c' :: "('d, 'e, 'f) com" shows "\<forall>c\<in>set (flatten c'). flatten c = [c]"
struct_spec_gb: shows "punit.struct_spec punit.gb_sel punit.add_pairs_canon punit.add_basis_canon punit.gb_red"
eval_And: fixes a :: "atom fm"   and b :: "atom fm"   and L :: "real list" shows "eval (And a b) L = (eval a L \<and> eval b L)"
simps_5: fixes t :: "'c local.term" shows "Dom \<^bold>\<l>\<^bold>[t\<^bold>] = Trg t \<^bold>\<star> Dom t"
inv_approx: fixes S :: "'a set"   and r :: "nat" assumes "independent_vertices E S"   and "card V \<le> card S * r" shows "approximation_miv r S"
finter_iff: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" shows "(c |\<in>| A |\<inter>| B) = (c |\<in>| A \<and> c |\<in>| B)"
lift_Suc_mono_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f n) (f (Suc n))"   and "n \<le> n'" shows "less_eq (f n) (f n')"
cospan_\<mu>\<nu>: shows "C.cospan \<mu>.leg0 \<nu>.leg1"
idom_mult_cancel_r: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Idomain R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "c \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<cdot>\<^sub>r\<^bsub>R\<^esub> c = b \<cdot>\<^sub>r\<^bsub>R\<^esub> c" shows "a = b"
simps: fixes xs :: "int list" shows "cuts xs = (if xs = [] then [] else let (c, xs) = take2 xs in c # cuts xs)"
D: shows "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s'"
mult_ge_zero: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(0::'a) \<le> a * b"
elims: fixes x :: "real_alg_2 list"   and y :: "real_alg_2" assumes "real_alg_2_list_hd_intern x = y" obtains xa :: "real_alg_2" and xs :: "real_alg_2 list" where "x = xa # xs" and "y = xa"   | "x = []" and "y = of_rat_2 0"
upper_comp: fixes y :: "'b" assumes "y \<in> carrier \<Y>" shows "\<pi>\<^sub>* (\<pi>\<^sup>* (\<pi>\<^sub>* y)) = \<pi>\<^sub>* y"
rsp: shows "(0, 1) \<approx> (0, 1)"
that: fixes g :: "real \<Rightarrow> 'a" assumes "arc g"   and "g ` {0..<1} \<subseteq> S"   and "pathstart g \<in> S"   and "pathfinish g \<in> V" shows "thesis"
map\<^sub>1_def: fixes f :: "'a" shows "map\<^sub>1 f = (map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f \<star>\<^sub>D counit (src\<^sub>C f)) \<cdot>\<^sub>D (map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 (src\<^sub>C f), map\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>C f), G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f), map\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D ((map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D (\<a>\<^sub>D[map\<^sub>0 (trg\<^sub>C f), \<tau>\<^sub>0 (trg\<^sub>C f), F f] \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D ((unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f))"
high_def: fixes interval :: "'a interval" shows "high interval = (case interval of Interval x1 x2 \<Rightarrow> x2)"
mod_eq_div_right_strict_mono_on: fixes k :: "nat"   and I :: "nat set" assumes "0 < k"   and "\<forall>x\<in>I. \<forall>y\<in>I. x mod k = y mod k" shows "SetInterval2.strict_mono_on (\<lambda>x. x div k) I"
wf_theory_translate_imp_exe_wf_theory: fixes a :: "exetheory" assumes "wf_theory (translate_theory a)" shows "exe_wf_theory a"
logically_equivalent_def: fixes P :: "'state"   and Q :: "'state" shows "S_transform.logically_equivalent P Q \<equiv> \<forall>x. P \<Turnstile>\<^sub>S x = Q \<Turnstile>\<^sub>S x"
put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'm, 'n, 'w option) RShadowRoot_scheme" shows "put shadow_root_ptr shadow_root = put (cast shadow_root_ptr) (cast shadow_root)"
assms_1: shows "monom_mult c t1 p = monom_mult c t2 p"
th: shows "norm (h x y) = norm (\<Sum>(i, j)\<in>Basis \<times> Basis. h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j))"
b1: shows "orthogonal_complement (orthogonal_complement M) = projection (orthogonal_complement M) -` {0::'a}"
rbt_lookup_RBT_Impl_diag: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less" shows "ord.rbt_lookup (less_prod less_eq less less) (RBT_Impl_diag t) = (\<lambda>(k, k'). if k = k' then ord.rbt_lookup less t k else None)"
infiter_step_mono: fixes c :: "'a::refinement_lattice" shows "mono ((;) (c::'a::refinement_lattice))"
plus_freeword_abs_eq: fixes xs :: "('b \<times> bool) list"   and ys :: "('b \<times> bool) list" assumes "proper_signed_list xs"   and "proper_signed_list ys" shows "Abs_freeword xs + Abs_freeword ys = Abs_freeword (prappend_signed_list xs ys)"
bi: shows "j_to_jb_index n_as i = (bi, li)"
complement_partition: fixes A :: "'a set" assumes "infinite UNIV" shows "(A \<in> infinite_complement_partition) = (- A \<notin> infinite_complement_partition)"
absorb1: fixes b :: "'a upred"   and a :: "'a upred" assumes "a \<sqsubseteq> b" shows "(a \<or> b) = a"
weight_le: fixes a :: "'v" assumes "a \<in> A \<Gamma>'" shows "weight \<Gamma>' a \<le> weight \<Gamma> a"
preserved_by_output_iso: fixes \<phi> :: "'a"   and g' :: "'a" assumes "\<guillemotleft>\<phi> : g' \<Rightarrow> g\<guillemotright>"   and "local.iso \<phi>" shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r (\<rho> \<cdot> \<phi>) f g'"
fset_of_list_elem: fixes x :: "'b"   and xs :: "'b list" shows "(x |\<in>| fset_of_list xs) = (x \<in> set xs)"
simps_8: fixes v :: "linear_poly"   and va :: "rat" shows "nonstrict_constr (GT v va) = False"
RedALengthNull: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" shows "extTA,P,t \<turnstile> \<langle>null\<bullet>length,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
eq_snd_iff: fixes b :: "'a"   and p :: "'b \<times> 'a" shows "(b = snd p) = (\<exists>a. p = (a, b))"
chain_map_of: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'd" shows "chain_map p g (frag_of f) = frag_of (simplex_map p g f)"
j_less_len_t: shows "Min {j \<in> {Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}} + 1..<length t}. prerecording_event t j} < length t"
derivation_well_ordered: fixes P :: "'a set" assumes "derivation P" shows "well_ordered_set P (\<sqsubseteq>)"
record_simps_1: fixes f :: "nat \<Rightarrow> nat"   and x :: "'b state" shows "clock (update_clock f x) = f (clock x)"
dgrad_p_set_exhaust: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" obtains m :: "nat" where "F \<subseteq> dgrad_p_set d m"
map_fun_def: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and h :: "'e \<Rightarrow> 'f" shows "map_fun f g h = g \<circ> h \<circ> f"
Gnn: shows "mat_of_rows n (map of_int_hom.vec_hom fs) \<in> carrier_mat m n"
complete_isometric_image: fixes e :: "real"   and s :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "0 < e"   and "subspace s"   and "bounded_linear f"   and "\<forall>x\<in>s. e * norm x \<le> norm (f x)"   and "complete s" shows "complete (f ` s)"
exhaust: fixes y :: "letter" obtains "y = A"   | "y = B"
lossless_exec_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'd) spmf"   and s :: "'d" assumes "lossless_gpv \<I> gpv"   and "\<And>s out. out \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (callee s out)"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s. \<I> \<turnstile>c callee s \<surd>" shows "lossless_spmf (exec_gpv callee gpv s)"
Abs_upsets_inverse: fixes y :: "'b set" assumes "y \<in> range \<Up>" shows "Rep_upsets (Abs_upsets y) = y"
bind_unbind: fixes t :: "lam"   and xs :: "name list"   and t' :: "lam" assumes "t \<mapsto> xs,t'" shows "t = bind xs t'"
filter_mset_add_mset: fixes P :: "'k \<Rightarrow> bool"   and x :: "'k"   and A :: "'k multiset" shows "filter_mset P (add_mset x A) = (if P x then add_mset x (filter_mset P A) else filter_mset P A)"
Nil: fixes r :: "('a \<times> 'b) set" shows "([], []) \<in> listrel r"
eq\<rho>E: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value"   and x :: "Vars.var"   and a :: "Arity" assumes "eq\<rho> ae \<rho>1 \<rho>2"   and "ae x = up\<cdot>a" shows "eq a (\<rho>1 x) (\<rho>2 x)"
len_xs: shows "length (map (\<lambda>i. Cn (Suc m) (r_code_constn n) [recf.Id (Suc m) i]) [0..<Suc m]) = Suc m"
Some: shows "applied_rule_rev Cp x p = \<lfloor>y\<rfloor>"
drop_f_join_eq1: fixes n :: "nat"   and I :: "nat set"   and xs :: "'b list" assumes "n < card (I \<down>< length xs)" shows "xs \<Join>\<^sub> I \<up> n = xs \<Join>\<^sub> (I \<down>\<ge> (I \<rightarrow> n))"
fundfold_comp_gallery_map: fixes Cs :: "'a set list"   and ss :: "'a permutation list" assumes "gallery Cs"   and "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (set ss)" shows "gallery (fold fst (map Spair ss) \<Turnstile> Cs)"
preorder_on_axioms: shows "Order_Predicates.preorder_on alts (R i)"
sigma_eqI: fixes M :: "'d set set"   and \<Omega> :: "'d set"   and N :: "'d set set" assumes "M \<subseteq> Pow \<Omega>"   and "N \<subseteq> Pow \<Omega>"   and "sigma_sets \<Omega> M = sigma_sets \<Omega> N" shows "sigma \<Omega> M = sigma \<Omega> N"
dualCompose: fixes p :: "signs \<Rightarrow> signs"   and q :: "signs \<Rightarrow> signs"   and r :: "signs \<Rightarrow> signs"   and P :: "signs \<Rightarrow> signs"   and Q :: "signs \<Rightarrow> signs"   and R :: "signs \<Rightarrow> signs" shows "dual p q r \<circ> dual P Q R = dual (p \<circ> P) (q \<circ> Q) (r \<circ> R)"
yun_factorization_1: fixes p :: "'a poly"   and c_bs :: "'a \<times> ('a poly \<times> nat) list" assumes "yun_factorization gcd p = c_bs" shows "square_free_factorization p c_bs"
set_tl_append': fixes xs :: "'c list"   and ys :: "'c list" shows "set (tl (xs @ ys)) \<subseteq> set (tl xs) \<union> set ys"
simps_2: fixes f :: "'b \<Rightarrow> 'b msstream \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'b msstream" shows "(case MSSCons x1 x2 of MSSCons x xa \<Rightarrow> f x xa) = f x1 x2"
m_closed: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (G Mod N)"   and "y \<in> carrier (G Mod N)" shows "x \<otimes>\<^bsub>G Mod N\<^esub> y \<in> carrier (G Mod N)"
discI: fixes enat :: "enat"   and x2 :: "enat" assumes "enat = eSuc x2" shows "enat \<noteq> 0"
Un_insert_left: fixes a :: "'f"   and B :: "'f set"   and C :: "'f set" shows "insert a B \<union> C = insert a (B \<union> C)"
Diagonal_to_Smith_row_i_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "Diagonal_to_Smith_row_i A i bezout = diagonal_to_Smith_i [i + 1..<min (nrows A) (ncols A)] A i bezout"
fin_dim_li_fin: fixes A :: "'c set" assumes "fin_dim"   and "lin_indpt A"   and "A \<subseteq> carrier V" shows "finite A"
degree_def: fixes p :: "'a poly" shows "Polynomial.degree p = (LEAST n. \<forall>i>n. poly.coeff p i = (0::'a))"
assms_2: shows "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|"
\<eta>_def: shows "\<eta> \<equiv> transformation_by_components.map (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map \<eta>o"
diff_union_swap2: fixes y :: "'b"   and M :: "'b multiset"   and x :: "'b" assumes "y \<in># M" shows "(x, M) - (y, {#}) = x, M - (y, {#})"
bezout_matrix_preserves_rest: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'e"   and n :: "'e"   and i :: "'e"   and A :: "(('d, 'f) vec, 'e) vec"   and k :: "'f"   and b :: "'f" assumes "is_bezout_ext bezout"   and "a \<noteq> n"   and "i \<noteq> n"   and "a \<noteq> i"   and "A $ i $ k \<noteq> (0::'d)"   and "is_zero_row_upt_k i (mod_type_class.to_nat k) A" shows "(bezout_matrix A i n k bezout ** A) $ a $ b = A $ a $ b"
mUnionM_add_mset: fixes x :: "'m"   and M :: "'m multiset" shows "mUnionM (add_mset x M) = munionM x (mUnionM M)"
phi_equiv_mappingE2': fixes g :: "'g"   and m\<^sub>1 :: "('val, ('node \<times> 'val) set) mapping"   and b :: "'val" assumes "g \<turnstile> m\<^sub>1 \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"   and "b \<in> snd ` Mapping.keys (phis g)"   and "\<forall>\<phi>\<in>Mapping.keys (phis g). b \<notin> set (the (Mapping.lookup (phis g) \<phi>))" shows "Mapping.lookup m\<^sub>1 b = None \<or> Mapping.lookup m\<^sub>1 b = Some {}"
finter_fminus: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |\<inter>| B |-| C = A |\<inter>| (B |-| C)"
tdec_b_def: shows "tdec_b \<equiv> [(W1, 1), (R, 2), (L, 14), (R, 3), (L, 4), (R, 3), (R, 5), (W0, 4), (R, 6), (W0, 5), (L, 7), (L, 8), (L, 11), (W0, 7), (W1, 8), (R, 9), (L, 10), (R, 9), (R, 5), (W0, 10), (L, 12), (L, 11), (R, 13), (L, 11), (R, 17), (W0, 13), (L, 15), (L, 14), (R, 16), (L, 14), (R, 0), (W0, 16)]"
C: shows "s\<^sub>0 \<Turnstile> s"
ideal_implies_subset: fixes A :: "'a set"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "ideal A R add mult \<zero> \<one>" shows "A \<subseteq> R"
transitive: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and s :: "real \<times> real"   and t :: "real \<times> real"   and S :: "(real \<times> real) set" assumes "distinct5 p q r s t"   and "in5 S p q r s t"   and "ccw t s p"   and "ccw t s q"   and "ccw t s r"   and "ccw t p q"   and "ccw t q r" shows "ccw t p r"
vars_term_ms_subst_mono: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "vars_term_ms s \<subseteq># vars_term_ms t" shows "vars_term_ms (s \<cdot> \<sigma>) \<subseteq># vars_term_ms (t \<cdot> \<sigma>)"
real_of_vreal_vreal_of_real: fixes a :: "real" shows "real_of_vreal (a\<^sub>\<real>) = a"
fminusE: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" assumes "c |\<in>| A |-| B" obtains "c |\<in>| A" and "c |\<notin>| B"
simI: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and C :: "'e" assumes "eqvt Rel"   and "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* \<Psi>; distinct (bn \<alpha>); bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
l3: shows "z \<notin> path_image (linepath ub (Complex (Re lb) (Im ub)))"
eq_fold: fixes M :: "'k multiset" shows "\<Prod>\<^sub># M = fold_mset (*) (1::'k) M"
prems_8: shows "set uus \<inter> \<Union> (FvarsT ` fst ` set (tx # txs)) = {}"
squarefree_prime_elem: fixes p :: "'b" assumes "prime_elem p" shows "squarefree p"
power4_eq_xxxx: fixes x :: "'a" shows "x ^ 4 = x \<cdot> x \<cdot> x \<cdot> x"
f_inj_on: fixes y :: "'index \<times> 'trap" shows "y \<in> set_spmf I \<longrightarrow> inj_on (F (fst y)) (range (fst y))"
assms_3: fixes xs' :: "V"   and x :: "V" assumes "xs = xs' #\<^sub>\<circ> x"   and "vfsequence xs'" shows "P"
\<omega>_simps_2: shows "src \<r>\<^sup>-\<^sup>1[r] = src w"
il_inf_right_unit: fixes x :: "'a" shows "x \<sqinter> top = x"
pp_ub1': fixes S :: "('c \<times> 'a) list"   and y :: "'a"   and a :: "'c"   and b :: "'a" assumes "snd ` set S \<subseteq> A"   and "y \<in> A"   and "(a, b) \<in> set S" shows "b \<sqsubseteq>\<^bsub>r\<^esub> map snd (filter (\<lambda>(p', t'). p' = a) S) \<Squnion>\<^bsub>f\<^esub> y"
streams_iff_snth: fixes s :: "'a stream"   and X :: "'a set" shows "(s \<in> streams X) = (\<forall>n. s !! n \<in> X)"