hcju/isabelleps huggingface dataset:
\<sigma>\<^sub>2_def: shows "\<sigma>\<^sub>2 = initial \<tau>\<^sub>2"
cStep: fixes P' :: "('a, 'b, 'c) psi"   and P'' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P' \<longmapsto> \<tau> \<prec> P''"   and "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "Prop P P'" shows "Prop P P''"
Confidentiality_Kas: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Key authK \<notin> analz (knows Spy evs)"
INT_anti_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Inter> (f ` B) \<subseteq> \<Inter> (g ` A)"
finsum_addf: fixes f :: "'e \<Rightarrow> 'c"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "f \<in> A \<rightarrow> carrier M"   and "g \<in> A \<rightarrow> carrier M" shows "(\<Oplus>\<^bsub>M\<^esub>x\<in>A. f x \<oplus>\<^bsub>M\<^esub> g x) = finsum M f A \<oplus>\<^bsub>M\<^esub> finsum M g A"
Rep_group_cases: fixes y :: "'d monoid" assumes "y \<in> {G. Group.group G}" obtains x :: "'d group" where "y = Rep_group x"
complex_roots_of_int_poly_def: fixes p :: "int poly" shows "complex_roots_of_int_poly p = (let ps = if 3 \<le> degree p then factors_of_int_poly p else [p] in concat (map complex_roots_of_int_poly_all ps))"
is_moebius_inv: fixes m :: "complex_homo \<Rightarrow> complex_homo" assumes "is_moebius m" shows "is_moebius (inv m)"
findnth_state_all_le2: fixes k :: "nat"   and ap :: "abc_inst list"   and n :: "nat"   and e :: "nat"   and a :: "action"   and b :: "nat" assumes "k < length ap"   and "ap ! k = Dec n e"   and "(a, b) \<in> set (shift (findnth n) (start_of (layout_of ap) k - Suc 0))" shows "b \<le> start_of (layout_of ap) (length ap)"
acyclic_star_inf_conv_iff: fixes w :: "'a" assumes "irreflexive w" shows "pd_kleene_allegory_class.acyclic w = (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star> = (1::'a))"
Pg_diff_def: fixes G :: "Pg"   and H :: "Pg" shows "G - H = G + - H"
card_partition: fixes C :: "'c set set"   and k :: "nat" assumes "finite C"   and "finite (\<Union> C)"   and "\<And>c. c \<in> C \<Longrightarrow> card c = k"   and "\<And>c1 c2. \<lbrakk>c1 \<in> C; c2 \<in> C; c1 \<noteq> c2\<rbrakk> \<Longrightarrow> c1 \<inter> c2 = {}" shows "k * card C = card (\<Union> C)"
set_bd_1: fixes x :: "('a, 'c, 'b) var0" shows "|set1_var0 x| \<le>o bd_pre_var0"
Lxy_snoc: fixes xs :: "'a list"   and x :: "'a"   and S :: "'a set" shows "Lxy (xs @ [x]) S = (if x \<in> S then Lxy xs S @ [x] else Lxy xs S)"
is_PureE: fixes generat :: "('g, 'h, 'i) generat" assumes "is_Pure generat" obtains x :: "'g" where "generat = Pure x"
x_def: shows "x \<in> rel_interior (domain f) \<and> f x < \<infinity>"
smalleraux2c: fixes a :: "'b"   and b :: "'b"   and l :: "'b set list"   and c :: "'b"   and d :: "'b"   and y :: "('b, 'c) Combinators"   and q :: "'c"   and p :: "'c" assumes "{a, b} \<in> set l"   and "{c, d} \<in> set l"   and "{a, b} \<noteq> {c, d}"   and "y = AllowPortFromTo a b q"   and "smaller (AllowPortFromTo c d p) y l" shows "\<not> smaller y (AllowPortFromTo c d p) l"
at_decomposeLS: fixes c :: "('g, 'h, 'i, 'j) com"   and ictxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" assumes "(c, ictxt, fctxt) \<in> decomposeLS s" shows "atC c \<subseteq> atCs (cPGM s)"
comp_closed_set_def: fixes X :: "'b set" shows "comp_closed_set X = (\<forall>x\<in>X. \<forall>y\<in>X. x \<cdot> y \<in> X)"
last_cs: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob" shows "last (IFC_def.cs prob \<pi> i) = \<pi> i"
\<R>\<^sub>1_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option" and \<S>' :: "'Var set \<times> 'Var set" and Pa :: "'BExp set" where "c\<^sub>2 = c\<^sub>1" and "\<turnstile> \<Gamma>',\<S>',Pa {c\<^sub>1} \<Gamma>,\<S>,P" and "tyenv_wellformed mds \<Gamma>' \<S>' Pa" and "mem\<^sub>1 =\<^bsub>\<Gamma>'\<^esub> mem\<^sub>2" and "pred Pa mem\<^sub>1" and "pred Pa mem\<^sub>2" and "\<forall>x\<in>dom \<Gamma>'. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>' x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
elims_3: fixes x :: "nat"   and xa :: "'b btree" assumes "\<not> almost_order x xa" obtains k :: "nat" and ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = k" and "xa = Node ts t" and "\<not> (length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t)"
sl_def: fixes P :: "'b prog"   and mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.sl P mxs mxl \<equiv> Err.sl (Opt.esl (Product.esl (stk_esl P mxs) (Err.esl (loc_sl P mxl))))"
vec_mat_Tensor_def: shows "vec_mat_Tensor \<equiv> \<lambda>uu uua. rec_list (\<lambda>xs. []) (\<lambda>ys yss yssa xs. vec_vec_Tensor xs ys # yssa xs) uua uu"
S_order2_add: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "s + s = 0"
abss_reps_id: fixes s' :: "'s \<times> ('c \<Rightarrow> real) set"   and s :: "'s \<times> ('c \<Rightarrow> real) set"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "s' \<in> \<S>"   and "s \<in> set_pmf (action cfg)"   and "cfg \<in> R_G.cfg_on s'" shows "abss (reps s) = s"
I_def: fixes f :: "complex poly"   and u :: "complex" shows "lindemann_weierstrass_aux.I f u = exp u * (\<Sum>j\<le>Polynomial.degree f. poly ((pderiv ^^ j) f) 0) - (\<Sum>j\<le>Polynomial.degree f. poly ((pderiv ^^ j) f) u)"
FOREACH_rule: fixes S :: "'g set"   and I :: "'g set \<Rightarrow> 'h \<Rightarrow> bool"   and \<sigma>0 :: "'h"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h nres"   and P :: "'h \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "FOREACH S f \<sigma>0 \<le> SPEC P"
deterministic_RP_model: fixes I :: "'a set" shows "(I \<Turnstile>s grounded_N0) = (I \<Turnstile>s grounded_R)"
module_hom_def: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'e, 'i) module_scheme"   and N :: "('g, 'f, 'j) module_scheme" shows "module_hom R M N = {f \<in> carrier M \<rightarrow> carrier N. (\<forall>m1 m2. m1 \<in> carrier M \<and> m2 \<in> carrier M \<longrightarrow> f (m1 \<oplus>\<^bsub>M\<^esub> m2) = f m1 \<oplus>\<^bsub>N\<^esub> f m2) \<and> (\<forall>r m. r \<in> carrier R \<and> m \<in> carrier M \<longrightarrow> f (r \<odot>\<^bsub>M\<^esub> m) = r \<odot>\<^bsub>N\<^esub> f m)}"
pr_rec1_scheme: fixes h :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat"   and a :: "nat" assumes "h \<in> PrimRec2"   and "f 0 = a"   and "\<forall>y. f (Suc y) = h y (f y)" shows "f \<in> PrimRec1"
i_aggregate_nth: fixes f :: "nat \<Rightarrow> 'd"   and k :: "nat"   and ag :: "'d list \<Rightarrow> 'd"   and n :: "nat" shows "i_aggregate f k ag n = ag (f \<Up> (n * k) \<Down> k)"
r_distr: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])"   and "z \<in> carrier (K [X])" shows "z \<otimes>\<^bsub>K [X]\<^esub> (x \<oplus>\<^bsub>K [X]\<^esub> y) = z \<otimes>\<^bsub>K [X]\<^esub> x \<oplus>\<^bsub>K [X]\<^esub> z \<otimes>\<^bsub>K [X]\<^esub> y"
iso_arcs_iso_inv: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and a :: "'f" assumes "digraph_isomorphism hom"   and "a \<in> arcs (app_iso hom G)" shows "iso_arcs hom (iso_arcs (inv_iso hom) a) = a"
compatible_iff_seq: fixes g :: "'c"   and f :: "'c" shows "(C.domains g \<inter> C.codomains f \<noteq> {}) = C.seq g f"
simps_2: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and xs :: "'c list" shows "isort_key f (x # xs) = insort_key f x (isort_key f xs)"
r_null: fixes x :: "'a mod_ring poly" assumes "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
translation_assoc: fixes b :: "'b"   and a :: "'b"   and S :: "'b set" shows "(+) b ` (+) a ` S = (+) (a + b) ` S"
ahm_\<alpha>_def: fixes hm :: "('d, 'e) hashmap" shows "ahm_\<alpha> hm = ArrayHashMap_Impl.ahm_\<alpha> (impl_of hm)"
qAbs_preserves_alpha2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "X #= X'"   and "qGoodAbs (qAbs xs x X) \<or> qGoodAbs (qAbs xs x X')" shows "qAbs xs x X $= qAbs xs x X'"
in_centered_ivlE: fixes r :: "real"   and t :: "real" assumes "r \<in> {- t..t}" obtains e :: "real" where "e \<in> {- 1..1}" and "r = e * t"
AIjeq: shows "\<Delta> ! j = (p \<leftrightarrow> q)"
map_le_map_add: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "f \<subseteq>\<^sub>m g ++ f"
b: shows "supp x = supp x - as \<union> supp x \<inter> as"
KL_div_pos_emp: fixes a :: "'b \<Rightarrow> real"   and e :: "'b \<Rightarrow> real" shows "0 \<le> KL_div {} a e"
drop_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "n < m" shows "drop m (xs[n := x]) = drop m xs"
Rel2Mon_homomorphism: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" assumes "single_valued X"   and "single_valued Y" shows "Rel2Mon (X O Y) = Rel2Mon X ;- Rel2Mon Y"
dist_triangle_less_add: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x1 :: "'a"   and y :: "'a"   and e1 :: "real"   and x2 :: "'a"   and e2 :: "real" assumes "class.metric_space dist uniformity open"   and "dist x1 y < e1"   and "dist x2 y < e2" shows "dist x1 x2 < e1 + e2"
prems: shows "steps0 (1, l, r) B (Suc stp) = (s', l', r')"
order_degree: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "Polynomial.order a p \<le> degree p"
Set_inter_code_2: fixes J :: "'b set"   and A'' :: "'b \<Rightarrow> bool" shows "J \<inter> Collect_set A'' = Collect_set (\<lambda>x. A'' x \<and> x \<in> J)"
iwlsFSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "iwlsFSw MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSwapCls MOD \<and> igAbsCongS MOD"
IfFalse: fixes e :: "exp"   and s :: "loc \<Rightarrow> nat"   and s' :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set"   and c1 :: "com"   and s1 :: "loc \<Rightarrow> nat"   and c0 :: "com" assumes "((e, s), Suc 0, s') \<in> eval"   and "(c1, s') -[eval]-> s1" shows "(IF e THEN c0 ELSE c1, s) -[eval]-> s1"
exhaust: fixes y :: "'a withERR" obtains "y = ERR"   | x2 :: "'a" where "y = OK x2"
Conseq: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<turnstile>\<^sub>t {P'} c {Q'}"   and "(\<forall>s t. (\<forall>z. P' z s \<longrightarrow> Q' z t) \<longrightarrow> (\<forall>z. P z s \<longrightarrow> Q z t)) \<and> (\<forall>s. (\<exists>z. P z s) \<longrightarrow> (\<exists>z. P' z s))" shows "C \<turnstile>\<^sub>t {P} c {Q}"
cases: fixes x :: "nat" obtains "x = 0"   | "x = Suc 0"   | va :: "nat" where "x = Suc (Suc va)"
bottom_least: fixes x :: "'b" shows "\<bottom> \<sqsubseteq> x"
hyps: fixes \<rho> :: "complex mat" shows "denote_while_n_iter M0 M1 DS (Suc n) \<rho> = DS (M1 * denote_while_n_iter M0 M1 DS n \<rho> * adjoint M1)"
simps_1: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" shows "rel_comp_pow r [] x y = False"
limit_nonemptyE: fixes x :: "nat \<Rightarrow> 'e" assumes "finite (range x)" obtains xa :: "'e" where "xa \<in> limit x"
subst_dnf_def: fixes \<A> :: "'b ltln fset set"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_dnf \<A> m = (\<Union>\<Phi>\<in>\<A>. subst_clause \<Phi> m)"
cases: fixes a :: "'a list" assumes "head_side a" obtains x :: "'a" and uu :: "'a list" where "a = x # uu"
nonzero_divide_mult_cancel_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "a \<noteq> zero" shows "divide a (times a b) = divide one b"
Least_submodule_def: fixes R :: "('f, 'g) Ring_scheme"   and M :: "('e, 'f, 'h) Module_scheme"   and T :: "'e set" shows "LSM\<^bsub>R\<^esub> M T = \<Inter> {N. submodule R M N \<and> T \<subseteq> N}"
avl_balR: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "avl l"   and "avl r"   and "height l - 1 \<le> height r \<and> height r \<le> height l + 2" shows "avl (balR l a r)"
get_ACS_def: fixes M :: "'a SecurityInvariant_configured list" shows "TopoS_Composition_Theory.get_ACS M \<equiv> filter (\<lambda>m. \<not> c_isIFS m) M"
27: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> (x \<sqinter> z \<squnion> x \<sqinter> (y \<squnion> z)) = x \<sqinter> (y \<squnion> z)"
gen_lset_code_2: fixes A :: "'a set"   and x :: "'a"   and xs :: "'a llist" shows "gen_lset A (LCons x xs) = gen_lset (insert x A) xs"
has_vector_derivative_def: fixes f :: "real \<Rightarrow> 'd"   and f' :: "'d"   and net :: "real filter" shows "(f has_vector_derivative f') net = (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
in_fold_Deriv: fixes v :: "'c list"   and w :: "'c list"   and L :: "'c list set" shows "(v \<in> fold Deriv w L) = (w @ v \<in> L)"
call_intra_edge_False: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "intra_kind (kind a')" shows "kind a' = (\<lambda>cf. False)\<^sub>\<surd>"
Subset_trans0: fixes a :: "tm"   and b :: "tm"   and c :: "tm" shows "{a SUBS b, b SUBS c} \<turnstile> a SUBS c"
map: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b list" shows "map_array f (Array x) = Array (map f x)"
summands_equal: fixes a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "a = b"   and "c = d" shows "a \<oplus>\<^bsub>W\<^esub> c = b \<oplus>\<^bsub>W\<^esub> d"
AnalBinds_lookup: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var" shows "(FBinds \<Gamma>\<cdot>ae) x = (case map_of \<Gamma> x of None \<Rightarrow> \<bottom> | Some e \<Rightarrow> fup\<cdot>(Texp e)\<cdot>(ae x))"
IDirProds_subgroup_id: fixes S :: "'d \<Rightarrow> 'a set"   and i :: "'d" assumes "subgroup (S i) G" shows "IDirProds G S {i} = S i"
netlimit_within_interior: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> interior S" shows "netlimit (at x within S) = x"
c2: assumes "ip \<in> kD rt"   and "sqn rt ip < \<pi>\<^sub>2 r" shows "P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"
permutation_of_list_permutes: fixes xs :: "('a \<times> 'a) list"   and A :: "'a set" assumes "list_permutes xs A" shows "permutation_of_list xs permutes A"
V_alt: fixes c :: "nat \<times> nat \<Rightarrow> 'b" shows "Graph.V c = fst ` Graph.E c \<union> snd ` Graph.E c"
neq_iff: fixes x :: "'a word"   and y :: "'a word" shows "(x \<noteq> y) = (x <s y \<or> y <s x)"
gen_lucas_lehmer_sequence_4_closed_form1: fixes n :: "nat" shows "real_of_int (gen_lucas_lehmer_sequence 4 n) = (2 + sqrt 3) ^ 2 ^ n + (2 - sqrt 3) ^ 2 ^ n"
to_nat_on_from_nat_into_infinite: fixes A :: "'b set"   and n :: "nat" assumes "countable A"   and "infinite A" shows "to_nat_on A (from_nat_into A n) = n"
finite_equiv_class: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "finite A"   and "r \<subseteq> A \<times> A"   and "X \<in> A // r" shows "finite X"
F_def: shows "F = (\<lambda>n x. MAX k\<in>{0..n}. birkhoff_sum f k x)"
scale_left_imp_eq: fixes a :: "complex"   and x :: "'d"   and y :: "'d" assumes "a \<noteq> 0"   and "a *\<^sub>C x = a *\<^sub>C y" shows "x = y"
map_ident: fixes t :: "'c llist" shows "lmap (\<lambda>x. x) t = t"
norm_triangle_ineq: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (plus x y) \<le> norm x + norm y"
intros_1: fixes x :: "'c"   and xs :: "'c llist" shows "Coinductive_List.lsetp (LCons x xs) x"
EQ_1: shows "C\<^sub>0 = c\<^sub>0"
OF_match_linear_ne_Undefined: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'b) flow_entry_match list"   and p :: "'d" shows "OF_match_linear \<gamma> ft p \<noteq> Undefined"
solver: fixes y :: "'a"   and p :: "'a"   and q :: "'a"   and r :: "'a" shows "(y \<in> set (depressed_solver p q r)) = (y ^ 4 + p * y\<^sup>2 + q * y + r = (0::'a))"
select_convs_3: fixes igWls :: "'f \<Rightarrow> 'c \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'f \<Rightarrow> 'h \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and igAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'h"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'c option) \<Rightarrow> ('e \<Rightarrow> 'h option) \<Rightarrow> 'c"   and igFresh :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSwapAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubst :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubstAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "igVar \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igVar"
lin_subspace: fixes x :: "'a" assumes "x \<in> V" shows "Subspace.subspace (lin x) V"
bl_to_bin_Nil: shows "bl_to_bin [] = 0"
foldr_option_Nil: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a option"   and e :: "'a" shows "foldr_option f [] e = Some e"
Ceq: shows "C = \<Inter> {T. C \<subseteq> T \<and> openin (top_of_set K) T \<and> compact T \<and> T \<subseteq> K}"
map_of_update_with_aux: fixes v :: "'d"   and k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and ps :: "('c \<times> 'd) list" shows "map_of (AList.update_with_aux v k f ps) = map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)"
monom_inv_def: fixes m :: "('a \<times> nat) list" shows "monom_inv m \<equiv> (\<forall>(x, n)\<in>set m. 1 \<le> n) \<and> distinct (map fst m) \<and> sorted (map fst m)"
IH2_5: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M2" shows "interpret_slp slp2 xs ! slp_index_lookup slp2 M2 f = interpret_floatarith f xs"
idtm1: shows "map (\<lambda>k. scalar_prod (row m k) cc) [0..<length r] = map (\<lambda>k. scalar_prod (tm ! k) cc) [0..<length r]"
surjective_on_implies_card_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" assumes "f ` A = B" shows "card ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) = card B"
compat_different_vars: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and i :: "nat"   and x :: "'a"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "fst (mems ! i) x = snd (mems ! i) x"   and "x \<notin> sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "mem\<^sub>1 x = mem\<^sub>2 x"
sum_le: shows "sum (content \<circ> snd) p \<le> content (cbox a b)"
lcm_B: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * a r\<rightarrow> c = a r\<rightarrow> b r\<rightarrow> c"
rep_eq: fixes x :: "'b \<Rightarrow> 'c" shows "comp_fun_commute_apply (sum_cfc x) = (+) \<circ> x"
distinct_2: fixes x2 :: "'b"   and x1 :: "'b" shows "Const x2 \<noteq> Var x1"
poly_cmult_distr: fixes a :: "'a"   and p :: "'a list"   and q :: "'a list" shows "a %* (p +++ q) = a %* p +++ a %* q"
reducedly_statically_complete: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and B :: "'a"   and N :: "'a set" assumes "reducedly_statically_complete_calculus Bot Inf entails Red_I Red_F"   and "B \<in> Bot"   and "calculus.reduc_saturated Inf Red_I Red_F N"   and "entails N {B}" shows "\<exists>B'\<in>Bot. B' \<in> N"
conv_radius_ratio_limit_ereal: fixes f :: "nat \<Rightarrow> 'b"   and c :: "ereal" assumes "\<forall>\<^sub>F n in sequentially. f n \<noteq> (0::'b)"   and "(\<lambda>n. ereal (norm (f n) / norm (f (Suc n)))) \<longlonglongrightarrow> c" shows "conv_radius f = c"
SPN_r_SPN_s_leg1_eq: shows "SPN_r_SPN_s.leg1 = rs.R\<^sub>1 \<odot> Maps.PRJ\<^sub>1 rs.R\<^sub>0 rs.S\<^sub>1"
diff_0_right: fixes minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"