hcju/isabelleps huggingface dataset:
simps_3: fixes t :: "dB"   and s :: "dB"   and k :: "nat" shows "Abs t[s/k] = Abs (t[lift s 0/k + 1])"
open_inter_closure_rel_interior: fixes S :: "'b set"   and A :: "'b set" assumes "convex S"   and "open A" shows "(A \<inter> closure S = {}) = (A \<inter> rel_interior S = {})"
llist_all2_all_lnthI: fixes xs :: "'e llist"   and ys :: "'f llist"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "llength xs = llength ys"   and "\<And>n. enat n < llength xs \<Longrightarrow> P (lnth xs n) (lnth ys n)" shows "llist_all2 P xs ys"
simps_4: fixes r :: "'b mrexp3"   and s :: "'b mrexp3"   and f1 :: "'b set"   and n :: "bool" shows "ok3 (Plus3 r s f1 n) = (ok3 r \<and> ok3 s \<and> (let rs = Plus (strip3 r) (strip3 s) in f1 = Collect (final1 rs) \<and> n = nullable rs))"
of_char_eq_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "char"   and d :: "char" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "(comm_semiring_1.of_char times one plus zero c = comm_semiring_1.of_char times one plus zero d) = (c = d)"
subspace_def: fixes S :: "'a set" shows "csubspace S = ((0::'a) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *\<^sub>C x \<in> S))"
minus_a_inv: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "domain R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<ominus>\<^bsub>R\<^esub> b = \<ominus>\<^bsub>R\<^esub> (b \<ominus>\<^bsub>R\<^esub> a)"
nat_le_eq_or_lt: fixes a :: "nat"   and b :: "nat" shows "(a \<le> b) = (a = b \<or> a < b)"
square_free_def: fixes p :: "'b poly" shows "square_free p = (p \<noteq> 0 \<and> (\<forall>q. 0 < degree q \<longrightarrow> \<not> q * q dvd p))"
agiop_giop: shows "iop (b_ag A) = mop A"
i_append_o: fixes g :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and f :: "nat \<Rightarrow> 'e" shows "g \<circ> xs \<frown> f = map g xs \<frown> (g \<circ> f)"
rep_eq: fixes x :: "'d conjugate_space" shows "norm x = norm (from_conjugate_space x)"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom UNIV smult M smult' T" shows "ModuleHom smult M smult' T"
consis: shows "reads_consistent pending_write \<O> m (x # xs)"
s_AuthV_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Auth e\<ggreater> V\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Auth e'a" and "\<lless>\<pi>, e\<ggreater> V\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | "\<pi>' = \<pi>" and "e' = Hash (hash e)" and "closed e" and "value e"
conseqPrePost: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P' :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q' :: "'e set"   and A' :: "'e set"   and P :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q',A'"   and "P \<subseteq> P'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
face_cycle_sets_def: shows "H.face_cycle_sets = H.face_cycle_set ` arcs (del_vert u)"
abs_inverse: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c set \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c set"   and c :: "'c set" assumes "quot_type R Abs Rep"   and "\<exists>x. R x x \<and> c = Collect (R x)" shows "Rep (Abs c) = c"
path_from_to_def: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w \<equiv> (Graph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
\<Gamma>'_sel_4: fixes x :: "'v" shows "weight \<Gamma>' x = (if x \<in> A \<Gamma> then weight \<Gamma> x - d_OUT g x else weight \<Gamma> x)"
pigeonhole_stream: fixes s :: "'a set"   and \<omega> :: "'a stream" assumes "alw (HLD s) \<omega>"   and "finite s" shows "\<exists>x\<in>s. alw (ev (HLD {x})) \<omega>"
simps_2: fixes I :: "interp"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "fml_sem I (Geq \<theta> \<eta>) = {\<omega>. term_sem I \<eta> \<omega> \<le> term_sem I \<theta> \<omega>}"
abs_real_def: fixes a :: "real" shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
simps: fixes \<Sigma> :: "'a set set" shows "delta \<Sigma> = \<delta> \<times> \<Delta>\<^sub>\<times> (semi_mojmir_def.step \<Sigma> \<delta>\<^sub>M \<circ> q\<^sub>0\<^sub>M \<circ> theG)"
ass_dvd': shows "A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) dvd ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))"
lemma3_5'_v: fixes r :: "('c \<times> 'c) set"   and \<tau> :: "'d \<times> ('c \<times> 'd) list"   and \<sigma> :: "'d \<times> ('c \<times> 'd) list"   and \<sigma>' :: "'d \<times> ('c \<times> 'd) list"   and \<tau>' :: "'d \<times> ('c \<times> 'd) list"   and \<upsilon> :: "'d \<times> ('c \<times> 'd) list"   and \<upsilon>' :: "'d \<times> ('c \<times> 'd) list"   and \<tau>'' :: "'d \<times> ('c \<times> 'd) list" assumes "trans r"   and "irrefl r"   and "D2 r (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "D2 r (\<tau>', \<upsilon>, \<upsilon>', \<tau>'')" shows "D2 r (\<tau>, (fst \<sigma>, snd \<sigma> @ snd \<upsilon>), (fst \<sigma>', snd \<sigma>' @ snd \<upsilon>'), \<tau>'')"
igSubstAllIPresIGWlsAll_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.igSubstAllIPresIGWlsAll Delta MOD \<equiv> FixSyn.igSubstIPresIGWls Delta MOD \<and> FixSyn.igSubstAbsIPresIGWlsAbs Delta MOD"
product_cone_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<pi> :: "'a \<Rightarrow> 'b" shows "product_cone J C D a \<pi> \<equiv> (category J \<and> category C) \<and> discrete_diagram J C D \<and> limit_cone J C D a \<pi>"
written_value: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and l :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and l' :: "nat"   and v :: "'var" assumes "path \<sigma> l = path \<sigma>' l'"   and "\<sigma>\<^bsup>l\<^esup> \<restriction> local.reads (path \<sigma> l) = \<sigma>'\<^bsup>l'\<^esup> \<restriction> local.reads (path \<sigma> l)"   and "v \<in> local.writes (path \<sigma> l)" shows "(\<sigma>\<^bsup>Suc l\<^esup>) v = (\<sigma>'\<^bsup>Suc l'\<^esup>) v"
F_hiding: fixes P :: "'b process"   and A :: "'b set" shows "F (P \ A) = {(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}"
less_than_def: shows "less_than = pred_nat\<^sup>+"
lemma_2_5_8_a: fixes b :: "'a"   and a :: "'a" shows "b * a \<le> a"
star_n_eq_iff: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" shows "(star_n X = star_n Y) = (\<forall>\<^sub>F n in \<U>. X n = Y n)"
well_analyzed_singleton_1: fixes t :: "('fun, 'var) Term.term" shows "well_analyzed [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)]"
distinct_51: shows "string_binop.LessEqOp \<noteq> string_binop.GreaterOp"
simps_14: fixes v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Times v va) One = False"
F2set2_JF1set_JF2set: fixes a' :: "'d JF1"   and a :: "'d JF2"   and c :: "'d" assumes "a' \<in> F2set2 (dtor2 a)"   and "c \<in> JF1set a'" shows "c \<in> JF2set a"
less_minus_one_simps_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less zero (uminus one)"
g_add_def: fixes ops :: "('b, 'c, 'a, 'd) map_basic_ops_scheme"   and m1 :: "'a"   and m2 :: "'a" shows "StdBasicMapDefs.g_add ops m1 m2 \<equiv> poly_map_iteratei_defs.iteratei (bmap_op_list_it ops) m2 (\<lambda>_. True) (\<lambda>(k, v). bmap_op_update ops k v) m1"
assms_4: fixes P :: "'state"   and Q :: "'state" assumes "P \<sim>\<cdot>[L (\<alpha>, F, f)] Q" shows "P \<Turnstile> x = Q \<Turnstile> x"
cone_next_dim: fixes X :: "'a vec set"   and y :: "'a vec" assumes "n + 1 = n + 1"   and "X \<subseteq> carrier_vec (n + 1)"   and "finite X"   and "\<forall>y\<in>X. y $ n = (0::'a)"   and "y \<in> carrier_vec (n + 1)"   and "y $ n = (0::'a)" shows "(vec_first y n \<in> m.cone {vec_first y n |y. y \<in> X}) = (y \<in> next_dim.cone X)"
terminating_path_root_end_forward_finite: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e" shows "backward_finite_path_root e (x\<^sup>T)"
dim_eq_card: fixes B :: "('c, 'd) vec set"   and V :: "('c, 'd) vec set" assumes "vec.span B = vec.span V"   and "vec.independent B" shows "vec.dim V = card B"
bdd_aboveI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and M :: "'a" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x M" shows "preorder.bdd_above less_eq A"
x0: shows "x0 \<in>\<^sub>i range_tm e t"
simps: fixes a :: "'b"   and r :: "('b \<times> 'c) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
convert_term_simps_2: fixes name :: "name" shows "convert_term (free name) = free name"
timpl_closure_Var_not_Fun: fixes x :: "'d"   and f :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set" shows "Var x \<notin> timpl_closure (Fun f T) TI"
facE: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>f v" obtains p :: "'b list" and s :: "'b list" where "v = p \<cdot> u \<cdot> s"
NSLIM_mult: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and l :: "'d"   and g :: "'c \<Rightarrow> 'd"   and m :: "'d" assumes "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l"   and "g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m" shows "(\<lambda>x. f x * g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l * m"
max_bag_card_def: shows "max_bag_card \<equiv> Max bag_cards"
Derives1_take: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b" shows "take i a = take i b"
inf_greatest: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y"   and "less_eq_prod x z" shows "less_eq_prod x (inf_prod y z)"
val_size_mem: fixes a :: "val"   and b :: "val"   and t :: "(val \<times> val) list" assumes "(a, b) \<in> set t" shows "val_size a + val_size b < fun_size t"
abs_Lcm_eq: fixes K :: "int set" shows "\<bar>Lcm K\<bar> = Lcm K"
assms_1: shows "\<forall>\<^sub>F x in nhds x. x \<in> A \<longrightarrow> f x = g x"
EF: shows "D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take i a) E (take j b) \<and> LeftDerivation (drop (Suc i) a) F (drop (Suc j) b)"
eq_fold': fixes A :: "'c DBMEntry set" shows "linordered_monoid.Max A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> linordered_monoid.max x z)) None A)"
Diff: fixes a :: "'c set"   and b :: "'c set" assumes "a \<in> sigma_sets UNIV (range lessThan)"   and "b \<in> sigma_sets UNIV (range lessThan)" shows "a - b \<in> sigma_sets UNIV (range lessThan)"
freshComp: fixes x :: "name"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" assumes "x \<sharp> \<Psi>"   and "x \<sharp> \<Psi>'" shows "x \<sharp> \<Psi> \<otimes> \<Psi>'"
the_inverseD_3: fixes g :: "'a" assumes "g \<in> G" shows "binop (\<ii> g) g = e"
optionFreshChain_3: fixes xvec :: "name list" shows "xvec \<sharp>* None"
prems_1: shows "\<forall>a\<in>set M. case a of (a, b) \<Rightarrow> SecurityInvariant_complies_formal_def a b"
eNorm_triangular_ineq: fixes N :: "'c quasinorm"   and x :: "'c"   and y :: "'c" shows "eNorm N (x + y) \<le> ennreal (defect N) * eNorm N x + ennreal (defect N) * eNorm N y"
reduced_GB_is_monic_set_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "is_monic_set (reduced_GB F)"
dim_poly_dim_vec_equiv: fixes p :: "linear_poly" shows "dim_vec (lpoly_to_vec p) = dim_poly p"
simps: fixes c :: "nat" shows "NSTD c = (stat c \<noteq> 0 \<or> left c \<noteq> 0 \<or> rght c \<noteq> 2 ^ lg (rght c + 1) 2 - 1 \<or> rght c = 0)"
tr: shows "(st, a, SubnetS s' t') \<in> automaton.trans (closed (pnet np (p1 \<parallel> p2)))"
weak_source_determination: fixes \<sigma> :: "'b"   and \<rho> :: "'b"   and v :: "'a" assumes "\<sigma> \<in> \<S>"   and "\<rho> \<in> \<S>"   and "get \<sigma> = get \<rho>"   and "put \<sigma> v = put \<rho> v" shows "\<sigma> = \<rho>"
total_conv_surjective: fixes x :: "'a" shows "times_top_class.total x = surjective (x\<^sup>T)"
ipset_prefix_match_def: fixes pfx :: "'b prefix_match"   and rg :: "'b word set" shows "ipset_prefix_match pfx rg = (let pfxrg = prefix_to_wordset pfx in (rg \<inter> pfxrg, rg - pfxrg))"
has_integral_affinity01: fixes f :: "real \<Rightarrow> 'a"   and i :: "'a"   and m :: "real"   and c :: "real" assumes "(f has_integral i) {0..1}"   and "m \<noteq> 0" shows "((\<lambda>x. f (m * x + c)) has_integral (1 / \<bar>m\<bar>) *\<^sub>R i) ((\<lambda>x. x / m - c / m) ` {0..1})"
fnz: shows "f (poincare_map {x \<in> S. x \<bullet> i = c} x) \<bullet> i \<noteq> 0"
isolated_verts_impl_correct: fixes G :: "'b list \<times> ('b \<times> 'b) list" shows "pre_digraph.isolated_verts (with_proj (list_digraph G)) = set (isolated_verts_impl G)"
Image_subsetI: fixes R :: "('b \<times> 'c) set"   and R' :: "('b \<times> 'c) set"   and X :: "'b set" assumes "R \<subseteq> R'" shows "R `` X \<subseteq> R' `` X"
Resolution_LSC_pre: fixes S :: "'b literal set set"   and R :: "'b literal set" assumes "S \<turnstile> R"   and "finite R"   and "finite S"   and "Ball S finite" shows "\<exists>S' R'. \<forall>\<Gamma>. set R' = R \<and> set (map set S') = S \<and> (disj_of_clause R', image_mset disj_of_clause (mset S') + \<Gamma> \<Rightarrow>\<^sub>n \<longrightarrow> image_mset disj_of_clause (mset S') + \<Gamma> \<Rightarrow>\<^sub>n)"
UN_equiv_class2: fixes A1 :: "'b set"   and r1 :: "('b \<times> 'b) set"   and A2 :: "'c set"   and r2 :: "('c \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd set"   and a1 :: "'b"   and a2 :: "'c" assumes "equiv A1 r1"   and "equiv A2 r2"   and "congruent2 r1 r2 f"   and "a1 \<in> A1"   and "a2 \<in> A2" shows "(\<Union>x1\<in>r1 `` {a1}. \<Union> (f x1 ` r2 `` {a2})) = f a1 a2"
obj_simps_4: fixes a :: "'a" assumes "S.obj a" shows "S.dom a = a"
invertible_is_bijective: fixes \<alpha> :: "'a \<Rightarrow> 'a" assumes "\<alpha> \<in> G \<rightarrow>\<^sub>E G" shows "transformation.invertible \<alpha> = bij_betw \<alpha> G G"
set_homE_2: fixes f :: "'b set_arrow"   and U :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "f \<in> Hom\<^bsub>set_cat U\<^esub> A B" shows "B \<subseteq> U"
relcompI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and s :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d" assumes "r a b"   and "s b c" shows "(r OO s) a c"
length_nns: fixes n :: "nat"   and ps :: "(real, 'b) vec list"   and p :: "(real, 'b) vec"   and kdt :: "'b kdt" shows "length (nearest_nbors n ps p kdt) = min n (KD_Tree.size_kdt kdt + length ps)"
sinks_aux_less_append: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and ys :: "'c list" shows "sinks_aux_less I D U (xs @ ys) = sinks_aux_less I D U xs \<union> sinks_aux_less I D (U \<union> sinks_aux_less I D U xs) ys"
succ_it_def: fixes succ_list_it :: "'e \<Rightarrow> 'd \<Rightarrow> (('c \<times> 'd) list \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'd \<Rightarrow> ('c \<times> 'd) list \<Rightarrow> ('c \<times> 'd) list) \<Rightarrow> ('c \<times> 'd) list \<Rightarrow> ('c \<times> 'd) list"   and g :: "'e"   and v :: "'d" shows "graph_succ_it_defs.succ_it succ_list_it g v \<equiv> it_to_it (succ_list_it g v)"
2: shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)"
perfect_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "perfect_map X Y f \<equiv> continuous_map X Y f \<and> proper_map X Y f \<and> f ` topspace X = topspace Y"
coboundedI1: fixes a :: "'d upred"   and c :: "'d upred"   and b :: "'d upred" assumes "c \<sqsubseteq> a" shows "c \<sqsubseteq> (a \<and> b)"
zeta_partial_sum_01_asymp_equiv: fixes s :: "real" assumes "s \<in> {0<..<1}" shows "sum_upto (\<lambda>n. real n powr - s) \<sim>[at_top] (\<lambda>x. x powr (1 - s) / (1 - s))"
good_item_simps_50: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xsa :: "'varSort"   and ys :: "'varSort"   and xa :: "'var"   and y :: "'var" assumes "good Y"   and "good X"   and "xsa \<noteq> ys \<or> xa \<noteq> y"   and "fresh xsa xa Y" shows "Abs xsa xa X $[Y / y]_ys = Abs xsa xa (X #[Y / y]_ys)"
inner_sum_left_Basis: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and b :: "'c"   and f :: "'c \<Rightarrow> real" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "b \<in> Basis" shows "inner (comm_monoid_add.sum plus zero (\<lambda>i. scaleR (f i) i) Basis) b = f b"
fls_X_inv_power_base_factor: fixes n :: "nat" shows "fls_base_factor (fls_X_inv ^ n) = 1"
i_Exec_Stream_nth_calc_Previous: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream trans_fun input c n = trans_fun (input n) ((i_Exec_Comp_Stream trans_fun input c)\<^bsup>\<leftarrow> c\<^esup> n)"
transp_on_reflclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A" shows "transp_on P\<^sup>=\<^sup>= A"
assoc_simps\<^sub>A\<^sub>W\<^sub>C_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "arr \<a>[f, g, h]"
strong_postdominate_prop_smaller_path: fixes n :: "'node"   and k :: "nat"   and n' :: "'node"   and as :: "'edge list"   and n'' :: "'node" assumes "\<forall>as nx. n -as\<rightarrow>* nx \<and> k \<le> length as \<longrightarrow> n' \<in> set (sourcenodes as)"   and "n -as\<rightarrow>* n''"   and "k \<le> length as" obtains as' :: "'edge list" and as'' :: "'edge list" where "n -as'\<rightarrow>* n'" and "length as' < k" and "n' -as''\<rightarrow>* n''" and "as = as' @ as''"
assms: shows "eval (r_universal 2) [u, u, x] \<down>"
simps_3: fixes f :: "'a \<Rightarrow> 'b"   and v :: "'a"   and va :: "'a list" shows "Common.merge f (v # va) [] = v # va"
semiring_hom_axioms: fixes a :: "'b" shows "semiring_hom (\<lambda>p. poly p a)"
lincomb_def: fixes a :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "V.module.lincomb a A = (\<Oplus>\<^bsub>V\<^esub>v\<in>A. a v \<odot>\<^bsub>V\<^esub> v)"
Inv_def: fixes xs :: "'b list"   and ys :: "'b list" shows "Inv xs ys = {(x, y). x < y in xs \<and> y < x in ys}"
NotAllD: fixes v :: "i"   and \<phi> :: "'e \<Rightarrow> \<o>" assumes "\<not> [\<^bold>\<forall>x. \<phi> x in v]" shows "\<exists>x. \<not> [\<phi> x in v]"
linI': fixes a :: "real"   and x :: "'b" shows "a \<cdot> x \<in> lin x"
vsubstAbs_eq_swapAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" assumes "goodAbs A"   and "freshAbs ys y1 A" shows "A $[y1 // y2]_ys = A $[y1 \<and> y2]_ys"
prime_prime_factors: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize p) = {p}"
dim_col: fixes A :: "'a Matrix.mat"   and i :: "nat" shows "dim_vec (Matrix.col A i) = dim_row A"
insert_ops_def: fixes list :: "('a \<times> 'a option) list" shows "insert_ops list \<equiv> spec_ops list set_option"
l2_\<phi>_def: fixes p1 :: "nat \<times> int"   and p2 :: "nat \<times> int" shows "l2_\<phi> p1 p2 = LBINT x. \<phi> p1 x * \<phi> p2 x"
seq_char: fixes g :: "Arr"   and f :: "Arr" shows "seq g f = (f = AA \<and> (g = AA \<or> g = AT) \<or> f = BB \<and> (g = BB \<or> g = BT) \<or> f = AT \<and> g = TT \<or> f = BT \<and> g = TT \<or> f = TT \<and> g = TT)"
enc_preserves_binary_pred_iff_source_target_rel_preserves_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "enc_preserves_binary_pred Pred = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_preserves_binary_pred Rel Pred)"
atLeast_Suc_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..<Suc n} = sum (g \<circ> Suc) {m..<n}"