hcju/isabelleps huggingface dataset:
real_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and real_of :: "'a \<Rightarrow> real" assumes "class.real_embedding plus zero minus uminus less_eq less sgn abs times one real_of" shows "real_of one = 1"
nrows_one_matrix: fixes n :: "nat" shows "nrows (one_matrix n) = n"
sinits_sscanl': fixes f :: "'a \<rightarrow> 'b \<rightarrow> 'a"   and z :: "'a"   and xs :: "[:'b:]" shows "smap\<cdot>(sfoldl\<cdot>f\<cdot>z)\<cdot>(sinits\<cdot>xs) = sscanl\<cdot>f\<cdot>z\<cdot>xs"
grounding_\<rho>_def: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and x :: "'b" assumes "Lambda_Free_Term.ground_heads gt_sym arity_sym arity_var ground_heads_var" shows "ground_heads.grounding_\<rho> arity_sym arity_var ground_heads_var x = (let s = Hd (Sym (SOME f. f \<in> ground_heads_var x)) in apps s (replicate (the_enat (arity.arity arity_sym arity_var s - arity_var x)) s))"
heqp: fixes z :: "real \<times> 'c" assumes "z \<in> {0..1} \<times> U" shows "(h \<circ> (\<lambda>z. (snd z, fst z))) z = p (k z)"
disc_2: shows "\<not> is_RELABEL' NONSAT_PUSH'"
simps_1: fixes S :: "char list \<Rightarrow> 'av"   and X :: "char list set" shows "top_on\<^sub>o (Some S) X = top_on\<^sub>s S X"
div_set_prime: fixes n :: "nat" assumes "prime n" shows "properdiv_set n = {1}"
cancel_separate_converse_idempotent: fixes x :: "'a" assumes "is_inj x"   and "is_p_fun x" shows "(x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>) ; (x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>) = x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
finite_quotient: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "finite A"   and "r \<subseteq> A \<times> A" shows "finite (A // r)"
AE_pair_iff: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "{x \<in> space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(AE x in M1. almost_everywhere M2 (P x)) = (AE x in M1 \<Otimes>\<^sub>M M2. P (fst x) (snd x))"
map_pmf_cong: fixes p :: "'c pmf"   and q :: "'c pmf"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "map_pmf f p = map_pmf g q"
fresh_pj: fixes p :: "perm"   and Vs :: "name set"   and a :: "atom"   and j :: "name" assumes "quote_perm p Vs"   and "a \<sharp> p"   and "j \<in> Vs" shows "a \<sharp> p \<bullet> j"
n_eq_lfp_F_inf: shows "n = lfp F_inf"
inv_closed: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "inv\<^bsub>DirProds G I\<^esub> x \<in> carrier (DirProds G I)"
Z_def: shows "Z \<equiv> {x \<in> X. \<exists>c. I x = Const c}"
intro: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme" assumes "category AA"   and "category BB" shows "two_cats AA BB"
cases: fixes x :: "'a literal" obtains P :: "char list" and ts :: "'a list" where "x = Pos P__ ts__"   | P :: "char list" and ts :: "'a list" where "x = Neg P ts"
FINITE_prob_dom: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" assumes "finite PROB" shows "finite (prob_dom PROB)"
add_anything: fixes b :: "assn"   and p :: "'b Heap"   and a :: "'b \<Rightarrow> assn" assumes "<b> p <a>" shows "<b * true> p <\<lambda>r. a r * true>\<^sub>t"
real_average_minus_second: fixes b :: "real"   and a :: "real" shows "(b + a) / 2 - a = (b - a) / 2"
If: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>3\<^sub>' {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s} c\<^sub>1 { Q}"   and "\<turnstile>\<^sub>3\<^sub>' {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b False s} c\<^sub>2 { Q}" shows "\<turnstile>\<^sub>3\<^sub>' {(\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1} IF b THEN c\<^sub>1 ELSE c\<^sub>2 { Q}"
F1set2_image_Lev2: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xb :: "'e"   and xd :: "bd_type_F"   and xa :: "'d" assumes "xe \<in> Lev2 s1 s2 n xb"   and "xe @ [Inl xd] \<in> Lev2 s1 s2 (Suc n) xb"   and "rv2 s1 s2 xe xb = Inl xa" shows "xd \<in> tobd_F12 s1 xa ` F1set2 (s1 xa)"
simps_1: fixes i :: "nat"   and p :: "'d"   and R\<^sub>i :: "nat set" shows "netmap (NodeS i p R\<^sub>i) = [i \<mapsto> p]"
set_lin_prev'_in': shows "set (linearize data_prev') \<union> set (linearize data_in') = set (linearize data_prev) \<union> set (linearize data_in)"
proj2_intersection_incident_1: fixes l :: "proj2_line"   and m :: "proj2_line" shows "proj2_incident (proj2_intersection l m) l"
dickson_less_pD2: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_less_p d m p q" shows "q \<in> dgrad_p_set d m"
subst_apply_fv_subset_strand_trm: fixes P :: "('c, 'd) strand_step \<Rightarrow> 'd set"   and ac :: "poscheckvariant"   and t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list"   and V :: "'d set"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q"   and "fv t \<subseteq> \<Union> (set (map P S)) \<union> V"   and "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv (t \<cdot> \<delta>) \<subseteq> \<Union> (set (map P (S \<cdot>\<^sub>s\<^sub>t \<delta>))) \<union> fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)"
Sbis_matchT_T_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>s t" shows "matchT_T t s"
fullchar: fixes xs :: "'a list"   and q :: "'a"   and i :: "nat"   and n :: "nat" assumes "index xs q < length xs" shows "(i < length xs) = (index xs q < i \<and> i < length xs \<or> index xs q = i \<or> index xs q - n \<le> i \<and> i < index xs q \<or> i < index xs q - n)"
icard_empty_iff_enat: fixes A :: "'a set" shows "(icard A = enat 0) = (A = {})"
OclNot_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not X \<equiv> \<lambda>\<tau>. case X \<tau> of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>\<bottom>\<rfloor> \<Rightarrow> \<lfloor>\<bottom>\<rfloor> | \<lfloor>\<lfloor>x\<rfloor>\<rfloor> \<Rightarrow> \<lfloor>\<lfloor>\<not> x\<rfloor>\<rfloor>"
l_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_tag_name type_wf get_tag_name get_tag_name_locs \<equiv> ((\<forall>element_ptr h h'. reads (get_tag_name_locs element_ptr) (get_tag_name element_ptr) h h') \<and> (\<forall>h element_ptr. type_wf h \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_tag_name element_ptr)) \<and> (\<forall>h element_ptr. h \<turnstile> ok get_tag_name element_ptr \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h) \<and> (\<forall>element_ptr h. pure (get_tag_name element_ptr) h)"
equivpI: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R"   and "symp R"   and "transp R" shows "equivp R"
moebius_coprime_sum: fixes n :: "nat" shows "moebius_mu n = (\<Sum>k | k \<in> {1..n} \<and> coprime k n. unity_root n (int k))"
fls_base_factor_X: shows "fls_base_factor fls_X = 1"
not_ex: fixes P :: "'b \<Rightarrow> bool" shows "(\<nexists>x. P x) = (\<forall>x. \<not> P x)"
list_order_def: fixes ops :: "('c \<times> ('c, 'd) list_op) list"   and x :: "'c"   and y :: "'c" shows "List_Spec.list_order ops x y \<equiv> \<exists>xs ys zs. fst (interp_ops ops) = xs @ [x] @ ys @ [y] @ zs"
def_hashmap_size_integer_def: shows "def_hashmap_size = (\<lambda>_. 16)"
prime_factorization_unit: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times x one" shows "factorial_semiring.prime_factorization zero times one normalize x = {#}"
lsorted_2: shows "\<forall>x. x \<in> Y \<longrightarrow> lsorted x"
equivpE: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "equivp R" obtains "reflp R" and "symp R" and "transp R"
typed_model'_axioms: shows "typed_model' arity public Ana \<Gamma>"
counit_simps_5: shows "trg \<epsilon>' = trg f"
uinvar_def: fixes v :: "'V"   and wl :: "'V set"   and res :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option"   and it :: "('W \<times> 'V) set"   and \<sigma> :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" shows "uinvar v wl res it \<sigma> \<equiv> let (wl', res') = \<sigma> in wl' = wl \<and> (\<forall>v'\<in>V. res' v' \<in> least_map path_weight' ({res v'} \<union> {Some (p @ [(v, w, v')]) |p w. res v = Some p \<and> (w, v') \<in> succ G v - it})) \<and> (\<forall>v'\<in>V. \<forall>p. res' v' = Some p \<longrightarrow> is_path v0 p v') \<and> res' v = res v"
wf_idecl_hiding: fixes G :: "prog"   and I :: "qtname"   and i :: "iface" assumes "wf_idecl G (I, i)" shows "table_of (imethods i) hiding methd G Object under \<lambda>new old. accmodi old \<noteq> Private entails \<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old \<and> is_static new = is_static old"
distinct_37: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x51 :: "'d"   and x52 :: "'a \<Rightarrow> 'b"   and x53 :: "('a, 'b, 'c, 'd) seqp" shows "x31 \<oplus> x32 \<noteq> {x51}broadcast(x52) . x53"
simps_14: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y2 :: "'b"   and x11 :: "'c \<times> sort_dir"   and x12 :: "'c sort_key" shows "\<not> rel_sort_key R (Final y2) (NonFinal x11 x12)"
grid_odd: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and ds :: "nat set" assumes "d < length b"   and "p ! d \<noteq> b ! d"   and "p \<in> grid b ds" shows "odd (ix p d)"
assms_4: shows "\<forall>m\<le>k - 1. ws ! m = poly p (unity_root k (int m))"
matrix_inv_matrix_change_of_basis: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "matrix_change_of_basis Y X = matrix_inv (matrix_change_of_basis X Y)"
ordIso_transitive: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r =o r'"   and "r' =o r''" shows "r =o r''"
ordinal_plus_left_cancel_less: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" shows "(w + x < w + y) = (x < y)"
tensor_unity_arr: fixes f :: "'c local.term set" assumes "arr f" shows "\<I> \<otimes>\<^sub>S f = f"
preserves_section_retraction: fixes e :: "'d"   and m :: "'d" assumes "Adj.Cop.ide (Adj.Cop.comp e m)" shows "Adj.Cop.ide (Adj.Cop.comp (Adj.Cop.map e) (Adj.Cop.map m))"
exists_split_characterisations_4: shows "(\<exists>x. ((top \<le> x \<and> x \<sqinter> x \<le> x) \<and> x \<sqinter> id x \<le> top) \<and> x \<squnion> id x = top) = bounded_distrib_allegory.transitively_orientable (\<sqinter>) (\<sqinter>) (\<le>) (\<squnion>) bot id (- top)"
odd_prime_gt_2_nat: fixes p :: "nat" assumes "odd p"   and "prime p" shows "2 < p"
that: fixes T :: "'b set" assumes "subspace T"   and "T \<subseteq> U"   and "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"   and "rel_frontier S homeomorphic sphere (0::'b) 1 \<inter> T" shows "thesis"
exec: shows "step0 (st, l, r) ap = (st', l', r')"
trg_def: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.trg \<mu> \<equiv> if Span.arr \<mu> then Span.mkObj (Maps.cod (Leg1 (Dom \<mu>))) else Span.null"
is_document_ptr_kind_obtains: fixes ptr :: "(_) object_ptr" assumes "is_document_ptr_kind ptr" obtains document_ptr :: "(_) document_ptr" where "ptr = cast document_ptr"
f_smooth_on: fixes k :: "enat"   and S :: "real set" shows "k-smooth_on S f"
dominatesI: fixes a :: "'b"   and u :: "'a"   and v :: "'a" assumes "arc_to_ends G a = (u, v)"   and "a \<in> arcs G" shows "u \<rightarrow> v"
simps_1: fixes a :: "'a fps" shows "compinv a 0 = fps_nth fps_X 0"
type: shows "type_definition Rep_bit0 Abs_bit0 {0..<int CARD('f bit0)}"
additiveI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P Q s. \<lbrakk>sound P; sound Q\<rbrakk> \<Longrightarrow> t (\<lambda>s. P s + Q s) s = t P s + t Q s" shows "Transformers.additive t"
simps_2: fixes ao :: "'a option"   and y :: "'a"   and ys :: "'a list" shows "Compl_set_less_eq_aux ao [] (y # ys) = (\<not> proper_interval ao (Some y) \<and> Compl_set_less_eq_aux (Some y) [] ys)"
S5n_rels_eq: fixes M :: "('b, 'c, 'd) KripkeStructure"   and w :: "'d"   and w' :: "'d"   and a :: "'b" assumes "S5n M"   and "(w, w') \<in> relations M a" shows "relations M a `` {w} = relations M a `` {w'}"
distinct_1: fixes int :: "int" shows "eint int \<noteq> \<infinity>"
increasing: fixes x :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "x \<in> Field leq" shows "(x, link x) \<in> leq"
monoid_incl_imp_submonoid: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "Group.monoid (G\<lparr>carrier := H\<rparr>)" shows "submonoid H G"
s_UnauthV: fixes s\<^sub>0 :: "Syntax.term"   and h :: "hash"   and \<pi> :: "Syntax.term list" assumes "closed s\<^sub>0"   and "hash s\<^sub>0 = h" shows "\<lless>s\<^sub>0 # \<pi>, Unauth (Hash h)\<ggreater> V\<rightarrow> \<lless>\<pi>, s\<^sub>0\<ggreater>"
drop_all_iff: fixes n :: "nat"   and xs :: "'c list" shows "(drop n xs = []) = (length xs \<le> n)"
big_e_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and j :: "nat" shows "big_e a b x j = map (\<lambda>i. eij a b i j - 1) (filter (\<lambda>i. dij a b i j \<le> x ! i) [0..<length x])"
listset_doubletonI: fixes a :: "'a"   and A :: "'a set"   and b :: "'a"   and B :: "'a set"   and ys :: "'a list" assumes "a \<in> A"   and "b \<in> B"   and "ys = [a, b]" shows "ys \<in> listset [A, B]"
le_infI: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a"   and "x \<le> b" shows "x \<le> a \<sqinter> b"
ps'_def: shows "ps' \<equiv> sturm_squarefree' p"
assms_2: shows "ys \<noteq> zs \<or> y \<notin> {z, z1}"
Int_subset_iff: fixes C :: "'f set"   and A :: "'f set"   and B :: "'f set" shows "(C \<subseteq> A \<inter> B) = (C \<subseteq> A \<and> C \<subseteq> B)"
add_left: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real" assumes "vnm_utility carrier le u"   and "\<And>x y. le x y \<Longrightarrow> f x \<le> f y" shows "vnm_utility carrier le (\<lambda>x. f x + u x)"
intros_11: fixes \<phi> :: "'a ltln"   and \<chi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<turnstile>\<^sub>s \<chi>"   and "\<psi> \<turnstile>\<^sub>s \<chi>" shows "\<phi> U\<^sub>n \<psi> \<turnstile>\<^sub>s \<chi>"
assms_1: shows "LeftDerivationIntro \<alpha> i r ix D j \<gamma>"
ffun_member_alt_def: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) ffun" shows "((x, y) \<in>\<^sub>f f) = (x \<in> fdom f \<and> f(x)\<^sub>f = y)"
upper_interval_of: fixes a :: "'a" shows "upper (interval_of a) = a"
npMulElmL: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "0 \<le> n" shows "x \<cdot>\<^sub>r x^\<^bsup>R n\<^esup> = x^\<^bsup>R Suc n\<^esup>"
inv_solve_left: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and c :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier P"   and "b \<in> carrier P"   and "c \<in> carrier P" shows "(a = \<ominus>\<^bsub>P\<^esub> b \<oplus>\<^bsub>P\<^esub> c) = (c = b \<oplus>\<^bsub>P\<^esub> a)"
ord_eq_le_trans: fixes a :: "'e"   and b :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
stutter_extend_def: fixes G :: "('d, 'e) graph_rec_scheme" shows "stutter_extend G \<equiv> \<lparr>g_V = g_V G, g_E = stutter_extend_edges (g_V G) (g_E G), g_V0 = g_V0 G, \<dots> = graph_rec.more G\<rparr>"
multi_member_split: fixes x :: "'k"   and M :: "'k multiset" assumes "x \<in># M" shows "\<exists>A. M = add_mset x A"
vsv_ex1_app2: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" shows "((r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A)"
stutter_equivE: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "\<sigma> \<approx> \<tau>" obtains f :: "nat \<Rightarrow> nat" and g :: "nat \<Rightarrow> nat" where "stutter_sampler f \<sigma>" and "stutter_sampler g \<tau>" and "\<sigma> \<circ> f = \<tau> \<circ> g"
k_i: fixes i :: "nat" assumes "flow_network \<Delta> (f_i i)"   and "value_flow \<Delta> (f_i i) = value_flow \<Delta> (f_aux i)" shows "flow (residual_network (f_i i)) (k_i i)"
subreachableI: fixes A :: "(('a \<Rightarrow> 'b) \<times> 'c, 'd) automaton"   and J :: "'a set"   and U :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" assumes "local_steps (automaton.trans A) J"   and "other_steps U J" shows "subreachable A U J"
not_eq_order_implies_strict: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term" assumes "a \<noteq> b"   and "b \<unrhd> a" shows "b \<rhd> a"
asig_of_par: fixes A :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and B :: "('d set \<times> 'd set \<times> 'd set) \<times> 'f set \<times> ('f \<times> 'd \<times> 'f) set" shows "asig_of (A || B) = asig_comp (asig_of A) (asig_of B)"
mergeSL: fixes r :: "('c \<times> 'c) set"   and L :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "wo_rel r"   and "adm_woL r L" shows "adm_wo r (wo_rel.mergeSL r S L)"
invertible_mat_four_block_mat_lower_right: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "invertible_mat A" shows "invertible_mat (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 n) (0\<^sub>m n 1) A)"
likelihood_def: fixes s :: "'a"   and os :: "'b list" shows "likelihood s os = emeasure (T' (I s)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>o\<^sub>0 xs \<omega>'. \<omega> = (s, o\<^sub>0) ## xs @- \<omega>' \<and> map snd xs = os}"
lin_independent_imp_independent_take: fixes ms :: "'v list"   and n :: "nat" assumes "set ms \<subseteq> V"   and "R_lin_independent ms" shows "R_lin_independent (take n ms)"
coprime_dvd_mult_right_iff: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "coprime a c" shows "(a dvd c * b) = (a dvd b)"
refl: fixes a :: "'d DBMEntry" shows "a \<preceq> a"
Diff_partition: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "A \<union> (B - A) = B"
distinct_27: fixes x41 :: "Vars.var"   and x42 :: "exp"   and x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp" shows "Lam [x41]. x42 \<noteq> (x61 ? x62 : x63)"
reduce_range_destination_def: fixes l :: "('b \<times> 'c wordinterval) list" shows "reduce_range_destination l \<equiv> let ps = remdups (map fst l); c = \<lambda>s. (wordinterval_Union \<circ> map snd \<circ> filter ((=) s \<circ> fst)) l in map (\<lambda>p. (p, c p)) ps"
Class_def: shows "Class = (\<lambda>a\<in>S. {b \<in> S. (b, a) \<in> E})"
set_map: fixes f :: "'e \<Rightarrow> 'c"   and v :: "('d, 'e) fmap" shows "fmran' (fmmap f v) = f ` fmran' v"
neg_minus_divide_le_eq: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq (uminus (divide b c)) a = less_eq (times a c) (uminus b)"
i_in_t2: shows "to_F \<iota> \<in> T2"
strict_prefix_simps_2: fixes x :: "'b"   and xs :: "'b list" shows "strict_prefix [] (x # xs) = True"
SKIP_le: fixes S :: "'b"   and c :: "'b acom" shows "(SKIP {S} \<le> c) = (\<exists>S'. c = SKIP {S'} \<and> S \<le> S')"
open_greaterThanLessThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order_topology less_eq less open" shows "open (ord.greaterThanLessThan less a b)"
sublist_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and i :: "nat"   and j :: "nat" shows "WB_Sort.sublist (map f xs) i j = map f (WB_Sort.sublist xs i j)"
filled_gap_0_init_tau: fixes f\<^sub>0 :: "nat \<Rightarrow> nat option"   and i :: "nat"   and t :: "nat" assumes "f\<^sub>0 = \<tau> i(gap i t \<mapsto> 0)" shows "f\<^sub>0 \<triangleright> (t + 1) = block i t"
simps_14: fixes x31 :: "com"   and x32 :: "com"   and x21 :: "char list"   and x22 :: "aexp" shows "x31;; x32 \<noteq> x21 ::= x22"
filtercomap_mono: fixes F :: "'a filter"   and F' :: "'a filter"   and f :: "'b \<Rightarrow> 'a" assumes "F \<le> F'" shows "filtercomap f F \<le> filtercomap f F'"
wf_Form_quot_dbfm: fixes A :: "dbfm"   and e :: "(name, hf) finfun" assumes "wf_dbfm A" shows "Form \<lbrakk>quot_dbfm A\<rbrakk>e"
sum_is_submodule: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and N1 :: "'d set"   and N2 :: "'d set" assumes "module R M"   and "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M" shows "LinearCombinations.submodule R (module.submodule_sum M N1 N2) M"
mem_sum_subgs: fixes H :: "'a set"   and K :: "'a set"   and h :: "'a"   and k :: "'a" assumes "M +> H"   and "M +> K"   and "h \<in> H"   and "k \<in> K" shows "h \<plusminus> k \<in> H \<minusplus> K"
MkI_eq_0_iff: fixes n :: "int" shows "(MkI\<cdot>n = 0) = (n = 0)"
inverse_nonpositive_iff_nonpositive: fixes a :: "'a" shows "(inverse a \<le> (0::'a)) = (a \<le> (0::'a))"
node\<^sub>i_cases: fixes xs :: "('b btree \<times> 'b) list"   and k :: "nat" shows "length xs \<le> k \<or> (\<exists>ls sub sep rs. split_half xs = (ls, (sub, sep) # rs))"
lexord_trans: fixes x :: "'g list"   and y :: "'g list"   and r :: "('g \<times> 'g) set"   and z :: "'g list" assumes "(x, y) \<in> lexord r"   and "(y, z) \<in> lexord r"   and "trans r" shows "(x, z) \<in> lexord r"
homeomorphism_def: fixes s :: "'d set"   and t :: "'e set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "homeomorphism s t f g = ((\<forall>x\<in>s. g (f x) = x) \<and> f ` s = t \<and> continuous_on s f \<and> (\<forall>y\<in>t. f (g y) = y) \<and> g ` t = s \<and> continuous_on t g)"
chain_tllist_ord_tSup: fixes b :: "'g"   and A :: "('f, 'g) tllist set"   and xs :: "('f, 'g) tllist" assumes "Complete_Partial_Order.chain (tllist_ord b) A"   and "xs \<in> A" shows "tllist_ord b xs (tSup b A)"