hcju/isabelleps huggingface dataset:
ZObisL_intro: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list" assumes "length cl = length dl"   and "\<And>n. \<lbrakk>n < length cl; n < length dl\<rbrakk> \<Longrightarrow> cl ! n \<approx>01 dl ! n" shows "ZObisL cl dl"
assms_2: shows "(y, \<sigma>') \<in> set_spmf (exec_gpv random_oracle \<A> \<sigma>)"
path_assoc: fixes p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and r :: "real \<Rightarrow> 'a" assumes "pathfinish p = pathstart q"   and "pathfinish q = pathstart r" shows "path (p +++ q +++ r) = path ((p +++ q) +++ r)"
hom_mult: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> hom G H"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
red_lA_incr: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "\<lfloor>dom l\<rfloor> \<squnion> \<A> e \<sqsubseteq> \<lfloor>dom l'\<rfloor> \<squnion> \<A> e'"
conv1: shows "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> map_converter id id f g 1\<^sub>C"
simple_path_join_loop_eq: fixes g2 :: "real \<Rightarrow> 'a"   and g1 :: "real \<Rightarrow> 'a" assumes "pathfinish g2 = pathstart g1"   and "pathfinish g1 = pathstart g2" shows "simple_path (g1 +++ g2) = (arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2})"
finite_ne_induct: fixes F :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite F"   and "F \<noteq> {}"   and "\<And>x. P {x}"   and "\<And>x F. \<lbrakk>finite F; F \<noteq> {}; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
that: fixes ns :: "nat list" assumes "DirProds (\<lambda>n. Z (ns ! n)) {..<length ns} \<cong> G"   and "successively (dvd) ns"   and "length ns \<le> card gen" shows "thesis"
ennreal_rat_dense: fixes x :: "ennreal"   and y :: "ennreal" assumes "x < y" shows "\<exists>r. x < ennreal (real_of_rat r) \<and> ennreal (real_of_rat r) < y"
Gromov_product_geodesic_segment: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and t :: "real" assumes "geodesic_segment_between G x y"   and "t \<in> {0..dist x y}" shows "Gromov_product_at x y (geodesic_segment_param G x t) = t"
joinable_def: fixes R :: "('c \<times> 'd) set" shows "joinable R = {(x, y). \<exists>z. (x, z) \<in> R \<and> (y, z) \<in> R}"
right_unique_rel_resumption: fixes R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "right_unique R1"   and "right_unique R2" shows "right_unique (Resumption.resumption.rel_resumption R1 R2)"
cobounded2: fixes b :: "'d"   and a :: "'d" shows "b \<le> a \<sqinter> b"
insertCI: fixes a :: "'c"   and B :: "'c set"   and b :: "'c" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> Set.insert b B"
dg_prod_2_op_dg_dg_Arr: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "(op_dg \<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr> = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
assms_1: shows "((f, l), p, f', l') \<in> trcl pr_algo_lts'"
fst_in_set_lemma: fixes x :: "'c"   and y :: "'d"   and l :: "('c \<times> 'd) list" assumes "(x, y) \<in> set l" shows "x \<in> fst ` set l"
PiM_component: fixes i :: "'s" assumes "i \<in> UNIV" shows "distr (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x))) (measure_pmf (K i)) (\<lambda>\<omega>. \<omega> i) = measure_pmf (K i)"
assms_1: shows "\<forall>S\<in>I. convex S \<and> rel_open S"
fct_gen_subalgebra_sets: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd" shows "sets (fct_gen_subalgebra M N X) = sigma_sets (space M) {X -` B \<inter> space M |B. B \<in> sets N}"
liveVal'_inv: fixes v :: "'val"   and vs :: "'val list" assumes "liveVal' g (v # vs)"   and "v \<noteq> chooseNext g" obtains vs' :: "'val list" where "step.liveVal' g (v # vs')"
dec_interp_not_Inr: fixes n :: "nat"   and FO :: "nat set"   and x :: "('a \<times> bool list) list"   and i :: "nat"   and P :: "nat set" assumes "dec_interp n FO x ! i = Inr P"   and "i \<in> FO"   and "i < n" shows "False"
rOutput: fixes M :: "'a"   and K :: "'a"   and N :: "'a" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop (pOutput M N) (K\<langle>N\<rangle>) P"
a2c_eqvt: fixes \<pi> :: "perm" shows "\<pi> \<bullet> a2c = a2c"
components_struct: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data' :: "nat \<times> 'g" assumes "struct_spec sel ap ab compl"   and "ps \<noteq> []"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "sps = sel gs bs ps data"   and "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)" shows "component_of_term ` Keys (args_to_set (gs, ab gs bs hs data', ap gs bs (ps -- sps) hs data')) = component_of_term ` Keys (args_to_set (gs, bs, ps))"
isLimOrd_succ: fixes r :: "('a \<times> 'a) set"   and i :: "'a" assumes "wo_rel r"   and "isLimOrd r"   and "i \<in> Field r" shows "succ r i \<in> Field r"
WT_fv: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E \<turnstile> e :: T" shows "fv e \<subseteq> dom E"
finsertE: fixes a :: "'m"   and b :: "'m"   and A :: "'m fset" assumes "a |\<in>| finsert b A" obtains "a = b"   | "a |\<in>| A"
addpre_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and npre :: "nat set" shows "addpre r npre \<equiv> let (dsn, dsk, flag, hops, nhip, pre) = r in (dsn, dsk, flag, hops, nhip, pre \<union> npre)"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'a" assumes "A.ide x" shows "\<epsilon> x = F (G x)"
projection_subset_elim: fixes Y :: "'a set"   and X :: "'a set"   and l :: "'a list" assumes "Y \<subseteq> X" shows "l \<upharpoonleft> X \<upharpoonleft> Y = l \<upharpoonleft> Y"
upper_density_large_k: fixes d :: "real" assumes "0 < d"   and "d \<le> 1" shows "\<exists>k. ennreal (1 - d) < emeasure M {x \<in> space M. upper_asymptotic_density {n. \<exists>l\<in>{k..n}. u n x - u (n - l) x \<le> - d * real l} < d}"
map2_val_compat'_imp_map2_val_compat: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and g :: "('b \<times> 'c) list \<times> 'a \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and ko :: "'a" assumes "oalist_raw.map2_val_compat' rep_key_order g" shows "ko.map2_val_compat (rep_key_order ko) (\<lambda>zs. fst (g (zs, ko)))"
sorted_list_of_fmap_is_insort_key_fst: fixes m :: "('d, 'c) fmap"   and k :: "'d"   and v :: "'c" assumes "m $$ k = None" shows "sorted_list_of_fmap (m(k $$:= v)) = insort_key fst (k, v) (sorted_list_of_fmap m)"
t_Var: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>" shows "\<Gamma> \<turnstile> Var x : T"
leD: fixes y :: "'b Word.word"   and x :: "'b Word.word" assumes "y \<le>s x" shows "\<not> x <s y"
wf: shows "is_density_expr (vs, vs', \<Gamma>, \<delta>) REAL f"
add_diff_cancel_left': fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and a :: "'r"   and b :: "'r" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
mset_ran_combine_eqs: fixes a :: "int \<Rightarrow> 'b"   and r\<^sub>1 :: "int set"   and b :: "int \<Rightarrow> 'b"   and r\<^sub>2 :: "int set" assumes "mset_ran a r\<^sub>1 = mset_ran b r\<^sub>1"   and "mset_ran a r\<^sub>2 = mset_ran b r\<^sub>2"   and "r\<^sub>1 \<inter> r\<^sub>2 = {}" shows "mset_ran a (r\<^sub>1 \<union> r\<^sub>2) = mset_ran b (r\<^sub>1 \<union> r\<^sub>2)"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list" assumes "\<And>f x xs. (xs \<noteq> [] \<Longrightarrow> P f xs) \<Longrightarrow> P f (x # xs)"   and "\<And>a. P a []" shows "P a0 a1"
assms_3: shows "simplex (mat_leqb_eqc A b c) = Inr X"
tendsto_mult_left: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter"   and c :: "'c" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
grid_shift_base: fixes ds :: "nat set"   and ds' :: "nat set"   and b :: "(nat \<times> int) list"   and dm :: "nat"   and p :: "(nat \<times> int) list" assumes "ds \<inter> ds' = {}"   and "b \<in> sparsegrid' dm"   and "p \<in> grid (base (ds' \<union> ds) b) (ds' \<union> ds)" shows "base ds' p \<in> grid (base (ds \<union> ds') b) ds"
lfp_while'': fixes f :: "'a set \<Rightarrow> 'a set" assumes "mono f"   and "finite (lfp f)" shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"
min_set_list_dnf_to_dnf_subst_dnf': fixes X :: "'b ltln list set"   and m :: "'b ltln \<Rightarrow> 'b ltln option" assumes "finite X" shows "min_set (list_dnf_to_dnf (subst_dnf' X m)) = min_set (subst_dnf (list_dnf_to_dnf X) m)"
cop_perp2__col: fixes A :: "'a"   and B :: "'a"   and Y :: "'a"   and Z :: "'a"   and X :: "'a" assumes "Coplanar A B Y Z"   and "X Y Perp A B"   and "X Z Perp A B" shows "Col X Y Z"
exp_base_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "set (exp_base x n) \<subseteq> carrier R"
\<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V'_def: shows "\<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V' \<equiv> {(aodv_init, \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PAodv)}"
A: fixes i :: "'b" assumes "i \<in> J" shows "A i \<in> events (t i)"
leg1_in_hom_2: shows "\<guillemotleft>(tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>"
simps_1: fixes x :: "float" shows "dest_Num_fa (Num x) = Some x"
filtration_space: fixes s :: "'b"   and t :: "'b" assumes "s \<le> t" shows "space (F s) = space (F t)"
AssertAllState''_precond: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "let s' = assert_bound_loop ats (init t) in \<triangle> (\<T> s') \<and> \<nabla> s' \<and> (\<not> \<U> s' \<longrightarrow> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s' \<and> \<diamond> s')"
iso_iff: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" shows "(h \<in> Group.iso G H) = (h \<in> hom G H \<and> h ` carrier G = carrier H \<and> inj_on h (carrier G))"
timpl_closure_set_supset: fixes FP :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and closure :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" assumes "\<forall>t\<in>FP. t \<in> closure"   and "\<forall>t\<in>closure. \<forall>(a, b)\<in>TI. \<forall>s\<in>set \<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle>. s \<in> closure" shows "timpl_closure_set FP TI \<subseteq> closure"
term_variants_P: fixes T :: "('c, 'd) Term.term list"   and S :: "('c, 'd) Term.term list"   and P :: "'c \<Rightarrow> 'c list"   and g :: "'c"   and f :: "'c" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)"   and "g \<in> set (P f)" shows "term_variants_pred P (Fun f T) (Fun g S)"
abs_eq: fixes g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "pure_dual g x = B (g x)"
rotater1_map2: fixes xs :: "'b list"   and ys :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "length xs = length ys" shows "rotater1 (map2 f xs ys) = map2 f (rotater1 xs) (rotater1 ys)"
get_M_Element_preserved8: fixes element_ptr :: "(_) element_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('j \<Rightarrow> 'j) \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme"   and v :: "'j"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'k" assumes "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
ring_simprules_26: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> y = \<ominus>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> y)"
least_def: fixes x :: "'c option"   and y :: "'c option" shows "least x y = (case x of None \<Rightarrow> y | Some x' \<Rightarrow> case y of None \<Rightarrow> x | Some y' \<Rightarrow> if x' \<le> y' then x else y)"
Rep_af_inverse: fixes x :: "'d af" shows "pure (??.Abstract_AF.af.Rep_af x) = x"
i_shrink_eq_Msg_iEx_iAll_cut_greater_conv2: fixes k :: "nat"   and m :: "'a message_af"   and s :: "nat \<Rightarrow> 'a message_af"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg" shows "((s \<div> k) t = m) = (\<diamond> t1 [\<dots>k - 1] \<oplus> t * k. s t1 = m \<and> (\<box> t2 ([\<dots>k - 1] \<oplus> t * k) \<down>> t1. s t2 = NoMsg))"
simps: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" shows "rel_pmf R p q = (\<exists>pq. (\<forall>x xa. (x, xa) \<in> set_pmf pq \<longrightarrow> R x xa) \<and> map_pmf fst pq = p \<and> map_pmf snd pq = q)"
ii_in_disc_o_x_axis: shows "ii\<^sub>h \<notin> disc o_x_axis"
tendsG: fixes x :: "'a" assumes "x \<notin> NG" shows "(\<lambda>n. G n x) \<longlonglongrightarrow> (if x \<in> S then g x else (0::'b))"
step: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "\<phi> \<omega>"   and "(\<phi> suntil \<psi>) (stl \<omega>)" shows "(\<phi> suntil \<psi>) \<omega>"
times_mat_def: fixes A :: "'c mat"   and B :: "'c mat" shows "A * B \<equiv> mat (dim_row A) (dim_col B) (\<lambda>(i, j). row A i \<bullet> col B j)"
le_funI: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "\<And>x. f x \<le> g x" shows "f \<le> g"
in_listsp_conv_set: fixes A :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "listsp A xs = (\<forall>x\<in>set xs. A x)"
fxM: shows "finite {y \<in> S. x \<le> y}"
rel_refl_strong: fixes x :: "'c pmf"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> Ra z z" shows "rel_pmf Ra x x"
first_edge_in_edges: fixes xs :: "'a list" assumes "tl xs \<noteq> []" shows "H.first_edge_of_walk xs \<in> H.edges_of_walk xs"
perp_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong B A C D"
flip_at_base_simps_1: fixes a :: "'b"   and b :: "'b" assumes "sort_of (atom a) = sort_of (atom b)" shows "(a \<leftrightarrow> b) \<bullet> a = b"
drf_lemma: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "P \<turnstile> (E, ws) \<surd>"   and "E \<in> \<E>"   and "correctly_synchronized P \<E>"   and "\<And>r. r \<in> read_actions E \<Longrightarrow> P,E \<turnstile> ws r \<le>hb r" shows "sequentially_consistent P (E, ws)"
app_pterm_def: fixes t :: "pterm"   and u :: "pterm" shows "app t u = t $\<^sub>p u"
ord_less_eq_trans: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a \<sqsubset> b"   and "b = c" shows "a \<sqsubset> c"
stutter_equivI: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> nat"   and \<tau> :: "nat \<Rightarrow> 'a" assumes "stutter_sampler f \<sigma>"   and "stutter_sampler g \<tau>"   and "\<sigma> \<circ> f = \<tau> \<circ> g" shows "\<sigma> \<approx> \<tau>"
sc_minus_am1: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and a :: "'f"   and m :: "'e" assumes "R module M"   and "a \<in> carrier R"   and "m \<in> carrier M" shows "-\<^sub>a\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> m = (-\<^sub>a\<^bsub>R\<^esub> a) \<cdot>\<^sub>s\<^bsub>M\<^esub> m"
intros_7: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and G :: "'b formula" assumes "F, \<Gamma> \<Rightarrow> \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta>"
mem_Times_iff: fixes x :: "'d \<times> 'e"   and A :: "'d set"   and B :: "'e set" shows "(x \<in> A \<times> B) = (fst x \<in> A \<and> snd x \<in> B)"
all_conj_distrib: fixes P :: "'i \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
StepStatus_expand: fixes ST :: "('d, 'e, 'f) status"   and TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and U :: "'f update" shows "Abs_status (HA ST, StepConf (HA ST) (Conf ST) TS, StepActEvent TS, U !!! Value ST) = StepStatus ST TS U"
afilter_sound: fixes s :: "char list \<Rightarrow> int"   and S :: "'av Abs_State.st option"   and e :: "aexp"   and a :: "'av" assumes "s \<in> \<gamma>\<^sub>o S"   and "aval e s \<in> \<gamma> a" shows "s \<in> \<gamma>\<^sub>o (afilter e a S)"
iter_def: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<equiv> lfp (\<lambda>x::'a::refinement_lattice. nil \<sqinter> c ; x)"
sorted_wrt_many: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "trans R" shows "Linorder_Relations.sorted_wrt R (x # y # xs) = ((x, y) \<in> R \<and> Linorder_Relations.sorted_wrt R (y # xs))"
right_residual_bl: fixes a :: "'a"   and x :: "'a"   and b :: "'a" shows "(a * x \<le> b) = (x \<le> a r\<rightarrow> b)"
3: shows "length zs_ws \<le> card {x. poly (p1 - p2) x = 0}"
n_split_omega_mult_pp: fixes xs :: "'a"   and xo :: "'a" assumes "xs \<sqinter> - - xo = xo"   and "xo \<sqinter> top = xo" shows "- (- top \<sqinter> top) \<sqinter> top \<sqinter> xo = xs \<sqinter> (- (- xo \<sqinter> top) \<sqinter> top) \<sqinter> top"
remove1_invar: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.invar bq" shows "SkewBinomialHeapStruc.invar (remove1 t bq)"
arc_imp_path: fixes g :: "real \<Rightarrow> 'c" assumes "arc g" shows "path g"
comp_ide_self: fixes a :: "('obj, 'obj) concrete_category.arr" assumes "ide a" shows "a \<cdot> a = a"
ltrm_eq_imp_lcf_eq: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "monom P (p (deg R p)) (deg R p) = monom P (q (deg R q)) (deg R q)" shows "p (deg R p) = q (deg R q)"
op_type_alt_simps_13: fixes op :: "boolean_binop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type" shows "boolean_binop_type op \<tau> \<sigma> \<rho> = (op = AndOp \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?] \<or> op = OrOp \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?] \<or> op = XorOp \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?] \<or> op = ImpliesOp \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?])"
arr_runit: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.arr C (monoidal_category.runit C T \<alpha> \<iota> a)"
simple_function_restrict_space_ennreal: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "simple_function (restrict_space M \<Omega>) f = simple_function M (\<lambda>x. f x * indicator \<Omega> x)"
sign_simps_36: fixes am :: "'d"   and bm :: "'d"   and ck :: "'d" shows "am * (bm + ck) = am * bm + am * ck"