hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'a set"   and a2 :: "'a list" assumes "local.independent a1 a2" obtains K :: "'a set" where "a1 = Ka__" and "a2 = []"   | u :: "'a" and K :: "'a set" and Us :: "'a list" where "a1 = K" and "a2 = u # Us" and "u \<in> carrier R" and "u \<notin> Span K Us" and "local.independent K Us"
density_eq: fixes A :: "'a set" assumes "A \<in> sets M" shows "set_nn_integral M A f = set_nn_integral M A f'"
rk2_fas_err_def: fixes ode_ops :: "ode_ops"   and rkp :: "floatarith"   and x0 :: "floatarith list"   and h :: "floatarith"   and cx :: "floatarith list"   and s2 :: "floatarith" shows "ode_operations.rk2_fas_err ode_ops rkp x0 h cx s2 = map (ode_operations.rk2_fas_err_nth ode_ops rkp x0 h cx s2) [0..<length x0]"
a_d_add_closure: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rdom (rel_ad x \<union> rel_ad y) = rel_ad x \<union> rel_ad y"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c list"   and b :: "'d list" shows "list_all2 R a b = ((a = []) = (b = []) \<and> (a \<noteq> [] \<longrightarrow> b \<noteq> [] \<longrightarrow> R (hd a) (hd b) \<and> list_all2 R (tl a) (tl b)))"
add_0_r: fixes a :: "'b"   and p :: "'b point" shows "ell_field_class.add a p Infinity = p"
hcomp_assoc: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" shows "(\<mu> \<star>\<^sub>D \<nu>) \<star>\<^sub>D \<tau> = \<mu> \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>"
ProP: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and j :: "'b"   and x :: "'c" assumes "Hintikka A H"   and "Nom j at i in' H"   and "Pro x at j in' H" shows "\<not> (\<^bold>\<not> Pro x) at i in' H"
Cauchy_product_sums: fixes a :: "nat \<Rightarrow> 'b"   and b :: "nat \<Rightarrow> 'b" assumes "summable (\<lambda>k. norm (a k))"   and "summable (\<lambda>k. norm (b k))" shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
lookup_cases: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'c)}}" obtains x :: "'b \<Rightarrow>\<^sub>0 'c" where "y = poly_mapping.lookup x"
enumerator_bound3: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "ls \<in> set (enumerator m nmax)"   and "1 < nmax" shows "last (butlast ls) < nmax - Suc 0"
id_func_def: fixes CC :: "('d, 'e, 'f) category_scheme" shows "id_func CC = \<lparr>om = \<lambda>A\<in>Ob\<^bsub>CC\<^esub>. A, am = \<lambda>f\<in>Ar\<^bsub>CC\<^esub>. f\<rparr>"
var_order_rel_alt_def: shows "var_order_rel = p2rel char.lexordp"
extensional'_arb: fixes f :: "'b \<Rightarrow> 's"   and A :: "'b set"   and x :: "'b" assumes "f \<in> extensional' A"   and "x \<notin> A" shows "f x = unity"
max_fun_diff_in: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rst.FINFUNC"   and "g \<in> rst.FINFUNC" shows "t.max_fun_diff f g \<in> {a \<in> Field t. f a \<noteq> g a}"
toplevel_summands_flatten_ACI_norm_image_Union: fixes r :: "'b rexp"   and s :: "'b rexp" shows "toplevel_summands (flatten PLUS (ACI_norm ` toplevel_summands r \<union> ACI_norm ` toplevel_summands s)) = ACI_norm ` toplevel_summands r \<union> ACI_norm ` toplevel_summands s"
image_mset_eq_implies_permutes: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd" assumes "finite A"   and "image_mset f (mset_set A) = image_mset f' (mset_set A)" obtains p :: "'c \<Rightarrow> 'c" where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
lb_ln_2: fixes y :: "float"   and prec :: "nat"   and x :: "float" assumes "Some y = lb_ln prec x" shows "0 < real_of_float x"
nat_of_name_def: fixes x :: "name" shows "nat_of_name x = nat_of (atom x)"
sig_red_singleD4: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'e"   and q :: "'c \<Rightarrow>\<^sub>0 'e"   and f :: "'c \<Rightarrow>\<^sub>0 'e"   and t :: "'d" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail p q f t" shows "ord.is_le_rel ord_term ord_term_strict sing_reg"
assms: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "\<And>t. t \<in> keys p \<Longrightarrow> P (except p {t})" shows "P p"
min_mult_left: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x" shows "x * min y z = min (x * y) (x * z)"
ab: fixes a :: "'a" assumes "a \<in> A" shows "a_to_b a \<in> B"
ccInf_insert: fixes A :: "'a set"   and a :: "'a" assumes "countable A" shows "Inf (insert a A) = inf a (Inf A)"
path_rtrancl_edgesD: fixes g :: "'a ugraph"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "path g u p v" shows "(u, v) \<in> (edges g)\<^sup>*"
simps_2: fixes x :: "char list"   and e :: "aexp"   and S :: "'a"   and x' :: "char list"   and e' :: "aexp"   and S' :: "'a" shows "(x ::= e {S} \<sqsubseteq> x' ::= e' {S'}) = (x = x' \<and> e = e' \<and> S \<sqsubseteq> S')"
a_2: shows "f \<in> \<Theta>(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p * ln u powr (p' - 1) / u powr (p + 1))))"
cmod_add_real_less: fixes z :: "complex"   and r :: "real" assumes "Im z \<noteq> 0"   and "r \<noteq> 0" shows "cmod (z + complex_of_real r) < cmod z + \<bar>r\<bar>"
list_product_assoc: fixes A :: "'b list set"   and B :: "'b list set"   and C :: "'b list set" shows "A \<otimes>\<^sub>l (B \<otimes>\<^sub>l C) = A \<otimes>\<^sub>l B \<otimes>\<^sub>l C"
arrI: fixes f :: "'a \<times> 'a" assumes "arr (fst f) \<and> arr (snd f) \<and> I.map (fst f) = I.map (snd f)" shows "VV.arr f"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'b list" assumes "i < \<^bold>|xs\<^bold>|" shows "xs = take i xs \<cdot> xs ! i # drop (Suc i) xs"
acc_of_list_def: fixes l :: "nat list" shows "acc_of_list l \<equiv> fold insert l {}"
\<psi>: shows "\<guillemotleft>(\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<sigma>.counit a) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<a>\<^sub>D[g, \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a]) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>.map\<^sub>0 a', g \<star>\<^sub>D \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a] : (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>.map\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>.map\<^sub>0 a' \<star>\<^sub>D g\<guillemotright>"
scaleR_collapse: fixes u :: "real"   and a :: "'c" shows "(1 - u) *\<^sub>R a + u *\<^sub>R a = a"
impl_of_update: fixes ks :: "'c list"   and v :: "'d"   and t :: "('c, 'd) trie" shows "impl_of (update ks v t) = update_trie ks v (impl_of t)"
rsquarefree_poly_roots_eq: fixes p :: "'a poly" assumes "rsquarefree p" shows "poly_roots p = mset_set {x. poly p x = (0::'a)}"
RSpan_contains_RSpans_Cons_right: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b"   and ms :: "'b list" assumes "RModule R smult M"   and "m \<in> M" shows "R_scalar_mult.RSpan R smult ms \<subseteq> R_scalar_mult.RSpan R smult (m # ms)"
cases: fixes x :: "'a list" obtains "x = []"   | uu_ :: "'a" where "x = [uu_]"   | xa :: "'a" and y :: "'a" and xs :: "'a list" where "x = xa # y # xs"
connected_root_iff4: fixes r :: "'a"   and x :: "'a" assumes "point r" shows "connected_root r x = (1 ; x = r\<^sup>T ; x\<^sup>+)"
eq: shows "image_mset (map ((`) \<sigma>)) (anonymous_profile R) = anonymous_profile R"
union_ring_is_upper_bound: fixes C :: "'c ring set"   and R :: "'c ring" assumes "\<And>R. R \<in> C \<Longrightarrow> field R"   and "\<And>R S. \<lbrakk>R \<in> C; S \<in> C\<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"   and "R \<in> C" shows "R \<lesssim> union_ring C"
False: shows "\<not> (\<alpha> = 0 \<or> \<beta> = 0)"
card_order_infinite_isLimOrd: fixes r :: "('b \<times> 'b) set" assumes "Card_order r"   and "infinite (Field r)" shows "isLimOrd r"
symmetric_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "symmetric x = (\<forall>y z. x y z = x z y)"
is_singleton_the_elem: fixes A :: "'e set" shows "is_singleton A = (A = {the_elem A})"
atLeast_borel: fixes a :: "'a" shows "{a..} \<in> sets borel"
homeomorphic_maps_compose: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'a"   and X'' :: "'c topology"   and g :: "'b \<Rightarrow> 'c"   and k :: "'c \<Rightarrow> 'b" assumes "homeomorphic_maps X Y f h \<and> homeomorphic_maps Y X'' g k" shows "homeomorphic_maps X X'' (g \<circ> f) (h \<circ> k)"
simps: fixes a1 :: "'c"   and a2 :: "'c"   and r :: "('c \<times> 'c) set" shows "((a1, a2) \<in> r\<^sup>*) = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>* \<and> (b, c) \<in> r))"
finite_outgoing: fixes u :: "nat" assumes "finite cf.V" shows "finite (cf.outgoing u)"
sWlsDisj_def: fixes SEM :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) semDom_scheme" shows "sWlsDisj SEM \<equiv> \<forall>s s' sX. sWls SEM s sX \<and> sWls SEM s' sX \<longrightarrow> s = s'"
fresh_HTuple: fixes x :: "atom"   and n :: "nat" shows "x \<sharp> HTuple n"
surjE: fixes f :: "'e \<Rightarrow> 'd"   and y :: "'d" assumes "surj f" obtains x :: "'e" where "y = f x"
nta_bisim_simps_sym_1: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta :: "('a, 'b, 'c) new_thread_action"   and t :: "'a"   and x :: "'d"   and m :: "'e" shows "nta_bisim bisim ta (NewThread t x m) = (\<exists>x' m'. ta = NewThread t x' m' \<and> bisim t (x', m') (x, m))"
dist_add_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "dist (plus a b) (plus a c) = dist b c"
heap_del_min: fixes t :: "'b tree" assumes "heap t"   and "braun t" shows "heap (del_min t)"
induct: fixes x1 :: "'d"   and x2 :: "'d"   and r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "(x1, x2) \<in> r\<^sup>+"   and "\<And>a b. (a, b) \<in> r \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>(a, b) \<in> r\<^sup>+; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
finite_restricted_funspace: fixes A :: "'a set"   and B :: "'b set"   and k :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B" shows "finite {f. f ` A \<subseteq> B \<and> {x. f x \<noteq> k x} \<subseteq> A}"
smap_szip_fst: fixes f :: "'c \<Rightarrow> 'b"   and s1 :: "'c stream"   and s2 :: "'d stream" shows "smap (\<lambda>x. f (fst x)) (szip s1 s2) = smap f s1"
C2: shows "card ((\<lambda>q. the (oldest_token q n)) ` ({token_run x n |x n. True} - Collect sink)) \<le> card ({token_run x n |x n. True} - Collect sink)"
unit_disc_fix_moebius_uminus: fixes M :: "moebius" assumes "unit_disc_fix M" shows "unit_disc_fix (- M)"
same_val_if_reduced: fixes t :: "'d ifex"   and X :: "'d set"   and s1 :: "'d \<Rightarrow> bool"   and s2 :: "'d \<Rightarrow> bool" assumes "reduced t X"   and "\<forall>x. x \<notin> X \<longrightarrow> s1 x = s2 x" shows "val_ifex t s1 = val_ifex t s2"
that: fixes \<delta> :: "real" assumes "0 < \<delta>"   and "\<And>T. \<lbrakk>T \<subseteq> S; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B\<in>\<C>. T \<subseteq> B" shows "thesis"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and oid :: "'a"   and ref :: "'a" shows "insert_spec (x # xs) (oid, Some ref) = (if x = ref then x # oid # xs else x # insert_spec xs (oid, Some ref))"
sccs_verts_conv_scc_of: shows "S.sccs_verts = S.scc_of ` pverts (mk_symmetric G)"
induct: fixes P :: "Core.term \<Rightarrow> bool"   and a0 :: "Core.term" assumes "\<And>uu_ uv_ uw_. P (Abs uu_ uv_ $ uw_)"   and "\<And>ux_ t. P t \<Longrightarrow> P (Abs ux_ t)"   and "\<And>v va u. \<lbrakk>P (Ct v va); P u\<rbrakk> \<Longrightarrow> P (Ct v va $ u)"   and "\<And>v va u. \<lbrakk>P (Fv v va); P u\<rbrakk> \<Longrightarrow> P (Fv v va $ u)"   and "\<And>v u. \<lbrakk>P (Bv v); P u\<rbrakk> \<Longrightarrow> P (Bv v $ u)"   and "\<And>v va u. \<lbrakk>P (v $ va); P u\<rbrakk> \<Longrightarrow> P (v $ va $ u)"   and "\<And>v va. P (Ct v va)"   and "\<And>v va. P (Fv v va)"   and "\<And>v. P (Bv v)" shows "P a0"
metric_Cauchy_iff2: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity X = (\<forall>j. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < inverse (real (Suc j)))"
join_distr_distr: fixes M :: "'c measure measure"   and R :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "sets M = sets (subprob_algebra R)"   and "f \<in> R \<rightarrow>\<^sub>M N" shows "join (distr M (subprob_algebra N) (\<lambda>M. distr M N f)) = distr (join M) N f"
DynCastRed: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (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"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow> \<langle>Cast C e',s'\<rangle>"
resr_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<rightarrow> y) \<cdot> (y \<rightarrow> z) \<le> x \<rightarrow> z"
bs: shows "bs = filter (\<lambda>(a, i). a \<noteq> 1) cs"
induct: fixes P :: "'b \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'b tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x AB CD. P x \<langle>AB, x, CD\<rangle>"   and "\<And>x b A B CD. x < b \<Longrightarrow> P x \<langle>\<langle>A, x, B\<rangle>, b, CD\<rangle>"   and "\<And>x b CD. x < b \<Longrightarrow> P x \<langle>\<langle>\<rangle>, b, CD\<rangle>"   and "\<And>x a b B CD. \<lbrakk>x < a; x < b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>\<langle>\<rangle>, a, B\<rangle>, b, CD\<rangle>"   and "\<And>x a b A B CD. \<lbrakk>x < a; x < b; A \<noteq> \<langle>\<rangle>; P x A\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle>"   and "\<And>a x b A CD. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>A, a, \<langle>\<rangle>\<rangle>, b, CD\<rangle>"   and "\<And>a x b B A CD. \<lbrakk>a < x; x < b; B \<noteq> \<langle>\<rangle>; P x B\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle>"   and "\<And>b x AB C D. b < x \<Longrightarrow> P x \<langle>AB, b, \<langle>C, x, D\<rangle>\<rangle>"   and "\<And>b x AB. b < x \<Longrightarrow> P x \<langle>AB, b, \<langle>\<rangle>\<rangle>"   and "\<And>b x c C AB D. \<lbrakk>b < x; x < c; C \<noteq> \<langle>\<rangle>; P x C\<rbrakk> \<Longrightarrow> P x \<langle>AB, b, \<langle>C, c, D\<rangle>\<rangle>"   and "\<And>b x c AB D. \<lbrakk>b < x; x < c\<rbrakk> \<Longrightarrow> P x \<langle>AB, b, \<langle>\<langle>\<rangle>, c, D\<rangle>\<rangle>"   and "\<And>b x c AB C. \<lbrakk>b < x; c < x\<rbrakk> \<Longrightarrow> P x \<langle>AB, b, \<langle>C, c, \<langle>\<rangle>\<rangle>\<rangle>"   and "\<And>a x c D AB C. \<lbrakk>a < x; c < x; D \<noteq> \<langle>\<rangle>; P x D\<rbrakk> \<Longrightarrow> P x \<langle>AB, a, \<langle>C, c, D\<rangle>\<rangle>" shows "P a0 a1"
nn_expl_cond_expect_const_exp: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> real"   and Y :: "'c \<Rightarrow> 'd"   and w :: "'c" assumes "\<forall>w\<in>space M. 0 \<le> X w" shows "\<integral>\<^sup>+ x. ennreal (expl_cond_expect M Y X w * indicat_real (Y -` {Y w} \<inter> space M) x) \<partial>M = \<integral>\<^sup>+ x. ennreal (expl_cond_expect M Y X x * indicat_real (Y -` {Y w} \<inter> space M) x) \<partial>M"
sf_less_sfD: fixes x :: "nat"   and y :: "nat" assumes "sf x < sf y" shows "x < y"
c: shows "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"
assms_2: fixes x :: "'a" assumes "P x" shows "x = a"
vrange_atE: fixes b :: "V"   and r :: "V"   and thesis :: "bool" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"   and "\<And>a. \<lbrakk>a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; r\<lparr>a\<rparr> = b\<rbrakk> \<Longrightarrow> thesis"   and "vsv r" shows "thesis"
abs_eq: fixes y :: "'e" shows "\<guillemotleft>y\<guillemotright> = mk\<^sub>e (\<lambda>b. y)"
iT_Min_4: fixes r :: "nat"   and m :: "nat" shows "iMin [ r, mod m ] = r"
poly_rel_mod: fixes ops :: "'a arith_ops_record"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "field_ops ops R" shows "(ring_ops.poly_rel R ===> ring_ops.poly_rel R ===> ring_ops.poly_rel R) (mod_field_poly_i ops) (mod)"
has_field_derivative_powr_of_int: fixes g :: "complex \<Rightarrow> complex"   and gd :: "complex"   and z :: "complex"   and S :: "complex set"   and n :: "int" assumes "(g has_field_derivative gd) (at z within S)"   and "g z \<noteq> 0" shows "((\<lambda>z. g z powr of_int n) has_field_derivative of_int n * g z powr (of_int n - 1) * gd) (at z within S)"
sinvar_monoI: shows "SecurityInvariant_withOffendingFlows.sinvar_mono (\<lambda>G nP. c_sinvar m G)"
card_of_Plus_commute: fixes A :: "'a set"   and B :: "'b set" shows "|A <+> B| =o |B <+> A|"
interior_open_segment: fixes a :: "'a"   and b :: "'a" shows "interior (open_segment a b) = (if 2 \<le> DIM('a) then {} else open_segment a b)"
Can_Inv: fixes t :: "'c local.term" assumes "Can t" shows "Can (Inv t)"
lNAct_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<equiv> LEAST n'. n = n' \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. latestAct_cond c t n k)"
add_closed: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "on_curve a b (local.add a p\<^sub>1 p\<^sub>2)"
eq_divide_eq_numeral1_1: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a = b / numeral w) = (if numeral w \<noteq> (0::'a) then a \<cdot> numeral w = b else a = (0::'a))"
glbound_in_ofD_glbound: fixes P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a"   and a :: "'a" assumes "glbound_in_of P x y b"   and "a \<in> P"   and "lbound_of x y a" shows "b \<succeq> a"
expand_fps_less_eq: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps" shows "(f \<le> g) = (\<forall>n. f $ n \<le> g $ n)"
less_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "(d1 \<sqsubset> d2) = (d1 = Low \<and> d2 = High)"
ibounded_by_def: shows "ind_cpa.ibounded_by = (\<lambda>(\<A>1, \<A>2) q. \<exists>q1 q2. interaction_any_bounded_by \<A>1 q1 \<and> (\<forall>cipher \<sigma>. interaction_any_bounded_by (\<A>2 cipher \<sigma>) q2) \<and> q1 + q2 \<le> q)"
ipresIGFreshAll_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'n, 'o, 'p, 'q, 'h, 'j, 'r) model_scheme"   and MOD' :: "('s, 't, 'n, 'u, 'v, 'q, 'i, 'k, 'w) model_scheme" shows "ipresIGFreshAll h hA MOD MOD' \<equiv> ipresIGFresh h MOD MOD' \<and> ipresIGFreshAbs hA MOD MOD'"
insert_var_def: fixes v :: "nat"   and f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and w :: "nat" shows "insert_var v f x w \<equiv> if w = v then x else if v < w then f (w - 1) else f w"
arrE: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)"
nextElem_congs_eq: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "xs \<cong> ys"   and "distinct xs"   and "x \<in> set xs" shows "nextElem xs (hd xs) x = nextElem ys (hd ys) x"
ltakes_one_iterates: fixes f :: "'b \<Rightarrow> 'b"   and p :: "'b" shows "ltake (enat (Suc 0)) (iterates f p) = LCons p LNil"
trans_less_add2: fixes i :: "nat"   and j :: "nat"   and m :: "nat" assumes "i < j" shows "i < m + j"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> certified_mail" shows "\<exists>A. Says A B X \<in> set evs"
Healthy_SUPREMUM: fixes A :: "'a upred set"   and H :: "'a upred \<Rightarrow> 'a upred" assumes "A \<subseteq> \<lbrakk>H\<rbrakk>\<^sub>H" shows "\<Or>(H ` A) = \<Or>A"
simps_6: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a"   and int :: "int" shows "rec_eint f1 f2 (eint int) = f1 int"
wf_cdecl'_def: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_cdecl' wf_md P = (\<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> (\<forall>m\<in>set ms. wf_overriding P D m)) \<and> (C = Thread \<longrightarrow> (\<exists>m. (run, [], Void, m) \<in> set ms)))"
cnf_of_big_or_of_literal_formulas_is: fixes fs :: "'b formula list" assumes "\<forall>f\<in>set fs. is_literal_formula f" shows "cnf \<^bold>\<Or>fs = {{literal_formula_to_literal f |f. f \<in> set fs}}"
sum_list_listset: fixes as :: "'a list"   and As :: "'a set list" assumes "as \<in> listset As" shows "sum_list as \<in> (\<Sum>A\<leftarrow>As. A)"
poly_poly_y_x: fixes p :: "'a poly poly"   and q :: "'a poly"   and y :: "'a" shows "poly (poly (poly_y_x p) q) y = poly (poly p [:y:]) (poly q y)"
The_onI': fixes U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>!x. x \<in> U \<and> P x" shows "P (the (THE x on U. P x))"
simps_1: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree"   and y21 :: "'b tree"   and y22 :: "'b"   and y23 :: "'b tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
tangent_space_linear_on: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and k :: "enat"   and p :: "'c" assumes "X \<in> manifold_eucl.tangent_space k p" shows "linear_on (manifold_eucl.diff_fun_space k) UNIV (*\<^sub>R) (*\<^sub>R) X"
c_merge: fixes xs :: "'a list"   and ys :: "'a list" shows "local.c_merge xs ys \<le> length xs + length ys"
iso_def: fixes f :: "'a arrow_of_spans_data" shows "V.iso f \<equiv> \<exists>g. V.inverse_arrows f g"
seqI: fixes f :: "(bool, 'arr list) concrete_category.arr"   and g :: "(bool, 'arr list) concrete_category.arr" assumes "arr f"   and "arr g"   and "local.dom g = cod f" shows "seq g f"