hcju/isabelleps huggingface dataset:
nat_dvd_iff: fixes z :: "int"   and m :: "nat" shows "(nat z dvd m) = (if 0 \<le> z then z dvd int m else m = 0)"
polys_rel_full_polys_rel: shows "polys_rel_full = Id \<times>\<^sub>r polys_rel"
restrict_to_halfspaces_invar_clw_ref: fixes X :: "'d set"   and a :: "'e \<Rightarrow> 'd set"   and sctns :: "'d sctn set"   and I :: "('f \<times> 'e) set" shows "sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>X. get_invar a X \<bind> (\<lambda>(X, i). restrict_to_halfspaces sctns X \<bind> (\<lambda>R. ASSERT (R \<subseteq> a i) \<bind> (\<lambda>_. RETURN (with_invar i (filter_empty_ivls (mk_coll R)) ::: clw_rel (\<langle>I, lvivl_rel\<rangle>invar_rel a)))))) (\<lambda>X Y. RETURN (Y \<union> X))) \<le> restrict_to_halfspaces_invar_clw sctns X"
basis_def: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme"   and A :: "'g set" assumes "vectorspace K V" shows "vectorspace.basis K V A = (\<not> module.lin_dep K V A \<and> module.span K V A = carrier V \<and> A \<subseteq> carrier V)"
a: shows "set (divisors_nat 6) = {1, 2, 3, 6}"
one_le_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq one a" shows "less_eq one (power.power one times a n)"
disjnt_iff: fixes A :: "'b set"   and B :: "'b set" shows "disjnt A B = (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
final_face_code_2: fixes vs :: "nat list" shows "final_face (Face vs Nonfinal) = False"
beukers_integral3_integrable: shows "set_integrable lborel D' (\<lambda>(w, x, y). (x * (1 - x) * y * (1 - y) * w * (1 - w)) ^ n / (1 - (1 - x * y) * w) ^ (n + 1))"
le_cube_hmset: fixes m :: "hmultiset" shows "m \<le> m * (m * m)"
induct: fixes P :: "('c \<times> 'd) list \<Rightarrow> bool"   and a0 :: "('c \<times> 'd) list" assumes "P []"   and "\<And>p ps. P (delete (fst p) ps) \<Longrightarrow> P (p # ps)" shows "P a0"
i: shows "{{v} |v. v \<in> V} = (\<lambda>v. {v}) ` V"
sup_least: fixes y :: "'g multiset"   and x :: "'g multiset"   and z :: "'g multiset" assumes "y \<subseteq># x"   and "z \<subseteq># x" shows "y \<union># z \<subseteq># x"
assms_3: fixes x :: "real" assumes "b \<le> x"   and "x \<le> a" shows "f x \<le> 0"
dim_eq: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W :: "('b, 'f, 'g) module_scheme"   and T :: "'d \<Rightarrow> 'f" assumes "linear_map K V W T"   and "vectorspace.fin_dim K V"   and "inj_on T (carrier V)"   and "T ` carrier V = carrier W" shows "vectorspace.dim K V = vectorspace.dim K W"
prems: shows "set_kdt (Node k v l r) \<union> set ps - set (nearest_nbors n ps p (Node k v l r)) \<noteq> {}"
row_def: fixes i :: "'f"   and A :: "(('d, 'e) vec, 'f) vec" shows "row i A = (\<chi>j. A $ i $ j)"
isACT_VAR: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and X :: "'a" assumes "guarded sys" shows "isACT sys (VAR X) = isACT sys (sys X)"
inclusion: fixes f :: "'c" assumes "Arr f" shows "C.arr f"
simps_10: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f1 :: "'m"   and f2 :: "'m"   and f3 :: "'n \<Rightarrow> 'm"   and x3 :: "'n" assumes "countable_bipartite_web \<Gamma>" shows "(case bi.Inner x3 of bi.SOURCE \<Rightarrow> f1 | bi.SINK \<Rightarrow> f2 | bi.Inner x \<Rightarrow> f3 x) = f3 x3"
no_Emptys_pass1: fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "no_Emptys (pass\<^sub>1 hs)"
injectivity: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" assumes "to_int_mod_ring x = to_int_mod_ring y" shows "x = y"
Example2_lemma2_aux2: fixes j :: "nat"   and s :: "nat"   and b :: "nat \<Rightarrow> 'b"   and t :: "'b" assumes "j \<le> s" shows "sum (b(s := t)) {0..<j} = sum b {0..<j}"
total_on_lexord_less_than_char_linear2: fixes xs :: "char list"   and ys :: "char list" assumes "xs \<noteq> ys" shows "((xs, ys) \<notin> lexord less_than_char) = ((ys, xs) \<in> lexord less_than_char)"
smult_conv_mult: fixes s :: "'a"   and p :: "'a mpoly" shows "MPoly_Type.smult s p = MPoly_Type.monom 0 s * p"
Range_converse: fixes r :: "('d \<times> 'e) set" shows "Range (r\<inverse>) = Domain r"
i_p: shows "enat i < llength Sts \<and> C \<in> lnth (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts) i"
axiom8_F: fixes x :: "('o, 'n, 'p) F" shows "|F'set3 x| \<le>o F'bd"
Max_ge: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set"   and x :: "'g" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
Diff_Un: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A - (B \<union> C) = (A - B) \<inter> (A - C)"
all_shared_acquired_or_owned: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "all_shared sb \<subseteq> all_acquired sb \<union> \<O>"
product_run_project_limit_fst: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c" assumes "finite (range (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w))" shows "project_transition_fst ` limit (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w) = limit (run\<^sub>t \<delta>\<^sub>1 q\<^sub>1 w)"
add_le_imp_le_right: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset" assumes "a + c \<subseteq># b + c" shows "a \<subseteq># b"
assms_1: shows "v1 \<Turnstile>\<^sub>t t"
assms_6: shows "a2 \<in> b2 \<rightarrow> carrier R"
If_starD: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and t :: "char list \<Rightarrow> int" assumes "(IF b THEN c1 ELSE c2, s) \<rightarrow>* (c, t)" shows "c = IF b THEN c1 ELSE c2 \<or> (c1, s) \<rightarrow>* (c, t) \<or> (c2, s) \<rightarrow>* (c, t)"
assms: shows "\<k> \<le> des.\<v> - 2"
geodesic_subsegment_exists: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment G"   and "x \<in> G"   and "y \<in> G" shows "\<exists>H\<subseteq>G. geodesic_segment_between H x y"
is_unit_Jacobi_2: fixes a :: "int" shows "Jacobi a (- 1) = 1"
path_component_sym: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "path_component S x y" shows "path_component S y x"
valid': shows "valid_unMultigraph (del_unEdge v w u G)"
d2: fixes x :: "real"   and y :: "real" assumes "g x = g y"   and "v \<le> x"   and "x \<le> u"   and "v \<le> y"   and "y \<le> u" shows "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
divisors_funD: fixes df :: "'a \<Rightarrow> 'a list"   and x :: "'a"   and d :: "'a" assumes "divisors_fun df"   and "x \<noteq> (0::'a)"   and "d dvd x" shows "d \<in> set (df x)"
chain_def: fixes Y :: "nat \<Rightarrow> 'c" shows "chain Y = (\<forall>i. Y i \<sqsubseteq> Y (Suc i))"
Healthy_fixed_points: fixes H :: "'a upred \<Rightarrow> 'a upred" shows "fps \<P> H = \<lbrakk>H\<rbrakk>\<^sub>H"
IH_2: fixes f :: "nat \<Rightarrow> real" assumes "\<And>x. x < count_weights False (remove_weights m2) \<Longrightarrow> f x = extract_weights False m2 x" shows "m2 = insert_weights False (remove_weights m2) f"
m1_inv0_finI: fixes x :: "'a m1r_state_scheme" assumes "finite (dom (runs x))" shows "x \<in> m1a_inv0_fin"
prems: shows "\<forall>x\<in>vars\<^sub>t (Fun f ts). \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
create_element_document_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list" assumes "h \<turnstile> ok create_element document_ptr tag" shows "document_ptr |\<in>| document_ptr_kinds h"
case_2: fixes f1 :: "'f"   and f2 :: "nat \<Rightarrow> 'f"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
runit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>MC.runit a : MC.tensor a MC.unity \<rightarrow> a\<guillemotright>"
less_eq_option_None: fixes x :: "'b option" shows "None \<le> x"
assms: shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). range_of \<Psi> v \<noteq> None"
bernoulli_fps_def: shows "bernoulli_fps = fps_X / (fps_exp (1::'a) - 1)"
trace_def: fixes A :: "(('a, 'b) vec, 'b) vec" shows "trace A = (\<Sum>i\<in>UNIV. A $ i $ i)"
fst_snd_flip: fixes xy :: "'d \<times> 'e" shows "fst xy = (snd \<circ> (\<lambda>(x, y). (y, x))) xy"
power_abs: 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 a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times a n) = power.power one times (abs a) n"
T2: shows "\<lfloor>\<P> (\<lambda>x w. \<forall>xa. \<P> xa w = (\<^bold>\<box>xa x) w)\<rfloor>"
Zorn_Lemma2: fixes A :: "'c set set" assumes "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U" shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
float_leI: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b"   and "valof a \<le> valof b" shows "a \<le> b"
dominates_path: fixes g :: "'g"   and n :: "'node"   and m :: "'node" assumes "old.dominates g n m"   and "invar g" obtains ns :: "'node list" where "g \<turnstile> n-ns\<rightarrow>m"
mono_step'_top: fixes C1 :: "ivl st option acom"   and C2 :: "ivl st option acom" assumes "C1 \<le> C2" shows "step_ivl \<top> C1 \<le> step_ivl \<top> C2"
accepting_pair\<^sub>G\<^sub>R_def: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and P :: "('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set"   and w :: "nat \<Rightarrow> 'e" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 P w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> fst P = {} \<and> (\<forall>I\<in>snd P. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {})"
map_comp: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'f"   and v :: "'g llist" shows "lmap g (lmap f v) = lmap (g \<circ> f) v"
pair_not_fst: fixes A :: "'b trm"   and B :: "'b trm"   and P :: "'b trm" shows "SimplyTyped.Pair A B \<noteq> Fst P"
assms: shows ":G:\<lbrakk>e\<^sub>L\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>R\<rbrakk>"
cla_empty_iff: fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> V" shows "(Z = {}) = (Closure\<^sub>\<alpha> Z = {})"
lt_Koszul_syz_in_Koszul_syz_sigs_aux: fixes i :: "nat"   and j :: "nat" assumes "distinct fs"   and "0 \<notin> set fs"   and "i < j"   and "j < length fs" shows "lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j)) - fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i))) \<in> set (Koszul_syz_sigs_aux fs 0)"
greaterThanLessThan_upt: fixes n :: "nat"   and m :: "nat" shows "{n<..<m} = set [Suc n..<m]"
ah_leq_il_left: fixes h1 :: "'b \<Rightarrow> 'b set"   and h2 :: "'b \<Rightarrow> 'b set"   and h1' :: "'b \<Rightarrow> 'b set" assumes "h1 [*] h2"   and "h1' \<le> h1" shows "h1' [*] h2"
ObjDg: shows "dom\<^bsub>C\<^esub> g \<in> obj\<^bsub>C\<^esub>"
tendsto_mult: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> a * b) F"
hyps: fixes es :: "expr list" assumes "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts''" shows "\<exists>Ts'. P,E,h \<turnstile> es [:] Ts' \<and> conformable Ts' Ts''"
valid_iff_nvalid: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A = (\<forall>n. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A)"
dvdE: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" obtains k :: "'a" where "a = b * k"
d_p2r: fixes P :: "'a \<Rightarrow> bool" shows "rdom \<lceil>P\<rceil> = \<lceil>P\<rceil>"
monotone_lzip: shows "monotone (rel_prod (\<sqsubseteq>) (\<sqsubseteq>)) (\<sqsubseteq>) (\<lambda>(x, y). lzip x y)"
inf_par_distrib2: fixes d :: "'a::refinement_lattice"   and c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice" shows "(d::'a::refinement_lattice) \<parallel> ((c\<^sub>0::'a::refinement_lattice) \<sqinter> (c\<^sub>1::'a::refinement_lattice)) = d \<parallel> c\<^sub>0 \<sqinter> d \<parallel> c\<^sub>1"
sublist_def: fixes xs :: "'c list"   and ys :: "'c list" shows "Sublist.sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
assms: shows "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>"
set2_F_bound: fixes x :: "('x, 'p, 'y, 'q, 'r, 's, 't, 'u, 'v, 'w) F" shows "|set2_F x| \<le>o bd_F"
asSort_castVar: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isVar s X" shows "asSort (fst (castVar s X)) = s"
M1: shows "mat_of_row y1 \<in> carrier_mat 1 nr1"
config_rand_set: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and init :: "'b list"   and qs :: "'b list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init"
Compr_image_eq: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set"   and P :: "'g \<Rightarrow> bool" shows "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
sdrop_stl: fixes n :: "nat"   and s :: "'a stream" shows "sdrop n (stl s) = stl (sdrop n s)"
assms_2: shows "\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)"
per_supset_rel_def: shows "local.per_supset_rel \<equiv> {(p1, p2). Restr p1 (Domain p2) = p2 \<and> p1 - Domain p2 \<times> Domain p2 \<subseteq> Id}"
simple_extension_isomorphism: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R" shows "K [X] Quot a_kernel (K [X]) R (\<lambda>p. local.eval p x) \<simeq> R\<lparr>carrier := simple_extension K x\<rparr>"
typ_ok_sig_imp_sortsT_ok_sig: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ"   and S :: "String.literal set" assumes "typ_ok_sig \<Sigma> T"   and "S \<in> SortsT T" shows "wf_sort (subclass (osig \<Sigma>)) S"
comp2lt_def: fixes cmp :: "'a \<Rightarrow> 'b \<Rightarrow> comp_res"   and a :: "'a"   and b :: "'b" shows "comp2lt cmp a b \<equiv> case cmp a b of LESS \<Rightarrow> True | _ \<Rightarrow> False"
hyps: fixes b :: "'a node ref option"   and x :: "'a" shows "<os_list xs b> os_rem x b <os_list (removeAll x xs)>\<^sub>t"
upt_generator_def: shows "local.upt_generator \<equiv> Generator (\<lambda>n. n < bound, \<lambda>n. (n, Suc n))"
inject: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "nat"   and x4 :: "('c, 'd) SkewBinomialTree list"   and y1 :: "'c"   and y2 :: "'d"   and y3 :: "nat"   and y4 :: "('c, 'd) SkewBinomialTree list" shows "(Node x1 x2 x3 x4 = Node y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
polyfun_eq_0: fixes c :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<forall>x. (\<Sum>i\<le>n. c i * x ^ i) = (0::'a)) = (\<forall>i\<le>n. c i = (0::'a))"
i: shows "has_bochner_integral lebesgue (\<lambda>x. indicat_real {a..c} x *\<^sub>R f x) i"
size_multiset_overloaded_eq: fixes x :: "'a multiset" shows "size x = sum (count x) (set_mset x)"
intfAPP_def: fixes f :: "interface \<Rightarrow> 'c"   and x :: "interface" shows "\<langle>x\<rangle>\<^sub>if \<equiv> f x"
cong_solve_int: fixes a :: "int"   and n :: "int" shows "\<exists>x. [a * x = gcd a n] (mod n)"
ln_Lambert_W: fixes x :: "real" assumes "0 < x" shows "ln (Lambert_W x) = ln x - Lambert_W x"
has_default_policy_runFw: fixes rs :: "'a simple_rule list"   and s :: "'a word"   and d :: "'a word"   and c :: "'b parts_connection_scheme" assumes "has_default_policy rs" shows "runFw s d c rs = Decision FinalAllow \<or> runFw s d c rs = Decision FinalDeny"
interv_sum_list_conv_sum_set_nat: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and f :: "nat \<Rightarrow> 'e"   and m :: "nat"   and n :: "nat" assumes "class.monoid_add plus zero" shows "sum_list (map f [m..<n]) = sum f (set [m..<n])"
Comp_in_comp_ARR: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "ARR f"   and "ARR g"   and "DOM f = COD g"   and "t \<in> f"   and "u \<in> g" shows "t \<^bold>\<cdot> u \<in> f \<cdot> g"
assms_3: shows "proj_list_3 ns' (ys, z, zs) = (ys', z', zs')"
simps_2: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and xs :: "'c list"   and x :: "'c" shows "ipurge_tr_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then ipurge_tr_aux I D U xs else ipurge_tr_aux I D U xs @ [x])"
monad_altc_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_altc return bind altc" shows "monad_altc (return_state return) (bind_state bind) (altc_state altc)"
prefix_par_Int2: fixes a :: "'b"   and A :: "'b set"   and P :: "'b process"   and Q :: "'b process" assumes "a \<in> A" shows "(a \<rightarrow> P \<lbrakk>A\<rbrakk> a \<rightarrow> Q) = (a \<rightarrow> (P \<lbrakk>A\<rbrakk> Q))"
assms: shows "S \<equiv> \<lambda>n. x ^ n /\<^sub>R fact n"
list_update_append: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat"   and x :: "'c" shows "(xs @ ys)[n := x] = (if n < length xs then xs[n := x] @ ys else xs @ ys[n - length xs := x])"
quorum_def: fixes Q :: "'a set" shows "proj.quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>project slices I p. Sl \<subseteq> Q"
fold_cong: fixes a :: "'b"   and b :: "'b"   and xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "a = b"   and "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "fold f xs a = fold g ys b"
negligible_uminus: fixes f :: "nat \<Rightarrow> real" shows "negligible (\<lambda>\<eta>. - f \<eta>) = negligible f"
sorted_hyp_var2: shows "\<forall>q1<length l. \<forall>q2<length l. l ! q1 < l ! q2 \<longrightarrow> q1 < q2"
eexp_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ereal" assumes "\<not> (\<exists>i\<in>I. \<exists>j\<in>I. f i = - \<infinity> \<and> f j = \<infinity>)" shows "eexp (sum f I) = (\<Prod>i\<in>I. eexp (f i))"
absorb_iff1: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (sup a b = a)"
finite_incoming: fixes u :: "nat" assumes "finite cf.V" shows "finite (cf.incoming u)"
distinct_3: fixes x1 :: "('a, 'b, 'c) val"   and x31 :: "('a, 'b, 'c) expr"   and x32 :: "('a, 'b, 'c) expr"   and x33 :: "('a, 'b, 'c) expr" shows "VE x1 \<noteq> Ite x31 x32 x33"
coeff_map_poly: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly"   and n :: "nat" assumes "f (0::'d) = (0::'c)" shows "coeff (map_poly f p) n = f (coeff p n)"
mapping_eq_iff: fixes m1 :: "('c, 'd) mapping"   and m2 :: "('c, 'd) mapping" shows "(m1 = m2) = (\<forall>k. Mapping.lookup m1 k = Mapping.lookup m2 k)"
eq_imp_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "eq_imp f g \<equiv> \<forall>s s'. (\<forall>x. f x s = f x s') \<longrightarrow> g s = g s'"
assms_7: shows "upw \<or> \<mu>_small fs i"
same: fixes p :: "(char list multiset \<times> int) multiset" shows "normalize_poly_p p p"
cond_bind_spmf: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf"   and A :: "'a set" shows "cond_spmf (p \<bind> f) A = cond_bind_spmf p f A \<bind> (\<lambda>x. cond_spmf (f x) A)"
n_de_morgan_var3: fixes x :: "'a"   and y :: "'a" shows "!(t x + t y) = !x \<cdot> !y"
Fract_of_int_quotient: fixes k :: "int"   and l :: "int" shows "Rat.Fract k l = rat_of_int k / rat_of_int l"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(nSome x1 = nSome y1) = (x1 = y1)"
compl_structD2: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h"   and sps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps" shows "component_of_term ` Keys (fst ` set (fst (compl gs bs (ps -- sps) sps data))) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, ps))"
D_def: shows "D = max C (exp 1 / (exp 1 - 1))"
inext_image: fixes n :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> nat" assumes "n \<in> I"   and "SetInterval2.strict_mono_on f I" shows "inext (f n) (f ` I) = f (inext n I)"
gen_power_0: fixes G :: "('a, 'b) cyclic_group_scheme"   and r :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> r = \<one>\<^bsub>G\<^esub>"   and "r < order G" shows "r = 0"
mat_mult2_01: fixes U :: "'b mat"   and V :: "'b mat"   and A :: "'b mat" assumes "U \<in> carrier_mat 2 2"   and "V \<in> carrier_mat 2 2"   and "A = U * V" shows "A $$ (0, 1) = U $$ (0, 0) * V $$ (0, 1) + U $$ (0, 1) * V $$ (1, 1)"
ennreal_top_mult: fixes a :: "ennreal" shows "\<top> * a = (if a = 0 then 0 else \<top>)"
that: fixes yvec1 :: "name list"   and y :: "name"   and yvec2 :: "name list" assumes "yvec = yvec1 @ y # yvec2"   and "length xvec1 = length yvec1"   and "length xvec2 = length yvec2"   and "\<lparr>\<nu>*(xvec1 @ xvec2)\<rparr>M \<prec>' P = \<lparr>\<nu>*(yvec1 @ yvec2)\<rparr>([(x, y)] \<bullet> N) \<prec>' ([(x, y)] \<bullet> Q)" shows "thesis"
m: shows "m \<in> {m. \<exists>lam. (m, lam) \<in> set n_as \<and> cmod lam = 1}"
Cons_2: fixes x :: "'a" assumes "x \<in> set (x__ # xs)" shows "r2 (f y) (f x) = r1 y x"
r_sigma: fixes i :: "nat"   and x :: "nat" shows "\<sigma> i x = \<phi> (the (eval r_limr [i])) x"
simps_2: fixes cs :: "'edge list"   and as :: "'edge list"   and a :: "'edge" shows "same_level_path_rev_aux cs (as @ [a]) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> c' \<in> get_return_edges a \<and> same_level_path_rev_aux cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> same_level_path_rev_aux (a # cs) as | _ \<Rightarrow> same_level_path_rev_aux cs as)"
neutr_eq_iff: fixes a :: "'g multiset"   and b :: "'g multiset" shows "({#} = a \<union># b) = (a = {#} \<and> b = {#})"
that: fixes i :: "nat" assumes "P (\<sigma> |\<^sub>s i)" shows "thesis"