hcju/isabelleps huggingface dataset:
append_Cons_nth_left: fixes i :: "nat"   and xs :: "'b list"   and u :: "'b"   and ys :: "'b list" assumes "i < length xs" shows "(xs @ u # ys) ! i = xs ! i"
that: fixes s1 :: "'a" assumes "s1 \<in> S"   and "s1 \<cdot> s \<cdot> fst x = s1 \<cdot> snd x \<cdot> r"   and "x \<in> R \<times> S" shows "thesis"
image_def: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
that: fixes s' :: "'state" assumes "(c, s) \<rightarrow>t s'"   and "s' \<approx> t'" shows "thesis"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Normal"   and "b \<in> Normal" shows "a \<cdot> b \<in> Normal"
tendsto_zero_mult_right_iff: fixes c :: "'a"   and a :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "(\<lambda>n. a n * c) \<longlonglongrightarrow> (0::'a) = a \<longlonglongrightarrow> (0::'a)"
cp_\<Sigma>: fixes \<eta> :: "nat" shows "chaum_ped_\<Sigma> (\<G> \<eta>)"
exhaust: fixes y :: "tag" obtains "y = insec"   | "y = auth"   | "y = confid"   | "y = secure"
Mid_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Midpoint B C \<or> A Midpoint C B" shows "A Midpoint B C"
map_upds_def: fixes m :: "'c \<Rightarrow> 'd option"   and xs :: "'c list"   and ys :: "'d list" shows "m(xs [\<mapsto>] ys) = m ++ map_of (rev (zip xs ys))"
substitutes_varE: fixes x :: "'a"   and y :: "'a"   and M :: "'a trm"   and X :: "'a trm" assumes "substitutes (Var x) y M X" shows "x = y \<and> M = X \<or> x \<noteq> y \<and> X = Var x"
eval_ring_hom: shows "eval R R id a \<in> ring_hom P R"
get_ancestors_locs_def: shows "Shadow_DOM.get_ancestors_locs \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs Shadow_DOM.get_parent_locs"
index_less: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "index xs x < n"
append_rows_mat_mul: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat" assumes "dim_col A = dim_col B" shows "(A @\<^sub>r B) * C = A * C @\<^sub>r B * C"
analz_insert_freshK2: fixes evs :: "event list"   and servK :: "nat"   and K :: "nat" assumes "evs \<in> kerbIV_gets"   and "servK \<notin> authKeys evs"   and "servK \<notin> range shrK"   and "K \<in> symKeys" shows "(Key K \<in> analz (insert (Key servK) (knows Spy evs))) = (K = servK \<or> Key K \<in> analz (knows Spy evs))"
hcomp_in_hom\<^sub>W\<^sub>C: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.in_hom V (H \<nu> \<mu>) (H (partial_magma.dom V \<nu>) (partial_magma.dom V \<mu>)) (H (partial_magma.cod V \<nu>) (partial_magma.cod V \<mu>))"
upper_triangular_upt_k_def: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "upper_triangular_upt_k A k = (\<forall>i j. j < i \<and> mod_type_class.to_nat j < k \<longrightarrow> A $ i $ j = (0::'b))"
l2_inv4_trans: shows "{l2_inv4 \<inter> l2_inv2 \<inter> l2_inv3} TS.trans l2 {> l2_inv4}"
f_1: shows "g \<in> L \<rightarrow>\<^sub>M prob_algebra M"
euclidean_eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (\<forall>b\<in>Basis. inner x b = inner y b)"
domains_simp: fixes f :: "'c" assumes "C.arr f" shows "C.domains f = {C.dom f}"
unfold_accept_eq: shows "\<MM>.accept\<^sub>R' = \<UU>.accept"
ap_gpv_id: fixes x :: "('f, 'g, 'h) gpv" shows "Generative_Probabilistic_Value.Done (\<lambda>x. x) \<diamondop> x = x"
supto_is_snil_conv_1: fixes i :: "int"   and j :: "int" shows "(supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j) = [::]) = (j < i)"
freshFrameDest_2: fixes xvec :: "name list"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "xvec \<sharp>* (\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>)"   and "A\<^sub>F \<sharp>* xvec" shows "xvec \<sharp>* \<Psi>\<^sub>F"
fc: shows "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
l2_inv5_def: shows "l2_inv5 \<equiv> {s. \<forall>A B M. Confid A B M \<in> chan s \<or> Secure A B M \<in> chan s \<longrightarrow> M \<in> dy_fake_msg (bad s) (ik s) (chan s)}"
rep_eq: fixes x :: "'a interval" shows "bounds_of_interval (- x) = (case bounds_of_interval x of (a, b) \<Rightarrow> (- b, - a))"
tick_count_is_fun: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "tick_count r c n = #\<^sub>\<le> r c n"
induct: fixes P :: "'a list \<Rightarrow> ('a \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a \<Rightarrow> nat"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat" assumes "\<And>f l up left. P [] f l up left"   and "\<And>x xs f l up left. \<lbrakk>f x < l \<Longrightarrow> P xs f l up left; \<lbrakk>\<not> f x < l; l < f x\<rbrakk> \<Longrightarrow> P (x # xs) f (l + 1) up left; \<lbrakk>\<not> f x < l; \<not> l < f x\<rbrakk> \<Longrightarrow> P xs f l up left\<rbrakk> \<Longrightarrow> P (x # xs) f l up left" shows "P a0 a1 a2 a3 a4"
lappend_ltake_ldrop: fixes n :: "enat"   and xs :: "'c llist" shows "lappend (ltake n xs) (ldrop n xs) = xs"
append_eq_append_conv: fixes xs :: "'a list"   and ys :: "'a list"   and us :: "'a list"   and vs :: "'a list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
betaupright_obtain_support: fixes va :: "'c zmultiset"   and vb :: "'c zmultiset"   and t :: "'c" assumes "beta_upright va vb"   and "0 < zcount va t" obtains s :: "'c" where "s < t" and "zcount va s < 0 \<or> zcount vb s < 0" and "nonpos_upto va s"
filter_set_def: fixes m :: "('c, 'd) mapping"   and X :: "'c set"   and t :: "'d" shows "filter_set m X t = Mapping.filter (filter_cond X m t) m"
approx_form_aux: fixes prec :: "nat"   and f :: "form"   and vs :: "float interval option list"   and ss :: "nat list"   and xs :: "real list" assumes "approx_form prec f vs ss"   and "bounded_by xs vs" shows "interpret_form f xs"
rel_prod2_simps: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and c :: "'d"   and b :: "'c" shows "rel_prod2 R a (c, b) = R a b"
equiv_domain_and_range_permutation: fixes A :: "'a set"   and B :: "'b set" shows "equiv (A \<rightarrow>\<^sub>E B) (domain_and_range_permutation A B)"
enn2real_positive_iff: fixes x :: "ennreal" shows "(0 < enn2real x) = (0 < x \<and> x < top_class.top)"
times_subset_iff: fixes A :: "'b set"   and C :: "'c set"   and B :: "'b set"   and D :: "'c set" shows "(A \<times> C \<subseteq> B \<times> D) = (A = {} \<or> C = {} \<or> A \<subseteq> B \<and> C \<subseteq> D)"
range_constant: fixes x :: "'b" shows "range (\<lambda>_. x) = {x}"
fraction_less_2p: fixes x :: "('a, 'b) IEEE.float" shows "fraction x < 2 ^ LENGTH('b)"
lens_obs_equiv: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "equivp (\<simeq>\<^bsub>x\<^esub>)"
cod_in_codomains: fixes f :: "'a" assumes "codomains f \<noteq> {}" shows "cod f \<in> codomains f"
l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs set_shadow_root set_shadow_root_locs known_ptr get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs remove_shadow_root remove_shadow_root_locs known_ptrs get_parent get_parent_locs"
assms_4: shows "f \<bullet> a \<noteq> \<zero>"
add_smult_distrib_vec: fixes a :: "'a"   and b :: "'a"   and v :: "'a vec" shows "(a + b) \<cdot>\<^sub>v v = a \<cdot>\<^sub>v v + b \<cdot>\<^sub>v v"
emeasure_countable_singleton: fixes X :: "'b set"   and M :: "'b measure" assumes "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "countable X" shows "emeasure M X = \<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space X"
select_convs_2: fixes prism_match :: "'b \<Rightarrow> 'a option"   and prism_build :: "'a \<Rightarrow> 'b"   and more :: "'c" shows "build\<^bsub>\<lparr>prism_match = prism_match, prism_build = prism_build, \<dots> = more\<rparr>\<^esub> = prism_build"
surj_onD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and b :: "'b" assumes "surj_on f A B"   and "b \<in> B" shows "\<exists>a\<in>A. b = f a"
inj_on_nat_add_square: fixes S :: "nat set" shows "inj_on (\<lambda>a. a + a * a) S"
free_group_is_group: fixes gens :: "'e set" shows "Group.group \<F>\<^bsub>gens\<^esub>"
prems: shows "subtract (Node l x d r) t\<^sub>2 = Some t"
assms_4: shows "\<omega> = \<omega>\<^bsub>G'\<^esub>"
C2: shows "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"
simps_4: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and i :: "int" shows "typeof\<^bsub>h\<^esub> (Intg i) = \<lfloor>Integer\<rfloor>"
leI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
int_nat_pow_rep: fixes k :: "int"   and n :: "nat" shows "[k] \<cdot> \<one> [^] n = [(k ^ n)] \<cdot> \<one>"
verit_minus_simplify_3: fixes b :: "'a" shows "(0::'a) - b = - b"
subidem_par: fixes x :: "'a" shows "x \<le> x \<parallel> x"
covering_space_open_map: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and T :: "'c set" assumes "covering_space c p S"   and "openin (top_of_set c) T" shows "openin (top_of_set S) (p ` T)"
freess_empty: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees [] = {||}"
derives_drop_prefixword_helper: fixes a :: "symbol list"   and b :: "symbol list"   and w :: "symbol list" assumes "derives a b"   and "is_word w"   and "is_prefix w a" shows "derives (drop (length w) a) (drop (length w) b)"
Q_min: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" assumes "Prim_Invar2_ctd_Presentation_Loc w g r Q \<pi>" shows "\<forall>u d. Q u = enat d \<longrightarrow> (\<exists>v. \<pi> u = Some v \<and> d = w {u, v} \<and> (\<forall>v'. (u, v') \<in> edges (Prim.rg g r) \<inter> (- Prim.S r (A Q \<pi>)) \<times> Prim.S r (A Q \<pi>) \<longrightarrow> d \<le> w {u, v'}))"
zexplode_def: fixes z :: "ZF" shows "zexplode z \<equiv> Abs_zet (explode z)"
replace_f: fixes a :: "'b negation_type" shows "\<not> has_disc disc2 (replace_f a)"
divide_eq_eq_numeral1_1: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / numeral w = a) = (if numeral w \<noteq> (0::'a) then b = a * numeral w else a = (0::'a))"
cutoff_branch_internal: fixes T :: "tree"   and red :: "bool list \<Rightarrow> bool" assumes "anybranch T red" shows "\<exists>T'. anyinternal T' (\<lambda>p. \<not> red p) \<and> anybranch T' red"
hash_dsz: shows "SIDE_GEN_ALGO (is_valid_def_hm_size TYPE('vi) node_def_hash_size)"
filterlim_at_top_dense: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "filterlim f at_top F = (\<forall>Z. \<forall>\<^sub>F x in F. Z < f x)"
nat_diff_left_cancel_less: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k - m < k - n" shows "n < m"
map_of_SomeD: fixes xs :: "('d \<times> 'c) list"   and k :: "'d"   and y :: "'c" assumes "table_of xs k = Some y" shows "(k, y) \<in> set xs"
s_ns_mul_ext: fixes A :: "'a multiset"   and B :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(A, B) \<in> s_mul_ext ns s" shows "(A, B) \<in> ns_mul_ext ns s"
bounded_linear_mult_left: fixes y :: "'d" shows "bounded_linear (\<lambda>x. x * y)"
lossless_map_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "lossless_spmf (map_spmf f p) = lossless_spmf p"
nprt_def: fixes x :: "'a" shows "nprt x = inf x (0::'a)"
Prefix_def: fixes L :: "'b list set" shows "Prefix L = {w. \<exists>u. w @ u \<in> L}"
strict_implies_not_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordering less_eq less"   and "less a b" shows "a \<noteq> b"
greaterThanLessThan_iff: fixes i :: "'b"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "(i \<in> ord.greaterThanLessThan less l u) = (less l i \<and> less i u)"
index_mult_mat_vec: fixes i :: "nat"   and A :: "'a mat"   and v :: "'a Matrix.vec" assumes "i < dim_row A" shows "(A *\<^sub>v v) $v i = Matrix.row A i \<bullet> v"
fds_nth_fds: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "fds f $ n = (if n = 0 then 0::'b else f n)"
LD_imp_D: fixes r :: "('c \<times> 'c) set"   and ars :: "('d \<times> 'c \<times> 'd) set"   and P :: "('d \<times> ('c \<times> 'd) list) \<times> 'd \<times> ('c \<times> 'd) list" assumes "trans r"   and "wf r"   and "\<forall>P. local_peak ars P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD ars r (fst P, snd P, \<sigma>', \<tau>'))"   and "peak ars P" shows "\<exists>\<sigma>' \<tau>'. DD ars r (fst P, snd P, \<sigma>', \<tau>')"
hd_inf_prefix_Nil: fixes xs :: "'b list"   and ys :: "'b list" assumes "hd xs \<noteq> hd ys" shows "xs \<sqinter> ys = []"
insert_def: fixes x :: "'d"   and dxs :: "'d dlist" shows "Dlist.insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"
path_split_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n (as @ a # as') n'" shows "CFG.path sourcenode targetnode valid_edge n as (sourcenode a)"
iiface_constrain_matches_Primitive: fixes \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and x :: "'c common_primitive"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" shows "(matches (common_matcher, \<alpha>) (MatchNot (iiface_constrain ipassmt (Match x))) a p = matches (common_matcher, \<alpha>) (MatchNot (Match x)) a p) = (matches (common_matcher, \<alpha>) (iiface_constrain ipassmt (Match x)) a p = matches (common_matcher, \<alpha>) (Match x) a p)"
num_ni_init: shows "num_ni (\<lambda>_. INIT) = 0"
qSwapAbs_preserves_alphaAbs: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and A' :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGoodAbs A \<or> qGoodAbs A'"   and "A $= A'" shows "A $[[z1 \<and> z2]]_zs $= A' $[[z1 \<and> z2]]_zs"
corthogonalI: fixes vs :: "'b vec list" assumes "\<And>i j. \<lbrakk>i < length vs; j < length vs\<rbrakk> \<Longrightarrow> (vs ! i \<bullet>c vs ! j = (0::'b)) = (i \<noteq> j)" shows "corthogonal vs"
Si_at_top_LBINT: shows "((\<lambda>t. LBINT x=0..\<infinity>. exp (- (x * t)) * (x * sin t + cos t) / (1 + x\<^sup>2)) \<longlongrightarrow> 0) at_top"
le_cases: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
a_get_element_by_id_def: fixes first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and ptr :: "(_) object_ptr"   and iden :: "char list" shows "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id first_in_tree_order get_attribute ptr iden = first_in_tree_order ptr (\<lambda>ptr. case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return None | Some element_ptr \<Rightarrow> Heap_Error_Monad.bind (get_attribute element_ptr ''id'') (\<lambda>id_opt. if id_opt = Some iden then return (Some element_ptr) else return None))"
not_diff_distrib: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not (minus a b) = plus (not a) b"
submodule_is_module: fixes N :: "'a vec set" assumes "LinearCombinations.submodule class_ring N V" shows "module class_ring (vs N)"
word_oa_dist: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "x AND y OR z = (x OR z) AND (y OR z)"
piecewise_C1_differentiable_diff: fixes f :: "real \<Rightarrow> 'b"   and S :: "real set"   and g :: "real \<Rightarrow> 'b" assumes "f piecewise_C1_differentiable_on S"   and "g piecewise_C1_differentiable_on S" shows "(\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
check_add_alt_def: shows "check_add A \<V> p q i r = SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> q \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars r \<subseteq> \<V>) \<bind> (\<lambda>b. if \<not> b then SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> q \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars r \<subseteq> \<V> \<and> the (fmlookup A p) + the (fmlookup A q) - r \<in> More_Modules.ideal polynomial_bool) else SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> q \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars r \<subseteq> \<V> \<and> the (fmlookup A p) + the (fmlookup A q) - r \<in> More_Modules.ideal polynomial_bool))"
assms_3: fixes s :: "'a" shows "right (f s) = f (RIGHT s)"
simps: fixes a :: "poly"   and n :: "nat"   and p :: "poly"   and k :: "nat"   and s :: "poly" shows "polydivide_aux a n p k s = (if s = 0\<^sub>p then (k, s) else let b = head s; m = Reflected_Multivariate_Polynomial.degree s in if m < n then (k, s) else let p' = (shift1 ^^ (m - n)) p in if a = b then polydivide_aux a n p k (s -\<^sub>p p') else polydivide_aux a n p (Suc k) (a *\<^sub>p s -\<^sub>p (b *\<^sub>p p')))"
gf_F_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a"   and z :: "real" shows "MC_syntax.gf_F K x y z = (\<Sum>n. MC_syntax.f K x y n * z ^ n)"
log_ln: fixes x :: "real" shows "ln x = log (exp 1) x"
steps_z_beta_alt: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and k :: "'d \<Rightarrow> nat"   and A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and l :: "'e"   and Z :: "('d \<Rightarrow> real) set"   and l' :: "'e"   and Z' :: "('d \<Rightarrow> real) set"   and l'' :: "'e"   and Z'' :: "('d \<Rightarrow> real) set" assumes "Regions X v n not_in_X"   and "Regions.step_z_beta X k v n not_in_X A l Z l' Z'"   and "Regions.steps_z_beta X k v n not_in_X A l' Z' l'' Z''" shows "Regions.steps_z_beta X k v n not_in_X A l Z l'' Z''"
subset_second_countable: fixes S :: "'b set" obtains \<B> :: "'b set set" where "countable \<B>" and "{} \<notin> \<B>" and "\<And>C. C \<in> \<B> \<Longrightarrow> openin (top_of_set S) C" and "\<And>T. openin (top_of_set S) T \<Longrightarrow> \<exists>\<U>\<subseteq>\<B>. T = \<Union> \<U>"
PiE: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'h \<Rightarrow> 'i set"   and x :: "'h" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
muaux_axioms: shows "muaux valid_mmuaux init_mmuaux add_new_mmuaux' length_mmuaux eval_mmuaux"
UP_smult_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "a \<in> carrier R"   and "p \<in> carrier (UP R)" shows "a \<odot>\<^bsub>UP R\<^esub> p \<in> carrier (UP R)"
remdups_append: fixes rs :: "'b list"   and rs2 :: "'b list" shows "remdups (rs @ rs2) = remdups (filter (\<lambda>r. r \<notin> set rs2) rs) @ remdups rs2"
u_facts_2: shows "u \<notin> Fvars \<phi>"
assms: shows "\<exists>algorithm. \<forall>A\<in>carrier_mat m n. is_SNF A (algorithm A)"
length_helper1: fixes p :: "nat"   and q :: "nat"   and m :: "bit list" shows "length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))))) @ sha1 (generate_M' (sha1 m) salt) @ BC) = length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))))) + 168"
hyps: fixes A :: "(('a, 'n) vec, 'n) vec" assumes "n < nrows A" shows "det (row_add_iterate A n i j) = det A"
"4_1": shows "f (f (f s l1) [a]) l''' = f (f (f s l1) [a]) l2"
some_list_ne: fixes cs :: "'a list" assumes "cs \<in> set css" shows "some_list cs \<noteq> []"
G_eq_Gamma_aux: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "G x = Gamma x"
mem_plus_V_E: fixes l :: "V"   and x :: "V"   and y :: "V" assumes "l \<in>\<^sub>\<circ> x + y" obtains "l \<in>\<^sub>\<circ> x"   | z :: "V" where "z \<in>\<^sub>\<circ> y" and "l = x + z"
terminates_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c" shows "IFC_def.terminates prob \<sigma> \<equiv> \<exists>i. IFC_def.path prob \<sigma> i = IFC_def.return prob"
induced_play_is_lprefix: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a" assumes "position p" shows "lprefix (llist_of p) (induced_play \<sigma> p)"
minus_1: fixes x :: "'a" shows "f (x - (1::'a)) = f x"