hcju/isabelleps huggingface dataset:
QR_lemma_11: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "card (QR.BuDuF p q) = card ({(int p - 1) div 2<..int p - 1} \<times> {0<..(int q - 1) div 2})"
fvi_less_nfv_regex: fixes \<phi> :: "Formula.formula Regex.regex" shows "\<forall>i\<in>fv_regex \<phi>. i < Formula.nfv_regex \<phi>"
Matrix_row_is_Legacy_row: fixes i :: "nat"   and A :: "complex Matrix.mat" assumes "i < dim_row A" shows "Matrix.row A i = vec_of_list (Matrix_Legacy.row (mat_to_cols_list A) i)"
intros_10: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "Aenc X Y \<in> synth H"
poly_asymp_equiv: fixes p :: "'a poly"   and F :: "'a filter" assumes "p \<noteq> 0"   and "F \<le> at_infinity" shows "poly p \<sim>[F] (\<lambda>x. lead_coeff p * x ^ degree p)"
RMWWrite: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and \<O>s :: "nat set list"   and i :: "nat"   and A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool" assumes "cond (\<theta>(t \<mapsto> m a))"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = {}"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> \<O>s ! j = {}"   and "a \<notin> read_only \<S>" shows "\<O>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
\<F>_eq_\<S>_generalized: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set" assumes "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>"   and "closed \<G> w" shows "\<forall>\<^sub>\<infinity>j. \<forall>\<psi>. G \<psi> \<in> \<G> \<longrightarrow> (\<forall>S. (S \<Turnstile>\<^sub>P \<F> \<psi> w \<G> j \<and> \<G> \<subseteq> S) = (\<forall>q. q \<in> mojmir_def.\<S> \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q} j \<longrightarrow> S \<Turnstile>\<^sub>P Rep q))"
add_neg_numeral_special_8: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) one = zero"
IH: assumes "as \<in> valid_plans PROB" shows "filter (\<lambda>a. varset_action a (prob_dom PROB)) as = as"
ide_char: fixes f :: "'a \<times> 'a" shows "H.VxV.ide f = (V.ide (fst f) \<and> V.ide (snd f))"
l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node_si get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_owner_document \<equiv> l_get_host get_host get_host_locs \<and> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node_si get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node_si get_host get_owner_document"
prod_cases3: fixes y :: "'g \<times> 'h \<times> 'i" obtains a :: "'g" and b :: "'h" and c :: "'i" where "y = (a, b, c)"
t_converg: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> U" shows "eval t [f \<triangleright> n] \<down>"
"case": fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and x :: "nat \<times> nat" shows "(case natpair.Pair x of natpair.Pair x \<Rightarrow> f x) = f x"
intro: fixes F :: "'b set set"   and T :: "'b set set" assumes "F \<in> Filters\<^bsub>T\<^esub>" shows "Topology.filter F T"
rightrec_base: fixes u :: "'b"   and v :: "'b"   and lc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and rc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat" shows "rightrec u v lc rc (real m / 2) = v"
expand_to_basis: fixes xs :: "'a vec list" assumes "lin_indpt_list xs" shows "\<exists>ys. set ys \<subseteq> set (unit_vecs n) \<and> lin_indpt_list (xs @ ys) \<and> length (xs @ ys) = n"
prems_1: shows "P,E,hp s \<turnstile> es [:] Ts"
vconn_implies_hpar_2: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom \<mu> f g" shows "Span.trg \<mu> = Span.trg f"
midpoint_idem: fixes x :: "'b" shows "midpoint x x = x"
simplicial_vertex_def: fixes i :: "'d"   and f :: "('d \<Rightarrow> 'e) \<Rightarrow> 'c" shows "simplicial_vertex i f = f (\<lambda>j. if j = i then 1::'e else (0::'e))"
assms: shows "\<forall>p. eval cid t t' n (\<gamma> p)"
n_def: shows "n = Suc (nat \<lceil>log 2 (real_of_int d)\<rceil>)"
set_sort: fixes xs :: "'a list" shows "set (local.sort xs) = set xs"
ev_1: shows "\<forall>\<^sub>F x in at_right 0. g0 x \<noteq> 0"
augment_flow_value: fixes f' :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "Flow cf s t f'" shows "Flow.val c s (augment f') = val + Flow.val cf s f'"
usedBy_uminus: fixes x :: "'a \<Longrightarrow> 'b"   and u :: "('c, 'b) uexpr" assumes "x \<natural> u" shows "x \<natural> - u"
jkbpC_jkbpCn_jAction_eq: fixes t :: "('a, 'es, 'as) BEState Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpCn n" shows "jAction SPR.MC t = jAction (SPR.MCn n) t"
prems: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "P n" shows "P (ZFC_in_HOL.succ n)"
Pow_lepoll_mono: fixes A :: "'a set"   and B :: "'b set" assumes "A \<lesssim> B" shows "Pow A \<lesssim> Pow B"
span_add_eq2: fixes y :: "'a"   and S :: "'a set"   and x :: "'a" assumes "y \<in> More_Modules.ideal S" shows "(x + y \<in> More_Modules.ideal S) = (x \<in> More_Modules.ideal S)"
fusion_absorption: fixes z :: "i" shows "(\<sigma>x. P x z) = z"
simps_4: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "eval (\<phi> or\<^sub>n \<psi>) = eval_or (eval \<phi>) (eval \<psi>)"
cgenideal_is_principalideal: fixes i :: "int" assumes "i \<in> UNIV" shows "principalideal (PIdl\<^bsub>\<Z>\<^esub> i) \<Z>"
generate_one: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.group G" shows "generate G {\<one>\<^bsub>G\<^esub>} = {\<one>\<^bsub>G\<^esub>}"
utp_order_fpl: fixes H :: "'a hrel \<Rightarrow> 'a hrel" shows "utp_order H = fpl \<P> H"
in_out_arcs_conv: fixes e :: "'b"   and G :: "('c, 'b) pre_digraph"   and v :: "'c" shows "(e \<in> out_arcs G v) = (e \<in> arcs G \<and> tail G e = v)"
coinduct: fixes X :: "'c \<Rightarrow> bool"   and x :: "'c"   and trsys :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and \<tau>move :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>s s'. x = s \<and> \<tau>trsys.silent_move trsys \<tau>move s s' \<and> (X s' \<or> \<tau>trsys.\<tau>diverge trsys \<tau>move s')" shows "\<tau>trsys.\<tau>diverge trsys \<tau>move x"
JN_transient: fixes i :: "'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd program"   and A :: "'d set" assumes "i \<in> I" shows "(JOIN I F \<in> transient A) = (\<exists>i\<in>I. F i \<in> transient A)"
path_finish_in_topspace: fixes X :: "'a topology"   and g :: "real \<Rightarrow> 'a" assumes "pathin X g" shows "g 1 \<in> topspace X"
uniqueness_algorithm_division_poly: fixes f :: "'a poly"   and g :: "'a poly"   and q1 :: "'a poly"   and r1 :: "'a poly"   and q2 :: "'a poly"   and r2 :: "'a poly" assumes "f = g * q1 + r1"   and "f = g * q2 + r2"   and "g \<noteq> 0"   and "r1 = 0 \<or> degree r1 < degree g"   and "r2 = 0 \<or> degree r2 < degree g" shows "q1 = q2 \<and> r1 = r2"
cosx_def: shows "cosx = (\<lambda>U\<in>rcosets H. U #> inv x)"
insert_Diff_single: fixes a :: "'d"   and A :: "'d set" shows "insert a (A - {a}) = insert a A"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'a list" shows "rotate n xs = rotate (n mod length xs) xs"
b: shows "l \<cdot> r = L \<cdot> [Inl N] \<cdot> R \<and> (N, rhs) \<in> set G \<and> (\<forall>A B. rhs = Branch A B \<longrightarrow> t = L \<cdot> [Inl A, Inl B] \<cdot> R) \<and> (\<forall>x. rhs = Leaf x \<longrightarrow> t = L \<cdot> [Inr x] \<cdot> R)"
set_bd_1: fixes x :: "('l, 'k, 'm, 'n) G" shows "|Gset1 x| \<le>o bd_G"
divide_less_eq_1_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less (divide b a) one = less a b"
Prec_FL_def: fixes Cl1 :: "'f \<times> 'l"   and Cl2 :: "'f \<times> 'l" shows "(Cl1 \<sqsubset> Cl2) = (fst Cl1 \<prec>\<cdot> fst Cl2 \<or> fst Cl1 \<doteq> fst Cl2 \<and> snd Cl1 \<sqsubset>L snd Cl2)"
rawpsubst2_fresh_switch: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "'b"   and t :: "'b"   and s :: "'b"   and x :: "'a"   and y :: "'a" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "r \<in> Syntax_Arith.atrm var Var num suc pls tms"   and "t \<in> trm"   and "s \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT s"   and "y \<notin> FvarsT t" shows "Generic_Syntax.rawpsubstT substT r [(s, y), (t, x)] = Generic_Syntax.rawpsubstT substT r [(t, x), (s, y)]"
ptrm_prm_unit_inaction: fixes a :: "'b"   and X :: "'b ptrm"   and b :: "'b" assumes "a \<notin> ptrm_fvs X"   and "b \<notin> ptrm_fvs X" shows "[a \<leftrightarrow> b] \<bullet> X \<approx> X"
span_insert_cong: fixes A :: "('t \<Rightarrow>\<^sub>0 'e) set"   and B :: "('t \<Rightarrow>\<^sub>0 'e) set"   and p :: "'t \<Rightarrow>\<^sub>0 'e" assumes "pmdl A = pmdl B" shows "pmdl (insert p A) = pmdl (insert p B)"
enab_n2: shows "\<turnstile> Enabled \<langle>n2\<rangle>_vars = ($pc2 = #a \<longrightarrow> #0 < $sem)"
hom_grading_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and d :: "'c \<Rightarrow> nat" assumes "class.monoid_add plus zero" shows "monoid_add.hom_grading plus d = (\<exists>f. monoid_add.hom_grading_fun plus d f)"
Controller_L_def: fixes y :: "nat \<Rightarrow> nat list"   and lIn :: "nat \<Rightarrow> bit"   and lOut :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list" shows "Controller_L y lIn lOut z \<equiv> z 0 = [Zero] \<and> (\<forall>t. if lIn t = Zero then if 300 < hd (y t) then z t = [] \<and> lOut t = Zero else z t = [One] \<and> lOut t = One else if hd (y t) < 700 then z t = [] \<and> lOut t = One else z t = [Zero] \<and> lOut t = Zero)"
subobjs_rel: fixes 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"   and Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "wf_prog wf_md P" shows "P,C \<turnstile> [C] \<sqsubseteq> Cs"
ksimplex_replace_1: fixes p :: "nat"   and n :: "nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "ksimplex p n s"   and "a \<in> s"   and "j < n"   and "\<forall>x\<in>s - {a}. x j = p" shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1"
subgroup_of_group_axioms: shows "subgroup_of_group H G (\<cdot>) \<one>"
bintr_Min: fixes n :: "nat" shows "take_bit n (- 1) = 2 ^ n - 1"
if_nz: fixes n :: "nat"   and thn :: "exp"   and els :: "exp" assumes "n \<noteq> 0" shows "EIf (ENat n) thn els \<longrightarrow> thn"
a: shows "R[a\<turnstile>c>b] = AndR <c>.M <d>.N e"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" and "f\<lparr>a\<rparr> = b"
homotopic_paths_sym: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" assumes "homotopic_paths s p q" shows "homotopic_paths s q p"
simps_2: fixes p :: "'d"   and args :: "'d \<Rightarrow> ('b + 'e, 'd) trm"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "FsubstFO ($\<phi> p args) \<sigma> = $\<phi> p (\<lambda>i. TsubstFO (args i) \<sigma>)"
Chn_in_hom: fixes \<tau> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<tau> : f \<Rightarrow> g\<guillemotright>" shows "\<guillemotleft>Chn \<tau> : Chn f \<rightarrow>\<^sub>C Chn g\<guillemotright>"
finite_monoid_imp_card_positive: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "submonoid H G"   and "finite (carrier G)" shows "0 < card H"
chine_in_homs_2: shows "\<guillemotleft>assoc\<nu>\<pi>\<rho>.chine : HH\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C H\<nu>H\<pi>\<rho>.chine\<guillemotright>"
surjective: fixes r :: "('g, 'h, 'i) Category_scheme" shows "r = \<lparr>Obj = obj\<^bsub>r\<^esub>, Mor = mor\<^bsub>r\<^esub>, Dom = Dom r, Cod = Cod r, Id = Category.Id r, Comp = (;;\<^bsub>r\<^esub>), \<dots> = Category.more r\<rparr>"
COMP_def: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "g \<cdot> f \<equiv> if (f \<noteq> Null \<and> Dom f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> Cod f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom f) (Cod f))) \<and> (g \<noteq> Null \<and> Dom g \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> Cod g \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> Map g \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom g) (Cod g))) \<and> Dom g = Cod f then MkArr (Dom f) (Cod g) (vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Map f) (Map g)) else Null"
design_support_def: shows "design_support \<equiv> set_mset \<B>"
sm: shows "(clinit, Static, [], Void, m) \<in> set ms"
C: fixes x :: "'a" assumes "x \<in> N <#> H" shows "inv x \<in> N <#> H"
aux_lemma: fixes a :: "'a"   and b :: "'a" assumes "set (support p a) \<inter> set (support p b) \<noteq> {}" shows "set (support p a) \<subseteq> set (support p b)"
no_lb: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.dlo less_eq less" shows "\<exists>l. less l x"
comp_eq_dest_lhs: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'e"   and c :: "'c \<Rightarrow> 'd"   and v :: "'c" assumes "a \<circ> b = c" shows "a (b v) = c v"
of_int_minus: fixes z :: "int" shows "of_int (- z) = - of_int z"
intro: fixes S :: "<'d, 'f> \<Longleftrightarrow> 'e" assumes "mwb_lens \<V>\<^bsub>S\<^esub>"   and "mwb_lens \<C>\<^bsub>S\<^esub>"   and "\<V>\<^bsub>S\<^esub> \<bowtie> \<C>\<^bsub>S\<^esub>"   and "pbij_lens (\<V>\<^bsub>S\<^esub> +\<^sub>L \<C>\<^bsub>S\<^esub>)" shows "psym_lens S"
assms: shows "\<forall>a. ListMem a as \<longrightarrow> fmdom' (fmrestrict_set vs (snd a)) = {}"
Rats_divide: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a / b \<in> \<rat>"
HInit_def: fixes s :: "state" shows "HInit s = (Init s \<and> chosen s = NotAnInput \<and> allInput s = range (inpt s))"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) atm" assumes "M.satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "M.int \<xi>'__ T1 = M.int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "intP p (map (M.int \<xi>) Tl)"
lm018: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c"   and N :: "'a set"   and G :: "'b set"   and a :: "('a \<times> 'b set) set" shows "summedBidVectorRel bids N G || a = {summedBid bids pair |pair. pair \<in> N \<times> (Pow G - {{}}) \<inter> a}"
wf_formula_def: fixes n :: "nat"   and \<phi> :: "'a formula" shows "wf_formula n \<phi> \<equiv> pre_wf_formula n \<phi> \<and> FOV \<phi> \<inter> SOV \<phi> = {}"
dom_lunit: fixes a :: "'a" assumes "ide a" shows "local.dom \<l>[a] = \<I> \<otimes> a"
span_add_eq2: fixes y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "y \<in> span S" shows "(x + y \<in> span S) = (x \<in> span S)"
closed_atLeastAtMost: fixes a :: "'b"   and b :: "'b" shows "closed {a..b}"
tok_id_hash_def: fixes n :: "nat"   and m :: "nat" shows "tok_id_hash n m \<equiv> insert (Spy, \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle>) (tok_id_pubk_prik n m)"
set_iterator_image_alt_def: fixes g :: "'f \<Rightarrow> 'e"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "set_iterator_image g it = (\<lambda>c f. it c (\<lambda>x. f (g x)))"
functor_category_axioms: shows "functor_category (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)"
a: shows "M[x\<turnstile>n>y] \<longrightarrow>\<^sub>l M'"
textbook_null_fun: fixes \<tau> :: "'b state \<times> 'b state" shows "I\<lbrakk>null\<rbrakk> \<tau> = null"
mult_eq': shows "(\<otimes>\<^bsub>units_of R\<^esub>) = (\<lambda>x y. x * y mod p)"
simps_1: fixes uu :: "'t"   and uv :: "'a"   and uw :: "nat" shows "find_sig_reducer [] uu uv uw = None"
a: shows "\<forall>w. O w a = (O w x \<or> O w y)"
set_list_update: fixes xs :: "'b list"   and i :: "nat"   and k :: "'b" shows "set (xs[i := k]) = (if i < length xs then insert k (set (take i xs) \<union> set (drop (Suc i) xs)) else set xs)"
cases: fixes a1 :: "'c net_state"   and a2 :: "'d node_action"   and a3 :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(a1, a2, a3) \<in> pnet_sos S T" obtains s :: "'c net_state" and R :: "nat set" and m :: "'d" and s' :: "'c net_state" and t :: "'c net_state" and H :: "nat set" and K :: "nat set" and t' :: "'c net_state" where "a1 = SubnetS sb__ t__" and "a2 = R__:*cast(m__)" and "a3 = SubnetS s'b__ t'__" and "(sb__, R__:*cast(m__), s'b__) \<in> S" and "(t__, H__\<not>K__:arrive(m__), t'__) \<in> T" and "H__ \<subseteq> R__" and "K__ \<inter> R__ = {}"   | s :: "'c net_state" and H :: "nat set" and K :: "nat set" and m :: "'d" and s' :: "'c net_state" and t :: "'c net_state" and R :: "nat set" and t' :: "'c net_state" where "a1 = SubnetS sc__ ta__" and "a2 = R:*cast(ma__)" and "a3 = SubnetS s'c__ t'a__" and "(sc__, Ha__\<not>Ka__:arrive(ma__), s'c__) \<in> S" and "(ta__, R:*cast(ma__), t'a__) \<in> T" and "Ha__ \<subseteq> R" and "Ka__ \<inter> R = {}"   | s :: "'c net_state" and H :: "nat set" and K :: "nat set" and m :: "'d" and s' :: "'c net_state" and t :: "'c net_state" and H' :: "nat set" and K' :: "nat set" and t' :: "'c net_state" where "a1 = SubnetS sd__ tb__" and "a2 = (H \<union> H')\<not>(K \<union> K'):arrive(m)" and "a3 = SubnetS s'd__ t'b__" and "(sd__, H\<not>K:arrive(m), s'd__) \<in> S" and "(tb__, H'\<not>K':arrive(m), t'b__) \<in> T"   | s :: "'c net_state" and i :: "nat" and d :: "nat" and s' :: "'c net_state" and t :: "'c net_state" where "a1 = SubnetS se__ tc__" and "a2 = ia__:deliver(da__)" and "a3 = SubnetS s'e__ tc__" and "(se__, ia__:deliver(da__), s'e__) \<in> S"   | t :: "'c net_state" and i :: "nat" and d :: "nat" and t' :: "'c net_state" and s :: "'c net_state" where "a1 = SubnetS sf__ td__" and "a2 = ib__:deliver(d)" and "a3 = SubnetS sf__ t'c__" and "(td__, ib__:deliver(d), t'c__) \<in> T"   | s :: "'c net_state" and s' :: "'c net_state" and t :: "'c net_state" where "a1 = SubnetS sg__ te__" and "a2 = \<tau>" and "a3 = SubnetS s'f__ te__" and "(sg__, \<tau>, s'f__) \<in> S"   | t :: "'c net_state" and t' :: "'c net_state" and s :: "'c net_state" where "a1 = SubnetS sh__ tf__" and "a2 = \<tau>" and "a3 = SubnetS sh__ t'd__" and "(tf__, \<tau>, t'd__) \<in> T"   | s :: "'c net_state" and i :: "nat" and i' :: "nat" and s' :: "'c net_state" and t :: "'c net_state" and t' :: "'c net_state" where "a1 = SubnetS si__ tg__" and "a2 = connect(ic__, i'__)" and "a3 = SubnetS s'g__ t'e__" and "(si__, connect(ic__, i'__), s'g__) \<in> S" and "(tg__, connect(ic__, i'__), t'e__) \<in> T"   | s :: "'c net_state" and i :: "nat" and i' :: "nat" and s' :: "'c net_state" and t :: "'c net_state" and t' :: "'c net_state" where "a1 = SubnetS s t" and "a2 = disconnect(i, i')" and "a3 = SubnetS s' t'" and "(s, disconnect(i, i'), s') \<in> S" and "(t, disconnect(i, i'), t') \<in> T"
finsum_empty: fixes f :: "'c \<Rightarrow> 'a mod_ring poly" shows "finsum (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}) f {} = \<zero>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub>"