hcju/isabelleps huggingface dataset:
nn_transfer_operator_intg: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * nn_transfer_operator g x \<partial>M = \<integral>\<^sup>+ x. f (T x) * g x \<partial>M"
simps_2: shows "sys (Suc 0) = ACT ''a'' (VAR 0)"
HFinite_stc_Infinitesimal_add: fixes x :: "complex star" assumes "x \<in> HFinite" shows "\<exists>e\<in>Infinitesimal. x = stc x + e"
mbisim_simulation_silent2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "\<tau>mred2 s2 s2'" shows "\<exists>s1'. \<tau>mRed1 s1 s1' \<and> s1' \<approx>m s2'"
scale_minus_right: fixes a :: "complex"   and x :: "'d" shows "a *\<^sub>C - x = - (a *\<^sub>C x)"
mono_mprefix_F_process: fixes P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process"   and A :: "'a set" assumes "\<forall>x. P x \<sqsubseteq>\<^sub>F Q x" shows "Mprefix A P \<sqsubseteq>\<^sub>F Mprefix A Q"
sorted_wrt_iff_nth_less: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" shows "sorted_wrt P xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> P (xs ! i) (xs ! j))"
gtrp2gtr: fixes s :: "'a"   and c :: "'a multiset"   and w :: "'b el_step list"   and s' :: "'a"   and c' :: "'a multiset"   and T :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((s, c), w, s', c') \<in> trcl (gtrp T)" shows "(add_mset s c, map le_rem_s w, add_mset s' c') \<in> trcl (gtr T)"
carrier_Fstack: fixes as :: "Arity list"   and S :: "stack_elem list" shows "carrier (Fstack as S) \<subseteq> fv S"
prim_invar2_ctd_alt: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" shows "prim_invar2_ctd Q \<pi> = (let A = A Q \<pi>; S = S A; cE = edges rg \<inter> (- S) \<times> S in prim_invar1 A \<and> \<pi> r = None \<and> Q r = \<infinity> \<and> (\<forall>(u, v)\<in>cE. Q u \<noteq> \<infinity>) \<and> (\<forall>u v. \<pi> u = Some v \<longrightarrow> v \<in> S \<and> (u, v) \<in> edges rg) \<and> (\<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> cE \<longrightarrow> d \<le> w {u, v'}))))"
priority_Min: fixes q :: "('c, 'd) pq" assumes "\<not> is_empty q" shows "priority q (PQ.min q) = Some (Min (the ` priority q ` set |q|))"
trans: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry"   and c :: "'d DBMEntry" assumes "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
simps_6: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x2 :: "nat" shows "(case R x2 of I x \<Rightarrow> f1 x | R x \<Rightarrow> f2 x) = f2 x2"
deg_comp: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" shows "deg_comp cmp = (\<lambda>x y. case comparator_of (deg_pp (fst (rep_nat_term x))) (deg_pp (fst (rep_nat_term y))) of Eq \<Rightarrow> cmp x y | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
as_mem_agree_valid_in_snapshot: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set" assumes "\<forall>a. ListMem a as \<longrightarrow> agree (fst a) s \<and> agree (snd a) s"   and "as \<in> valid_plans PROB" shows "as \<in> valid_plans (snapshot PROB s)"
AA33: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<diamond>\<box>\<diamond>\<langle>A\<rangle>_v = \<box>\<diamond>\<langle>A\<rangle>_v"
index_def: fixes xs :: "'a list" shows "index xs = (\<lambda>a. find_index (\<lambda>x. x = a) xs)"
normf_upper: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "continuous_on S f"   and "x \<in> S" shows "\<bar>f x\<bar> \<le> normf f"
prefixes_eq_snoc: fixes ys :: "'a list"   and xs :: "'a list list"   and x :: "'a list" shows "(prefixes ys = xs @ [x]) = ((ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs @ [z] \<and> xs = prefixes zs)) \<and> x = ys)"
select_convs_4: fixes Obj :: "'b set"   and Mor :: "'a set"   and Dom :: "'a \<Rightarrow> 'b"   and Cod :: "'a \<Rightarrow> 'b"   and Id :: "'b \<Rightarrow> 'a"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'c" shows "Cod \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Cod"
\<tau>diverge_bisim_inv: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 \<approx> s2" shows "s1 -\<tau>1\<rightarrow> \<infinity> = s2 -\<tau>2\<rightarrow> \<infinity>"
group_of_Units: shows "Group_Theory.group additive.Units (+) \<zero>"
filter_Q_J_neg_same: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and Q_I_neg :: "(nat set \<times> 'b option list set) set"   and n :: "nat" assumes "getIJ getIJ"   and "2 \<le> card V"   and "(I, J) = getIJ Q Qn V"   and "Q_I_neg = New_max.filterQueryNeg I Qn"   and "rwf_query n V Q Qn" shows "New_max.filterQuery J (Qn - Q_I_neg) = Qn - Q_I_neg"
from_list_correct_1: fixes l :: "(('v \<times> 'v) \<times> nat) list" assumes "valid_wgraph_repr l" shows "invar (from_list l)"
\<psi>_bigo: shows "\<psi> \<in> O(\<lambda>x. x)"
map_single: fixes c :: "'c"   and f :: "'c \<Rightarrow> 'd"   and x :: "'e" assumes "c = (0::'c) \<Longrightarrow> f (0::'c) = (0::'d)" shows "Poly_Mapping.map f (monomial c x) = monomial (f c) x"
is_bisimulation_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "nominal_ts.is_bisimulation satisfies transition R" shows "nominal_ts.is_bisimulation satisfies transition (p \<bullet> R)"
in_set_ConsE: fixes xs :: "'c list"   and A :: "'c set"   and As :: "'c list set" assumes "xs \<in> set_Cons A As" obtains y :: "'c" and ys :: "'c list" where "xs = y # ys" and "y \<in> A" and "ys \<in> As"
has_integral_spike_finite: fixes S :: "'c set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "(f has_integral y) T" shows "(g has_integral y) T"
rem_condless_valid_2: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "sat_precond_as s (rem_condless_act s [] as)"
cod_trg_eq_dom_trg: shows "assoc\<nu>\<pi>\<rho>.cod.trg = assoc\<nu>\<pi>\<rho>.dtrg"
gram_schmidt_triv_def: fixes n :: "nat"   and ws :: "rat vec list" shows "gram_schmidt_triv n ws = rev (gram_schmidt_sub_triv n [] ws)"
distinct_5: fixes x1 :: "real mpoly"   and x4 :: "real mpoly" shows "Less x1 \<noteq> Neq x4"
span_clauses_2: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale" shows "(0::'b) \<in> module.span scale S"
assms_5: shows "A \<turnstile> ((\<^bold>@ i p) # (\<^bold>\<diamond> Nom i) # ps, a) # branch"
dist_metric_completion_limit': fixes x :: "nat \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> 'b" assumes "Cauchy x"   and "Cauchy y" shows "(\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> dist (abs_metric_completion x) (abs_metric_completion y)"
Can_implies_Arr: fixes t :: "'a local.term" assumes "Can t" shows "Arr t"
sim_disj: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula"   and G :: "'b formula" assumes "(F::'b formula) \<triangleright> (\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (H::'b formula)"   and "(G::'b formula) \<triangleright> (\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (H::'b formula)"   and "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<or> (G::'b formula)" shows "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (H::'b formula)"
img_fst: fixes a :: "'d"   and b :: "'e"   and S :: "('d \<times> 'e) set" assumes "(a, b) \<in> S" shows "a \<in> fst ` S"
compact_simplex: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "compact S"
Mapping_lookup_map_values: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a option"   and m :: "('b, 'c) mapping"   and k :: "'b" shows "Mapping.lookup (map_values f m) k = Mapping.lookup m k \<bind> f k"
enqueue_new_threads_def: shows "Round_Robin.enqueue_new_threads = foldl Round_Robin.enqueue_new_thread"
IS: fixes k :: "nat"   and xs :: "char list"   and x :: "char" assumes "\<forall>j\<ge>k. k \<le> MAX_STR \<longrightarrow> length xs = k \<longrightarrow> List.member (fin_string_upto j) (Abs_fin_string xs)" shows "\<forall>j\<ge>Suc k. Suc k \<le> MAX_STR \<longrightarrow> length (x # xs) = Suc k \<longrightarrow> List.member (fin_string_upto j) (Abs_fin_string (x # xs))"
get_scdom_component_owner_document_same: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_scdom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and thesis :: "bool"   and ptr :: "(_) object_ptr" assumes "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed type_wf known_ptr known_ptrs get_scdom_component get_dom_component"   and "\<And>owner_document. \<lbrakk>h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document; cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r owner_document \<in> set sc\<rbrakk> \<Longrightarrow> thesis"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "thesis"
Cons_1: assumes "Ind (set u) (set v)" shows "u @ v =\<^sub>F v @ u"
assms_2: assumes "f \<noteq> Null"   and "Dom f \<in> Obj"   and "Cod f \<in> Obj"   and "Map f \<in> Hom (Dom f) (Cod f)" shows "T"
negL: shows "\<forall>L\<in>#SDA'. is_neg L"
r: shows "ball z r \<inter> frontier S \<subseteq> V"
to_nat_plus_one_less_card: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "\<forall>a. plus a one \<noteq> zero \<longrightarrow> mod_type.to_nat Rep a + 1 < CARD('d)"
contour_integrable_on_def: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" shows "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
optimize_matches: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and f :: "'c match_expr \<Rightarrow> 'c match_expr"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state" assumes "\<forall>m a. matches \<gamma> (f m) a p = matches \<gamma> m a p" shows "approximating_bigstep_fun \<gamma> p (optimize_matches f rs) s = approximating_bigstep_fun \<gamma> p rs s"
dom_ord: fixes x :: "'a"   and y :: "'a" shows "(d x \<le> d y) = (d x \<cdot> d y = d x)"
cases: fixes a1 :: "'tp"   and a2 :: "('a, 'b) lit list" assumes "a1 \<turnstile> a2" obtains \<sigma> :: "'tp" and c :: "('a, 'b) lit list" where "a1 = \<sigma>'__" and "a2 = ca__" and "infTp \<sigma>'__"   | c :: "('a, 'b) lit list" and \<sigma> :: "'tp" where "a1 = \<sigma>" and "a2 = c" and "\<forall>x\<in>nvC c. TE.tpOfV x \<noteq> \<sigma>"
order_trans: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" assumes "x \<subseteq># y"   and "y \<subseteq># z" shows "x \<subseteq># z"
case_prodD': fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and c :: "'c" assumes "(case (a, b) of (c, d) \<Rightarrow> R c d) c" shows "R a b c"
andalso_thms_2: fixes y :: "tr" shows "(FF andalso y) = FF"
Span_append_eq_set_add: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R" shows "Span K (Us @ Vs) = Span K Us <+> Span K Vs"
\<iota>_in_hom': shows "\<guillemotleft>\<iota> : cod \<iota> \<otimes> cod \<iota> \<rightarrow> cod \<iota>\<guillemotright>"
conjugate_square_eq_0_1: fixes x :: "'c" shows "(x * conjugate x = (0::'c)) = (x = (0::'c))"
assms_2: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
type_eq_type: fixes v :: "val"   and T :: "ty"   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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "typeof v = \<lfloor>T\<rfloor>" shows "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>"
timpl_closure_Var_in_iff: fixes x :: "'d"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "(Var x \<in> timpl_closure t TI) = (t = Var x)"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "C.ide (g \<cdot>\<^sub>C f)" obtains "C.seq g f" and "C.seq f g" and "g \<cdot>\<^sub>C f = C.dom f" and "g \<cdot>\<^sub>C f = C.cod g"
rep_eq: fixes x :: "\<o>" shows "eval\<o> (\<^bold>\<not>x) = (\<lambda>s w. s = dj \<and> \<not> eval\<o> x dj w \<or> s \<noteq> dj \<and> I_NOT s (eval\<o> x s) w)"
deg_aboveI: fixes n :: "nat"   and p :: "nat \<Rightarrow> 'a" assumes "\<And>m. n < m \<Longrightarrow> coeff P p m = \<zero>"   and "p \<in> carrier P" shows "deg R p \<le> n"
sel: fixes x2 :: "'b" shows "the \<lfloor>x2\<rfloor> = x2"
finite_dimensional_vector_space_pair_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'e set"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B2 :: "'f set" shows "finite_dimensional_vector_space_pair s1 B1 s2 B2 \<equiv> finite_dimensional_vector_space s1 B1 \<and> finite_dimensional_vector_space s2 B2"
abs_summable_Re: fixes f :: "'b \<Rightarrow> complex"   and A :: "'b set" assumes "f abs_summable_on A" shows "(\<lambda>x. Re (f x)) abs_summable_on A"
fcrbe_def: fixes \<Gamma> :: "'e Gamma"   and \<rho> :: "'e V_rec \<Rightarrow> 'e set"   and ur :: "('s \<times> 's) set" shows "fcrbe \<Gamma> \<rho> ur \<equiv> \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>s''\<in>S\<^bsub>SES\<^esub>. reachable SES s \<and> s v\<longrightarrow>\<^bsub>SES\<^esub> s'' \<and> En \<rho> s c \<longrightarrow> (\<exists>s'\<in>S\<^bsub>SES\<^esub>. \<exists>\<delta>. (\<forall>d\<in>set \<delta>. d \<in> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub>) \<and> s ([c] @ \<delta> @ [v])\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> (s'', s') \<in> ur)"
eq_fold: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "abel_semigroup_set f z" shows "abel_semigroup_set.F f z g A = Finite_Set.fold (f \<circ> g) z A"
r_def: shows "r = {(f, g). f \<in> Field_r \<and> g \<in> Field_r \<and> f \<le> g}"
AllowedActs_extend: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program" assumes "Extend h" shows "AllowedActs (extend h F) = project_act h -` AllowedActs F"
inner_in_Sum: fixes b :: "'a"   and x :: "'a" assumes "b \<in> Basis" shows "x \<bullet> b = (\<Sum>i\<in>Basis. x \<bullet> i * (i \<bullet> b))"
vmatch'_rel: fixes Q :: "value \<Rightarrow> value \<Rightarrow> bool"   and t\<^sub>1 :: "value"   and t\<^sub>2 :: "value"   and p :: "pat" assumes "value_struct_rel Q"   and "Q t\<^sub>1 t\<^sub>2" shows "rel_option (fmrel Q) (??.Value.vmatch' p t\<^sub>1) (??.Value.vmatch' p t\<^sub>2)"
cis_multiple_2pi: fixes n :: "real" assumes "n \<in> \<int>" shows "cis (2 * pi * n) = 1"
A: shows "\<tau> \<Turnstile> OclAllInstances_generic pre_post H->includes\<^sub>S\<^sub>e\<^sub>t(x)"
hseq_char': fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "B.hseq \<nu> \<mu> = (\<mu> \<star> \<nu> \<noteq> B.null)"
filtrationE1: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and t :: "'c" assumes "Filtration.filtration M F" shows "subalgebra M (F t)"
factoringlemma_withconstant: fixes c :: "real"   and S0 :: "'a list set"   and A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)" assumes "1 \<le> c"   and "\<forall>e\<in>S0. distinct e"   and "\<forall>e\<in>S0. 0 < length e"   and "List_Factoring.pairwise A"   and "\<forall>s0\<in>S0. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. \<forall>(x, y)\<in>{(x, y). x \<in> set s0 \<and> y \<in> set s0 \<and> x < y}. T\<^sub>p_on_rand' A (fst A (Lxy s0 {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy s0 {x, y}, is))) (Lxy qs {x, y}) \<le> c * real (T\<^sub>p_opt (Lxy s0 {x, y}) (Lxy qs {x, y})) + b"   and "\<And>is s q. \<forall>((free, paid), uu_)\<in>set_pmf (snd A (s, is) q). paid = []"   and "\<And>init qs x. \<lbrakk>distinct init; set qs \<subseteq> set init; x < length qs\<rbrakk> \<Longrightarrow> finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))" shows "\<forall>s0\<in>S0. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. T\<^sub>p_on_rand' A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs \<le> c * real (T\<^sub>p_opt s0 qs) + b"
chamber_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexAutomorphism X f"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp X (f ` C)"
recv_marker_means_cs_Done: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "t ! i = RecvMarker cid p q"   and "i < length t" shows "snd (cs (distributed_system.s channel trans send recv init t (i + 1)) cid) = Done"
standard_decomp_map_times: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and X :: "'x set"   and s :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "standard_decomp k ps"   and "valid_decomp X ps"   and "s \<noteq> 0" shows "standard_decomp (k + poly_deg s) (map (apfst ((*) s)) ps)"
or_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "(or a b = zero) = (a = zero \<and> b = zero)"
is_len_cong_is_len: fixes A :: "'p"   and B :: "'p"   and l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'p"   and D :: "'p" assumes "TarskiLen A B l"   and "Cong A B C D" shows "TarskiLen C D l"
sup_state_refl: fixes P :: "'c prog"   and s :: "ty list \<times> ty err list" shows "P \<turnstile> s \<le>\<^sub>i s"
inv_image_codomain_eq: fixes c :: "('b, 'c) chart" shows "inv_chart c ` codomain c = domain c"
bind_SE'_def: fixes f :: "'c \<Rightarrow> ('e \<times> 'c) option"   and g :: "'c \<Rightarrow> ('d \<times> 'c) option" shows "f ;- g = _bind_SE _ f g"
is_map_from_is_homomorphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "Set_Theory.map (idim.im_sheaf_morphisms U V) (idim.im_sheaf U) (idim.im_sheaf V)"
list_exhaust2: fixes y :: "'c list"   and ya :: "'d list" obtains "y = []" and "ya = []"   | x21 :: "'d" and x22 :: "'d list" where "y = []" and "ya = x21__ # x22__"   | x21 :: "'c" and x22 :: "'c list" where "y = x21a__ # x22a__" and "ya = []"   | x21 :: "'c" and x22 :: "'c list" and x21a :: "'d" and x22a :: "'d list" where "y = x21 # x22" and "ya = x21a # x22a"
new_tv_Nil: fixes n :: "nat" shows "new_tv n []"
List_Null: fixes h :: "ref \<Rightarrow> ref"   and ps :: "ref list" shows "List Null h ps = (ps = [])"
representation_basis: fixes basis :: "'b set"   and b :: "'b" assumes "cindependent basis"   and "b \<in> basis" shows "crepresentation basis b = (\<lambda>v. if v = b then 1 else 0)"
rtranclp_into_tranclp1: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
normalize_quot_mult_coprime: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "coprime a b"   and "coprime c d"   and "unit_factor b = (1::'b)"   and "unit_factor d = (1::'b)" shows "normalize_quot (a * c, b * d) = (fst (normalize_quot (a, d)) * fst (normalize_quot (c, b)), snd (normalize_quot (a, d)) * snd (normalize_quot (c, b)))"
sort_oalist_id: fixes ko :: "'a key_order"   and xs :: "('a \<times> 'b) list" assumes "ko.oalist_inv_raw ko xs" shows "sort_oalist_ko' ko xs = xs"