hcju/isabelleps huggingface dataset:
of_int_code_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
simps_8: fixes x1 :: "'d"   and x2 :: "'d stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
cap_simps_6: fixes xo :: "'v vertex" shows "cap (xo, SOURCE) = 0"
take_append: fixes n :: "nat"   and xs :: "'k list"   and ys :: "'k list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
S_list_pgallery_decomp2: fixes ss :: "'a permutation list"   and Cs :: "'a set list" assumes "set ss = (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "Cs \<noteq> []"   and "pgallery (C0 # Cs)" shows "\<exists>rs s ts. ss = rs @ s # ts \<and> (\<exists>C\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> C \<in> g \<turnstile> \<C>) \<and> (\<forall>r\<in>set rs. \<forall>C\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> C \<in> f \<turnstile> \<C>)"
assms: shows "f \<in> stock_measure (val_type x) \<rightarrow>\<^sub>M subprob_algebra M"
unique_servKeys: fixes A :: "agent"   and K :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and K' :: "nat"   and B' :: "agent"   and Ts' :: "msg"   and X' :: "msg" assumes "Says Tgs A \<lbrace>Crypt K \<lbrace>Key servK, Agent B, Ts\<rbrace>, X\<rbrace> \<in> set evs"   and "Says Tgs A' \<lbrace>Crypt K' \<lbrace>Key servK, Agent B', Ts'\<rbrace>, X'\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "A = A' \<and> B = B' \<and> K = K' \<and> Ts = Ts' \<and> X = X'"
step: fixes x :: "'a" assumes "x \<le> ccpo_class.fixp f"   and "P x" shows "P (f x)"
update_convs_1: fixes NonceS' :: "rat option \<Rightarrow> rat option"   and NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "NonceS_update NonceS' \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = \<lparr>NonceS = NonceS' NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr>"
assms_3: shows "h (f (deg R f)) \<noteq> \<zero>\<^bsub>S\<^esub>"
ofilter_under_UNION: fixes A :: "'a set" assumes "local.ofilter A" shows "A = \<Union> (local.under ` A)"
list_all3_conv_all_nth: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and zs :: "'f list" shows "list_all3 P xs ys zs = (length xs = length ys \<and> length ys = length zs \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i) (zs ! i)))"
gamma_i_component_smooth: shows "\<forall>x\<in>{0..1} - s. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
UN_I: fixes a :: "'k"   and A :: "'k set"   and b :: "'l"   and B :: "'k \<Rightarrow> 'l set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
functor_abbrevE: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and A :: "('a, 'c, 'e) Category_scheme"   and B :: "('b, 'd, 'f) Category_scheme" assumes "Ftor F : A \<longrightarrow> B" obtains "Functor F" and "CatDom F = A" and "CatCod F = B"
exec_step_xcpt_h: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and vars :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xcp :: "nat"   and ins :: "bytecode"   and T :: "ty"   and mxs :: "nat"   and xt :: "ex_table"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and loc :: "val list" assumes "fst (exec_step P h stk vars C M pc ics frs sh) = \<lfloor>xcp\<rfloor>"   and "instrs_of P C M = ins"   and "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"   and "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]" shows "\<exists>obj. h xcp = \<lfloor>obj\<rfloor>"
nn_integral_ge_point: fixes x :: "'a"   and A :: "'a set"   and p :: "'a \<Rightarrow> ennreal" assumes "x \<in> A" shows "p x \<le> integral\<^sup>N (count_space A) p"
eq_on_def: fixes s :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow> 'c"   and X :: "'b set" shows "(s = t on X) = (\<forall>x\<in>X. s x = t x)"
simps_4: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and x :: "'b"   and \<tau> :: "PreSimplyTyped.type"   and A :: "'b ptrm" shows "ptrm_infer_type \<Gamma> (PFn x \<tau> A) = (case ptrm_infer_type (\<Gamma>(x \<mapsto> \<tau>)) A of None \<Rightarrow> None | Some \<sigma> \<Rightarrow> Some (TArr \<tau> \<sigma>))"
wt_ge_\<delta>\<^sub>h: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm" assumes "legal_zpassign A" shows "zhmset_of \<delta>\<^sub>h \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
the_elemI: fixes A :: "'a set" assumes "is_singleton A" shows "the_elem A \<in> A"
permutations_of_set_altdef: fixes A :: "'a set" assumes "finite A" shows "permutations_of_set A = permutations_of_multiset (mset_set A)"
one_neq_iffs_3: fixes x :: "one" shows "(x \<noteq> \<bottom>) = (x = ONE)"
mod_poly_one_main_i: fixes n :: "nat"   and X :: "'a list"   and x :: "'i list"   and y :: "'i list"   and Y :: "'a list"   and N :: "nat" assumes "n \<le> length X"   and "list_all2 R x X"   and "list_all2 R y Y"   and "n = N" shows "list_all2 R (mod_poly_one_main_i ops x y n) (mod_poly_one_main_list X Y N)"
2: shows "(H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) \<le> (H * d)\<^sup>\<star> * (H * a * top)"
size_partial_fraction_decomposition: fixes i :: "nat"   and xs :: "('b \<times> nat) list"   and x :: "'b"   and y :: "'b" assumes "i < length xs"   and "fst (xs ! i) \<noteq> (0::'b)"   and "x \<in> set (snd (partial_fraction_decomposition y xs) ! i)" shows "euclidean_size x < euclidean_size (fst (xs ! i))"
subtract_Some_set_of: fixes t\<^sub>1 :: "'b tree"   and t\<^sub>2 :: "'b tree"   and t :: "'b tree" assumes "subtract t\<^sub>1 t\<^sub>2 = Some t" shows "set_of t\<^sub>1 \<subseteq> set_of t\<^sub>2"
cases: fixes x :: "'a ifex \<times> 'a ifex \<times> 'a ifex" obtains i :: "'a ifex" and t :: "'a ifex" and e :: "'a ifex" where "x = (i, t, e)"
not_empty: shows "vote_set (votes sa) {p} \<noteq> {}"
mset_subset_of_empty: fixes A :: "'a multiset" shows "(A \<subset># \<Empt>) = False"
plus_stream_stl: fixes x :: "'b stream"   and y :: "'b stream" shows "stl (x + y) = stl x + stl y"
Class_eq: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Ring_Congruence" shows "additive.Class a = additive.Class b"
filter_set: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "Set.filter P (set xs) = set (filter P xs)"
compatAbsSubst_def: fixes xs :: "'varSort"   and s :: "'c"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd" shows "compatAbsSubst xs s f \<equiv> \<forall>x X x' X'. (\<forall>Y. wls (asSort xs) Y \<longrightarrow> X #[Y / x]_xs = X' #[Y / x']_xs) \<longrightarrow> f x X = f x' X'"
inf_greatest: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y"   and "less_eq x z" shows "less_eq x (inf y z)"
emb_preserve_mono: fixes J :: "'i set"   and L :: "'i set"   and X :: "('i \<Rightarrow> 'a) set"   and Y :: "('i \<Rightarrow> 'a) set" assumes "J \<subseteq> L"   and "L \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)"   and "Y \<in> sets (Pi\<^sub>M J M)"   and "emb L J X \<subseteq> emb L J Y" shows "X \<subseteq> Y"
ERETURN_rule: fixes \<Psi> :: "'c \<Rightarrow> bool"   and x :: "'c"   and \<Phi> :: "'d \<Rightarrow> bool" assumes "\<Psi> x" shows "ERETURN x \<le> ESPEC \<Phi> \<Psi>"
pochhammer_product: fixes m :: "nat"   and n :: "nat"   and z :: "'a" assumes "m \<le> n" shows "pochhammer z n = pochhammer z m * pochhammer (z + of_nat m) (n - m)"
eq: shows "(P,E(V \<mapsto> T'),h \<turnstile> blocks (Vs, Ts, vs, e) : T) = (P,E(V \<mapsto> T', Vs [\<mapsto>] Ts),h \<turnstile> e : T \<and> (\<exists>Ts'. map P \<turnstile> typeof\<^bsub>h\<^esub> vs = map Some Ts' \<and> P \<turnstile> Ts' [\<le>] Ts))"
oSuc_less_limit_ordinal: fixes x :: "ordinal"   and w :: "ordinal" assumes "limit_ordinal x" shows "(oSuc w < x) = (w < x)"
comp_char: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" shows "g \<cdot> f = (if seq g f then if f = Zero then g else if g = One then f else null else null)"
enum_alt_inj: fixes enum_alt :: "nat \<Rightarrow> 'b option"   and x :: "nat"   and y :: "nat" assumes "class.enumeration_alt enum_alt"   and "enum_alt x = enum_alt y" shows "x = y \<or> enum_alt x = None"
Pi_anti_mono: fixes A' :: "'e set"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "A' \<subseteq> A" shows "Pi A B \<subseteq> Pi A' B"
quorum_is_quorum_of_some_slice: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and Q :: "'b set" assumes "stellar slices W"   and "stellar.quorum_of slices W p Q"   and "p \<in> W" obtains S :: "'b set" where "S \<in> slices p" and "S \<subseteq> Q" and "\<And>p'. p' \<in> S \<inter> W \<Longrightarrow> stellar.quorum_of slices W p' Q"
dist_blinfun_def: fixes a :: "'a \<Rightarrow>\<^sub>L 'b"   and b :: "'a \<Rightarrow>\<^sub>L 'b" shows "dist a b = norm (a - b)"
add_numeral_special_3: shows "II \<sqinter> II = numeral.numeral II (\<sqinter>) (num.Bit0 num.One)"
append_Cons_eq_iff: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and xs' :: "'b list"   and ys' :: "'b list" assumes "x \<notin> set xs"   and "x \<notin> set ys" shows "(xs @ x # ys = xs' @ x # ys') = (xs = xs' \<and> ys = ys')"
cwiseext_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "cwiseext gt ys xs" shows "cwiseext gt' ys xs"
nodes_def: shows "nodes = Helper ## map Assumption assumptions @- map Conclusion conclusions @- smap Rule rules"
ideal_inc_one: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I"   and "1\<^sub>r\<^bsub>R\<^esub> \<in> I" shows "I = carrier R"
nths_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and I :: "nat set" shows "nths (map f xs) I = map f (nths xs I)"
Amicable_pair_def: fixes m :: "nat"   and n :: "nat" shows "m Amic n \<equiv> m = aliquot_sum n \<and> n = aliquot_sum m"
matcheq_matchNone: fixes m :: "'a match_expr"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b" assumes "\<not> has_primitive m" shows "matcheq_matchNone m = (\<not> Semantics.matches \<gamma> m p)"
inner_Basis_scale: fixes b1 :: "'a"   and c :: "real"   and v :: "'a" assumes "b1 \<in> S" shows "local.inner_Basis (c *\<^sub>R b1) v = c * local.inner_Basis b1 v"
laplace_integrand_absolutely_integrable_on_Ici_iff: fixes M :: "real"   and c :: "real"   and f :: "real \<Rightarrow> complex"   and s :: "complex"   and a :: "real" assumes "exponential_order M c f"   and "c < Re s" shows "complex_set_integrable lebesgue {a..} (laplace_integrand f s) = (\<forall>k>a. complex_set_integrable lebesgue {a..k} (laplace_integrand f s))"
\<sigma>_K: fixes v :: "'a" assumes "v \<in> K \<union> W1" shows "\<sigma> v = succ v"
monom_def: shows "monom \<equiv> map_fun id (map_fun id MPoly) Poly_Mapping.single"
step: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi" assumes "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"   and "\<Psi> \<rhd> P \<leadsto>[bisim] Q"   and "\<forall>\<Psi>''. \<Psi> \<otimes> \<Psi>'' \<rhd> P \<sim> Q"   and "\<Psi> \<rhd> Q \<sim> P" shows "\<Psi> \<rhd> P \<sim> Q"
Fun_lam: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>" shows "Fun (\<Lambda> a b c f) = (\<lambda>x\<^bold>\<in>local.set a. UP (happ (hlam (ide_to_hf a) (ide_to_hf b) (ide_to_hf c) (arr_to_hfun f)) (DOWN x)))"
elims_2: fixes x :: "'b pltl" assumes "next_free x" obtains "x = false\<^sub>p"   | p :: "'b \<Rightarrow> bool" where "x = atom\<^sub>p(p)"   | \<phi> :: "'b pltl" and \<psi> :: "'b pltl" where "x = \<phi>'__ implies\<^sub>p \<psi>__" and "next_free \<phi>'__ \<and> next_free \<psi>__"   | \<phi> :: "'b pltl" and \<psi> :: "'b pltl" where "x = \<phi> U\<^sub>p \<psi>" and "next_free \<phi> \<and> next_free \<psi>"
functional_converse_relation: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_unique R"   and "right_total R" shows "\<forall>y. \<exists>!x. R x y"
FVar_lemma2: fixes state :: "state"   and field :: "'a member_scheme"   and statDeclC :: "qtname"   and a :: "val"   and fn :: "vname"   and w :: "val" assumes "error_free state" shows "error_free (assign (\<lambda>v. supd (upd_gobj (if static field then Inr statDeclC else Inl (the_Addr a)) (Inl (fn, statDeclC)) v)) w state)"
minVertex_zero1: fixes z :: "nat" shows "minVertex (Face [0..<Suc z] Final) = 0"
Await: fixes s :: "'b"   and b :: "'b set"   and P :: "'b com"   and t :: "'b" assumes "s \<in> b"   and "(Some P, s) -c*\<rightarrow> (None, t)" shows "(Some (Await b P), s) -c\<rightarrow> (None, t)"
ab: shows "types P \<subseteq> types (class_add P (C, cdec))"
comm_gr: shows "comm_group (H\<lparr>one := h \<one>\<rparr>)"
dvd_normalize_iff: fixes a :: "'a"   and b :: "'a" shows "(a dvd normalize b) = (a dvd b)"
i_contract_ideal: fixes f :: "'a \<Rightarrow> 'c"   and J :: "'c set" assumes "f \<in> rHom R R'"   and "ideal R' J" shows "ideal R (i_contract f R R' J)"
assms_2: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
meet_join_absorb: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> (x \<squnion> y) = x"
ab: shows "\<Union> (range S) \<subseteq> cbox a b"
mono_fun_fun_cnv: fixes ordA :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ordB :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and F :: "('d \<Rightarrow> 'e) \<Rightarrow> 'c \<Rightarrow> 'f" assumes "\<And>d. monotone (fun_ord ordA) ordB (\<lambda>x. F x d)" shows "monotone (fun_ord ordA) (fun_ord ordB) F"
Bool: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and L :: "Vars.var list" shows "\<Gamma> : Bool b \<Down>\<^bsub>L\<^esub> \<Gamma> : Bool b"
atom_ordering_irrefl: shows "\<forall>x y. (x, y) \<in> atom_ordering \<longrightarrow> (y, x) \<notin> atom_ordering"
sound_mode_use\<^sub>A: shows "abs.sound_mode_use (cms\<^sub>A, mem\<^sub>A_of mem\<^sub>C)"
refl_on_def: fixes A :: "'e set"   and r :: "('e \<times> 'e) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
le_mask_iff_lt_2n: fixes n :: "nat"   and w :: "'b word" shows "(n < LENGTH('b)) = ((w \<le> mask n) = (w < 2 ^ n))"
mono_cSup: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "bdd_above A"   and "A \<noteq> {}"   and "mono f" shows "Sup (f ` A) \<le> f (Sup A)"
HInter_iff: fixes A :: "hf"   and x :: "hf" assumes "A \<noteq> 0" shows "(x \<^bold>\<in> \<Sqinter>A) = (\<forall>y. y \<^bold>\<in> A \<longrightarrow> x \<^bold>\<in> y)"
nprv_def: 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 prv :: "'c \<Rightarrow> bool"   and F :: "'c set"   and \<phi> :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv" shows "Deduct_with_False_Disj.nprv cnj imp fls prv F \<phi> \<equiv> prv (imp (Syntax_with_Connectives_False.scnj cnj imp fls F) \<phi>)"
integrable_real_indicator: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "integrable M (indicat_real A)"
monoid_axioms: shows "monoid (+) (zero_class.zero::'a)"
lookup_pair_sort_oalist': fixes xs :: "('a \<times> 'b) list"   and ko :: "'a nat_term_order" assumes "distinct (map fst xs)" shows "lookup_pair_ko_ntm ko (sort_oalist_ko_ntm' ko xs) = lookup_dflt xs"
bind_ask2: fixes m :: "'m"   and f :: "'a \<times> 's \<Rightarrow> 'r \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
fun_apply_def: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "f(x)\<^sub>a = f x"
WTrtNew: 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 E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "is_class P C" shows "P,E,h \<turnstile> new C : Class C"
linear_scale_real: fixes f :: "real \<Rightarrow> real"   and r :: "real"   and b :: "real" assumes "linear f" shows "f (r * b) = r * f b"
2: fixes "is" :: "nat list" assumes "is \<lhd> replicate (2 * N_half) (last rs)" shows "polyfun {..<weight_space_dim} (\<lambda>x. Tensor.lookup (A x) is)"
shift_append: fixes xs :: "(action \<times> nat) list"   and ys :: "(action \<times> nat) list"   and n :: "nat" shows "shift (xs @ ys) n = shift xs n @ shift ys n"
in_\<R>_dma': fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>A :: "'Com\<^sub>A"   and mds\<^sub>A :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and c\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and x\<^sub>A :: "'Var\<^sub>A" assumes "preserves_modes_mem \<R>"   and "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R>" shows "dma\<^sub>A mem\<^sub>A x\<^sub>A = dma\<^sub>C mem\<^sub>C (var\<^sub>C_of x\<^sub>A)"
fg_same_round_same_state: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and m :: "nat"   and p :: "'f"   and n :: "nat" assumes "fg_run A rho HOs SHOs coords"   and "round (rho m) p = round (rho n) p" shows "state (rho m) p = state (rho n) p"
fold_fun_left_comm: fixes A :: "'e set"   and x :: "'e"   and z :: "('e, 'f) mapping" assumes "finite A" shows "Mapping.delete x (Finite_Set.fold Mapping.delete z A) = Finite_Set.fold Mapping.delete (Mapping.delete x z) A"
a: shows "\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
postList_preList: fixes C2 :: "acom"   and upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "postList (map (\<lambda>(x, e, v). (x, preT C2 e, fune C2 v)) upds) l s = preList upds C2 l s"
stutter_no_time: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and m :: "nat"   and c :: "clock" assumes "dilating f sub r"   and "\<And>k. f n < k \<and> k \<le> m \<Longrightarrow> \<nexists>k\<^sub>0. f k\<^sub>0 = k"   and "f n < m" shows "time (Rep_run r m c) = time (Rep_run r (f n) c)"
subcocycle_integrable_2: fixes u :: "nat \<Rightarrow> (int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> real"   and n :: "nat" assumes "I.subcocycle u" shows "I.random_variable borel (u n)"
A72_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA93 \<and> C \<noteq> sA82 \<longrightarrow> sA72 \<notin> Sources level1 C"
comp_arr_dom: fixes f :: "('obj, 'obj) concrete_category.arr"   and a :: "('obj, 'obj) concrete_category.arr" assumes "arr f"   and "local.dom f = a" shows "f \<cdot> a = f"
assms: shows "intersect_segment_xline prec (p0, p1) x = None"
ctr_transfer_1: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R None None"
imp_conjL: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(P \<and> Q \<longrightarrow> R) = (P \<longrightarrow> Q \<longrightarrow> R)"
Abs1_eq_iff_3: fixes a :: "'f"   and x :: "'e"   and b :: "'f"   and y :: "'e" shows "([[atom a]]lst. x = [[atom b]]lst. y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = (a \<leftrightarrow> b) \<bullet> y \<and> atom a \<sharp> y)"
one_mat_def: fixes n :: "nat" shows "1\<^sub>m n \<equiv> mat n n (\<lambda>(i, j). if i = j then 1::'b else (0::'b))"
f: shows "\<forall>x. eval f [x] \<down>= (if x \<in> {x. eval r_phi [x, x] \<down>} then 1 else 0)"
reduced_labelling_unique: fixes r :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> nat" assumes "r \<le> n"   and "\<forall>i<r. x i = 0"   and "r = n \<or> x r \<noteq> 0" shows "reduced n x = r"
in_roofed_mono: fixes x :: "'k"   and G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set"   and S' :: "'k set" assumes "x \<in> roofed_gen G B S"   and "S \<subseteq> S'" shows "x \<in> roofed_gen G B S'"
simps: fixes x :: "'a"   and t :: "'a tree" shows "local.insert x t = (case ins (size t) 0 x t of Same \<Rightarrow> t | Bal t' \<Rightarrow> t')"
until_and_left_distrib: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi>\<^sub>1 :: "'b ltln"   and \<phi>\<^sub>2 :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n (\<phi>\<^sub>1 and\<^sub>n \<phi>\<^sub>2) U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<phi>\<^sub>1 U\<^sub>n \<psi> and\<^sub>n \<phi>\<^sub>2 U\<^sub>n \<psi>"
join_iro_def: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option"   and p :: "value option list" shows "join_iro i r p = (\<lambda>x. case x of Ip n \<Rightarrow> Some (i ! n) | Op n \<Rightarrow> p ! n | Rg n \<Rightarrow> r $ n)"
valid_y: shows "\<tau> \<Turnstile> \<upsilon> y"
MGUI: fixes t :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and u :: "'b trm" assumes "t \<lhd> \<sigma> = u \<lhd> \<sigma>"   and "\<And>\<theta>. t \<lhd> \<theta> = u \<lhd> \<theta> \<Longrightarrow> \<exists>\<gamma>. \<theta> \<doteq> \<sigma> \<lozenge> \<gamma>" shows "MGU \<sigma> t u"
sturm_signs: fixes p :: "real poly"   and i :: "nat"   and x :: "real" assumes "rsquarefree p"   and "i < length (sturm p) - 2"   and "poly (sturm p ! (i + 1)) x = 0" shows "poly (sturm p ! (i + 2)) x * poly (sturm p ! i) x < 0"
beta_subst_bv1: fixes s :: "Core.term"   and t :: "Core.term"   and lev :: "nat"   and x :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "subst_bv1 s lev x \<rightarrow>\<^sub>\<beta> subst_bv1 t lev x"
GSubmodule_GSpan_single: fixes m :: "'c" assumes "m \<in> V" shows "RModule FG (\<cdot>) (GSpan [m]) \<and> GSpan [m] \<subseteq> V"