hcju/isabelleps huggingface dataset:
sheaf_spec_morphisms_are_maps: fixes V :: "('a \<times> 'a) set set set"   and U :: "('a \<times> 'a) set set set" assumes "local.local.is_zariski_open V"   and "V \<subseteq> U" shows "Set_Theory.map (local.local.sheaf_spec_morphisms U V) (local.local.sheaf_spec U) (local.local.sheaf_spec V)"
open_preimage_inner: fixes S :: "real set"   and i :: "'b" assumes "open S"   and "i \<in> Basis" shows "open {x. x \<bullet> i \<in> S}"
zero_le_one: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
Failures_implies_Traces: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set"   and s :: "'a event list"   and X :: "'a event set" assumes "is_process P"   and "(s, X) \<in> FAILURES P" shows "s \<in> TRACES P"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "F.satC \<xi> \<equiv> list_ex (F.satL \<xi>)"
src_cod: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>C (C.cod \<mu>) = src\<^sub>C \<mu>"
IH: shows "\<forall>m<n. odd m \<longrightarrow> [(m - 1) div 2 - m div 4 = (m\<^sup>2 - 1) div 8] (mod 2)"
rel2p_nres_RETURN: fixes A :: "('g \<times> 'h) set"   and x :: "'g"   and y :: "'h" shows "rel2p (\<langle>A\<rangle>nres_rel) (RETURN x) (RETURN y) = rel2p A x y"
apartments_have_many_foldings2: fixes A :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<sim> D"   and "C \<noteq> D"   and "C \<in> A"   and "D \<in> A" shows "OpposedThinChamberComplexFoldings A (restrict1 (canonical_retraction A D \<circ> canonical_retraction (supapartment C (some_third_chamber C D (C \<inter> D))) C) (\<Union> A)) (restrict1 (canonical_retraction A C \<circ> canonical_retraction (supapartment D (some_third_chamber C D (C \<inter> D))) D) (\<Union> A)) C"
transpose_column_length: fixes xs :: "'g list list"   and i :: "nat" assumes "sorted (rev (map length xs))"   and "i < length xs" shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
set_restriction_subset: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "set_restriction f" shows "f A \<subseteq> A"
msum_comm: fixes N1 :: "'c set"   and N2 :: "'c set" assumes "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M" shows "submodule_sum N1 N2 = submodule_sum N2 N1"
update_matchF_def: fixes n :: "nat"   and I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and rels :: "event_data option list set list"   and nt :: "nat"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list" shows "update_matchF n I mr mrs rels nt aux = fst (foldr (update_matchF_step I mr mrs nt) aux (update_matchF_base n I mr mrs rels nt))"
ivl_rel_br: fixes a :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "\<langle>br a I\<rangle>ivl_rel = br (\<lambda>(x, y). set_of_ivl (a x, a y)) (\<lambda>(x, y). I x \<and> I y)"
less_imp_not_eq: fixes y :: "'a list"   and x :: "'a list" assumes "y <lex x" shows "(x = y) = False"
e: shows "{i. i < length qs \<and> qs ! i \<in> {y, x}} = {i. i < length qs} \<inter> {i. qs ! i \<in> {y, x}}"
sup_right_top: fixes x :: "'c" shows "top_homomorphism ((\<squnion>) x)"
many_strongly_connected_implies_9: fixes x :: "'a" assumes "many_strongly_connected x" shows "x\<^sup>T ; x \<le> x\<^sup>+"
4: fixes u :: "'b" assumes "comp k k' = Gt" shows "sorted_wrt lt (k' # map fst (update_by_pair (k, u) xs))"
act_annir: fixes e :: "'b"   and Ea :: "'b set"   and x :: "'a" assumes "e \<in> Ea" shows "Dla x e \<and> \<alpha> x e = e"
are_perspective_from_point_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and R :: "Points" shows "are_perspective_from_point A B C A' B' C' R \<equiv> distinct7 A B C A' B' C' R \<and> triangle A B C \<and> triangle A' B' C' \<and> distinct3l (line A A') (line B B') (line C C') \<and> meet_3_in (line A A') (line B B') (line C C') R"
not_ctran_None: fixes s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and i :: "nat" assumes "(None, s) # xs \<in> cptn"   and "i < length xs" shows "((None, s) # xs) ! i -e\<rightarrow> xs ! i"
Rep_game: fixes x :: "game" shows "Rep_game x \<in> game"
UN_simps_9: fixes C :: "'b \<Rightarrow> 'a set"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "\<Union> (C ` \<Union> (B ` A)) = (\<Union>x\<in>A. \<Union> (C ` B x))"
simps: fixes base :: "nat"   and e :: "nat"   and n :: "nat" assumes "2 \<le> base" shows "encode base e n = (if n = 0 then 0 else of_nat (n mod base) * \<omega>^ (encode base 0 e) + encode base (e + 1) (n div base))"
isAbr_simps_2: fixes s :: "'d" shows "isAbr (Abrupt s) = True"
le_of_int_ceiling: fixes x :: "'a" shows "x \<le> of_int \<lceil>x\<rceil>"
strand_vars_split_1: fixes S :: "('e, 'd) strand_step list"   and S' :: "('e, 'd) strand_step list" shows "vars\<^sub>s\<^sub>t (S @ S') = vars\<^sub>s\<^sub>t S \<union> vars\<^sub>s\<^sub>t S'"
rotate1_map: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "rotate1 (map f xs) = map f (rotate1 xs)"
None_eq_map_option_iff: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d option" shows "(None = map_option f x) = (x = None)"
vimage_def: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'b set" shows "f -` B \<equiv> {x. f x \<in> B}"
cross3_simps_12: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b * (a * c) = a * (b * c)"
simps_12: fixes var :: "nat"   and i :: "nat"   and a :: "atom fm" shows "freeIn var (AllN i a) = freeIn (var + i) a"
base_x_6: shows "\<forall>i\<in>R. \<forall>j\<in>R. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
Units_l_cancel: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and z :: "nat \<times> nat" assumes "x \<in> Units S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "(x \<otimes>\<^bsub>S\<^esub> y = x \<otimes>\<^bsub>S\<^esub> z) = (y = z)"
fGOver_def: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme" shows "M fgover R = (\<exists>H. finite_generator R M H)"
cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "L F f = L F g"
barrier_inv_def: shows "barrier_inv \<equiv> enough_flushs"
associative: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "monoid M composition unit"   and "a \<in> M"   and "b \<in> M"   and "c \<in> M" shows "composition (composition a b) c = composition a (composition b c)"
typedef_bi_unique: fixes Rep :: "'b \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and T :: "'d \<Rightarrow> 'b \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "bi_unique T"
matroid_ax_1a: fixes X :: "Points set" shows "0 \<le> rk X"
subset_oc_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(open_segment a b \<subseteq> closed_segment c d) = (a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
fls_conv_base_factor_shift_subdegree: fixes f :: "'a fls" shows "f = fls_shift (- fls_subdegree f) (fls_base_factor f)"
zero_order_4: fixes n :: "'k multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
obs_intra_the_element: fixes m :: "'node"   and n :: "'node"   and S :: "'node SDG_node set" assumes "m \<in> obs_intra n \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "(THE m. m \<in> obs_intra n \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) = m"
lt_zero: shows "extended_ord.lpp 0 = extended_ord.punit.min_term"
birkhoff_sum_of_invariants: fixes f :: "'a \<Rightarrow> real"   and x :: "'a"   and n :: "nat" assumes "f \<in> borel_measurable Invariants"   and "x \<in> space M" shows "birkhoff_sum f n x = real n * f x"
p\<^sub>0_simps: shows "trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)"
pfun_lens_def: fixes i :: "'b" shows "pfun_lens i = \<lparr>lens_get = \<lambda>s. s(i)\<^sub>p, lens_put = \<lambda>s. pfun_upd s i\<rparr>"
P''Trans: shows "\<Psi> \<rhd> P'' \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"
quotient_ClassE: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "A \<in> Partition"   and "\<And>a. a \<in> G \<Longrightarrow> P (Class a)" shows "P A"
6: fixes x :: "'a"   and y :: "'a" shows "- - x \<le> - (- x \<sqinter> y)"
hfun_sigma_ord_iff: fixes s :: "hf" shows "hfun_sigma_ord s = (OrdDom s \<and> hfun_sigma s)"
has_bochner_integral_scaleR_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> real"   and x :: "real" assumes "c \<noteq> (0::'c) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x *\<^sub>R c) (x *\<^sub>R c)"
tendsto_mult_right: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter"   and c :: "'c" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. f x * c) \<longlongrightarrow> l * c) F"
cases: fixes K :: "'c \<Rightarrow> 'c pmf"   and a1 :: "'c"   and a2 :: "'c stream" assumes "MC_syntax.enabled K a1 a2" obtains \<omega> :: "'c stream" and s :: "'c" where "a1 = s" and "a2 = \<omega>" and "MC_syntax.enabled K (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> set_pmf (K s)"
vv': assumes "v \<noteq> v'" shows "fst (splitAt v (verticesFrom f v')) = v' # tl (fst (splitAt v (verticesFrom f v')))"
raw_match'_dom_res_subset_tvsT: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'" shows "dom subs' \<subseteq> tvsT T \<union> dom subs"
prefixToLevelAppend: fixes level :: "nat"   and a :: "('a \<times> bool) list"   and b :: "('a \<times> bool) list" shows "prefixToLevel level (a @ b) = (if level < currentLevel a then prefixToLevel level a else a @ prefixToLevel_aux b level (currentLevel a))"
IdI: fixes a :: "'g" shows "(a, a) \<in> Id"
qfN_div_prime: fixes p :: "int"   and N :: "int"   and q :: "int"   and a :: "int"   and b :: "int" assumes "prime (p\<^sup>2 + N * q\<^sup>2) \<and> p\<^sup>2 + N * q\<^sup>2 dvd a\<^sup>2 + N * b\<^sup>2" shows "\<exists>u v. a\<^sup>2 + N * b\<^sup>2 = (u\<^sup>2 + N * v\<^sup>2) * (p\<^sup>2 + N * q\<^sup>2) \<and> (\<exists>e. a = p * u + e * N * q * v \<and> b = p * v - e * q * u \<and> \<bar>e\<bar> = 1)"
thetaSeqWT_WretrT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaSeqWT tval aval indis \<subseteq> PL_Indis.WretrT tval aval indis (PL_Indis.thetaSeqWT tval aval indis \<union> PL_Indis.WbisT tval aval indis)"
resumption_ord_antisym: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "resumption_ord r' r" shows "r = r'"
steps_z_norm_sound: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D"   and "[D']\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<exists>u\<in>[D]\<^bsub>v,n\<^esub>. \<exists>u'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l', u'\<rangle>"
simps_11: fixes a :: "atom" shows "push_forall (AllQ (fm.Atom a)) = (if freeIn 0 (fm.Atom a) then fm.Atom (lowerAtom 0 1 a) else AllQ (fm.Atom a))"
mirror_point_aform_val: fixes X :: "'a \<times> 'a pdevs"   and e :: "nat \<Rightarrow> real" shows "mirror_point (fst X) (aform_val e X) = aform_val (- e) X"
BoundO: fixes a :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a[x] \<prec> Q'"   and "a \<noteq> x" shows "\<exists>P'. P \<longmapsto> a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
cbox_point_def: fixes p\<^sub>0 :: "(real, 'a) vec"   and p\<^sub>1 :: "(real, 'a) vec" shows "cbox p\<^sub>0 p\<^sub>1 = {p. \<forall>k. p\<^sub>0 $ k \<le> p $ k \<and> p $ k \<le> p\<^sub>1 $ k}"
if_seq_eval\<^sub>w_helper\<^sub>A: fixes B :: "bexp"   and T :: "(Eg1.var, aexp, bexp) Stmt"   and E :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem :: "Eg1.var \<Rightarrow> nat"   and TAIL :: "(Eg1.var, aexp, bexp) Stmt" assumes "eval_abv\<^sub>A \<langle>Stmt.If B T E, mds, mem\<rangle>\<^sub>A \<langle>if ev\<^sub>B mem B then T else E, mds, mem\<rangle>\<^sub>A" shows "eval_abv\<^sub>A \<langle>Stmt.If B T E ;; TAIL, mds, mem\<rangle>\<^sub>A \<langle>if ev\<^sub>B mem B then T ;; TAIL else E ;; TAIL, mds, mem\<rangle>\<^sub>A"
Brouwer_degree2_compose: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "continuous_map (nsphere p) (nsphere p) f"   and "continuous_map (nsphere p) (nsphere p) g" shows "Brouwer_degree2 p (g \<circ> f) = Brouwer_degree2 p g * Brouwer_degree2 p f"
subfieldE_3: fixes K :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subfield K R" shows "K \<subseteq> carrier R"
hash_rt_F\<^sub>m_def: fixes h :: "'a \<Rightarrow> 'c"   and rhm :: "'b \<Rightarrow> 'd" shows "hash_rt_F\<^sub>m h rhm \<equiv> hash_blindable id \<circ> map_blindable\<^sub>m (map_prod h (map rhm)) id"
Inf_mono: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and B :: "'b::type set"   and A :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "\<And>b::'b::type. b \<in> (B::'b::type set) \<Longrightarrow> \<exists>a::'b::type\<in>A::'b::type set. (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) a b" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Inf::'b::type set \<Rightarrow> 'b::type) (A::'b::type set)) (Inf (B::'b::type set))"
hyps: shows "list_all (\<lambda>a. GE.int \<xi> (gT a) = Ik.int \<xi> a) Tl"
assms: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V"   and "v \<noteq> w" shows "v \<rightarrow> w"
co_in1c: shows "connected (inside (path_image c1 \<union> path_image c))"
inorder_splay_maxD: fixes t :: "'b tree"   and l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "splay_max t = \<langle>l, a, r\<rangle>"   and "Sorted_Less.sorted (inorder t)" shows "inorder l @ [a] = inorder t \<and> r = \<langle>\<rangle>"
notin_set_remove1: fixes x :: "'g"   and xs :: "'g list"   and y :: "'g" assumes "x \<notin> set xs" shows "x \<notin> set (remove1 y xs)"
cinner_CBasis: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and CBasis :: "'a set"   and u :: "'a"   and v :: "'a" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "u \<in> CBasis"   and "v \<in> CBasis" shows "cinner u v = (if u = v then 1 else 0)"
case4: assumes "x \<le> 0"   and "y \<le> 0" shows "x * y \<le> 4611686018427387904"
rel_refl: fixes R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('c, 'a, 'b) sp\<^sub>\<mu>" assumes "\<And>x. R1a x x"   and "\<And>x. R2a x x" shows "rel_sp\<^sub>\<mu> R1a R2a x x"
plus_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f + g = floatarith.Add f g"
aux_alpha_ignoreSecond_def: fixes K :: "('d, 'e, 'f, 'g, 'h) qTerm \<times> ('d, 'e, 'f, 'g, 'h) qTerm + ('d, 'e, 'f, 'g, 'h) qAbs \<times> ('d, 'e, 'f, 'g, 'h) qAbs" shows "aux_alpha_ignoreSecond K \<equiv> case K of Inl (X, Y) \<Rightarrow> termIn X | Inr (A, B) \<Rightarrow> absIn A"
norm_eq_sqrt_inner: fixes x :: "'b" shows "norm x = sqrt (inner x x)"
Cauchy_def: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
set_of_add_inc_left: fixes A :: "'b interval"   and A' :: "'b interval"   and B :: "'b interval" assumes "set_of A \<subseteq> set_of A'" shows "set_of (A + B) \<subseteq> set_of (A' + B)"
ideal_not_empty: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'b set" assumes "class.ring plus zero minus uminus times"   and "ring.ideal plus zero uminus times I" shows "I \<noteq> {}"
ratfps_Gcd: fixes A :: "'b ratfps set" assumes "A - {0} \<noteq> {}" shows "Gcd A = ratfps_of_poly (monom (1::'b) (Inf (ratfps_subdegree ` (A - {0}))))"
HInv4a2_def: fixes s :: "state"   and p :: "Proc" shows "HInv4a2 s p = (\<forall>D\<in>MajoritySet. \<exists>d\<in>D. mbal (disk s d p) \<le> mbal (dblock s p) \<and> bal (disk s d p) \<le> bal (dblock s p))"
coeff_mult_semiring_closed: fixes R :: "'a set"   and p :: "'a poly"   and q :: "'a poly"   and i :: "nat" assumes "(0::'a) \<in> R"   and "\<And>x y. \<lbrakk>x \<in> R; y \<in> R\<rbrakk> \<Longrightarrow> x + y \<in> R"   and "\<And>x y. \<lbrakk>x \<in> R; y \<in> R\<rbrakk> \<Longrightarrow> x * y \<in> R"   and "\<And>i. poly.coeff p i \<in> R"   and "\<And>i. poly.coeff q i \<in> R" shows "poly.coeff (p * q) i \<in> R"
sound_weaken_appR: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and SGS :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list" assumes "sound (SG, C)" shows "sound (SG @ SGS, C)"
nth_map: fixes n :: "nat"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
reduce_aforms_def: fixes prec :: "nat"   and C :: "(real \<times> real pdevs) list \<Rightarrow> nat \<Rightarrow> real list \<Rightarrow> bool"   and X :: "(real \<times> real pdevs) list" shows "reduce_aforms prec C X = summarize_aforms prec C (degree_aforms X) X"
timesD1: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf" assumes "\<langle>a, b\<rangle> \<^bold>\<in> A * B" shows "a \<^bold>\<in> A"
normal_inter: fixes H :: "'a set"   and K :: "'a set"   and H1 :: "'a set" assumes "subgroup H G"   and "subgroup K G"   and "H1 \<lhd> G\<lparr>carrier := H\<rparr>" shows "H1 \<inter> K \<lhd> G\<lparr>carrier := H \<inter> K\<rparr>"
new_packets_to_simple_firewall_overapproximation: fixes \<gamma> :: "'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and rs :: "'a common_primitive rule list"   and \<Gamma> :: "char list \<Rightarrow> 'a common_primitive rule list option" assumes "matcher_agree_on_exact_matches \<gamma> common_matcher"   and "simple_ruleset rs" shows "_Collect p (\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New) \<subseteq> _Collect p (simple_fw (to_simple_firewall (upper_closure (optimize_matches abstract_for_simple_firewall (upper_closure (packet_assume_new rs))))) p = Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New)"
rcoset_equiv: fixes H :: "'g set" assumes "Subgroup H" shows "equiv G (rcoset_rel H)"
strategy_attracts_viaE: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy_attracts_via G p \<sigma> v0 A W" shows "visits_via P A W"
limsup_le_extended_Gromov_product_inside: fixes v :: "nat \<Rightarrow> 'a"   and eta :: "'a Gromov_completion"   and e :: "'a"   and x :: "'a" assumes "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> eta" shows "limsup (\<lambda>xa. ereal (Gromov_product_at e x (v xa))) \<le> extended_Gromov_product_at e (to_Gromov_completion x) eta + ereal (deltaG TYPE('a))"
arr_cod_iff_arr: fixes f :: "'s" shows "S.arr (S.cod f) = S.arr f"
allAllocationsEquivalence: fixes \<Omega> :: "'d set"   and N :: "'c set" shows "allAllocations N \<Omega> = allAllocations' N \<Omega> \<and> allAllocations N \<Omega> = allAllocations'' N \<Omega>"
\<Sigma>_protocol_def: shows "chaum_ped_sigma.\<Sigma>_protocol \<equiv> chaum_ped_sigma.completeness \<and> chaum_ped_sigma.special_soundness \<and> chaum_ped_sigma.HVZK"
word_rev_tf: fixes bl :: "bool list" shows "to_bl (of_bl bl) = rev (takefill False LENGTH('a) (rev bl))"
not_integrable_integral_eq: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "\<not> integrable M f" shows "integral\<^sup>L M f = (0::'e)"
hom_prod: fixes f :: "'b \<Rightarrow> rat"   and X :: "'b set" shows "of_rat (prod f X) = (\<Prod>x\<in>X. of_rat (f x))"
hyps: assumes "identify_block A j = i" shows "i \<le> j \<and> (i = 0 \<or> A $$ (i - 1, i) \<noteq> (1::'a)) \<and> (\<forall>k\<ge>i. k < j \<longrightarrow> A $$ (k, Suc k) = (1::'a))"
maintain_invar2_first_step_1: fixes r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and d :: "nat"   and w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph" assumes "Prim2.prim_invar2_init r Q \<pi>"   and "Q u = enat d" shows "Prim2.prim_invar2_ctd w g r (Prim2.Q' w g r Q \<pi> u) (Prim2.\<pi>' w g r Q \<pi> u)"
simps: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "Inf.inf r a = (\<exists>x y. a = x \<and> r x y \<and> Inf.inf r y)"
vpa_to_vpra: fixes cs :: "'edge list"   and as :: "'edge list"   and cs' :: "'edge list"   and n :: "'node"   and n' :: "'node"   and n'' :: "'node" assumes "valid_path_aux cs as"   and "valid_path_aux (upd_cs cs as) cs'"   and "n -as\<rightarrow>* n'"   and "valid_call_list cs n"   and "valid_return_list cs' n''" shows "valid_path_rev_aux cs' as \<and> valid_path_rev_aux (upd_rev_cs cs' as) (rev cs)"
assms_2: fixes x :: "V" assumes "a \<in>\<^sub>\<circ> x"   and "x \<in>\<^sub>\<circ> A" shows "R"