hcju/isabelleps huggingface dataset:
fix_asset_price: fixes Mkt :: "('d, 'c) discrete_market"   and pr :: "nat \<Rightarrow> 'd \<Rightarrow> real" shows "\<exists>x Mkt2. x \<notin> stocks Mkt \<and> coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr"
inv_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "local.iso \<nu>"   and "local.iso \<mu>"   and "src \<nu> = trg \<mu>" shows "local.inv (\<nu> \<star> \<mu>) = local.inv \<nu> \<star> local.inv \<mu>"
that: fixes P' :: "ccs" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "(P', Q') \<in> Rel" shows "thesis"
run_ended_trans: fixes s :: "l1_state"   and R :: "fid_t"   and s' :: "l1_state" assumes "run_ended (progress s R)"   and "(s, s') \<in> TS.trans l1" shows "run_ended (progress s' R)"
partial_summation_sum_upto_strong: shows "((\<lambda>t. A t * f' t) has_integral A x * f x - sum_upto (\<lambda>n. a n * f (real n)) x) {0..x}"
document_ptr_kinds_def: fixes heap :: "(_) heap" shows "document_ptr_kinds heap = the |`| cast |`| ffilter is_document_ptr_kind (object_ptr_kinds heap)"
le_int_of_uint64: fixes x :: "uint64"   and y :: "uint64" shows "(x \<le> y) = (int_of_uint64 x \<le> int_of_uint64 y)"
is_rewrite_ordD4: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and a :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and b :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "is_rewrite_ord rword"   and "rword a b"   and "rword b a" shows "fst a = fst b"
inj_on_def: fixes f :: "'q \<Rightarrow> 'r"   and A :: "'q set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
scaleR_columns_Gram_Schmidt_matrix: fixes i :: "'a"   and j :: "'a"   and A :: "((real, 'a) vec, 'b) vec" assumes "i \<noteq> j"   and "rank A = ncols A" shows "column j (Gram_Schmidt_matrix A) \<bullet> column i (Gram_Schmidt_matrix A) = 0"
dgrad_p_set_leI_insert_keys: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dgrad_p_set_le d F G"   and "dgrad_set_le d (pp_of_term ` keys f) (pp_of_term ` Keys G)" shows "dgrad_p_set_le d (insert f F) G"
VA: shows "Vagree \<nu> \<nu>' (FVT (Times \<theta>\<^sub>1 \<theta>\<^sub>2))"
le_infI: fixes x :: "'b upred"   and a :: "'b upred"   and b :: "'b upred" assumes "a \<sqsubseteq> x"   and "b \<sqsubseteq> x" shows "(a \<and> b) \<sqsubseteq> x"
Complexs_inverse: fixes a :: "'b" assumes "a \<in> \<complex>" shows "inverse a \<in> \<complex>"
that: fixes t :: "'t"   and loc :: "'a" assumes "t \<in>#\<^sub>z a loc"   and "\<forall>t' loc'. t' \<in>#\<^sub>z a loc' \<longrightarrow> \<not> t' < t" shows "thesis"
P'_def: shows "P' \<equiv> \<lambda>x y. P x y \<and> Q (h x) (h y)"
newton_seq_props_induct: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "ns k \<in> carrier Zp \<and> val_Zp (f' \<bullet> ns k) = val_Zp (f' \<bullet> a) \<and> eint 2 * val_Zp (f' \<bullet> a) + eint (2 ^ k) * t \<le> val_Zp (f \<bullet> ns k)"
finite_True_in_row: fixes n :: "nat"   and w :: "('a \<times> bool list) list" shows "\<forall>i<n. finite {p. snd ((w @- sconst (any, replicate n False)) !! p) ! i}"
rep_abs_rsp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and x1 :: "'a"   and x2 :: "'a" assumes "Quotient3 R Abs Rep"   and "R x1 x2" shows "R x1 (Rep (Abs x2))"
concat_map_concat: fixes xsss :: "'a list list list" shows "concat (map concat xsss) = concat (concat xsss)"
dim_kronecker_2: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_col (kronecker_product A B) = dim_col A * dim_col B"
LeastI2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(Const x1 = Const y1) = (x1 = y1)"
sel_1: fixes x1 :: "'g" shows "projl (Inl x1) = x1"
indep_system_axioms: shows "indep_system \<E> (indep_in \<E>)"
countable_card_le_natLeq: fixes A :: "'b set" shows "countable A = (|A| \<le>o natLeq)"
inj_on_opp_chambers'': fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "f ` C = f ` D" shows "C = D"
cauchy_schwarz_col: fixes v :: "complex Matrix.vec"   and w :: "complex Matrix.vec" assumes "dim_vec v = dim_vec w"   and "(cmod \<langle>v|w\<rangle>)\<^sup>2 = Re (\<langle>v|v\<rangle> * \<langle>w|w\<rangle>)" shows "\<exists>l. v = l \<cdot>\<^sub>v w \<or> w = l \<cdot>\<^sub>v v"
3: shows "\<^bold>\<not> (E (a \<cdot> (dom a)))"
ele_n_prodTr0: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set"   and a :: "'a" assumes "\<forall>k\<le>Suc n. ideal R (J k)"   and "a \<in> i\<Pi>\<^bsub>R,Suc n\<^esub> J" shows "a \<in> i\<Pi>\<^bsub>R,n\<^esub> J \<and> a \<in> J (Suc n)"
emeasure_empty: fixes M :: "'a measure" shows "emeasure M {} = 0"
a_l_coset_def: fixes G :: "('c, 'd) ring_scheme" shows "(<+\<^bsub>G\<^esub>) = (<#\<^bsub>add_monoid G\<^esub>)"
disjnt_sym: fixes A :: "'d set"   and B :: "'d set" assumes "disjnt A B" shows "disjnt B A"
map_update: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and k :: "nat"   and y :: "'f" shows "map f (xs[k := y]) = (map f xs)[k := f y]"
y: shows "_Collect (p_src p) ((newpkt::('i, 'pkt_ext) tagged_packet_scheme \<Rightarrow> bool) p \<and> (common_matcher, in_doubt_allow),p\<lparr>p_iiface := iface_sel (iface::iface)\<rparr>\<turnstile> \<langle>(preprocess::'i common_primitive rule list \<Rightarrow> 'i common_primitive rule list) (rs::'i common_primitive rule list), Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow) \<subseteq> ipcidr_union_set (set (the ((ipassmt::iface \<Rightarrow> ('i word \<times> nat) list option) iface)))"
simps: fixes p :: "real poly"   and qs :: "real poly list" shows "calculate_data p qs = (let len = length qs in if len = 0 then case reduce_system p ([1], base_case_info) of (a, b, c) \<Rightarrow> (a, b, map (drop 1) c) else if len \<le> 1 then reduce_system p (qs, base_case_info) else let q1 = take (len div 2) qs; left = calculate_data p q1; q2 = drop (len div 2) qs; right = calculate_data p q2 in Let (combine_systems p (q1, left) (q2, right)) (reduce_system p))"
length_nths: fixes xs :: "'b list"   and I :: "nat set" shows "length (nths xs I) = card {i. i < length xs \<and> i \<in> I}"
mathias: fixes xs' :: "'c list"   and xs :: "'c list"   and lo :: "nat"   and i :: "nat"   and hi :: "nat"   and x :: "'c" assumes "mset xs' = mset xs"   and "lo \<le> i"   and "i \<le> hi"   and "xs' ! i = x"   and "hi < length xs"   and "\<And>i. i < lo \<Longrightarrow> xs' ! i = xs ! i"   and "\<And>j. \<lbrakk>hi < j; j < length xs\<rbrakk> \<Longrightarrow> xs' ! j = xs ! j" shows "\<exists>j\<ge>lo. j \<le> hi \<and> xs ! j = x"
valid_decompD_3: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "valid_decomp X ps"   and "(h, U) \<in> set ps" shows "U \<subseteq> X"
bound_on_all_plans_bounds_MPLS_s_i: fixes s :: "('c, 'd) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and x :: "nat" assumes "s \<in> valid_states PROB"   and "x \<in> MPLS_s PROB s" shows "\<exists>as. x = Inf (PLS s as) \<and> as \<in> valid_plans PROB"
hyps: fixes rv1 :: "'a \<times> nat" assumes "RedBlack prb"   and "SubRel.ces rv1 res rv2 (subs prb)" shows "Graph.ces (fst rv1) (ui_es res) (fst rv2)"
intro: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "semilat (A, r, f)" shows "Semilat A r f"
fst_set_zip_map_fst: fixes xs :: "('a \<times> 'b) list"   and ys :: "'c list" assumes "length xs = length ys" shows "fst ` set (zip (map fst xs) ys) = fst ` set xs"
interval_inner_leI_2: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and a :: "'e"   and b :: "'e"   and i :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "x \<in> ord.atLeastAtMost less_eq a b"   and "less_eq zero i" shows "inner x i \<le> inner b i"
Rep_traffic: fixes x :: "traffic" shows "Rep_traffic x \<in> {ts. (\<forall>c. fst (snd ts) c \<sqinter> fst (snd (snd ts)) c = bot) \<and> (\<forall>c. 1 \<le> |fst (snd ts) c|) \<and> (\<forall>c. |fst (snd ts) c| \<le> 2) \<and> (\<forall>c. |fst (snd (snd ts)) c| \<le> 1) \<and> (\<forall>c. |fst (snd ts) c| + |fst (snd (snd ts)) c| \<le> 2) \<and> (\<forall>c. fst (snd (snd ts)) c \<noteq> bot \<longrightarrow> (\<exists>n. Rep_nat_int (fst (snd ts) c) \<union> Rep_nat_int (fst (snd (snd ts)) c) = {n, n + 1})) \<and> (\<forall>c. 0 < fst (snd (snd (snd (snd ts)))) c) \<and> (\<forall>c. 0 < snd (snd (snd (snd (snd ts)))) c)}"
isoE: fixes f :: "'a \<times> 'a" assumes "CC.iso f" obtains g :: "'a \<times> 'a" where "CC.inverse_arrows f g"
srtc_def: fixes a :: "'a"   and b :: "'a" shows "a \<leftrightarrow>\<^sup>* b \<equiv> (\<leftrightarrow>)\<^sup>*\<^sup>* a b"
index_mat_addrow_mat_3: fixes n :: "nat"   and a :: "'b"   and k :: "nat"   and l :: "nat" shows "dim_col (addrow_mat n a k l) = n"
simps_2: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat"   and p :: "'c point" assumes "ell_field R" shows "ell_field.point_mult R a (Suc n) p = ell_field.add R a p (ell_field.point_mult R a n p)"
Inf_Assertion_def: fixes A :: "'c Assertion set" shows "\<Sqinter>A = - \<Squnion>(uminus ` A)"
R_def: fixes x :: "nat"   and y :: "nat" shows "R x y = (x \<noteq> 0 \<longrightarrow> y = 0)"
next_initial_ps2_vertex: fixes v :: "node"   and S :: "node fset"   and G :: "diagram" shows "initial_ps2 ({|v|} |\<union>| S) G = initial_ps2 S G \<ominus> {|v|} ++\<^sub>f [ {|v|} |=> Bot ]"
single_gen_carrTr: fixes a :: "'a" assumes "a \<in> carrier A" shows "A +> {x. \<exists>n. x = n\<triangleright>a\<^bsub>A\<^esub>}"
simps_3: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a dres" shows "dRETURN x \<bind> f = f x"
Its_Suc: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and s' :: "char list \<Rightarrow> int"   and n :: "nat" assumes "bval b s"   and "(c, s) \<Rightarrow> s'"   and "Its b c s' n" shows "Its b c s (Suc n)"
simps_4: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c, 'd) Tree"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c, 'd) Tree" shows "(tAct f1 \<alpha>1 t1 =\<^sub>\<alpha> tAct f2 \<alpha>2 t2) = (f1 = f2 \<and> (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)))"
I_def: fixes s :: "'b" shows "I s = map_pmf (Pair s) (pmf_of_set \<O>\<^sub>s)"
is_asig_def: fixes triple :: "'d set \<times> 'd set \<times> 'd set" shows "is_asig triple \<equiv> inputs triple \<inter> outputs triple = {} \<and> outputs triple \<inter> internals triple = {} \<and> inputs triple \<inter> internals triple = {}"
MakeFormP_Ex: fixes v :: "tm"   and t :: "tm"   and x :: "tm"   and y :: "tm" shows "{AbstFormP v Zero t x} \<turnstile> MakeFormP (Q_Ex x) t y"
zero_le: fixes x :: "hf" shows "0 \<le> x"
similar_mat_trans: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat" assumes "similar_mat A B"   and "similar_mat B C" shows "similar_mat A C"
times_le_mono: fixes x :: "'a"   and y :: "'a"   and u :: "'a"   and v :: "'a" assumes "x \<le> y"   and "u \<le> v" shows "u * x \<le> v * y"
ground_subst_fv_subset: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and t :: "('d, 'c) Term.term" assumes "ground (subst_range \<theta>)" shows "fv (t \<cdot> \<theta>) \<subseteq> fv t"
bfs_correct: fixes src :: "nat"   and dst :: "nat" shows "bfs src dst \<le> bfs_spec src dst"
reduced_labelling_1: fixes n :: "nat"   and x :: "nat \<Rightarrow> nat" shows "reduced n x \<le> n"
dim_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C" shows "card (f ` C) = card C"
poly_rel_dupe_monic_i: fixes D :: "'a mod_ring poly"   and d :: "'i list"   and h :: "'i list"   and H :: "'a mod_ring poly"   and s :: "'i list"   and S :: "'a mod_ring poly"   and t :: "'i list"   and T :: "'a mod_ring poly"   and u :: "'i list"   and U :: "'a mod_ring poly" assumes "monic D"   and "poly_rel d D"   and "poly_rel h H"   and "poly_rel s S"   and "poly_rel t T"   and "poly_rel u U" shows "rel_prod poly_rel poly_rel (dupe_monic_i ff_ops d h s t u) (dupe_monic D H S T U)"
pdevs_apply_inject: fixes x :: "'b pdevs"   and y :: "'b pdevs" shows "(pdevs_apply x = pdevs_apply y) = (x = y)"
TRel_def: shows "TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel)"
gtr2gtrp: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "(add_mset s c, w, c') \<in> trcl (ntr fg)" obtains s' :: "'a list" and ce' :: "'a list multiset" and ww :: "('b, 'c) label list el_step list" where "c' = add_mset s' ce'" and "w = map le_rem_s ww" and "((s, c), ww, s', ce') \<in> trcl (ntrp fg)"
gdpr_abs_att: shows "\<turnstile>\<^sub>V [\<N>\<^bsub>(Igdpr, sgdpr)\<^esub>] \<oplus>\<^sub>\<and>\<^bsup>(Igdpr, sgdpr)\<^esup>"
induct: fixes x :: "'a"   and T :: "('a, 'b) TS_scheme"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> reach T"   and "\<And>s. s \<in> init T \<Longrightarrow> P s"   and "\<And>s t. \<lbrakk>(s, t) \<in> TS.trans T; s \<in> reach T; P s\<rbrakk> \<Longrightarrow> P t" shows "P x"
body_lc: fixes t :: "sterm"   and s :: "sterm"   and p :: "sterm" assumes "body t"   and "lc s"   and "lc p" shows "lc (t\<^bsup>[s,p]\<^esup>)"
zero_interval_def: shows "0 \<equiv> Interval (0::'a) (0::'a)"
p5: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All j' (SyntaxN.All sj (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) ki IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP ki (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))))))(j::=Var j)"
of_nat_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m * n) = times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
fds_truncate_add_strong: fixes m :: "nat"   and f :: "'b fds"   and g :: "'b fds" shows "fds_truncate m (f + g) = fds_truncate m f + fds_truncate m g"
selector_pushout: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and selector :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and G'' :: "nat \<times> ('d \<times> nat \<times> nat) set"   and R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph"   and f :: "('e \<times> nat) set"   and t :: "'f itself" assumes "valid_selector Rs selector"   and "selector G'' = Some (R, f)"   and "graph (graph_of G'')" shows "pushout_step t R (graph_of G'') (LG (on_triple (extend (fst G'') R f) `` edges (snd R) \<union> snd G'') {0..<max (fst G'') (nextMax (Range (extend (fst G'') R f)))})"
prod_mset_zero_iff: fixes A :: "'a multiset" shows "(\<Prod>\<^sub># A = (0::'a)) = ((0::'a) \<in># A)"
wf_rexp_of': fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "alphabet.wf (wf_atom \<Sigma>) n (formula.rexp_of' n \<phi>)"
circ_mult_1: fixes x :: "'b"   and y :: "'b" shows "(x * y)\<^sup>\<star> = Rf (x * (y * x)\<^sup>\<star>) y"
target_alt_def_1: fixes io :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f" assumes "length io = length tr"   and "length io = 0" shows "target (io || tr) q = q"
xt1_6: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<in> U"   and "x \<in> U"   and "z \<in> U"   and "y \<le>\<^sub>o\<^sub>w x"   and "z \<le>\<^sub>o\<^sub>w y" shows "z \<le>\<^sub>o\<^sub>w x"
dense: fixes 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 x :: "'b"   and y :: "'b" assumes "class.time plus zero minus uminus less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
homologous_rel_singular_subdivision: fixes p :: "nat"   and X :: "'b topology"   and T :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X T c" shows "homologous_rel p X T (singular_subdivision p c) c"
divide_less_0_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (R a)"
\<rho>'_ide_simp: fixes a :: "'d" assumes "ide a" shows "\<rho>' a = \<r>\<^sup>-\<^sup>1[a]"
compat2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "delta_gt \<delta> x y"   and "z \<le> y" shows "delta_gt \<delta> x z"
comp_char: fixes g :: "'a"   and f :: "'a" shows "hom.comp g f = (if hom.arr f \<and> hom.arr g \<and> seq g f then g \<cdot> f else null)"
cofinite_interior_infinite: fixes X :: "'a cofinite set" assumes "infinite X"   and "infinite (- X)" shows "interior X = {}"
relcomp3_I: fixes t :: "'b"   and u :: "'c"   and A :: "('b \<times> 'c) set"   and s :: "'c"   and B :: "('c \<times> 'b) set"   and v :: "'b" assumes "(t, u) \<in> A"   and "(s, t) \<in> B"   and "(u, v) \<in> B" shows "(s, v) \<in> B O A O B"
a_get_tag_name_def: fixes element_ptr :: "(_) element_ptr" shows "get_tag_name element_ptr = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name"
groupWIs_complete: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "\<Union> (wordinterval_list_to_set ` set (groupWIs c rs)) = UNIV"
game2_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "game2 \<A> \<equiv> key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (oracle_encrypt2 key b \<oplus>\<^sub>O oracle_decrypt2 key) \<A> Map_empty \<bind> (\<lambda>(b', D). return_spmf (b = b'))))"
update: shows "GEN_OP upd op_map_update (A \<rightarrow> nat_rel \<rightarrow> \<langle>A, nat_rel\<rangle>Rm \<rightarrow> \<langle>A, nat_rel\<rangle>Rm)"
hd_lcls: shows "fst ` lhd (lmap lclss_of_state Sts) = lhd (lmap (\<lambda>St. N_of_state St \<union> P_of_state St \<union> Q_of_state St) Sts)"
assms: fixes G :: "('a, 'x) labeled_graph"   and h\<^sub>1 :: "('c \<times> 'x) set"   and h\<^sub>2 :: "('b \<times> 'x) set" assumes "graph_homomorphism (snd R) G h\<^sub>1"   and "graph_homomorphism G\<^sub>1 G h\<^sub>2"   and "f\<^sub>1 O h\<^sub>2 \<subseteq> h\<^sub>1" shows "\<exists>h. graph_homomorphism G\<^sub>2 G h \<and> h\<^sub>2 \<subseteq> h"
\<eta>_is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map GF.map \<eta>"
synthesized_call_def: fixes P :: "'a prog"   and h :: "'heap" shows "synthesized_call P h = (\<lambda>(a, M, vs). \<exists>T Ts Tr D. typeof_addr h a = \<lfloor>T\<rfloor> \<and> P \<turnstile> class_type_of T sees M: Ts\<rightarrow>Tr = Native in D)"
linear_reflexive: fixes x :: "'a" assumes "linear x" shows "reflexive x"
JF1col_0: fixes x :: "'c JF1" shows "JF1col 0 x = {}"
\<Z>_\<alpha>_\<alpha>\<omega>: shows "\<alpha> + \<omega> \<in>\<^sub>\<circ> \<alpha> + \<omega> + \<omega>"
if_eq: fixes B :: "bool"   and T :: "'a"   and E :: "'a" shows "(if \<not> B then T else E) = (if B then E else T)"
YMapInv_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and x :: "ZF" shows "YMapInv C X F x \<equiv> MakeNT (YMapInv' C X F x)"
subtopologyI: fixes S :: "'b set set"   and T :: "'b set set" assumes "\<And>s. s open\<^bsub>S\<^esub> \<Longrightarrow> \<exists>t. t open\<^bsub>T\<^esub> \<and> s = t \<inter> carrier\<^bsub>S\<^esub>"   and "\<And>t. t open\<^bsub>T\<^esub> \<Longrightarrow> t \<inter> carrier\<^bsub>S\<^esub> open\<^bsub>S\<^esub>" shows "subtopology S T"
path0E: fixes l1 :: "'vtx"   and l2 :: "'vtx" assumes "path l1 l2 []" obtains "l2 = l1"
pdiv_5: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> p"   and "K2 \<in> p"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
tmp_def: shows "tmp = (\<Union>as\<in>rel2. case Mapping.lookup a1_map (proj_tuple maskL as) of None \<Rightarrow> if \<not> pos then {(tp - len, as)} else {} | Some tp' \<Rightarrow> if pos then {(max (tp - len) tp', as)} else {(max (tp - len) (tp' + 1), as)}) \<union> (if left I = 0 then {tp} \<times> rel2 else {})"
3: shows "\<forall>m t. m < ord \<and> real_of_float (lower (I ! 0)) \<le> t \<and> t \<le> real_of_float (upper (I ! 0)) \<longrightarrow> ((\<lambda>x. interpret_floatarith (deriv_rec f m) ((map real_of_float xs)[0 := x])) has_real_derivative interpret_floatarith (deriv_rec f (Suc m)) ((map real_of_float xs)[0 := t])) (at t)"
max_def: fixes a :: "'b word"   and b :: "'b word" shows "signed.max a b = (if a \<le>s b then b else a)"
hd_addc: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and e' :: "expr" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "randomfree e'"   and "free_vars e' \<subseteq> V'" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Add $$ <e ,  e'> \<Rightarrow> (\<lambda>\<rho> x. f \<rho> (op_sem Add <|x, expr_sem_rf \<rho> (Minus $$ e')|>))"
create_gba_def: fixes \<phi> :: "'b ltlr" shows "create_gba \<phi> \<equiv> create_graph\<^sub>T \<phi> \<bind> (\<lambda>nds. RETURN (create_gba_from_nodes \<phi> nds))"
left_commute: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b + (a + c) = a + (b + c)"
assms: shows "pseudo_proj_True n x = pseudo_proj_True n w"