hcju/isabelleps huggingface dataset:
filtercomap_inf: fixes f :: "'c \<Rightarrow> 'd"   and F1 :: "'d filter"   and F2 :: "'d filter" shows "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)"
zero_fun_apply: fixes x :: "'b" shows "0 x = (0::'a)"
pcr_cr_eq: shows "pcr_cblinfun (=) (=) = cr_cblinfun"
X: shows "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))"
List_hd_not_in_tl: fixes h :: "'c \<Rightarrow> 'c ref"   and a :: "'c"   and as :: "'c list" assumes "List h (h a) as" shows "a \<notin> set as"
pair_wf_digraph_def: fixes G :: "'b pair_pre_digraph" shows "pair_wf_digraph G \<equiv> (\<forall>e. e \<in> parcs G \<longrightarrow> fst e \<in> pverts G) \<and> (\<forall>e. e \<in> parcs G \<longrightarrow> snd e \<in> pverts G)"
set_map_pmf: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf" shows "set_pmf (map_pmf f M) = f ` set_pmf M"
i_last_message_hold_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" shows "f \<Up> (n * k) \<longmapsto> k = f \<longmapsto> k \<Up> (n * k)"
iso_verts_head: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and e :: "'b" assumes "digraph_isomorphism hom"   and "e \<in> arcs G" shows "iso_head hom (iso_arcs hom e) = iso_verts hom (head G e)"
prems: shows "\<forall>op\<in>set (a # ops). v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)"
refl_ext: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "(x \<doteq> x) = (if \<upsilon> x then true else invalid endif)"
size_list_estimation': fixes x :: "'a"   and xs :: "'a list"   and y :: "nat"   and f :: "'a \<Rightarrow> nat" assumes "x \<in> set xs"   and "y \<le> f x" shows "y \<le> size_list f xs"
filter_eq_replicate_mset: fixes x :: "'b"   and D :: "'b multiset" shows "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
Subst_def: fixes p :: "hf"   and q :: "hf" shows "Subst p q \<equiv> \<exists>v u. SubstForm v u p q"
admI: fixes P :: "'b \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i)\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)" shows "adm P"
match_tcp_flags_conjunct_option_def: fixes f1 :: "ipt_tcp_flags"   and f2 :: "ipt_tcp_flags" shows "match_tcp_flags_conjunct_option f1 f2 = (case match_tcp_flags_conjunct f1 f2 of TCP_Flags fmask c \<Rightarrow> if c \<subseteq> fmask then Some (TCP_Flags fmask c) else None)"
Keys_to_list_eq_pps_to_list: fixes ps :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.punit.Keys_to_list ps = pp_pm.punit.pps_to_list (Keys (set ps))"
star_of_one: shows "star_of (1::'b) = 1"
least_closed: fixes L :: "('a, 'b) gorder_scheme"   and l :: "'a"   and A :: "'a set" assumes "least L l A" shows "l \<in> carrier L"
whiletrue: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = True" shows "SRabbr (while\<^bsub>\<iota>\<^esub> b do c od) m [] (Some (c;while\<^bsub>\<iota>\<^esub> b do c od)) m"
simps_2: fixes h :: "'b \<Rightarrow> int"   and e :: "'b"   and t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" shows "remove h e (T t1 x t2) = (if h e < h x then T (remove h e t1) x t2 else if h x < h e then T t1 x (remove h e t2) else if t1 = Tip then t2 else let (t1p, r) = wrmrm h t1 in T t1p r t2)"
univ_poly_units: fixes K :: "'a list set" assumes "subfield K (K [X])" shows "Units (K [X]\<^bsub>K [X]\<^esub>) = {[k] |k. k \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}}"
Lv_pos: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "0 < Lv K v"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a"   and pdf_type :: "pdf_type" shows "P (case pdf_type of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = (\<not> (pdf_type = UNIT \<and> \<not> P f1 \<or> pdf_type = BOOL \<and> \<not> P f2 \<or> pdf_type = INTEG \<and> \<not> P f3 \<or> pdf_type = REAL \<and> \<not> P f4 \<or> (\<exists>x51 x52. pdf_type = PRODUCT x51 x52 \<and> \<not> P (f5 x51 x52))))"
connected_component_trans: fixes S :: "'c set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "connected_component S x y"   and "connected_component S y z" shows "connected_component S x z"
le_cases3: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.linorder less_eq less" obtains "less_eq x y" and "less_eq y z"   | "less_eq y x" and "less_eq x z"   | "less_eq x z" and "less_eq z y"   | "less_eq z y" and "less_eq y x"   | "less_eq y z" and "less_eq z x"   | "less_eq z x" and "less_eq x y"
subst_mono: fixes t :: "('c, 'd) Term.term"   and u :: "('c, 'd) Term.term"   and s :: "'d \<Rightarrow> ('c, 'e) Term.term" assumes "t \<sqsubseteq> u" shows "t \<cdot> s \<sqsubseteq> u \<cdot> s"
cong_mod_right: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b (modulo c a) a = unique_euclidean_semiring.cong modulo b c a"
A6_NSources_L1Acc: shows "\<forall>C\<in>AbstrLevel level1. C \<notin> Acc level1 sA6 \<longrightarrow> sA6 \<notin> Sources level1 C"
rep_eq: fixes x :: "'a['c, 'b]"   and xa :: "'a['c, 'b]" shows "fromQ (x + xa) = \<lparr>mag = mag (fromQ x) + mag (fromQ xa), dim = QD('c), unit_sys = unit\<rparr>"
list_of_pdevs_perm_filter_nonzero: fixes X :: "'b pdevs" shows "map snd (list_of_pdevs X) <~~> filter ((\<noteq>) (0::'b)) (dense_list_of_pdevs X)"
list_all3_list_all2I: fixes Q :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list" assumes "list_all3 (\<lambda>x y. Q x) xs ys zs" shows "list_all2 Q xs zs"
process_rule_2_traces: fixes xs :: "'b list"   and xs' :: "'b list"   and P :: "'b process" assumes "xs @ xs' \<in> traces P" shows "xs \<in> traces P"
r_child_lower_bound_alt: fixes l :: "int"   and x :: "int"   and p :: "int" assumes "l \<le> x"   and "x \<le> p" shows "x < r_child l p"
homeomorphic_unicoherent: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "unicoherent S" shows "unicoherent T"
prems_1: fixes i :: "'a"   and j :: "'a" assumes "i \<in> insert x F"   and "j \<in> insert x F"   and "i \<noteq> j" shows "coprime (f i) (f j)"
substEnv_compose1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y1 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y2 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y1"   and "good Y2" shows "rho &[Y1 / y]_ys &[Y2 / y]_ys = rho &[Y1 #[Y2 / y]_ys / y]_ys"
P: shows "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp tpD tpFD tpCD"
cycle_free_loop_dest: fixes i :: "nat"   and n :: "nat"   and xs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "linordered_monoid.cycle_free m n" shows "Le (0::'c) \<preceq> linordered_monoid.len m i i xs"
intros_2: fixes c :: "('c, 'd) lit list"   and \<sigma> :: "'tp" assumes "\<forall>x\<in>nvC c. TE.tpOfV x \<noteq> \<sigma>" shows "\<sigma> \<turnstile> c"
ideD_1: fixes a :: "'c" assumes "Cop.ide a" shows "Cop.arr a"
norm_eq: fixes x :: "'g"   and y :: "'h" shows "(norm x = norm y) = (x \<bullet> x = y \<bullet> y)"
isnonconstant_pnormal_iff: fixes p :: "poly"   and bs :: "'a list" assumes "isnonconstant p" shows "pnormal (polypoly bs p) = (\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> (0::'a))"
gbg_to_idx_ext_def: fixes ecnv :: "('a, 'c) gb_graph_rec_scheme \<Rightarrow> 'b"   and A :: "('a, 'c) gb_graph_rec_scheme" shows "gbg_to_idx_ext ecnv A = F_to_idx_impl (gbg_F A) \<bind> (\<lambda>(num_acc, acc). RETURN \<lparr>g_V = g_V A, g_E = g_E A, g_V0 = g_V0 A, igbg_num_acc = num_acc, igbg_acc = acc, \<dots> = ecnv A\<rparr>)"
0: fixes u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Nml (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) \<and> Dom (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Dom u \<and> Cod (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Cod t"
tl_Nil: fixes xs :: "'b list" shows "(tl xs = []) = (xs = [] \<or> (\<exists>x. xs = [x]))"
assms_7: fixes s2 :: "'s2"   and x :: "'a" assumes "bad2 s2"   and "x \<in> outs_\<I> \<I>" shows "lossless_spmf (oracle2 s2 x)"
matrix_vector_mult_def: fixes m :: "(('a, 'd) vec, 'c) vec"   and x :: "('a, 'd) vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. vec_nth (vec_nth m i) j * vec_nth x j"
size\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) strand_step list" shows "size\<^sub>s\<^sub>t S \<equiv> size_list (\<lambda>x. Max (insert 0 (size ` trms\<^sub>s\<^sub>t\<^sub>p x))) S"
arsinh_real_def: fixes x :: "real" shows "arsinh x = ln (x + sqrt (x\<^sup>2 + 1))"
simps_1: fixes r :: "('a \<times> 'a) set" shows "r|[]| = {#}"
xor_protocol_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "xor_protocol A B = (let (a1, b1) = A; (a2, b2) = B in return_spmf (a1 \<oplus> a2, b1 \<oplus> b2))"
closedD: fixes F :: "'b program"   and T :: "'b set"   and B :: "'b set"   and L :: "'b set set"   and act :: "('b \<times> 'b) set"   and M :: "'b set" assumes "closed F T B L"   and "act \<in> Acts F"   and "B \<subseteq> M"   and "T \<inter> M \<in> L" shows "T \<inter> (B \<union> wp act M) \<in> L"
bbw_lcs_3: fixes y :: "int"   and x :: "int"   and z :: "int" shows "y XOR x XOR z = x XOR y XOR z"
expl_cond_expect_def: fixes M :: "'b measure"   and Y :: "'b \<Rightarrow> 'c"   and X :: "'b \<Rightarrow> real" shows "expl_cond_expect M Y X = img_dce M Y X \<circ> Y"
add_sheaf_spec_def: fixes U :: "nat set set"   and s :: "nat set \<Rightarrow> (nat \<times> nat) set"   and s' :: "nat set \<Rightarrow> (nat \<times> nat) set" shows "cring0.add_sheaf_spec U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.add_rel ({0}\<setminus>\<pp>) {0} ring0.subtraction ring0.subtraction 0 (s \<pp>) (s' \<pp>)"
test_preserve: fixes p :: "'a"   and x :: "'a" assumes "test p"   and "p \<cdot> x \<le> p \<cdot> x \<cdot> p" shows "p \<cdot> x\<^sup>\<dagger> = (p \<cdot> x)\<^sup>\<dagger> \<cdot> p"
extf_min_empty: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and ss :: "('b, 'a) tm list" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf" shows "\<not> extf f gt [] ss"
toplevel_summands_PLUS_strong: fixes xs :: "'b rexp list" assumes "xs \<noteq> []"   and "list_all (\<lambda>x. \<nexists>r s. x = Plus r s) xs" shows "toplevel_summands (PLUS xs) = set xs"
lupd_def: fixes x :: "vname"   and v :: "val"   and s :: "state" shows "lupd(x\<mapsto>v) s \<equiv> s\<lparr>locals := locals s(x \<mapsto> v)\<rparr>"
lcm_real_def: shows "lcm = normalization_euclidean_semiring_class.lcm"
simps_9: fixes f :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "map_option f (Some x2) = Some (f x2)"
Int_space_eq1: fixes x :: "'a set" assumes "x \<in> M" shows "\<Omega> \<inter> x = x"
nn_integral_count_space_indicator: fixes X :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = \<integral>\<^sup>+ x. f x * indicator X x \<partial>count_space UNIV"
random_linorder_by_prios: fixes A :: "'b set"   and a :: "real"   and b :: "real"   and B :: "'b set" assumes "finite A"   and "a < b"   and "A \<subseteq> B" shows "distr (Pi\<^sub>M A (\<lambda>_. uniform_measure lborel {a..b})) (count_space (Pow (B \<times> B))) (linorder_from_keys A) = uniform_measure (count_space (Pow (B \<times> B))) (linorders_on A)"
vcons_eqD_1: fixes xs\<^sub>1 :: "V"   and x\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>2 :: "V" assumes "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2"   and "vfsequence xs\<^sub>1"   and "vfsequence xs\<^sub>2" shows "xs\<^sub>1 = xs\<^sub>2"
prob_space_restr_to_subalg: fixes M :: "'c measure"   and F :: "'c measure" assumes "subalgebra M F"   and "prob_space M" shows "prob_space (restr_to_subalg M F)"
simps_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and e :: "'d"   and es :: "'d list"   and v :: "'c" shows "pre_digraph.cas G u (e # es) v = (tail G e = u \<and> pre_digraph.cas G (head G e) es v)"
solves_odeI: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set" assumes "(y has_vderiv_on (\<lambda>t. f t (y t))) T"   and "\<And>t. t \<in> T \<Longrightarrow> y t \<in> X" shows "(y solves_ode f) T X"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set"   and R :: "('procS, 'procT) Proc" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q"   and "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> R" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> R"
fls_subdegree_leI': fixes f :: "'a fls"   and n :: "int"   and m :: "int" assumes "fls_nth f n \<noteq> (0::'a)"   and "n \<le> m" shows "fls_subdegree f \<le> m"
subspace: fixes A :: "'b set" assumes "closed_csubspace A" shows "csubspace A"
cond_L7: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and c :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  (P \<triangleleft> c \<triangleright>  Q)) = (P \<triangleleft> b \<or> c \<triangleright>  Q)"
coset_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier G" shows "M #> \<one> = M"
UP_m_comm: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
comp_inv_arr': fixes f :: "'d" assumes "D.iso f" shows "D.inv f \<cdot>\<^sub>D f = D.dom f"
bhv2cnf_def: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.bhv2cnf tCMP active c t n \<equiv> dynamic_component.lActive active c t + (n - (the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1))"
b_def: shows "b = dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>\<lparr>i\<rparr>"
assms: shows "(a, b) \<in> A\<^sup>*"
cleb: fixes m :: "nat"   and n :: "nat" shows "c (real m / 2 ^ n) \<le> b (real m / 2 ^ n)"
zero_discr_def: shows "0 \<equiv> Discr (0::'b)"
ent_disjI2: fixes P :: "assn"   and Q :: "assn"   and R :: "assn" assumes "P \<or>\<^sub>A Q \<Longrightarrow>\<^sub>A R" shows "Q \<Longrightarrow>\<^sub>A R"
eventually_nhds: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" shows "eventually P (nhds a) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
ExprTypable_with_smallerd_implies_dH_indistinguishable: fixes H :: "(Dom \<times> ('c, 'd) Expr) set"   and e :: "('c, 'd) Expr"   and d' :: "Dom"   and d :: "Dom" assumes "H \<turnstile>\<^bsub>\<E>\<^esub> e : d'"   and "d' \<le> d" shows "e \<equiv>\<^bsub>d,H\<^esub> e"
dependent_def: fixes P :: "'c set" shows "cdependent P = (\<exists>a\<in>P. a \<in> cspan (P - {a}))"
\<delta>_vars_bound: shows "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
map_ran_conv: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and al :: "('c \<times> 'b) list"   and k :: "'c" shows "map_of (map_ran f al) k = map_option (f k) (map_of al k)"
assms_2: fixes x :: "'a" assumes "x \<in> space M" shows "(\<lambda>i. f i x) \<longlonglongrightarrow> g x"
Lcm_least: fixes A :: "'a set"   and a :: "'a" assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a" shows "Lcm A dvd a"
r_tuple_encode_prim: fixes n :: "nat" shows "prim_recfn (Suc n) (r_tuple_encode n)"
has_integral_componentwise_iff: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and A :: "'a set" shows "(f has_integral y) A = (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y \<bullet> b) A)"
AAS_def: shows "AAS = {AAs \<in> lists (mset ` AS). length AAs \<le> max_ary}"
inducts: fixes xs :: "'d list"   and x :: "'d list"   and P :: "'d list \<Rightarrow> bool" assumes "cyclic1 xs x"   and "P (rotate1 xs)" shows "P x"
CR_def: shows "CR \<equiv> part_circlepath 0 R 0 pi"
OAlist_list_of_oalist_ntm: fixes xs :: "('a, 'b) oalist_ntm" shows "OAlist_ntm (list_of_oalist_ntm xs) = xs"
ipp_def: fixes i :: "'e \<Rightarrow> 'e"   and a :: "'e" shows "\<^sub>i- a \<equiv> i a"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sup>s <\<nu>x>P \<parallel> Q"
domain_eq: shows "Domainp (pcr_antichain (=)) = (\<lambda>A. finite A \<and> incomparable A)"
axioms_1: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "continuous f S T" shows "func f S T"
marked_E: fixes e :: "'v \<times> 'v" assumes "e \<in> \<^bold>E" shows "tailmarked e \<or> headmarked e"
a2: shows "r \<rightarrow>\<^sub>\<beta> s"
path_linepath: fixes a :: "'c"   and b :: "'c" shows "path (linepath a b)"
image_function_of_distinct: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A" shows "function_of A enum ` {xs. set xs \<subseteq> B \<and> length xs = card A \<and> distinct xs} \<subseteq> {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A}"
simps_5: fixes r :: "'a rexp"   and s :: "'a rexp" shows "lang (Times r s) = lang r @@ lang s"
lt_spair: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rep_list p \<noteq> 0"   and "punit.lt (rep_list p) \<oplus> lt q \<prec>\<^sub>t punit.lt (rep_list q) \<oplus> lt p" shows "lt (spair p q) = (lcs (punit.lt (rep_list p)) (punit.lt (rep_list q)) - punit.lt (rep_list p)) \<oplus> lt p"
while_elim': fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(While e c, mem) \<leadsto>\<^sub>s (c', mem')" obtains "c' = Stmt.If e (c ;; While e c) Stop" and "mem' = mem"
rcs_eq: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e"   and b :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(b \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a \<in> H) = (H \<bullet>\<^bsub>G\<^esub> a = H \<bullet>\<^bsub>G\<^esub> b)"
map_comp_simps_1: fixes m2 :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and m1 :: "'c \<Rightarrow> 'e option" assumes "m2 k = None" shows "(m1 \<circ>\<^sub>m m2) k = None"
supp_eqvt: fixes p :: "perm"   and x :: "'a" shows "p \<bullet> supp x = supp (p \<bullet> x)"
ereal_MInfty_lessI: fixes a :: "ereal" assumes "a \<noteq> - \<infinity>" shows "- \<infinity> < a"
rel_distinct_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y2 :: "'a" shows "\<not> rel_option R (Some y2) None"
sets_Sup_measure': fixes M :: "'a measure set"   and A :: "'a measure" assumes "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A"   and "M \<noteq> {}" shows "sets (local.Sup_measure' M) = sets A"
pos_val: fixes z :: "int" assumes "z \<in> range pos" shows "pos (val z) = z"
acc_le_listI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order r"   and "acc r" shows "acc (Listn.le r)"
simps_7: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and I :: "\<I>"   and \<phi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Next I \<phi>) j = Monitor.progress \<sigma> \<phi> j - 1"
blocks_nempty: fixes bl :: "'b set" assumes "bl \<in># \<B>'" shows "bl \<noteq> {}"
rref_condition1: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"
fds_fds_nth: fixes f :: "'b fds" shows "fds (fds_nth f) = f"
morphism_locally_ringed_spaces_def: fixes X :: "'c set"   and is_open\<^sub>X :: "'c set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'c set \<Rightarrow> 'd set"   and \<rho>\<^sub>X :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and one_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and Y :: "'e set"   and is_open\<^sub>Y :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'e set \<Rightarrow> 'f set"   and \<rho>\<^sub>Y :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and d :: "'f"   and add_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and one_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and f :: "'c \<Rightarrow> 'e"   and \<phi>\<^sub>f :: "'e set \<Rightarrow> 'f \<Rightarrow> 'd" shows "morphism_locally_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> morphism_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<and> morphism_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
simps_1: fixes I :: "float interval" shows "subdivide_interval 0 I = [I]"
UTM_uhalt_lemma': fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and args :: "nat list" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <args>) tp stp)"   and "args \<noteq> []" shows "\<forall>stp. \<not> is_final (steps0 (Suc 0, [], <code tp # args>) UTM stp)"
rn_rev_price_rev_borel_adapt: fixes cash_flow :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "cash_flow \<in> borel_measurable (G matur)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "n \<le> matur" shows "rn_rev_price N cash_flow matur n \<in> borel_measurable (G (matur - n))"
nlistsp_nlists_eq: fixes A :: "'c set"   and n :: "nat" shows "nlistsp A n = (\<lambda>x. x \<in> nlists A n)"
Abs_cpx_sqr_mat_cases: fixes x :: "cpx_sqr_mat" obtains y :: "complex mat" where "x = Abs_cpx_sqr_mat y" and "y \<in> {M |M. square_mat M}"
ge: shows "HMA_Connect.spectral_radius A * v $h i \<le> ev * v $h i"