hcju/isabelleps huggingface dataset:
Collect_minI_ex2: fixes I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "{k \<in> I. P k} \<noteq> {}" shows "\<exists>x\<in>I. P x \<and> (\<forall>y\<in>I. y < x \<longrightarrow> \<not> P y)"
l5_3: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Bet A B D"   and "Bet A C D" shows "Bet A B C \<or> Bet A C B"
R9_3: shows "pmf (sds R14) a = pmf (sds R35) a"
csk: shows "cs\<^bsup>\<pi>\<^esup> k = xs @ [x]"
generate_maskedDB_def: fixes EM :: "bit list"   and emBits :: "nat"   and hLen :: "nat" shows "generate_maskedDB EM emBits hLen = take (roundup emBits 8 * 8 - hLen - 8) EM"
zet_def': shows "zet = {uu_. \<exists>A f z. uu_ = A \<and> inj_on f A \<and> f ` A = explode z}"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "partial_order L"   and "upper_semilattice_axioms L" shows "upper_semilattice L"
le_log2_of_power: fixes n :: "nat"   and m :: "nat" assumes "2 ^ n \<le> m" shows "real n \<le> log 2 (real m)"
hyp3: shows "fst (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]) \<le> nrows A"
laplace_expansion_column: fixes A :: "'a mat"   and n :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "j < n" shows "Determinant.det A = (\<Sum>i<n. A $$ (i, j) * Determinant.cofactor A i j)"
discI: fixes list :: "'e list"   and x21 :: "'e"   and x22 :: "'e list" assumes "list = x21 # x22" shows "list \<noteq> []"
ap_pure_list: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "[f] \<diamondop> xs = map f xs"
abs_summable_on_uminus: assumes "(\<lambda>x. - f x) abs_summable_on A" shows "(\<lambda>x. - (- f x)) abs_summable_on A"
pmdl_filter_syzygy_basis: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_pot_ord"   and "distinct bs"   and "is_Groebner_basis (set gs)"   and "pmdl (set gs) = pmdl (set (init_syzygy_list bs))" shows "pmdl (set (filter_syzygy_basis (length bs) gs)) = syzygy_module_list bs"
conversionI: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> (A\<^sup>\<leftrightarrow>)\<^sup>*" shows "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
Arg2pi_times_of_real2: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "Arg2pi (z * complex_of_real r) = Arg2pi z"
add_mono_thms_linordered_semiring_4: fixes i :: "'g"   and j :: "'g"   and k :: "'g"   and l :: "'g" assumes "i = j \<and> k = l" shows "i + k = j + l"
len_bit0: fixes uu :: "'a bit0 itself" shows "len_of uu = 2 * LENGTH('a)"
Compose: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi"   and O :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel"   and "(\<Psi>', T, U) \<in> Rel'"   and "(\<Psi>', U, O) \<in> Rel" shows "(\<Psi>', S, O) \<in> Rel'"
leq_nat_non_empty: fixes m :: "nat"   and n :: "nat" shows "m \<le> n \<longrightarrow> Abs_nat_int {m..n} \<noteq> bot"
cong_less_unique_nat: fixes m :: "nat"   and a :: "nat" assumes "0 < m" shows "\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m)"
compare_is_comparator_of: fixes compare :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.compare_order compare less_eq less" shows "compare = linorder.comparator_of less"
ccInf_superset_mono: fixes A :: "'a set"   and B :: "'a set" assumes "countable A"   and "B \<subseteq> A" shows "Inf A \<le> Inf B"
generatePolygon_def: fixes n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" shows "generatePolygon n v f g \<equiv> let enumeration = enumerator n |vertices f|; enumeration = filter (\<lambda>is. \<not> containsDuplicateEdge g f v is) enumeration in Let (map (indexToVertexList f v) enumeration) (map (subdivFace g f))"
ramsey2_full: fixes q1 :: "nat"   and q2 :: "nat"   and r :: "nat" shows "\<exists>N. partn_lst {..<N} [q1, q2] r"
precise_pure: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "precise (pure R)"
graphEqImage: fixes x :: "'a"   and X :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> X" shows "graph X f ,, x = f x"
shift1_isnpolyh: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0"   and "p \<noteq> 0\<^sub>p" shows "isnpolyh (shift1 p) 0"
prog_1: fixes i :: "nat" assumes "i < k" shows "0 < steps i"
fps_plus_def: shows "(+) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
assms: fixes P :: "'a \<Rightarrow> ('b \<times> 'b) set"   and P' :: "'a \<Rightarrow> ('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "profile A Is P"   and "profile A Is P'"   and "x \<in> A"   and "y \<in> A"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P i)\<^esub>\<prec> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P' i)\<^esub>\<prec> y"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P i)\<^esub>\<approx> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P' i)\<^esub>\<preceq> y"   and "\<exists>k\<in>Is. x \<^bsub>(P k)\<^esub>\<approx> y \<and> x \<^bsub>(P' k)\<^esub>\<prec> y \<or> y \<^bsub>(P k)\<^esub>\<prec> x \<and> x \<^bsub>(P' k)\<^esub>\<preceq> y"   and "x \<^bsub>(scf P)\<^esub>\<preceq> y" shows "x \<^bsub>(scf P')\<^esub>\<prec> y"
univ_poly_is_domain: fixes K :: "'a list set" assumes "subring K (A [X])" shows "domain (K [X]\<^bsub>A [X]\<^esub>)"
init_sim: fixes a0 :: "'b" assumes "a0 \<in> s1.b.V0" shows "\<exists>b0. b0 \<in> s2.b.V0 \<and> (a0, b0) \<in> R2"
higher_pderiv_smult: fixes n :: "nat"   and c :: "'a"   and p :: "'a poly" shows "(pderiv ^^ n) (Polynomial.smult c p) = Polynomial.smult c ((pderiv ^^ n) p)"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C 'b"   and xa :: "'a \<Rightarrow>\<^sub>C 'b" shows "dist x xa = (SUP x. dist (apply_bcontfun x x) (apply_bcontfun xa x))"
strip_list_comb_const: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and name :: "name"   and xs :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (term.list_comb app (const name) xs) = (const name, xs)"
rel_upd3: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and m :: "'b \<Rightarrow> bool"   and q :: "'b"   and t :: "bool" assumes "(a, b) \<notin> (r | m(q := t))"   and "(a, b) \<in> (r | m)" shows "a = q"
LIM_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "LIM_wrt \<psi> \<equiv> {U. \<exists>s. learn_lim \<psi> U s}"
eq_iff_16: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Unauth term = Unauth terma) = (term = terma)"
exprTypeable: fixes e :: "expr"   and \<Gamma> :: "char list \<Rightarrow> secLevel option" assumes "fv e \<subseteq> dom \<Gamma>" obtains T :: "secLevel" where "\<Gamma> \<turnstile> e : T"
simps_3: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i :: "nat" shows "fw m n k (Suc i) 0 = fw_upd (fw m n k i n) k (Suc i) 0"
lappend_LNil_LNil: shows "lappend LNil LNil = LNil"
zip_Nil: fixes ys :: "'e list" shows "[] || ys = []"
MakeFtor_def: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" shows "MakeFtor F \<equiv> \<lparr>CatDom = CatDom F, CatCod = CatCod F, MapM = restrict ((##) F) (mor\<^bsub>CatDom F\<^esub>), \<dots> = Functor.more F\<rparr>"
emeasure_eq_1_AE: fixes S :: "'a set" assumes "S \<in> events"   and "AE x in M. x \<in> S" shows "emeasure M S = 1"
bij_betw_from_nat_into: fixes S :: "'c set" assumes "countable S"   and "infinite S" shows "bij_betw (from_nat_into S) UNIV S"
that: assumes "y = f x"   and "x \<in> A" shows "thesis"
subset_Diff_insert: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b"   and C :: "'b set" shows "(A \<subseteq> B - (x \<triangleright> C)) = (A \<subseteq> B - C \<and> x \<notin> A)"
ineq_4: shows "Suc l < (if Suc j = mm then Suc j else Suc (Suc j))"
is_ortho_set_cindependent: fixes A :: "'c set" assumes "is_ortho_set A" shows "cindependent A"
asym: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "ord_strict_p a b"   and "ord_strict_p b a" shows "False"
powser_continuous_suminf: fixes r :: "real"   and a :: "nat \<Rightarrow> 'b"   and \<xi> :: "'b" assumes "ereal r < conv_radius a" shows "continuous_on (cball \<xi> r) (\<lambda>x. \<Sum>i. a i * (x - \<xi>) ^ i)"
separated_byI: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and B :: "'b set" assumes "x \<in> A"   and "y \<in> B" shows "separated_by {A, B} x y"
ennreal_def: shows "ennreal \<equiv> map_fun id e2ennreal ((\<squnion>) 0 \<circ> ereal)"
Cod_dom: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Cod (local.dom f) = Dom f"
homeomorphic_local_compactness: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "locally compact S = locally compact T"
slice_kind_Return_in_slice: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ins :: "'var list"   and outs :: "'var list" assumes "sourcenode a \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs" shows "slice_kind S a = Q\<hookleftarrow>\<^bsub>p\<^esub>\<lambda>cf cf'. rspp (targetnode a) (HRB_slice S) outs cf' cf"
fst_image_times: fixes A :: "'d set"   and B :: "'e set" shows "fst ` (A \<times> B) = (if B = {} then {} else A)"
locallyE: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and w :: "'b set"   and x :: "'b" assumes "locally P S"   and "openin (top_of_set S) w"   and "x \<in> w" obtains u :: "'b set" and v :: "'b set" where "openin (top_of_set S) u" and "P v" and "x \<in> u" and "u \<subseteq> v" and "v \<subseteq> w"
psubst_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "good X"   and "goodEnv rho"   and "goodEnv rho'" shows "X #[rho] #[rho'] = X #[rho &[rho']]"
distinct_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 # x22 \<noteq> []"
diff_fresh_vsubst: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "good X"   and "y \<noteq> y1" shows "fresh ys y (X #[y1 // y]_ys)"
open_poly_mapping_def: fixes U :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "open U \<equiv> \<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U"
cases: fixes a1 :: "(nat \<Rightarrow> nat) \<times> 'b memref list \<times> nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and a2 :: "(nat \<Rightarrow> nat) \<times> 'b memref list \<times> nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)" assumes "a1 \<rightarrow>\<^sub>f a2" obtains \<O>' :: "nat set" and volatile :: "bool" and \<O> :: "nat set" and A :: "nat set" and R :: "nat set" and \<S>' :: "nat \<Rightarrow> bool option" and \<S> :: "nat \<Rightarrow> bool option" and W :: "nat set" and L :: "nat set" and \<R>' :: "nat \<Rightarrow> bool option" and \<R> :: "nat \<Rightarrow> bool option" and m :: "nat \<Rightarrow> nat" and a :: "nat" and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" and v :: "nat" and rs :: "'b memref list" where "a1 = (ma__, Write\<^sub>s\<^sub>b volatile__ a__ sop v__ A__ L__ R__ W__ # rs__, \<O>__, \<R>__, \<S>''__)" and "a2 = (ma__(a__ := v__), rs__, \<O>', \<R>', \<S>')" and "\<O>' = (if volatile__ then \<O>__ \<union> A__ - R__ else \<O>__)" and "\<S>' = (if volatile__ then \<S>''__ \<oplus>\<^bsub>W__\<^esub> R__ \<ominus>\<^bsub>A__\<^esub> L__ else \<S>''__)" and "\<R>' = (if volatile__ then Map.empty else \<R>__)"   | m :: "nat \<Rightarrow> nat" and volatile :: "bool" and a :: "nat" and t :: "nat" and v :: "nat" and rs :: "'b memref list" and \<O> :: "nat set" and \<R> :: "nat \<Rightarrow> bool option" and \<S> :: "nat \<Rightarrow> bool option" where "a1 = (mb__, Read\<^sub>s\<^sub>b volatile a t v # rsa__, \<O>''__, \<R>''__, \<S>'''__)" and "a2 = (mb__, rsa__, \<O>''__, \<R>''__, \<S>'''__)"   | m :: "nat \<Rightarrow> nat" and p :: "'b" and p' :: "'b" and "is" :: "instr list" and rs :: "'b memref list" and \<O> :: "nat set" and \<R> :: "nat \<Rightarrow> bool option" and \<S> :: "nat \<Rightarrow> bool option" where "a1 = (mc__, Prog\<^sub>s\<^sub>b p p' is # rsb__, \<O>'''__, \<R>'''__, \<S>''''__)" and "a2 = (mc__, rsb__, \<O>'''__, \<R>'''__, \<S>''''__)"   | m :: "nat \<Rightarrow> nat" and A :: "nat set" and L :: "nat set" and R :: "nat set" and W :: "nat set" and rs :: "'b memref list" and \<O> :: "nat set" and \<R> :: "nat \<Rightarrow> bool option" and \<S> :: "nat \<Rightarrow> bool option" where "a1 = (m, Ghost\<^sub>s\<^sub>b A L R W # rs, \<O>, \<R>, \<S>)" and "a2 = (m, rs, \<O> \<union> A - R, augment_rels (dom \<S>) R \<R>, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
space\<^sub>N_def: fixes N :: "'b quasinorm" shows "space\<^sub>N N = {f. eNorm N f < \<infinity>}"
osum_antisym: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "antisym r"   and "antisym r'" shows "antisym (r +o r')"
l_span_closed2: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> linear_span R M (carrier R) H" shows "l_comb R M n s f \<in> linear_span R M A H"
prefix_of_map_\<Gamma>_D: fixes \<pi>' :: "'c prefix"   and f :: "'d set \<Rightarrow> 'c set"   and \<sigma> :: "'d trace" assumes "prefix_of \<pi>' (map_\<Gamma> f \<sigma>)" shows "\<exists>\<pi>''. \<pi>' = pmap_\<Gamma> f \<pi>'' \<and> prefix_of \<pi>'' \<sigma>"
is_iso_classI: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.ide f" shows "Maps.is_iso_class (Maps.iso_class f)"
p: shows "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
set_proj_orig_basis: fixes n :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (proj_orig_basis n bs) = proj_poly_syz n ` set bs"
limit_mat_denote_while_n: fixes M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and \<rho> :: "complex mat" assumes "well_com (While M S)"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "limit_mat (matrix_sum d (\<lambda>k. denote_while_n (M 0) (M 1) (denote S) k \<rho>)) (denote (While M S) \<rho>) d"
matrix_inv_unique: fixes A :: "(('b, 'c) vec, 'c) vec"   and B :: "(('b, 'c) vec, 'c) vec" assumes "A ** B = mat (1::'b)"   and "B ** A = mat (1::'b)" shows "matrix_inv A = B"
assms_3: shows "(\<Sigma>, \<Sigma>') \<in> br_step \<delta>"
run_one_step_basic_load_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x191 :: "t"   and x192 :: "(tp \<times> sx) option"   and x193 :: "nat"   and x194 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Load x191 x192 x193 x194) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
induct: fixes P :: "Ferrack.num \<Rightarrow> bool"   and a0 :: "Ferrack.num" assumes "\<And>n. P (Bound n)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>c a. P a \<Longrightarrow> P (Mul c a)"   and "\<And>n c a. P a \<Longrightarrow> P (CN n c a)"   and "\<And>v. P (C v)" shows "P a0"
cr_streamI: fixes xs :: "'b llist" assumes "\<not> lfinite xs" shows "cr_stream xs (stream_of_llist xs)"
fully_faithful_functor_R: shows "fully_faithful_functor Right_a.comp Right_a.comp Right_a.R"
E_wfD2: fixes G :: "'a graph" assumes "wf_graph G" shows "\<forall>e\<in>edges G. fst e \<in> nodes G \<and> snd e \<in> nodes G"
fresh_at_def': fixes t :: "('form, 'rule, 'a, 'b) itnode rose_tree"   and "is" :: "nat list" shows "fresh_at t is = (if is = [] then {} else freshenLC (iAnnot (tree_at t (butlast is))) ` a_fresh (inPorts' (iNodeOf (tree_at t (butlast is))) ! last is))"
is_volatile_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_volatile (Ghost\<^sub>s\<^sub>b A L R W) = False"
pseudofunctor_def: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi> :: "'r \<times> 'r \<Rightarrow> 's" shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<and> weak_arrow_of_homs V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) \<and> (composite_functor (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) V\<^sub>C V\<^sub>D (\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>)) F \<and> composite_functor (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) (subcategory.comp (product_category.comp V\<^sub>D V\<^sub>D) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>D (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>D (snd \<mu>\<nu>) \<and> src\<^sub>D (fst \<mu>\<nu>) = trg\<^sub>D (snd \<mu>\<nu>))) V\<^sub>D (weak_arrow_of_homs.FF V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) (\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<and> natural_isomorphism (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) V\<^sub>D ((\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<circ> weak_arrow_of_homs.FF V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) (F \<circ> (\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<Phi> \<and> pseudofunctor_axioms V\<^sub>C H\<^sub>C \<a>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D F \<Phi>"
Suc_in_imp_neq_iMODb: fixes n :: "nat"   and S :: "nat set"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "n \<in> S"   and "Suc n \<in> S"   and "m \<noteq> Suc 0" shows "S \<noteq> [ r, mod m, c ]"
causal_program_history_empty: fixes "is" :: "instr list" shows "causal_program_history is []"
lossless_response: fixes r :: "nat"   and w :: "nat"   and c :: "nat" shows "lossless_spmf (response r w c)"
2: shows "proots_count (map_poly complex_of_real P) circle_01_diam = proots_count (map_poly complex_of_real (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:])) {x. 0 < Re x}"
summable_iff_convergent: fixes f :: "nat \<Rightarrow> 'a" shows "summable f = convergent (\<lambda>n. sum f {..<n})"
exhaust: fixes y :: "'b tag_const" obtains x :: "'b" where "y = \<tau>\<^sub>c\<^sub>s\<^sub>t x"
tendsto_PInfty_eq_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" shows "((\<lambda>z. ereal (f z)) \<longlongrightarrow> \<infinity>) F = filterlim f at_top F"
sub_which2: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<subseteq> B \<or> B \<subseteq> A"   and "x \<notin> A"   and "x \<in> B" shows "A \<subseteq> B"
that: fixes a :: "'a"   and b :: "real" assumes "z + a \<in> affine hull insert z S"   and "a \<noteq> (0::'a)"   and "a \<bullet> z \<le> b"   and "\<And>x. x \<in> S \<Longrightarrow> b \<le> a \<bullet> x" shows "thesis"
normal_def: fixes H :: "'g set" shows "normal H \<equiv> \<forall>g\<in>G. LCoset_rel H `` {g} = RCoset_rel H `` {g}"
someI_ex: fixes P :: "'d \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
cfis: shows "f\<inverse> O s \<subseteq> ov"
add_match_distrib: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m1 :: "'c match_expr"   and m2 :: "'c match_expr"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m1 (add_match m2 rs), s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m2 (add_match m1 rs), s\<rangle> \<Rightarrow> t"
only_if: shows "\<forall>D\<in>\<G>_F C. D \<in> Red_F_G (\<G>_Fset N) \<or> (\<exists>E\<in>N - local.Red_F_\<G> N. Prec_F_g D E C \<and> D \<in> \<G>_F E)"
cont: fixes N :: "'a stream measure"   and M :: "'a stream measure" assumes "R N M" shows "\<exists>N' M' p. (\<forall>y\<in>set_pmf p. R (N' y) (M' y)) \<and> (\<forall>x. N' x \<in> space (prob_algebra (stream_space (count_space UNIV)))) \<and> (\<forall>x. M' x \<in> space (prob_algebra (stream_space (count_space UNIV)))) \<and> N = measure_pmf p \<bind> (\<lambda>y. distr (N' y) (stream_space (count_space UNIV)) ((##) y)) \<and> M = measure_pmf p \<bind> (\<lambda>y. distr (M' y) (stream_space (count_space UNIV)) ((##) y))"
induced_map_recurrent_typical_1: fixes A :: "'a set" assumes "A \<in> sets MA" shows "AE z in restrict_space MA A. z \<in> A.recurrent_subset A"
selectlike_union: fixes X :: "(variable \<Rightarrow> real) set"   and Y :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "selectlike (X \<union> Y) \<nu> V = selectlike X \<nu> V \<union> selectlike Y \<nu> V"
divide_poly: fixes g :: "'b poly"   and f :: "'b poly" assumes "g \<noteq> 0" shows "f * g div g = f"
inner_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "x \<bullet> y = fst x \<bullet> fst y + snd x \<bullet> snd y"
Max_start: fixes s :: "'b" assumes "s \<in> \<S>" shows "(MAX t\<in>\<S>. indicator {t} s) = 1"
poly_sgnx_values_1: fixes p :: "real poly"   and a :: "real" assumes "p \<noteq> 0" shows "sgnx (poly p) (at_left a) = 1 \<or> sgnx (poly p) (at_left a) = - 1"
no_recursive_calls_helper: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and chain :: "char list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"   and "matches \<gamma> m p"   and "\<Gamma> chain = Some [Rule m (Call chain)]" shows "False"
Union_components: fixes u :: "'b set" shows "\<Union> (components u) = u"