hcju/isabelleps huggingface dataset:
sel: fixes r :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "the_gpv (r \<bind> f) = map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. r \<bind> f)))) (the_gpv r \<bind> case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))"
possible_traces_def: fixes SES :: "('d, 'c) SES_rec" shows "possible_traces SES \<equiv> {t. enabled SES s0\<^bsub>SES\<^esub> t}"
fps_right_inverse: fixes f :: "'b fps"   and y :: "'b" assumes "f $ 0 * y = (1::'b)" shows "f * fps_right_inverse f y = 1"
has_integral_integral: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" shows "(f integrable_on s) = (f has_integral integral s f) s"
sourcesI: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "weak_composition.weak_unit V H a"   and "H \<mu> a \<noteq> partial_magma.null V" shows "a \<in> weak_composition.sources V H \<mu>"
convex_on_convex_hull_bound: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and b :: "real" assumes "convex_on (convex hull S) f"   and "\<forall>x\<in>S. f x \<le> b" shows "\<forall>x\<in>convex hull S. f x \<le> b"
assms_1: shows "{A, B, P, Q} \<subseteq> carrier_mat n n"
monoid_homomorphism_axioms: shows "monoid_homomorphism (\<rho> z' w) (\<FF> z') +\<^bsub>z'\<^esub> \<zero>\<^bsub>z'\<^esub> (\<FF> w) +\<^bsub>w\<^esub> \<zero>\<^bsub>w\<^esub>"
gcd_eq_0_iff: fixes a :: "'a"   and b :: "'a" shows "(gcd a b = (0::'a)) = (a = (0::'a) \<and> b = (0::'a))"
T_monotonic: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "MonotProblem (ProblemIkTpart.TE_wtFsym wtFsym resOf) wtPsym (ProblemIkTpart.TE_arOf arOf) (ProblemIkTpart.TE_resOf resOf) parOf (ProblemIkTpart.tPB wtFsym arOf resOf \<Phi> prot protFw)"
antimonoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f y \<le> f x" shows "order.antimono less_eq f"
some1_equality: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "\<exists>!x. P x"   and "P a" shows "(SOME x. P x) = a"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and M :: "'b ltln set" shows "\<phi> and\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2 = \<phi>[M]\<^sub>\<Sigma>\<^sub>2 and\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2"
gh: fixes x :: "'c" assumes "x \<in> A \<inter> B" shows "g x = h x"
simps_6: fixes h :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and \<phi> :: "'b fm"   and xs :: "'c list" shows "interpret h (Neg \<phi>) xs = (\<not> interpret h \<phi> xs)"
bilinear_mem: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m1 :: "'g"   and m2 :: "'i"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m1 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "bilinear_map f R M1 M2 N" shows "f (m1, m2) \<in> carrier N"
lem_def: fixes x :: "'g"   and c :: "('d, 'e, 'f) com" shows "lem x c = c"
perm_1: fixes pi :: "(name \<times> name) list"   and x3 :: "name"   and x1 :: "name"   and x2 :: "pi" shows "pi \<bullet> x3<\<nu>x1> \<prec> x2 = (pi \<bullet> x3)<\<nu>(pi \<bullet> x1)> \<prec> (pi \<bullet> x2)"
value_plus_flow: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>"   and "countable_network.wf_residual_network \<Delta>"   and "flow \<Delta> f"   and "flow (countable_network.residual_network \<Delta> f) g" shows "value_flow \<Delta> (f \<oplus>\<^bsub>\<Delta>\<^esub> g) = value_flow \<Delta> f + value_flow \<Delta> g"
Nset_Suc: fixes n :: "nat" shows "{j. j \<le> Suc n} = insert (Suc n) {j. j \<le> n}"
complete_monitor: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a option list"   and i :: "nat" assumes "prefix_of \<pi> \<sigma>"   and "wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v"   and "\<And>\<sigma>. prefix_of \<pi> \<sigma> \<Longrightarrow> MFOTL.sat \<sigma> (map the v) i \<phi>" shows "\<exists>\<pi>'. prefix_of \<pi>' \<sigma> \<and> (i, v) \<in> M \<pi>'"
pbij_lens_weak: fixes x :: "'d \<Longrightarrow> 'e" assumes "pbij_lens x" shows "weak_lens x"
is_dghm_axioms: shows "\<GG> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
fls_uminus_subdegree: fixes f :: "'b fls" shows "fls_subdegree (- f) = fls_subdegree f"
X_def: shows "X = val_process Mkt (delta_hedging N der matur)"
nth_concat_split: fixes i :: "nat"   and xs :: "'c list list" assumes "i < length (concat xs)" shows "\<exists>j k. j < length xs \<and> k < length (xs ! j) \<and> concat xs ! i = xs ! j ! k"
neighborhoods_def: fixes x :: "'a" shows "stk1.neighborhoods x \<equiv> {U. is_open U \<and> x \<in> U}"
UNION_fun_upd: fixes A :: "'b \<Rightarrow> 'a set"   and i :: "'b"   and B :: "'a set"   and J :: "'b set" shows "\<Union> (A(i := B) ` J) = \<Union> (A ` (J - {i})) \<union> (if i \<in> J then B else {})"
var: shows "\<forall>var\<in>V. v var = v' var"
le_\<I>_def: fixes \<I> :: "('d, 'e) \<I>"   and \<I>' :: "('d, 'e) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
plossless_parallel_converter2: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>1' :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and \<I>2 :: "('m, 'n) \<I>"   and \<I>2' :: "('o, 'p) \<I>"   and conv2 :: "('m, 'n, 'o, 'p) converter" assumes "plossless_converter \<I>1 \<I>1' conv1"   and "plossless_converter \<I>2 \<I>2' conv2" shows "plossless_converter (\<I>1 \<oplus>\<^sub>\<I> \<I>2) (\<I>1' \<oplus>\<^sub>\<I> \<I>2') (conv1 |\<^sub>= conv2)"
mask_integer_def: shows "mask \<equiv> map_fun id integer_of_int mask"
in_hom_def: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" shows "VoV.in_hom f a b \<equiv> a \<in> VoV.domains f \<and> b \<in> VoV.codomains f"
V1: shows "wf_nfa (quantify_nfa v N) n"
Gram_Schmidt_column_k_iarrays_def: fixes A :: "real iarray iarray"   and k :: "nat" shows "Gram_Schmidt_column_k_iarrays A k = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>a b. (if b = k then column_iarray b A - (\<Sum>x\<in>set (map (\<lambda>n. column_iarray n A) [0..<b]). (column_iarray b A \<bullet>i x / (x \<bullet>i x)) *\<^sub>R x) else column_iarray b A) !! a)"
lfinite_llist_of: fixes xs :: "'c list" shows "lfinite (llist_of xs)"
FunctorCodDom_1: fixes f :: "'c" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "dom\<^bsub>CatCod F\<^esub> (F ## f) = F @@ dom\<^bsub>CatDom F\<^esub> f"
power_Suc2: fixes a :: "('a \<times> 'a) set"   and n :: "nat" shows "rel_dioid.power a (Suc n) = rel_dioid.power a n O a"
Int_absorb2: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "A \<inter> B = A"
assms_5: shows "\<forall>(v, a)\<in>dom s'. \<forall>(v, a')\<in>dom s'. s' (v, a) = Some True \<and> s' (v, a') = Some True \<longrightarrow> (v, a) = (v, a')"
continuous_on_of_real_o_iff: fixes S :: "'a set"   and g :: "'a \<Rightarrow> real" shows "continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g"
rk_ax_pasch: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" shows "rk {A, B, C, D} \<le> 3 \<longrightarrow> (\<exists>J. rk {A, B, J} = 2 \<and> rk {C, D, J} = 2)"
division_of_self: fixes a :: "'a"   and b :: "'a" assumes "cbox a b \<noteq> {}" shows "{cbox a b} division_of cbox a b"
trans_O_iff: fixes A :: "('b \<times> 'b) set" shows "trans A = (A O A \<subseteq> A)"
insert_seq_no_start: fixes ops :: "('b \<times> 'b option) list"   and ref :: "'b"   and xs :: "('b \<times> 'b option) list" assumes "insert_ops ops"   and "insert_seq (Some ref) xs"   and "insert_ops xs"   and "set xs \<subseteq> set ops"   and "ref \<notin> set (interp_ins ops)" shows "\<forall>i\<in>set (map fst xs). i \<notin> set (interp_ins ops)"
ssubst_Fvar: fixes z :: "char list"   and u :: "sterm"   and x :: "char list" shows "[z \<rightarrow> u] Fvar x = (if z = x then u else Fvar x)"
cancel_compls2: fixes L\<^sub>1 :: "'a literal set"   and L\<^sub>2 :: "'a literal set" assumes "L\<^sub>1\<^sup>C = L\<^sub>2\<^sup>C" shows "L\<^sub>1 = L\<^sub>2"
sort_descending_key_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "linorder.sort_descending_key less_eq f xs \<equiv> rev (linorder.sort_key less_eq f xs)"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) * c = a * c + b * c"
size_union: fixes M :: "'k multiset"   and N :: "'k multiset" shows "size (M + N) = size M + size N"
aa_Suc_n: shows "aa (Suc (card X)) = poly_deg f"
iprev_nth_atMost: fixes a :: "nat"   and n :: "nat" assumes "a \<le> n" shows "{..n} \<leftarrow> a = n - a"
range_subset_lower: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b"   and x :: "'b" assumes "{a..b} \<subseteq> {c..d}"   and "x \<in> {a..b}" shows "c \<le> a"
axioms_1: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2" shows "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1"
list_app_singletonE: fixes rs\<^sub>1 :: "'c list"   and rs\<^sub>2 :: "'c list"   and x :: "'c" assumes "rs\<^sub>1 @ rs\<^sub>2 = [x]" obtains "rs\<^sub>1 = [x]" and "rs\<^sub>2 = []"   | "rs\<^sub>1 = []" and "rs\<^sub>2 = [x]"
connected_def: fixes v :: "Vertices"   and w :: "Vertices" shows "connected v w \<equiv> \<exists>xs. walk xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
cospan_diagram_axioms_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f0 :: "'a"   and f1 :: "'a" shows "cospan_diagram_axioms C f0 f1 \<equiv> partial_magma.arr C f0 \<and> partial_magma.arr C f1 \<and> partial_magma.cod C f0 = partial_magma.cod C f1"
swap_delete_add_edge: fixes a :: "'c"   and b :: "'d"   and c :: "'c"   and x :: "'c"   and y :: "'d"   and z :: "'c"   and H :: "('c, 'd, 'e) graph_scheme" assumes "(a, b, c) \<noteq> (x, y, z)" shows "delete_edge a b c (add_edge x y z H) = add_edge x y z (delete_edge a b c H)"
dropWhile_filter: fixes xs :: "(nat \<times> 'a) list"   and nt :: "nat"   and c :: "enat" assumes "sorted (map fst xs)"   and "\<forall>t\<in>fst ` set xs. t \<le> nt" shows "dropWhile (\<lambda>(t, X). c < enat (nt - t)) xs = filter (\<lambda>(t, X). enat (nt - t) \<le> c) xs"
simps_3: fixes x31a :: "'b E.term"   and x32a :: "'b E.term"   and y31a :: "'b E.term"   and y32a :: "'b E.term" shows "(x31a \<^bold>\<star> x32a = y31a \<^bold>\<star> y32a) = (x31a = y31a \<and> x32a = y32a)"
hom: shows "(\<lambda>x. finprod G x I) \<in> hom (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) (G\<lparr>carrier := J\<rparr>)"
right_cancel: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "(b + a = c + a) = (b = c)"
cb01: shows "cball (0::'a) 1 \<subseteq> S"
exec_all_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>' = ((\<sigma>, \<sigma>') \<in> {(\<sigma>, \<sigma>'). exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>}\<^sup>*)"
cod_simp: fixes f :: "'a \<times> 'a" assumes "VxV.arr f" shows "VxV.cod f = (cod (fst f), cod (snd f))"
filter_compliant_stateful_ACS_def: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS G M = TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G M [] (edgesL G)"
hom_mult: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "eval R R id x (x \<otimes>\<^bsub>P\<^esub> y) = eval R R id x x \<otimes> eval R R id x y"
simps_4: fixes i :: "int" shows "typeof (Intg i) = \<lfloor>Integer\<rfloor>"
map_of_sorted_list: fixes m :: "('c, 'd) fmap" shows "map_of (sorted_list_of_fmap m) = ($$) m"
loop_has_cd': fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and j :: "nat"   and i :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "j < i"   and "\<pi> i = \<pi> j"   and "\<pi> j \<noteq> IFC_def.return prob" shows "\<exists>k\<in>{j..<i}. IFC_def.is_cdi prob i \<pi> k"
evaln: shows "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"
dgrad_p_set_red_single_pp: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m"   and "red_single p q f t" shows "d t \<le> m"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "is_root (Hp x hs) = no_Emptys hs"
Fiber_equality: fixes a :: "'a"   and b :: "'a" assumes "a \<in> M"   and "b \<in> M" shows "(Class a = Class b) = (\<eta> a = \<eta> b)"
disjoint_iff_fnot_equal: fixes A :: "'c fset"   and B :: "'c fset" shows "(A |\<inter>| B = {||}) = fBall A (\<lambda>x. fBall B ((\<noteq>) x))"
ide_char: fixes a :: "'c \<times> 'c" shows "C'.VV.ide a = (C'.VV.arr a \<and> C.VxV.ide a)"
f6: shows "(r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (s'' \<otimes> s' \<otimes> (r'' \<otimes> r) \<oplus> s'' \<otimes> s \<otimes> (r'' \<otimes> r') |\<^bsub>rel\<^esub> s'' \<otimes> s \<otimes> (s'' \<otimes> s'))"
distinct_15: fixes x3 :: "'a \<times> nat"   and x4 :: "'a \<times> nat" shows "Formal_out x3 \<noteq> Actual_in x4"
distrib_right': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) ; z = x ; z + y ; z"
rotate_map: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "rotate n (map f xs) = map f (rotate n xs)"
GSubspace_is_FGModule: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "FGModule G (\<cdot>) U"
assms_1: shows "(u, v) \<in> (E - UNIV \<times> R)\<^sup>*"
fresh: fixes \<sigma> :: "'b \<Rightarrow> bool list option"   and x :: "'b" assumes "\<sigma> x = None" shows "oracle \<sigma> x = spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs)))"
convex_imp_contractible: fixes S :: "'b set" assumes "convex S" shows "contractible S"
open_generated_order: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool" assumes "class.order_topology less_eq less open" shows "open = generate_topology (range (ord.lessThan less) \<union> range (ord.greaterThan less))"
funcset_to_empty_iff: fixes A :: "'b set" shows "A \<rightarrow> {} = (if A = {} then UNIV else {})"
acquired_reads_all_acquired: fixes pending_write :: "bool"   and sb :: "'b memref list"   and \<O> :: "nat set" shows "acquired_reads pending_write sb \<O> \<subseteq> \<O> \<union> all_acquired sb"
four_block_carrier_mat: fixes A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and D :: "'a mat"   and nr2 :: "nat"   and nc2 :: "nat"   and B :: "'a mat"   and C :: "'a mat" assumes "A \<in> carrier_mat nr1 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "four_block_mat A B C D \<in> carrier_mat (nr1 + nr2) (nc1 + nc2)"
assms_4: shows "sh\<^sub>2 D = \<lfloor>(sfs, Processing)\<rfloor>"
cnj_a_def: shows "cnj_a = - inner_prod x y / cnj (inner_prod y y)"
scaleR_left_distrib: fixes a :: "real"   and b :: "real"   and x :: "'d" shows "(a + b) *\<^sub>R x = a *\<^sub>R x + b *\<^sub>R x"
hyps: fixes xs :: "'a list" shows "list_diff (x # xs) ys = (if x \<in> set ys then list_diff xs ys else x # list_diff xs ys)"
simps_1: fixes t :: "'b tree" shows "T_merge \<langle>\<rangle> t = 1"
jk: shows "homeomorphism (rel_frontier U - {c}) V j k"
into_WW_imp_ge: fixes y :: "nat"   and x :: "nat"   and ns :: "nat list" assumes "y \<in> list.set (into_WW x ns)" shows "x \<le> y"
ring_tOp_closed: fixes R :: "('h, 'i) Ring_scheme"   and x :: "'h"   and y :: "'h" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> carrier R"
eventually_lt_imp_eventually_le: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x < g x" shows "\<forall>\<^sub>F x in F. f x \<le> g x"
card_of_mono1: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "|A| \<le>o |B|"
IH: fixes s :: "'var \<Rightarrow> 'node option"   and ms :: "'node list"   and m :: "'node"   and v :: "'val" assumes "g \<turnstile> ns\<Down>s"   and "g \<turnstile> last ns-ms\<rightarrow>m"   and "v \<in> allUses g m"   and "\<forall>n\<in>set (tl ms). var g v \<notin> var g ` allDefs g n" shows "s (var g v) = s' v"
curry_case_prod: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "curry (\<lambda>(x, y). f x y) = f"
simps_7: fixes t :: "'c local.term" shows "Dom \<^bold>\<r>\<^bold>[t\<^bold>] = Dom t \<^bold>\<otimes> \<^bold>\<I>"
not_min_galleryI_betw: fixes x :: "'a set"   and ys :: "'a set list"   and y :: "'a set"   and xs :: "'a set list" assumes "gallery (x # ys @ [y])"   and "order.greater (length xs) (length ys)" shows "\<not> min_gallery (x # xs @ [y])"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "quasi_ordered_set A less_eq" shows "transitive A less_eq"
quasi_isometry_betweenD_3: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and Y :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "lambda C -quasi_isometry_between X Y f"   and "y \<in> Y" shows "\<exists>x\<in>X. dist (f x) y \<le> C"
sum_upto_sum_divisors: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x :: "real" shows "sum_upto (\<lambda>n. \<Sum>d | d dvd n. f n d) x = sum_upto (\<lambda>k. sum_upto (\<lambda>d. f (d * k) k) (x / real k)) x"
fact_Suc: fixes n :: "nat" shows "fact (Suc n) = of_nat (Suc n) * fact n"
separatingD: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and S :: "'a set"   and x :: "'a"   and p :: "'a list"   and y :: "'a" assumes "separating_gen G A B S"   and "path G x p y"   and "x \<in> A"   and "y \<in> B" shows "(\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S"
scaleR_diff_right: fixes a :: "real"   and x :: "'c"   and y :: "'c" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
STL4: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> \<box>F \<longrightarrow> \<box>G"
Sample_L_def: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and buffer_in :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus"   and buffer_out :: "nat \<Rightarrow> ECall_Info list" shows "Sample_L req dt a1 stop lose st_in buffer_in ack i1 vc st_out buffer_out \<equiv> (\<forall>t. buffer_out t = (if dt t = [] then buffer_in t else dt t)) \<and> tiTable_SampleT req a1 stop lose st_in buffer_in ack i1 vc st_out"
cmp_lex'_def: shows "cmp_lex' \<equiv> \<lambda>uu uua. rec_list (\<lambda>cmp m. case m of [] \<Rightarrow> EQUAL | a # list \<Rightarrow> LESS) (\<lambda>a l la cmp m. case m of [] \<Rightarrow> GREATER | b # m \<Rightarrow> case cmp a b of LESS \<Rightarrow> LESS | EQUAL \<Rightarrow> la cmp m | GREATER \<Rightarrow> GREATER) uua uu"
dynProcModifyReturn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return c Q,A"