hcju/isabelleps huggingface dataset:
map_of_zip_upto_length_eq_nth: fixes i :: "nat"   and B :: "'b list"   and d :: "nat" assumes "i < length B"   and "d = length B" shows "map_of (zip [0..<d] B) i = Some (B ! i)"
fps_shift_uminus: fixes n :: "nat"   and f :: "'a fps" shows "fps_shift n (- f) = - fps_shift n f"
AllN: fixes xs :: "real list" shows "eval (QE_dnf' opt step \<phi>) xs = eval \<phi> xs"
return_stateT_def: shows "Monad_Overloading.return = return_state Monad_Overloading.return"
sams_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "SAMS A B C D E F" shows "SAMS A B C F E D"
mod_add_self2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a b) b = modulo a b"
coeff_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "coeff (monom m a) m = a"
inject_2: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list"   and y41 :: "nat"   and y42 :: "recf"   and y43 :: "recf list" shows "(Cn x41 x42 x43 = Cn y41 y42 y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
deduct_if_synth: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "M \<turnstile>\<^sub>c t" shows "M \<turnstile> t"
isPrefix_def: fixes p :: "'b list"   and t :: "'b list" shows "isPrefix p t = (\<exists>s. p @ s = t)"
minBA: fixes x :: "'a" assumes "x \<in> B"   and "x \<notin> A" shows "ab \<le> x \<and> (x \<le> ab \<longrightarrow> ab = x)"
lfinite_eq_range_llist_of: fixes xs :: "'b llist" shows "lfinite xs = (xs \<in> range llist_of)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) (\<Phi>'.Y a') \<Phi>\<Psi>'.map"
False: shows "\<not> (A $h 0 $h 0 = (0::'a) \<and> A $h 0 $h 1 \<noteq> (0::'a))"
reduction_doesnt_break_length_signs: fixes signs :: "rat list list"   and qs :: "real poly list"   and p :: "real poly"   and subsets :: "nat list list" assumes "\<forall>x\<in>set signs. length x = length qs"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "\<forall>x\<in>set (reduction_signs p qs signs subsets (M_mat signs subsets)). length x = length qs"
Nil_lenlex_iff1: fixes ns :: "'g list"   and r :: "('g \<times> 'g) set" shows "(([], ns) \<in> lenlex r) = (ns \<noteq> [])"
simps_2: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" shows "C M2 M1 \<Omega> V m (Suc 0) = V"
dvd_mult2: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b" shows "dvd.dvd times a (times b c)"
inject: fixes x :: "nat"   and ya :: "nat" shows "(LnChain x = LnChain ya) = (x = ya)"
K_subset: fixes x :: "'s" assumes "x \<in> S" shows "set_pmf (K x) \<subseteq> S"
row_append: fixes A :: "'b list list"   and B :: "'b list list"   and i :: "nat" shows "row (A @ B) i = row A i @ row B i"
graph_extD2: fixes H :: "'b set"   and h :: "'b \<Rightarrow> real"   and H' :: "'b set"   and h' :: "'b \<Rightarrow> real" assumes "graph H h \<subseteq> graph H' h'" shows "H \<subseteq> H'"
initial_upper_bound_def: fixes n :: "nat"   and r :: "rat" shows "initial_upper_bound n r = rat_of_int (root_rat_ceiling n r)"
BIT_init_def: fixes init :: "'c list" shows "BIT_init init = map_pmf (\<lambda>l. (l, init)) (bv (length init))"
prop_equiv_card: fixes P :: "'b ltln set" assumes "finite P" shows "card {abs_ltln\<^sub>P \<psi> |\<psi>. prop_atoms \<psi> \<subseteq> P} \<le> 2 ^ 2 ^ card P"
Inv_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "Inv a = a"
n_value_in_aug_inf: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "n_val K v x \<in> Z\<^sub>\<infinity>"
fmapU_ErrorT: fixes f :: "udom \<rightarrow> udom"   and m :: "udom\<cdot>'f error\<cdot>'e" shows "fmapU\<cdot>f\<cdot>(ErrorT\<cdot>m) = ErrorT\<cdot>(fmap\<cdot>(fmap\<cdot>f)\<cdot>m)"
sums_mult2: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a" assumes "f sums a" shows "(\<lambda>n. f n * c) sums (a * c)"
nat_mirror_eq_conv: fixes a :: "nat"   and l :: "nat"   and r :: "nat"   and b :: "nat" assumes "a \<le> l + r"   and "b \<le> l + r" shows "(nat_mirror a l r = nat_mirror b l r) = (a = b)"
entropy_chain_rule: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e" assumes "simple_function M X"   and "simple_function M Y" shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y | X)"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and B :: "('a, 'b, 'c) boundOutput"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> ROut M B"   and "guarded P"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) M B A\<^sub>P (\<Psi>\<^sub>P \<otimes> \<one>)"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (!P) M B [] (\<one>)"
integrable_integral: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a set" assumes "f integrable_on i" shows "(f has_integral integral i f) i"
simps_10: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G (\<phi> U \<psi>) w = \<G>\<^sub>F\<^sub>G \<phi> w \<union> \<G>\<^sub>F\<^sub>G \<psi> w"
unch_pair: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'd" shows "|~ Unchanged (x, y) = (Unchanged x \<and> Unchanged y)"
distproj_in_proj_set_2: fixes x :: "'b"   and A :: "'b set" assumes "proj_set x A \<noteq> {}" shows "distproj x A \<in> A"
Input_def: fixes x3 :: "name"   and x1 :: "name"   and x2 :: "pi" shows "x3<x1>.x2 \<equiv> Abs_pi (Input_Rep x3 ([x1].Rep_pi x2))"
psubst_new: fixes c :: "'a"   and p :: "('a, 'b) form"   and x :: "'a" assumes "new c p" shows "psubst (id(c := x)) p = p"
ds_dom_llp_strong: fixes x :: "'a"   and y :: "'a" shows "(x = d y \<cdot> x) = (d x \<sqsubseteq> d y)"
not_None_eq: fixes x :: "'x option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
abelian_monoidE_5: fixes R :: "('d, 'e) ring_scheme"   and x :: "'d"   and y :: "'d" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
vec_setI: fixes v :: "'b vec"   and i :: "nat"   and a :: "'b" assumes "v $ i = a"   and "i < dim_vec v" shows "a \<in> set\<^sub>v v"
simplicial_chain_def: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow>\<^sub>0 'd" shows "simplicial_chain p S c \<equiv> Poly_Mapping.keys c \<subseteq> Collect (simplicial_simplex p S)"
S_transform_def: fixes x :: "('d, 'f, 'e) formula" shows "S_transform x = Abs_formula (S_transform_formula x)"
equivalence_cancel_left: fixes e :: "'d"   and \<mu> :: "'d"   and \<mu>' :: "'d" assumes "D.equivalence_map e"   and "D.par \<mu> \<mu>'"   and "src\<^sub>D e = trg\<^sub>D \<mu>"   and "e \<star>\<^sub>D \<mu> = e \<star>\<^sub>D \<mu>'" shows "\<mu> = \<mu>'"
pred_restrict_space: fixes S :: "'c set"   and M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "S \<in> sets M" shows "pred (restrict_space M S) P = pred M (\<lambda>x. x \<in> S \<and> P x)"
ordinal_0_times: fixes x :: "ordinal" shows "0 * x = 0"
ord_cases: fixes a :: "int"   and b :: "int" obtains "ord a b = LESS" and "a < b"   | "ord a b = EQUAL" and "a = b"   | "ord a b = GREATER" and "b < a"
line_convex_combination2: fixes u :: "real"   and a :: "'b"   and b :: "'b"   and i :: "real" shows "(1 - u) *\<^sub>R a + u *\<^sub>R {a -- b}\<^bsub>i\<^esub> = {a -- b}\<^bsub>i * u\<^esub>"
alpha_Tree_rel_def: shows "Validity.alpha_Tree_rel \<equiv> {(x, y). x =\<^sub>\<alpha> y}"
assms: fixes x :: "nat" assumes "x \<in> set xs" shows "max_Var_floatariths f \<le> x"
greaterThanAtMost_empty_iff2: fixes k :: "'a \<times> ('a, 'b) operation"   and l :: "'a \<times> ('a, 'b) operation" shows "({} = hb.greaterThanAtMost k l) = (\<not> hb k l)"
simps_2: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "equal_base.list_distinct equal (x # xs) = (\<not> equal_base.list_member equal xs x \<and> equal_base.list_distinct equal xs)"
vrat_ls_def: shows "vrat_ls = ZFC_in_HOL.set {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> rat_of_vrat a < rat_of_vrat b}"
I_qelim: fixes qe :: "'a list \<Rightarrow> 'a fm"   and as :: "'a list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> is_dnf_qe qe as" shows "is_dnf_qe (qelim qe) as"
simps: fixes \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and \<Delta> :: "('sf, 'sc, 'sz) formula list"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "Ssubst (\<Gamma>, \<Delta>) \<sigma> = (map (\<lambda>\<phi>. Fsubst \<phi> \<sigma>) \<Gamma>, map (\<lambda>\<phi>. Fsubst \<phi> \<sigma>) \<Delta>)"
dvd_gcdD2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd gcd b c" shows "a dvd c"
ttree_rest_substitute: fixes f :: "'a \<Rightarrow> 'a ttree"   and S :: "'a set"   and T :: "'a set"   and t :: "'a ttree" assumes "\<And>x. carrier (f x) \<inter> S = {}" shows "ttree_restr S (substitute f T t) = ttree_restr S t"
add_closed': shows "\<forall>x\<in>U\<^sub>M. \<forall>y\<in>U\<^sub>M. x +\<^sub>M y \<in> U\<^sub>M"
rcut1: fixes A :: "fmla"   and B :: "fmla"   and H :: "fmla set"   and C :: "fmla" assumes "{A} \<turnstile> B"   and "insert B H \<turnstile> C" shows "insert A H \<turnstile> C"
i3: shows "\<forall>d\<in>D. \<not> hasRead s q d qq"
domain_subid: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and domain_op :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.domain_pre_dioid_one plus times one less_eq less domain_op"   and "less_eq x one" shows "less_eq x (domain_op x)"
top_conj_1: fixes x :: "'e"   and P :: "bool" shows "(top x \<and> P) = P"
word_eq_numeral_iff_iszero_10: fixes y :: "num" shows "(0 = numeral y) = iszero (numeral y)"
i_take_drop_map: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> n \<up> m = map f [m..<n]"
subcls_antisym: fixes G :: "prog" assumes "ws_prog G" shows "antisym ((subcls1 G)\<^sup>*)"
length_app: fixes rf :: "recf" shows "length (get_fstn_args (arity rf - Suc 0) (arity rf - Suc 0) @ [Cn (arity rf - Suc 0) (constn 0) [recf.id (arity rf - Suc 0) 0]]) = Suc (arity rf - Suc 0)"
sop_closed: fixes s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and i :: "'d \<Rightarrow> 'd"   and z :: "'d"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_one R s (aug_pm_set z i (s_set R s A))"   and "sop_z R s z" shows "\<forall>r\<in>carrier R. \<forall>x\<in>addition_set f (aug_pm_set z i (s_set R s A)). r \<^sub>s\<cdot> x \<in> addition_set f (aug_pm_set z i (s_set R s A))"
sets_Collect_countable_Ex': fixes I :: "'c set"   and P :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> {x \<in> \<Omega>. P i x} \<in> M"   and "countable I" shows "{x \<in> \<Omega>. \<exists>i\<in>I. P i x} \<in> M"
irreducible_monic_factor: fixes p :: "'b poly" assumes "0 < degree p" shows "\<exists>q r. irreducible q \<and> p = q * r \<and> monic q"
insert_body_split: fixes xs :: "'a list"   and e :: "'a" shows "\<exists>p s. xs = p @ s \<and> insert_body xs e = p @ e # s"
inf_prefix_length_def: fixes xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length xs ys \<equiv> equal_pair_count (zip xs ys)"
lexLessState_def: fixes Vbl :: "nat set" shows "lexLessState Vbl \<equiv> {(state1, state2). (getM state1, getM state2) \<in> lexLessRestricted Vbl}"
disc_12: shows "\<not> is_Tensor \<^bold>\<I>"
cod_simp_3: shows "partial_magma.cod (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) parallel_pair.j0 = quiver.One"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "F.A.seq g f" shows "F.B.seq (F g) (F f)"
CommuteWithIntervalCut4_def: fixes "fun" :: "'b set \<Rightarrow> 'b set" shows "CommuteWithIntervalCut4 fun \<equiv> \<forall>t fun2 I. fun2 = (\<lambda>I. I \<down>< t) \<or> fun2 = (\<lambda>I. I \<down>\<le> t) \<or> fun2 = (\<lambda>I. I \<down>> t) \<or> fun2 = (\<lambda>I. I \<down>\<ge> t) \<longrightarrow> fun (fun2 I) = fun2 (fun I)"
swap_image_eq: fixes a :: "'d"   and A :: "'d set"   and b :: "'d"   and f :: "'d \<Rightarrow> 'e" assumes "a \<in> A"   and "b \<in> A" shows "Fun.swap a b f ` A = f ` A"
FG: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (c::'a::refinement_lattice) ; x ; (d::'a::refinement_lattice) \<sqinter> d)"
enn2real_mono: fixes a :: "ennreal"   and b :: "ennreal" assumes "a \<le> b"   and "b < top_class.top" shows "enn2real a \<le> enn2real b"
convex_hull_translation: fixes a :: "'b"   and S :: "'b set" shows "convex hull (+) a ` S = (+) a ` (convex hull S)"
trivial_group: shows "trivial_group (Z 1) = (\<exists>a. carrier (Z 1) = {a})"
len: shows "p + q - gcd p q \<le> \<^bold>|w\<^bold>|"
amalgamate_def: fixes i :: "nat"   and j :: "nat" shows "amalgamate i j \<equiv> smn 1 (encode r_amalgamation) [i, j]"
index_mat_multrow_5: fixes mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "nat"   and a :: "'c"   and A :: "'c mat" shows "dim_col (mat_multrow_gen mul k a A) = dim_col A"
OUT: fixes x :: "'v" shows "d_OUT f (Inl x) = f (Inl x, Inr x)"
MOST_finite_Ball_distrib: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A" shows "(\<forall>\<^sub>\<infinity>y. \<forall>x\<in>A. P x y) = (\<forall>x\<in>A. \<forall>\<^sub>\<infinity>y. P x y)"
lossless1: fixes pk :: "'grp" shows "lossless_gpv \<I>_full (\<A>\<^sub>1 pk)"
t: shows "t \<subseteq> insert p (set_mset A \<union> (\<lambda>c. (Var c)\<^sup>2 - Var c) ` {c. c \<noteq> x'})"
2: shows "\<forall>i<length xl. \<xi> (xl ! i) = al ! i"
lookup_env_restr_not_there: fixes x :: "'c"   and S :: "'c set"   and m :: "'c \<Rightarrow> 'd" assumes "x \<notin> S" shows "(m f|` S) x = \<bottom>"
filterlim_at_infinity_imp_norm_at_top: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "filterlim f at_infinity F" shows "LIM x F. norm (f x) :> at_top"
s: shows "mk_all_list (xs @ [x]) B = (case x of (x, ty) \<Rightarrow> mk_all x ty) (mk_all_list xs B)"
streams_mono: fixes s :: "'b stream"   and A :: "'b set"   and B :: "'b set" assumes "s \<in> streams A"   and "A \<subseteq> B" shows "s \<in> streams B"
tran_ss_ssa: shows "\<forall>p<length ss. ss ! p \<sqsubseteq>\<^bsub>r\<^esub> ssa ! p"
nth_take_lemma: fixes k :: "nat"   and xs :: "'d list"   and ys :: "'d list" assumes "k \<le> length xs"   and "k \<le> length ys"   and "\<And>i. i < k \<longrightarrow> xs ! i = ys ! i" shows "take k xs = take k ys"
ThinishChamberComplex_axioms: shows "ThinishChamberComplex X"
soundnessForUNSAT: fixes F :: "Literal list list"   and M :: "Literal list"   and F0 :: "Literal list list" assumes "equivalentFormulae (F @ val2form M) F0"   and "formulaFalse F M" shows "\<not> satisfiable F0"
iso_imp_comm_group: fixes H :: "('e, 'f) monoid_scheme" assumes "H \<cong> H"   and "Group.monoid H" shows "comm_group H"
path_weightp_distinct_member: fixes s :: "'sum"   and l1 :: "'loc"   and l2 :: "'loc" assumes "s \<in> {s. path_weightp l1 l2 s}" shows "\<exists>u. u \<in> {s. path_weightp_distinct l1 l2 s} \<and> u \<le> s"
totient_primorial: fixes x :: "real" shows "real (totient (primorial x)) = real (primorial x) * (\<Prod>p\<in>{p. prime p \<and> real p \<le> x}. 1 - 1 / real p)"
assms_1: fixes s :: "'a" assumes "s \<in> start A" shows "P s"
product_topology_def: fixes T :: "'b \<Rightarrow> 'c topology"   and I :: "'b set" shows "product_topology T I = topology_generated_by {Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
Cons_less_Cons: fixes a :: "'b"   and x :: "'b list"   and b :: "'b"   and y :: "'b list" shows "(a # x < b # y) = (a < b \<or> a = b \<and> x < y)"
invariants_def: fixes E :: "'d set"   and \<phi> :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'e" shows "invariants E \<phi> g = {x \<in> E. \<phi> g x = x}"
ord_isom_less_forall: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "\<forall>a\<in>carrier D. \<forall>b\<in>carrier D. a \<prec> b = f a \<prec>\<^bsub>E\<^esub> f b"
vlrestriction_vdomain: fixes U :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda U f \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda U f) = VLambda U f"
H_not_eq_coset: fixes x :: "nat" assumes "euler_witness (int x) p" shows "H \<noteq> H #>\<^bsub>G\<^esub> x"
face_of_def: fixes T :: "'b set"   and S :: "'b set" shows "(T face_of S) = (T \<subseteq> S \<and> convex T \<and> (\<forall>a\<in>S. \<forall>b\<in>S. \<forall>x\<in>T. x \<in> open_segment a b \<longrightarrow> a \<in> T \<and> b \<in> T))"
fmadd_idempotent: fixes m :: "('c, 'd) fmap" shows "m ++\<^sub>f m = m"
int_simps_33: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "quasi_ordered_set A less_eq" shows "reflexive A less_eq"
dgrad_p_set_le_rep_list: fixes d :: "'a \<Rightarrow> nat"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "dgrad_set_le d (pp_of_term ` keys r) (Keys (set fs))" shows "punit.dgrad_p_set_le d {rep_list r} (set fs)"
simplicial_chain_simplicial_cone: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and v :: "nat \<Rightarrow> real"   and T :: "(nat \<Rightarrow> real) set" assumes "simplicial_chain p S c"   and "\<And>x u. \<lbrakk>0 \<le> u; u \<le> 1; x \<in> S\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * v i + u * x i) \<in> T" shows "simplicial_chain (Suc p) T (simplicial_cone p v c)"
add_right_imp_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.cancel_semigroup_add plus"   and "plus b a = plus c a" shows "b = c"
assms_1: shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"
trans_greater: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Greater"   and "cmp b c = Greater" shows "cmp a c = Greater"
simps_1: fixes is_true :: "'a \<Rightarrow> bool"   and is_false :: "'a \<Rightarrow> bool"   and box_ubx1 :: "'b \<Rightarrow> 'a"   and unbox_ubx1 :: "'a \<Rightarrow> 'b option"   and box_ubx2 :: "'c \<Rightarrow> 'a"   and unbox_ubx2 :: "'a \<Rightarrow> 'c option"   and d :: "'a" assumes "unboxedval is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2" shows "unboxedval.norm_unboxed box_ubx1 box_ubx2 (OpDyn d) = d"
plus_flow_simps: fixes G :: "('a, 'b) graph_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" shows "(f \<oplus>\<^bsub>G\<^esub> g) (x, y) = (if edge G x y then f (x, y) + g (x, y) - g (y, x) else 0)"
a0n_anyP_observable: fixes P :: "'b a0n_state set" shows "observable (obs a0n) P"
rel_set_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
bfs2_op_def: fixes am :: "nat \<Rightarrow> nat list"   and cf :: "nat \<times> nat \<Rightarrow> 'capacity" shows "bfs2_op am cf \<equiv> Graph.bfs2 cf (rg_succ2 am cf) s t"
list_update_splice: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "n < length xs + length ys"   and "length ys \<le> length xs"   and "length xs \<le> length ys + 1" shows "(splice xs ys)[n := x] = (if even n then splice (xs[n div 2 := x]) ys else splice xs (ys[n div 2 := x]))"
assms_3: shows "\<GG> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
mult_rel_class_of: fixes U :: "'c set"   and x :: "'d"   and V :: "'c set"   and y :: "'d"   and W :: "'c set" assumes "(U, x) \<in> Sigma (target.neighborhoods (f x)) \<O>\<^sub>Y"   and "(V, y) \<in> Sigma (target.neighborhoods (f x)) \<O>\<^sub>Y"   and "W \<in> target.neighborhoods (f x)"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "dlY.mult_rel (dlY.class_of U x) (dlY.class_of V y) = dlY.class_of W (mult_str\<^sub>Y W (\<rho>\<^sub>Y U W x) (\<rho>\<^sub>Y V W y))"
Restrict_Int: fixes A :: "'d set"   and B :: "'d set"   and r :: "('d \<times> 'e) set" shows "Restrict A (Restrict B r) = Restrict (A \<inter> B) r"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "nat set \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> 'a"   and f5 :: "'b \<Rightarrow> 'a"   and f6 :: "nat \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'a"   and seq_action :: "'b seq_action" shows "P (case seq_action of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = ((\<forall>x1. seq_action = broadcast x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. seq_action = groupcast x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32. seq_action = unicast x31 x32 \<longrightarrow> P (f3 x31 x32)) \<and> (\<forall>x4. seq_action = \<not>unicast x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x5. seq_action = send x5 \<longrightarrow> P (f5 x5)) \<and> (\<forall>x6. seq_action = deliver x6 \<longrightarrow> P (f6 x6)) \<and> (\<forall>x7. seq_action = receive x7 \<longrightarrow> P (f7 x7)) \<and> (seq_action = \<tau>\<^sub>s \<longrightarrow> P f8))"
execlass_leq_code: fixes cs :: "('a \<times> 'b) list"   and c1 :: "'a"   and c2 :: "'b" shows "class_leq (set cs) c1 c2 = execlass_leq cs c1 c2"
lin_dep_def: fixes S :: "'c set" shows "w.lin_dep S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R \<and> w.lincomb a A = \<zero>\<^bsub>md N\<^esub> \<and> v \<in> A \<and> a v \<noteq> \<zero>)"
simps_22: fixes x8 :: "pdf_type"   and x1 :: "nat" shows "Fail x8 \<noteq> Var x1"
tests_def: shows "tests = concat (map (\<lambda>b0. concat (map (\<lambda>bn. if coprime b0 bn then [rat_of_int b0 / rat_of_int bn] else []) (dp (lead_coeff ip)))) (df (coeff ip 0)))"
negligible_standard_hyperplane: fixes k :: "'a"   and c :: "real" assumes "k \<in> Basis" shows "negligible {x. x \<bullet> k = c}"
add_block_wf_cond: fixes b :: "'a set" assumes "b \<subseteq> \<V>" shows "incidence_system \<V> (add_block b)"