hcju/isabelleps huggingface dataset:
subst2: fixes x :: "name"   and M :: "'a"   and xvec :: "name list"   and Tvec :: "'b list" assumes "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> Tvec" shows "x \<sharp> M[xvec::=Tvec]"
distinct_load_tmps_prog_step: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')"   and "valid ts" shows "distinct_load_tmps is' \<and> load_tmps is' \<inter> load_tmps is = {} \<and> load_tmps is' \<inter> read_tmps sb = {}"
measure_exclude: fixes A :: "'b set"   and M :: "'b pmf"   and B :: "'b set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M"   and "measure_pmf.prob M A = measure_pmf.prob M (space (measure_pmf M))"   and "A \<inter> B = {}" shows "measure_pmf.prob M B = 0"
has_contour_integral_rmul: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "(f has_contour_integral i) g" shows "((\<lambda>x. f x * c) has_contour_integral i * c) g"
ipp_cond1_def: fixes A :: "'e set"   and i :: "'e \<Rightarrow> 'e" shows "ipp_cond1 A i = (\<forall>x\<in>A. \<^sub>i- (\<^sub>i- x) = x)"
Amk_zero: shows "A $ m $ mod_type_class.from_nat k \<noteq> (0::'a)"
nuc_Sup_closed: fixes f :: "'b \<Rightarrow> 'b" assumes "nucleus f" shows "f \<circ> Sup \<circ> (`) f = f \<circ> Sup"
Un_subset_iff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "((A::'b set) \<union> (B::'b set) \<subseteq> (C::'b set)) = (A \<subseteq> C \<and> B \<subseteq> C)"
basis_def: fixes A :: "'a vec set" shows "KD.basis A = (KD.lin_indpt A \<and> KD.span A = mat_kernel D \<and> A \<subseteq> mat_kernel D)"
ranI: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a" assumes "m a = Some b" shows "b \<in> ran m"
lb: fixes n :: "nat" assumes "0 < n" shows "real n < k n"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (K [X]); y \<in> carrier (K [X])\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>K [X]\<^esub> y = y \<otimes>\<^bsub>K [X]\<^esub> x" shows "Group.comm_monoid (K [X])"
degree: shows "degree (finite_fourier_poly ws) \<le> k - 1"
kmin_def: fixes f :: "nat \<Rightarrow> real"   and P :: "nat set" shows "kmin f P = (SOME S. S \<subseteq> P \<and> card S = khl \<and> (\<forall>i\<in>S. \<forall>j\<in>P - S. f i \<le> f j))"
dvd_triv_left: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times a b)"
linorder_ow_axioms: shows "linorder_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2"
entails_concl_compact_union: fixes EE :: "'b literal multiset set"   and CC :: "'b literal multiset set"   and DD :: "'b literal multiset set" assumes "finite EE"   and "CC \<union> DD \<TTurnstile>e EE" shows "\<exists>CC'\<subseteq>CC. finite CC' \<and> CC' \<union> DD \<TTurnstile>e EE"
g_def: fixes p :: "'a" shows "g p = (if p \<in> W then \<phi> p *\<^sub>R f' p else (0::'e))"
iso_verts_tail: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and e :: "'d" assumes "H.digraph_isomorphism hom"   and "e \<in> arcs (app_iso hom G)" shows "iso_tail hom (iso_arcs hom e) = iso_verts hom (tail (app_iso hom G) e)"
assms_2: shows "homeomorphism (cbox b c) (cbox v w) f2 g2"
subcomplex_opp_half_apartment: shows "order.greater_eq X folding.Y \<and> SimplicialComplex folding.Y"
mem_G'_1: shows "(u, v) \<in> parcs G'"
sort_mergesort_by_rel: shows "sort = mergesort_by_rel (\<le>)"
mult: fixes a :: "'a"   and H :: "'a set"   and b :: "'a" assumes "a \<in> additive.generate H"   and "b \<in> additive.generate H" shows "a + b \<in> additive.generate H"
expunge_delete_tag_causality: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and mo :: "'a"   and ir :: "'a"   and e2 :: "'b"   and pre :: "('a \<times> ('a, 'b) operation) event list"   and k :: "nat" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Expunge e1 mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)"   and "pre @ [Broadcast (ir, Delete is e2)] prefix of k" shows "Deliver (i, Expunge e2 mo i) \<in> set (history k)"
orthogonal_complement_def: fixes W :: "('a, 'cols) vec set" shows "COLS.v.orthogonal_complement W = {x. \<forall>y\<in>W. COLS.v.orthogonal y x}"
simps_10: shows "ReleaseAcquire \<noteq> Unlock"
factorize_polys_map_square_free_prod_list: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list" assumes "factorize_polys ps = (fs, data)"   and "fss = map real_of_rat_poly fs" shows "square_free (prod_list fss)"
sexp_In0I: fixes M :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp" shows "In0 M \<in> sexp"
fmdom_notD: fixes x :: "'e"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "x |\<notin>| fmdom m" shows "fmlookup m x = None"
assms_4: shows "prim_invar_impl Qi \<pi>i"
less_mask_eq: fixes x :: "'b word"   and n :: "nat" assumes "x < 2 ^ n" shows "x && mask n = x"
monom_mult_in_pmdl: fixes b :: "'t \<Rightarrow>\<^sub>0 'd"   and B :: "('t \<Rightarrow>\<^sub>0 'd) set"   and c :: "'d"   and t :: "'a" assumes "b \<in> B" shows "monom_mult c t b \<in> pmdl B"
MakeFaceFinal_minGraphProps': fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "minGraphProps' (makeFaceFinal f g)"
S4: shows "c_assoc_value ls (c_pair (c_snd (c_snd (c_fst key))) (c_snd key)) = univ_for_pr (c_pair (c_snd (c_snd (c_fst key))) (c_snd key))"
compl_closed: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "\<^bold>-\<^bsub>L\<^esub> x \<in> carrier L"
w: shows "vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s"
P: shows "pre @ [Broadcast (r, Expunge e2 i r)] prefix of k"
cring_monom_degree: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier R"   and "p = monom P a m"   and "a [^] n \<noteq> \<zero>" shows "deg R (p [^]\<^bsub>P\<^esub> n) = n * m"
word_oa_dist: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "x AND y OR z = (x OR z) AND (y OR z)"
fd_def: fixes x :: "'a"   and y :: "'a" shows "fd x y = domain_op (x \<cdot> y)"
span_pr: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "span (pr1 a b) (pr0 a b)"
less_linord_helper_def: fixes a :: "('a, 'b) linord_helper"   and b :: "('a, 'b) linord_helper" shows "(a < b) = (a \<noteq> b \<and> linord_helper_less_eq1 a b)"
assms_1: shows "insert_ops (op_list @ [(oid, oper)])"
map_fst_zip_take: fixes xs :: "'a list"   and ys :: "'b list" shows "map fst (zip xs ys) = take (min (length xs) (length ys)) xs"
ran_def: fixes m :: "'d \<Rightarrow> 'c option" shows "ran m = _Collect b (\<exists>a. m a = Some b)"
is_nan_times: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "IEEE.is_nan (x * y)"
i_f_join_i_expand_iMODb: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "f \<odot> k \<Join>\<^sub> [ n * k, mod k, d ] = f \<Join>\<^sub> [n\<dots>,d]"
comp_ide_self: fixes a :: "'b" assumes "D.ide a" shows "a \<cdot>\<^sub>D a = a"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b" assumes "prebicategory V H \<a>"   and "weak_composition V H"   and "prebicategory_with_units_axioms V H \<i>" shows "prebicategory_with_units V H \<a> \<i>"
unop_slp_of_fa_induction_step: fixes fa1 :: "floatarith"   and xs :: "real list"   and Unop :: "floatarith \<Rightarrow> floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M' :: "(floatarith, nat) mapping"   and slp' :: "floatarith list"   and unop :: "real \<Rightarrow> real" assumes "\<And>M slp M' slp'. \<lbrakk>slp_of_fa fa1 M slp = (M', slp'); \<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M; \<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp; \<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs\<rbrakk> \<Longrightarrow> subterms fa1 \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"   and "(case slp_of_fa fa1 M slp of (M1, slp1) \<Rightarrow> slp_of_fa_un Unop fa1 M slp M1 slp1) = (M', slp')"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs"   and "\<And>a b. subterms (Unop a) = insert (Unop a) (subterms a)"   and "\<And>a b xs. interpret_floatarith (Unop a) xs = unop (interpret_floatarith a xs)" shows "insert (Unop fa1) (subterms fa1) \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"
conditional_entropy_eq_ce_with_hypothesis: fixes X :: "'key \<times> 'message list \<Rightarrow> 'a"   and Y :: "'key \<times> 'message list \<Rightarrow> 'b" shows "\<H>( X | Y ) = - (\<Sum>y\<in>Y ` msgs. prob (Y -` {y} \<inter> msgs) * (\<Sum>x\<in>X ` msgs. prob ((\<lambda>x. (X x, Y x)) -` {(x, y)} \<inter> msgs) / prob (Y -` {y} \<inter> msgs) * log b (prob ((\<lambda>x. (X x, Y x)) -` {(x, y)} \<inter> msgs) / prob (Y -` {y} \<inter> msgs))))"
semiring_of_sets_axioms_def: fixes M :: "'c set set" shows "semiring_of_sets_axioms M \<equiv> {} \<in> M \<and> (\<forall>a b. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> a \<inter> b \<in> M) \<and> (\<forall>a b. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> (\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union> C))"
measurable_def: fixes A :: "'d measure"   and B :: "'e measure" shows "A \<rightarrow>\<^sub>M B = {f \<in> space A \<rightarrow> space B. \<forall>y\<in>sets B. f -` y \<inter> space A \<in> sets A}"
indep_sets_cong: fixes I :: "'b set"   and J :: "'b set"   and F :: "'b \<Rightarrow> 'a set set"   and G :: "'b \<Rightarrow> 'a set set" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> F i = G i" shows "indep_sets F I = indep_sets G J"
prems_4: fixes d :: "nat" assumes "Suc dm \<le> d"   and "d < length b" shows "p ! d = b ! d"
simps_2: fixes xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "wellformed_policy2Pr (DenyAll # xs) = wellformed_policy2Pr xs"
FirstOccurrenceExists: fixes msg :: "('p, 'v) message" assumes "enabled (last exec) msg"   and "\<exists>p. isReceiverOf p msg" shows "\<exists>n. firstOccurrence msg n"
antisym: fixes a :: "'a"   and b :: "'a" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> a" shows "a = b"
assms_3: shows "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts"
dim_A: shows "mat dim dim (\<lambda>(i, j). lim (\<lambda>n. f n $$ (i, j))) \<in> carrier_mat dim dim"
unsigned_push_bit_eq: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and w :: "'c word" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.unsigned one plus zero (push_bit n w) = take_bit LENGTH('c) (push_bit n (semiring_1.unsigned one plus zero w))"
prod_cases6: fixes y :: "'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g" obtains a :: "'b" and b :: "'c" and c :: "'d" and d :: "'e" and e :: "'f" and f :: "'g" where "y = (a, b, c, d, e, f)"
upper_bits_unset_is_l2p: fixes n :: "nat"   and p :: "'b word" assumes "n < LENGTH('b)" shows "(\<forall>n'\<ge>n. n' < LENGTH('b) \<longrightarrow> \<not> p !! n') = (p < 2 ^ n)"
induced_automorphism_on_simplices_fg: fixes x :: "'a set"   and v :: "'a" assumes "x \<in> f \<turnstile> X"   and "v \<in> x" shows "\
 v = g v"
tendsto_unique: fixes "open" :: "'b set \<Rightarrow> bool"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.t2_space open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open a) F"   and "filterlim f (topological_space.nhds open b) F" shows "a = b"
satisfies_bounded_nFEx: fixes \<AA> :: "'i"   and b :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b nFEx b \<phi>) = (\<AA> \<Turnstile>\<^sub>b FEx b \<phi>)"
from_hma\<^sub>m_def: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>m a = Matrix.mat CARD('f) CARD('e) (\<lambda>(i, j). a $h Bij_Nat.from_nat i $h Bij_Nat.from_nat j)"
assms_1: shows "\<exists>e. e |\<in>| x1 \<and> e |\<notin>| x2"
2: shows "distributed M lborel (\<lambda>x. - \<nu> + Y x) (\<lambda>x. ennreal (normal_density 0 \<tau> x))"
coarser_partitions_inv_without: fixes P :: "'b set set"   and elem :: "'b" assumes "is_non_overlapping P"   and "elem \<in> \<Union> P" shows "P \<in> coarser_partitions_with elem (partition_without elem P)"
L\<^sub>2\<^sub>3: shows "instance_of\<^sub>l\<^sub>s L\<^sub>2 L\<^sub>3"
assms_2: fixes x :: "'a" assumes "x \<in> s" shows "(f has_field_derivative (0::'a)) (at x within s)"
process_ret_Undecided_sound: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr" assumes "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), Undecided\<rangle> \<Rightarrow> Undecided" shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> Undecided"
isomorphicE: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" assumes "S.VV.isomorphic a a'" obtains f :: "'a \<times> 'a" where "S.VV.in_hom f a a' \<and> S.VV.iso f"
comp_cod_arr: fixes f :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "B.VV.arr f"   and "B.VV.cod f = b" shows "B.VV.comp b f = f"
ord_works: fixes a :: "nat"   and n :: "nat" shows "[a ^ ord n a = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> [a ^ m \<noteq> 1] (mod n))"
mod_eq_self_iff_div_eq_0: 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 divide plus minus zero times one modulo euclidean_size" shows "(modulo a b = a) = (divide a b = zero)"
l_cancel: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x +++ y = x +++ z) = (y = z)"
hyps: assumes "xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last xs = hd ys \<and> last xs \<noteq> (0::'a)" shows "changes (xs @ ys) = changes xs + changes ys"
sets_into_space: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and x :: "'b set" assumes "subset_class \<Omega> M"   and "x \<in> M" shows "x \<subseteq> \<Omega>"
diff_fun_onE: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c" assumes "W'.src.diff_fun_on A f" obtains W :: "'a set" and f' :: "'a \<Rightarrow> 'c" where "A \<subseteq> W" and "W \<subseteq> W'.src.carrier" and "open W" and "diff_fun k (W'.src.charts_submanifold W) f'" and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x"
simps_1: fixes TI :: "('e set \<times> 'e set) list"   and x :: "'h"   and y :: "'h" shows "equal_mod_timpls TI (Var x) (Var y) = (x = y)"
fg: shows "group_isomorphisms (reduced_homology_group (int p) (nsphere p)) (reduced_homology_group (1 + int p) (nsphere (Suc p))) f g"
mono_fun: fixes S1 :: "'b st"   and S2 :: "'b st"   and x :: "char list" assumes "S1 \<le> S2" shows "fun S1 x \<le> fun S2 x"
set_less_eq_UNIV: fixes A :: "'a set" shows "A \<sqsubseteq> UNIV"
Guard: 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 g :: "'e set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(g \<inter> P) Guard f g c Q,A"
eigen_vector_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "('a, 'b) Finite_Cartesian_Product.vec"   and ev :: "'a" shows "eigen_vector A v ev = (v \<noteq> 0 \<and> A *v v = ev *s v)"
G: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G"
vconn_implies_hpar_1: fixes \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "src\<^sub>D \<mu> = src\<^sub>D f"
cases: fixes x :: "('c \<times> 'c) set \<times> ('d \<Rightarrow> 'c) \<times> 'c \<times> 'd list" obtains uu_ :: "('c \<times> 'c) set" and uv_ :: "'d \<Rightarrow> 'c" and uw_ :: "'c" where "xa__ = (uu_, uv_, uw_, [])"   | I :: "('c \<times> 'c) set" and D :: "'d \<Rightarrow> 'c" and u :: "'c" and xs :: "'d list" and x :: "'d" where "xa__ = (I, D, u, xs @ [x])"
pochhammer_minus': fixes b :: "'b"   and k :: "nat" shows "pochhammer (b - of_nat k + (1::'b)) k = (- (1::'b)) ^ k * pochhammer (- b) k"
bounded_iff: fixes b :: "'d multiset"   and c :: "'d multiset"   and a :: "'d multiset" shows "(b \<union># c \<subseteq># a) = (b \<subseteq># a \<and> c \<subseteq># a)"
transp_preserved_f: fixes f :: "('a, 'b, 'c) prog" shows "transp (preserved f)"
trans: fixes b :: "nat list"   and a :: "nat list"   and c :: "nat list" assumes "b \<le>\<^sub>v a"   and "c \<le>\<^sub>v b" shows "c \<le>\<^sub>v a"
order_le_err: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "order (Err.le r) = order r"
comp_monotone: fixes w :: "'b list"   and r :: "'b list"   and u :: "'b list" assumes "w \<bowtie> r"   and "u \<le>p w" shows "u \<bowtie> r"
fmimage_alt_def: fixes m :: "('b, 'a) fmap"   and S :: "'b fset" shows "fmimage m S = fmran (fmrestrict_fset S m)"
inv_del: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "invh t"   and "invc t" shows "invh (del x t) \<and> (color t = Red \<longrightarrow> bheight (del x t) = bheight t \<and> invc (del x t)) \<and> (color t = Black \<longrightarrow> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
chart_in_codomain: fixes x :: "'a"   and c :: "('a, 'b) chart" assumes "x \<in> domain c" shows "apply_chart c x \<in> codomain c"
F_subset: shows "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
distr_plus_ring_count_space: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and c :: "'a" assumes "has_density M (count_space UNIV) f" shows "has_density (distr M (count_space UNIV) ((+) c)) (count_space UNIV) (\<lambda>x. f (x - c))"
rep_eq: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and xa :: "'b" shows "mapping_of (MPoly_Type.monom x xa) = Poly_Mapping.single x xa"
abs_eq: fixes x :: "nat" shows "Number x = Ab (cNumber x)"
funcset_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<forall>x\<in>A. f x = g x" shows "f = g"
run_fail_env: fixes fail :: "'d"   and r :: "'e" shows "run_env (fail_env fail) r = fail"
blin': fixes v :: "(real, 'sz) vec" shows "(\<lambda>v'. sterm_sem I \<theta>\<^sub>1 v * frechet I \<theta>\<^sub>2 v v' + frechet I \<theta>\<^sub>1 v v' * sterm_sem I \<theta>\<^sub>2 v) \<in> {f. bounded_linear f}"
simps_1: fixes e :: "'c \<times> 'd \<times> bool" shows "insert_body [] e = [e]"
var_disjoint_def: fixes Cs :: "('c, 'd) Term.term literal multiset list" shows "var_disjoint Cs = (\<forall>\<sigma>s. length \<sigma>s = length Cs \<longrightarrow> (\<exists>\<tau>. \<forall>i<length Cs. \<forall>S. S \<subseteq># Cs ! i \<longrightarrow> subst_cls S (\<sigma>s ! i) = subst_cls S \<tau>))"
ch: shows "x \<in> convex hull set (map fst (half_segments_of_aform X))"
rel_cset_iff: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d cset"   and b :: "'e cset" shows "rel_cset R a b = ((\<forall>t. cin t a \<longrightarrow> (\<exists>u. cin u b \<and> R t u)) \<and> (\<forall>t. cin t b \<longrightarrow> (\<exists>u. cin u a \<and> R u t)))"
close_world_imp_val: fixes M :: "object atom formula set"   and \<phi> :: "object atom formula" assumes "wm_basic M"   and "M \<^sup>c\<TTurnstile>\<^sub>= \<phi>" shows "valuation M \<Turnstile> \<phi>"
nonneg_linorder_cases: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less_eq zero a"   and "less_eq zero b" obtains "less a b"   | "a = b"   | "less b a"
add_diff_cancel_left': fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and a :: "'f"   and b :: "'f" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
fix_subset: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" shows "fix f A \<subseteq> A"
norm_exp_imaginary: fixes z :: "complex" assumes "cmod (exp z) = 1" shows "Re z = 0"
rel_compile_fundef: fixes get_arity :: "'fun \<Rightarrow> nat option"   and load_oracle :: "nat \<Rightarrow> Unboxed.type option"   and fd1 :: "('b, 'm, 'fun, 'd, 'e) Inca.instr fundef"   and fd2 :: "('b, 'm, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef" assumes "compile_fundef get_arity load_oracle fd1 = Ok fd2" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
simps_2: fixes f :: "'c" shows "\<^bold>\<langle>f\<^bold>\<rangle>\<^bold>\<down> = \<^bold>\<langle>f\<^bold>\<rangle>"
short_exact_sequence_def: fixes A :: "'b monoid"   and B :: "'b monoid"   and C :: "'b monoid"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" shows "short_exact_sequence A B C f g \<equiv> \<exists>T1 T2 e1 e2. exact_seq ([T1, A, B, C, T2], [e1, f, g, e2]) \<and> trivial_group T1 \<and> trivial_group T2"
nhds_prod: fixes a :: "'a"   and b :: "'b" shows "nhds (a, b) = nhds a \<times>\<^sub>F nhds b"
moebius_translation_inv: fixes v1 :: "complex" shows "moebius_inv (moebius_translation v1) = moebius_translation (- v1)"
a_ipz: fixes z :: "int" shows "\<infinity> + ant z = \<infinity>"