hcju/isabelleps huggingface dataset:
coeff_sum_list: fixes xs :: "nat list"   and f :: "nat \<Rightarrow> rat"   and v :: "nat" assumes "distinct xs" shows "coeff (\<Sum>x\<leftarrow>xs. f x *R lp_monom 1 x) v = (if v \<in> set xs then f v else 0)"
neg_eq_bottom_iff: fixes x :: "tr" shows "(neg\<cdot>x = \<bottom>) = (x = \<bottom>)"
to_depressed_cubic: fixes a :: "'a"   and x :: "'a"   and y :: "'a"   and b :: "'a"   and e :: "'a"   and c :: "'a"   and f :: "'a"   and d :: "'a" assumes "a \<noteq> (0::'a)"   and "x = y - b / ((3::'a) * a)"   and "e = (c - b\<^sup>2 / ((3::'a) * a)) / a"   and "f = (d + (2::'a) * b ^ 3 / ((27::'a) * a\<^sup>2) - b * c / ((3::'a) * a)) / a" shows "(a * x ^ 3 + b * x\<^sup>2 + c * x + d = (0::'a)) = (y ^ 3 + e * y + f = (0::'a))"
hyps_1: shows "x \<rightarrow>\<^sub>s y \<and> (\<forall>r. y \<rightarrow>\<^sub>\<beta> r \<longrightarrow> x \<rightarrow>\<^sub>s r)"
parts_msg_def: fixes X :: "msg" shows "parts_msg X \<equiv> parts {X}"
set_lebesgue_integral_def: fixes M :: "'d measure"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b" shows "set_lebesgue_integral M A f \<equiv> LINT x|M. indicat_real A x *\<^sub>R f x"
inconsistent_conjoin: fixes G :: "'b fm list"   and p :: "'b fm" assumes "set G \<union> {p} \<notin> mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>})" shows "\<turnstile> (\<^bold>\<not> conjoin G p)"
fBNsep: fixes z :: "('b, 'c) Combinators list"   and b :: "'b"   and c :: "'b"   and a :: "('b, 'c) Combinators" assumes "\<And>a. a \<in> set z \<Longrightarrow> {b, c} \<noteq> first_bothNet a"   and "a \<in> set (separate z)" shows "{b, c} \<noteq> first_bothNet a"
\<section>: fixes u :: "real" assumes "0 \<le> u"   and "u \<le> x" shows "\<exists>y. ((\<lambda>x. x + sin x) has_real_derivative 1 + cos u) (at u)"
iso_assoc': fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.iso \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h]"
mononom_of_vars_def: shows "mononom_of_vars \<equiv> \<lambda>(xs, n). (+) (Const n * poly_of_vars xs)"
is_zariski_open_empty: shows "cring0.is_zariski_open {}"
fs: fixes x :: "'b" assumes "x \<in> R" shows "f x \<le> Sup (f ` R)"
nsum_cmp_eq: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and h1 :: "nat \<Rightarrow> nat"   and h2 :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "f \<in> {j. j \<le> n} \<rightarrow> carrier A"   and "h1 \<in> {j. j \<le> n} \<rightarrow> {j. j \<le> n}"   and "h2 \<in> {j. j \<le> n} \<rightarrow> {j. j \<le> n}"   and "i \<le> n" shows "\<Sigma>\<^sub>e A cmp f (cmp h2 h1) i = \<Sigma>\<^sub>e A cmp (cmp f h2) h1 i"
hyps: fixes m :: "nat" shows "ltake (enat n) (iterates Suc m) = llist_of [m..<n + m]"
Snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" shows "Y \<in> parts H"
PBQ: shows "(P, S, Q) = Smith_2x2_eucl A"
3: shows "(\<zero> \<oplus> \<zero>) \<odot>\<^bsub>M\<^esub> m = \<zero> \<odot>\<^bsub>M\<^esub> m \<oplus>\<^bsub>M\<^esub> \<zero> \<odot>\<^bsub>M\<^esub> m"
subspace_diff: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "subspace S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
length_tl: fixes xs :: "'c list" shows "length (tl xs) = length xs - 1"
IN: shows "(\<Squnion>n. d_IN (h n) b) = weight \<Gamma> b"
profile_mono_schemeV_axioms: shows "profile_mono_schemeV f g"
VAsub: fixes a :: "'a" assumes "x1a = Inr a" shows "FVT (\<sigma> a) \<subseteq> (\<Union>i\<in>{i. Inr i \<in> SIGT ($f x1a x2a)}. FVT (\<sigma> i))"
elims_2: fixes x :: "primfun"   and xa :: "primfun" assumes "x \<le> xa" obtains x :: "nat" and y :: "nat" where "xa__ = LnChain x" and "xa = LnChain y" and "x \<le> y"
has_field_derivative_iff_has_vector_derivative: fixes f :: "real \<Rightarrow> real"   and y :: "real"   and F :: "real filter" shows "(f has_real_derivative y) F = (f has_vector_derivative y) F"
trivial_ker_imp_inj: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h"   and "a_kernel R S h = {\<zero>\<^bsub>R\<^esub>}" shows "inj_on h (carrier R)"
f'_def: shows "f' = fps_shift (subdegree g) f"
cl10_d_var1: fixes x :: "'a" shows "d (x \<sqinter> nc) = 1\<^sub>\<sigma> \<sqinter> x \<cdot> nc"
word_mod_by_0: fixes k :: "'b word" shows "k mod 0 = k"
intros_5: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "Lt a \<prec> Le b"
null_sets_distr_iff: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and A :: "'d set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(A \<in> null_sets (distr M N f)) = (f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N)"
less_eq_value_def: fixes v1 :: "value"   and v2 :: "value" shows "(v1 \<le> v2) = (v1 < v2 \<or> v1 = v2)"
j_xs: shows "\<forall>j\<in>set [0..<min (dim_row A) (dim_col A) - 1]. j < min CARD('c) CARD('b)"
induct: fixes P :: "('c \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) list \<Rightarrow> bool"   and a0 :: "('c \<times> 'b) list"   and a1 :: "'b"   and a2 :: "('c \<times> 'b) list" assumes "\<And>x prev. P [] x prev"   and "\<And>sub sep xs x prev. (sep < x \<Longrightarrow> P xs x (prev @ [(sub, sep)])) \<Longrightarrow> P ((sub, sep) # xs) x prev" shows "P a0 a1 a2"
\<theta>_vars_S_bvars_disj: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "(subst_domain \<theta> \<union> range_vars \<theta>) \<inter> set X = {}"
k: fixes k' :: "nat" assumes "k < k'" shows "\<not> root_unity k' dvd p"
simps: fixes a :: "'a set" shows "finite a = (a = {} \<or> (\<exists>A a. a = insert a A \<and> finite A))"
lcm_right_idem: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "lcm (lcm a b) b = lcm a b"
linear_implies_no_abs: fixes t :: "term" assumes "linear t" shows "no_abs t"
wellformed1_sorted: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "wellformed_policy1 p" shows "wellformed_policy1 (FWNormalisationCore.sort p l)"
bet__lt1213: fixes B :: "'p"   and C :: "'p"   and A :: "'p" assumes "B \<noteq> C"   and "Bet A B C" shows "A B Lt A C"
f_p_3: shows "\<forall>i<n. S (CAs0f i) \<cdot> \<eta>s0f i = S_M S M (CAs ! i)"
t_def: fixes s :: "'c list"   and r :: "'c"   and a :: "nat \<times> nat list" shows "t s r a = (let (mf, sws) = a in index (swaps sws s) r + 1 + length sws)"
strict_implies_order: fixes b :: "'b zmultiset"   and a :: "'b zmultiset" assumes "b \<subset>#\<^sub>z a" shows "b \<subseteq>#\<^sub>z a"
assms_2: fixes u :: "'item list" assumes "u \<le>\<^sub>F\<^sub>I v" shows "u \<preceq>\<^sub>F w (l u)"
instant_index_increase_generalized: fixes n :: "nat"   and n\<^sub>k :: "nat"   and \<rho> :: "'c run"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" assumes "n < n\<^sub>k"   and "\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g" shows "\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, n\<^sub>k \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, n\<^sub>k \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
wncarr: shows "wn \<in> carrier (relative_homology_group (int n) (nsphere n) (lower n))"
tl_of_cptn_is_cptn: fixes x :: "'b com option \<times> 'b"   and xs :: "('b com option \<times> 'b) list" assumes "x # xs \<in> cptn"   and "xs \<noteq> []" shows "xs \<in> cptn"
assms: shows "\<downharpoonright>\<alpha>\<downharpoonleft> = pi"
trace_and_start_determines_end: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and c' :: "('d, 'e, 'f) configuration"   and d' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv c t c'"   and "distributed_system.trace channel trans send recv c t d'" shows "c' = d'"
w_in_hom_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.in_hom V w w w"
Singleton: fixes x :: "ZF"   and y :: "ZF" shows "Elem x (Singleton y) = (x = y)"
monomials_swap_poly: fixes b :: "nat"   and i :: "nat"   and mp :: "'b mpoly" shows "monomials (swap_poly b i mp) = swap0 b i ` monomials mp"
axioms_1: fixes G :: "('a, 'b) ring_scheme" assumes "abelian_group G" shows "abelian_monoid G"
paid_A_def: shows "paid_A = map (filter (\<lambda>x. Suc x < length init)) paid_A'"
bdd_below_V_def: fixes X :: "'a set" shows "bdd_below_V X \<equiv> X \<noteq> {}"
finite_measure_distr: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and M' :: "'d measure" assumes "finite_measure M"   and "f \<in> M \<rightarrow>\<^sub>M M'" shows "finite_measure (distr M M' f)"
fp_def: shows "fp = (\<lambda>x. ennreal (f x))"
reduced_GB_pmdl_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (local.punit.component_of_term ` Keys F)"   and "F \<subseteq> punit.dgrad_p_set d m" shows "local.punit.pmdl (punit.reduced_GB F) = local.punit.pmdl F"
HashMap_inverse: fixes y :: "('a, 'b) ArrayHashMap_Impl.hashmap" assumes "y \<in> {hm. ArrayHashMap_Impl.ahm_invar hm}" shows "impl_of (HashMap y) = y"
field_differentiable_within_sin: fixes z :: "'a"   and S :: "'a set" shows "sin field_differentiable at z within S"
1: shows "s # slist \<in> state_set (state_list s as)"
ide_is_iso: fixes a :: "'b" assumes "B.ide a" shows "B.iso a"
IH: shows "(\<forall>n. (\<forall>i<n. inpt_st (\<lambda>a. case a of (u, v) \<Rightarrow> \<lambda>a. case a of (u', v') \<Rightarrow> r u u' OO r' v v') (u i) (u (Suc i)) (x i)) \<longrightarrow> inpt_st r (fst (u n)) (fst (u (Suc n))) (x n) \<and> (\<forall>y. r (fst (u n)) (fst (u (Suc n))) (x n) y \<longrightarrow> inpt_st r' (snd (u n)) (snd (u (Suc n))) y)) \<longrightarrow> (\<forall>i<n. inpt_st r ((fst \<circ> u) i) ((fst \<circ> u) (Suc i)) (x i) \<and> (\<forall>y. r (fst (u i)) (fst (u (Suc i))) (x i) y \<longrightarrow> inpt_st r' (snd (u i)) (snd (u (Suc i))) y))"
simps_5: fixes x :: "three" shows "f1 x Two = Two"
smap_strict: fixes f :: "'b \<rightarrow> 'a" shows "smap\<cdot>f\<cdot>\<bottom> = \<bottom>"
right_idem: fixes a :: "'a"   and b :: "'a" shows "(a AND b) AND b = a AND b"
asc: shows "\<forall>i. f i < f (Suc i)"
trivial_limit_at_left_real: fixes x :: "'a" shows "at_left x \<noteq> bot"
replace_f: fixes a :: "'b negation_type" shows "\<not> has_disc_negated disc2 neg (replace_f a)"
commute: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "dbm_add a b = dbm_add b a"
orientable_1_implies_5: fixes x :: "'a" assumes "orientable_1 x" shows "orientable_5 x"
metric_LIMSEQ_D: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b"   and r :: "real" assumes "X \<longlonglongrightarrow> L"   and "0 < r" shows "\<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
new_f12_norm: shows "normFace f12 \<notin> set (normFaces (faces g))"
beta_reduces_fn: fixes A :: "'a trm"   and A' :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "A \<rightarrow>\<beta>\<^sup>* A'" shows "Fn x T A \<rightarrow>\<beta>\<^sup>* Fn x T A'"
parallel_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<parallel> ys) = (\<not> xs \<le>p ys \<and> \<not> ys \<le>p xs)"
poly_weak_mono: fixes p :: "('c monom \<times> 'd) list" assumes "\<And>v. v \<in> poly_vars p \<Longrightarrow> poly_weak_mono p v" shows "poly_weak_mono_all p"
encode_operator_precondition_is_cnf: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and op :: "'b strips_operator" shows "is_cnf (encode_operator_precondition \<Pi> k op)"
connected_refl: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat" shows "Graph.connected c v v"
RedWhile: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and b :: "expr"   and c :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "redp P E (while (b) c) s (if (b) (c;; while (b) c) else unit) s"
Extend_empty_satisfies_bounded: fixes \<AA> :: "'i"   and P :: "'v"   and k :: "'k"   and \<phi> :: "('a, 'k) aformula" assumes "Length \<AA> = 0"   and "len P = 0" shows "(Extend k 0 \<AA> P \<Turnstile>\<^sub>b \<phi>) = (\<AA> \<Turnstile>\<^sub>b \<phi>)"
inj_monoid_add_hom_axioms: shows "inj_monoid_add_hom (inv (inv hom))"
wt_inter: fixes \<Gamma> :: "(nat \<times> ty) list"   and x :: "nat"   and e :: "exp"   and A :: "funty"   and B :: "funty" assumes "\<Gamma> \<turnstile> ELam x e : TFun A"   and "\<Gamma> \<turnstile> ELam x e : TFun B" shows "\<Gamma> \<turnstile> ELam x e : TFun (A \<sqinter> B)"
interaction_bound_bind: fixes "consider" :: "'d \<Rightarrow> bool"   and p :: "('g, 'd, 'f) gpv"   and f :: "'g \<Rightarrow> ('e, 'd, 'f) gpv" shows "interaction_bound consider (p \<bind> f) \<le> interaction_bound consider p + (\<Squnion>x\<in>results'_gpv p. interaction_bound consider (f x))"
that: fixes lc :: "rat"   and q :: "int poly" assumes "p = Polynomial.smult lc (of_int_poly q)"   and "0 < lc"   and "content q = 1" shows "thesis"
is_acc_def: fixes ma :: "'a igb_graph_rec \<Rightarrow> 'd"   and m :: "'c"   and r :: "nat \<Rightarrow> 'a \<times> nat" shows "gb_graph.is_acc (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<equiv> \<forall>A\<in>gbg_F (b_graph.to_gbg_ext (degeneralize_ext ma) m). \<exists>\<^sub>\<infinity>i. r i \<in> A"
subst_term_stepwise: fixes insts :: "((variable \<times> typ) \<times> Core.term) list"   and t :: "Core.term" assumes "distinct (map fst insts)"   and "\<And>x. x \<in> \<Union> (fv ` snd ` set insts) \<Longrightarrow> x \<notin> fst ` set insts" shows "subst_term insts t = fold (\<lambda>single. subst_term [single]) insts t"
lm142: fixes X :: "'b set" assumes "trivial X" shows "card (Pow X) \<in> {1, 2}"
IfFalse: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and ps' :: "char list \<Rightarrow> int option"   and c2 :: "com"   and x :: "nat"   and y :: "nat"   and c1 :: "com" assumes "vars b \<subseteq> dom ps"   and "dom ps' = dom ps"   and "\<not> pbval b ps"   and "(c2, ps) \<Rightarrow>\<^sub>A x \<Down> ps'"   and "y = x + 1" shows "(IF b THEN c1 ELSE c2, ps) \<Rightarrow>\<^sub>A y \<Down> ps'"
im_sheaf_def: fixes V :: "'a set" shows "idim.im_sheaf V \<equiv> \<FF> (identity S \<^sup>\<inverse> S V)"
S: shows "m \<bind> f = \<Union> (f ` m)"
elims: fixes x :: "('e, 'f, 'd) interp"   and xa :: "('e, 'd) ODE"   and y :: "('d + 'd) set" assumes "semBV x xa = y" obtains I :: "('e, 'f, 'd) interp" and ODE :: "('e, 'd) ODE" where "x = I" and "xa = ODE" and "y = Inl ` ODE_vars I ODE \<union> Inr ` ODE_vars I ODE"
euclidean_size_gauss_int_def: shows "euclidean_size = gauss_int_norm"
ind_add_str_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and U :: "'d set"   and V :: "'d set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "ind_sheaf.ind_add_str add_str U V \<equiv> add_str (U \<inter> V)"
abelian_groupE_6: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_group R"   and "x \<in> carrier R" shows "\<exists>y\<in>carrier R. y \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
PDE: shows "P \<turnstile> D \<prec>\<^sup>1 E"
constrainsI: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "\<And>act s s'. \<lbrakk>act \<in> Acts F; (s, s') \<in> act; s \<in> A\<rbrakk> \<Longrightarrow> s' \<in> A'" shows "F \<in> A co A'"
wtC_def: shows "local.wtC \<equiv> list_all local.wtL"
finite_vote_set: fixes r :: "nat"   and v_hist :: "nat \<Rightarrow> 'a \<Rightarrow> 'b option"   and S :: "'a set" assumes "\<forall>r'\<ge>r. v_hist r' = Map.empty"   and "finite S" shows "finite (vote_set v_hist S)"
index_zero_vec_1: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "0\<^sub>v n $ i = (0::'c)"
subgroup_generated_by_image: fixes S :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set set" assumes "S \<subseteq> carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))" shows "carrier (subgroup_generated (relative_homology_group (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0}) (gn ` S)) = gn ` carrier (subgroup_generated (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) S)"
rtrancl3p_converse_induct: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and bs :: "'b list"   and a'' :: "'a"   and P :: "'a \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> bool" assumes "rtrancl3p r a bs a''"   and "\<And>a. P a [] a"   and "\<And>a b a' bs a''. \<lbrakk>rtrancl3p r a' bs a''; r a b a'; P a' bs a''\<rbrakk> \<Longrightarrow> P a (b # bs) a''" shows "P a bs a''"
a: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) ((F\<^sup>\<flat>) (x::'a::refinement_lattice)) \<sqsubseteq> F (y::'a::refinement_lattice)"
step_5: assumes "s \<subseteq> r"   and "F \<subseteq> G" shows "(y, b) \<in> rtrancl_on G r"
cons: shows "\<forall>v\<in>V - {s, t}. sum f (incoming v) = sum f (outgoing v)"
f_tendsto_1: shows "(f \<longlongrightarrow> f x) (at_left x)"
eval_until_length: fixes I :: "\<I>"   and nt :: "nat"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and res :: "event_data option list set list"   and auxlist' :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "eval_until I nt auxlist = (res, auxlist')" shows "length auxlist = length res + length auxlist'"
leading_coeff_0_iff: fixes p :: "'a Polynomial.poly" shows "(lead_coeff p = (0::'a)) = (p = 0)"
removeAll_filter_not_eq: fixes x :: "'c" shows "removeAll x = filter ((\<noteq>) x)"
A: shows "finite {y. [(x, y)] \<bullet> M \<noteq> M}"
graph_map_dom: fixes R :: "('c \<times> 'd) set" shows "dom (graph_map R) = fst ` R"
infty_plus_unbox: fixes a :: "'a"   and b :: "'a" shows "Num a + Num b = Num (a + b)"
simps_5: fixes \<phi> :: "'a ltln" shows "rewrite_syn_imp (X\<^sub>n \<phi>) = mk_next (rewrite_syn_imp \<phi>)"
f1: shows "x' \<in> (r |\<^bsub>rel\<^esub> s)"
assms_3: shows "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
assms_3: shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> Vset \<alpha>"
lhs_step_RES: fixes X :: "'a set"   and m :: "'a nres" assumes "\<And>x. x \<in> X \<Longrightarrow> RETURN x \<le> m" shows "RES X \<le> m"
hom_add: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e"   and y :: "'e" assumes "abelian_group_hom G H h"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> h y"
ideal_generated_dvd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and d :: "'b"   and d' :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times {a, b} = ring.ideal_generated plus zero uminus times {d}"   and "dvd.dvd times d' a"   and "dvd.dvd times d' b" shows "dvd.dvd times d' d"
generate_sincl: fixes A :: "'a set" shows "A \<subseteq> generate (G\<lparr>carrier := generate G {a}\<rparr>) A"
fsmult_smult_comm: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and r :: "('b, 'a) aezfun"   and v :: "'c"   and a :: "'b" assumes "FGModule G smult V"   and "r \<in> aezfun_setspan G"   and "v \<in> V" shows "aezfun_scalar_mult.fsmult smult a (smult r v) = smult r (aezfun_scalar_mult.fsmult smult a v)"
wp_loop_nguard: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "healthy (wp body)"   and "sound P"   and "\<not> G s" shows "wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P s = P s"
degree_polysub_samehead: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "head p = head q"   and "Reflected_Multivariate_Polynomial.degree p = Reflected_Multivariate_Polynomial.degree q" shows "Reflected_Multivariate_Polynomial.degree (p -\<^sub>p q) < Reflected_Multivariate_Polynomial.degree p \<or> p -\<^sub>p q = 0\<^sub>p"
suminf_split_initial_segment: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "summable f" shows "suminf f = (\<Sum>n. f (n + k)) + sum f {..<k}"
dep: shows "\<forall>a\<in>set as. depends\<^sub>R a"
legendre_identity: fixes x :: "real" shows "sum_upto (\<lambda>x. ln (real x)) x = prime_sum_upto (\<lambda>p. real (legendre_aux x p) * ln (real p)) x"
ccBind_eq: fixes v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccBind v e\<cdot>(ae, G) = (if v--v\<notin>G \<or> \<not> isVal e then \<G>\<^sup>\<bottom>\<^bsub>ae v\<^esub> e G|` fv e else fv e\<^sup>2)"
differentiable_on_openD: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and x :: "'c" assumes "f differentiable_on X"   and "open X"   and "x \<in> X" shows "f differentiable at x"