hcju/isabelleps huggingface dataset:
BT_lemma_2_21: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat"   and j :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2"   and "3 \<le> i"   and "i \<le> k"   and "j < n i" shows "Polynomial.smult (subresultant_prs_locale.\<alpha> \<delta> f i ^ (n (i - 1) - j)) (subresultant j (F (i - 2)) (F (i - 1))) = Polynomial.smult ((- 1) ^ ((n (i - 2) - j) * (n (i - 1) - j)) * f (i - 1) ^ (\<delta> (i - 2) + \<delta> (i - 1)) * \<beta> i ^ (n (i - 1) - j)) (subresultant j (F (i - 1)) (F i))"
arsinh_minus_real: fixes x :: "real" shows "arsinh (- x) = - arsinh x"
m_def: shows "m \<equiv> \<lambda>N x. Max {m. m \<le> N \<and> x \<in> \<Union> (B m ` {..N})}"
add_assoc: fixes a :: "int"   and b :: "int"   and p\<^sub>1 :: "int point"   and p\<^sub>2 :: "int point"   and p\<^sub>3 :: "int point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3" shows "local.add a p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3) = local.add a (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
prems_1: fixes g :: "real \<Rightarrow> complex" assumes "valid_path g"   and "pathstart g = a"   and "pathfinish g = b"   and "path_image g \<subseteq> s - insert p pts"   and "f contour_integrable_on g" shows "thesis"
summable_Dirichlet_L': fixes s :: "complex" assumes "\<chi>\<^sub>0 \<noteq> \<chi>\<^sub>0"   and "0 < Re s" shows "summable (\<lambda>n. \<chi>\<^sub>0 n * of_nat n powr - s)"
s: shows "punit.lt (rep_list p) = s + punit.lt (rep_list g)"
cmp_extend_def: fixes D :: "'b set"   and cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res" shows "cmp_extend D cmp \<equiv> cmp_combine D cmp UNIV univ_cmp"
p_preserves_inv_imps_work_sum: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t" assumes "next_propagate' c0 c1 loc t"   and "inv_imps_work_sum c0" shows "inv_imps_work_sum c1"
intros: 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 C :: "char list"   and Cs :: "char list list" assumes "Subobjs P C Cs" shows "init_obj P C (Cs, init_class_fieldmap P (last Cs))"
update_delete: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "update k v (delete k m) = update k v m"
beta_cfun: fixes f :: "'c \<Rightarrow> 'd"   and u :: "'c" assumes "cont f" shows "(\<Lambda> x. f x)\<cdot>u = f u"
join_meet_distrib2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "y \<sqinter> z \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
carrier_stalk_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 I :: "'d set set"   and x :: "'d" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.carrier_stalk is_open \<FF> \<rho> x \<equiv> dlim \<FF> \<rho> (topological_space.neighborhoods is_open x)"
map_id: fixes t :: "'a option" shows "map_option id t = t"
distinct_set_remove_mid: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "distinct (xs @ [x] @ ys)" shows "set (xs @ ys) = set (xs @ [x] @ ys) - {x}"
strict2: fixes x :: "'a state \<times> 'a state \<Rightarrow> Set('b)" shows "x->including\<^sub>S\<^sub>e\<^sub>t(invalid) = invalid"
subcls_induct: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and Q :: "String.literal \<Rightarrow> bool"   and C :: "String.literal" assumes "wf_prog wf_md P"   and "\<And>C. \<forall>D. (subcls1 P)\<^sup>+\<^sup>+ C D \<longrightarrow> Q D \<Longrightarrow> Q C" shows "Q C"
genby_closed: fixes S :: "'g set" assumes "order.greater_eq G S" shows "order.greater_eq G \<langle>S\<rangle>"
Id_on_Gr: fixes A :: "'e set" shows "Id_on A = BNF_Def.Gr A id"
replicating_portfolio_def: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and der :: "'a \<Rightarrow> real"   and matur :: "nat" shows "replicating_portfolio pf der matur = (stock_portfolio Mkt pf \<and> trading_strategy pf \<and> self_financing Mkt pf \<and> AEeq M (cls_val_process Mkt pf matur) der)"
inv_closed: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C" shows "inv\<^bsub>C\<^esub> x \<in> carrier C"
bind_eq_Some_conv: fixes f :: "'c option"   and g :: "'c \<Rightarrow> 'b option"   and x :: "'b" shows "(Option.bind f g = \<lfloor>x\<rfloor>) = (\<exists>y. f = \<lfloor>y\<rfloor> \<and> g y = \<lfloor>x\<rfloor>)"
XN: shows "complex_of_real (cmod (inner_prod v (X N *\<^sub>v v) - inner_prod v (A *\<^sub>v v))) < - inner_prod v (A *\<^sub>v v)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g' f \<phi> \<psi>'"
gcd_mult_add: fixes n :: "nat"   and k :: "nat"   and m :: "nat" assumes "0 < n" shows "gcd (n * k + m) n = gcd m n"
eventuallyI: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "P (\<sigma> |\<^sub>s i)" shows "(\<diamond>P) \<sigma>"
del_def: shows "del \<equiv> \<lambda>k i j. enum (DF k i) j - enum (DF k i) (j - Suc 0)"
simps_3: fixes uu :: "const" shows "uu \<squnion> Any = Any"
product_topology_basis': fixes I :: "'a set"   and U :: "'a \<Rightarrow> 'b set"   and x :: "'a \<Rightarrow> 'c" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> open (U i)" shows "open {f. \<forall>i\<in>I. f (x i) \<in> U i}"
tauStepTransitionChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<noteq> P'" shows "P \<Longrightarrow>\<tau> \<prec> P' = P \<Longrightarrow>\<^sub>\<tau> P'"
rel_setD2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set"   and y :: "'d" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
invar_step: fixes F :: "'a set set"   and C :: "'a set" assumes "F \<noteq> {}"   and "invar C F" shows "invar (C \<union> some F) (F - {e' \<in> F. some F \<inter> e' \<noteq> {}})"
Cons: fixes i :: "nat"   and j :: "nat" shows "(i < length v * row_length M \<and> j < length M) \<and> mat (row_length M) (length M) M \<longrightarrow> vec_mat_Tensor v M ! j ! i = v ! (i div row_length M) * M ! j ! (i mod row_length M)"
sat_MatchP_point: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex" assumes "Formula.sat \<sigma> V v i (formula.MatchP I r)" obtains j :: "nat" where "j \<le> i" and "mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I" and "Formula.sat \<sigma> V v i (formula.MatchP (point (\<tau> \<sigma> i - \<tau> \<sigma> j)) r)"
wf_cdecl_fdecl: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and f :: "fdecl" assumes "wf_cdecl G (C, c)"   and "f \<in> set (cfields c)" shows "wf_fdecl G (pid C) f"
inner_eq_zero_iff: fixes x :: "'a" shows "(inner x x = 0) = (x = (0::'a))"
inv_on_f_range: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "y \<in> f ` A" shows "inv_on f A y \<in> A"
grounding_of_cls_def: fixes C :: "('a, 'b) Term.term literal multiset" shows "grounding_of_cls C = {subst_cls C \<sigma> |\<sigma>. is_ground_subst \<sigma>}"
Fix_def: fixes f :: "'b \<Rightarrow> 'b" shows "Fix f = {x. f x = x}"
mk_eqcl_fixpt_fst_snd_nth: fixes i :: "nat"   and M :: "nat bdd list \<times> bool list"   and v :: "nat" assumes "i < length (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))))"   and "wf_dfa M v" shows "snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! i < length (fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M)))) \<and> fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! i) = i"
iter_hcomp_def: fixes X :: "'b fset"   and f :: "'b \<Rightarrow> interface" shows "iter_hcomp X f \<equiv> ffold ((\<otimes>) \<circ> f) \<epsilon> X"
simps: fixes s :: "'s" shows "local.unstream s = (case generator g s of Done \<Rightarrow> [] | Skip x \<Rightarrow> local.unstream x | Yield x s' \<Rightarrow> x # local.unstream s')"
flush_data_cache_privilege: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "get_S (cpu_reg_val PSR s) = 0"   and "s' = flush_data_cache s" shows "get_S (cpu_reg_val PSR s') = 0"
underS_incr2: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set"   and a :: "'c" assumes "r \<subseteq> r'" shows "underS r a \<subseteq> underS r' a"
mirror_ex: fixes p :: "fm"   and bbs :: "bool list"   and bs :: "int list" assumes "iszlfm p" shows "(\<exists>x. Ifm bbs (x # bs) (mirror p)) = (\<exists>x. Ifm bbs (x # bs) p)"
Ex2_cofinitary: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> Ex2"   and "f \<noteq> id" shows "Fix f = {}"
ennreal_minus_if: fixes a :: "real"   and b :: "real" shows "ennreal a - ennreal b = ennreal (if 0 \<le> b then if b \<le> a then a - b else 0 else a)"
hyps_7: assumes "list_all is_NegChecks (unlabel A)" shows "list_all is_NegChecks (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
ax_valids_def: fixes G :: "prog"   and A :: "'b triple set"   and ts :: "'c triple set" shows "G,A|\<Turnstile>ts = (\<forall>n. G|\<Turnstile>n:A \<longrightarrow> G|\<Turnstile>n:ts)"
Right_Coset_Class_unit: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and E :: "('c \<times> 'c) set"   and g :: "'c" assumes "group_congruence G composition unit E"   and "g \<in> G" shows "coset_notation.Right_Coset composition (group_congruence.Normal G unit E) g = equivalence.Class G E g"
inverse_positive_imp_positive: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero (inverse a)"   and "a \<noteq> zero" shows "less zero a"
vsv_vimageI1: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "a \<in>\<^sub>\<circ> A" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr> `\<^sub>\<circ> A"
m_o_def: fixes opt :: "'a st option"   and X :: "char list set" shows "m\<^sub>o opt X = (case opt of None \<Rightarrow> h * card X + 1 | Some S \<Rightarrow> m\<^sub>s S X)"
fresh_10: fixes a :: "name"   and x1 :: "pi" shows "a \<sharp> !x1 = a \<sharp> x1"
Arg_lt_pi: fixes z :: "complex" shows "(0 < Arg z \<and> Arg z < pi) = (0 < Im z)"
tllist_all2_tmap1: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'k \<Rightarrow> 'g"   and g :: "'l \<Rightarrow> 'i"   and xs :: "('k, 'l) tllist"   and ys :: "('h, 'j) tllist" shows "tllist_all2 P Q (tmap f g xs) ys = tllist_all2 (\<lambda>x. P (f x)) (\<lambda>x. Q (g x)) xs ys"
right_gpv_inline: fixes callee :: "'b \<Rightarrow> 'g \<Rightarrow> ('h \<times> 'b, 'd, 'f) gpv"   and gpv :: "('a, 'g, 'h) gpv"   and s :: "'b" shows "right_gpv (inline callee gpv s) = inline (\<lambda>s q. right_gpv (callee s q)) gpv s"
comm_semiring_hom_axioms: shows "comm_semiring_hom of_rat"
write_bytes_def: shows "write_bytes \<equiv> map_fun Rep_mem (map_fun id (map_fun Rep_bytes Abs_mem)) (\<lambda>m n bs. take n m @ bs @ drop (n + length bs) m)"
Suc: shows "order a ([:- a, 1::'a:] ^ n) = n"
Sup_upper2: fixes u :: "'b"   and A :: "'b set"   and v :: "'b" assumes "u \<in> A"   and "u \<le> v" shows "\<Sqinter> A \<le> v"
from_nat_0: shows "mod_type_class.from_nat 0 = (0::'a)"
6: shows "w\<^sup>+\<^sup>T \<sqinter> prim_E w v e\<^sup>T * top \<sqinter> - v\<^sup>T = bot"
binary_functor_transformation_axioms: shows "binary_functor_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>"
subset_eq_diff_conv: fixes A :: "'b multiset"   and C :: "'b multiset"   and B :: "'b multiset" shows "(A - C \<subseteq># B) = (A \<subseteq># B + C)"
assms_2: shows "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
att_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.att prob = ifc_problem.att prob"
cloS: shows "closedin (top_of_set (S \<union> T)) S"
le_Sup_iff_less: fixes x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(x \<le> Sup (f ` A)) = (\<forall>y<x. \<exists>i\<in>A. y \<le> f i)"
fresh_4: fixes a :: "tyvrs"   and x2 :: "tyvrs"   and x1 :: "ty" shows "a \<sharp> TVarB x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
intro: assumes "\<And>x y. PP x y = (P x y \<and> x \<noteq> y)"   and "\<And>y x. PP y x \<Longrightarrow> \<exists>z. P z x \<and> \<not> O z y" shows "MM2_axioms"
chainE: fixes Y :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "chain Y" shows "Y i \<sqsubseteq> Y (Suc i)"
lm056: fixes P :: "('a \<times> 'b) set" shows "fst ` P = snd ` P\<inverse>"
isMaxim_unique: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "isMaxim A x"   and "isMaxim A y" shows "x = y"
subst_cls_list_Cons: fixes C :: "'a literal multiset"   and Cs :: "'a literal multiset list"   and \<sigma> :: "'s" shows "(C # Cs) \<cdot>cl \<sigma> = C \<cdot> \<sigma> # Cs \<cdot>cl \<sigma>"
lexordp_trans: fixes xs :: "'d word list"   and ys :: "'d word list"   and zs :: "'d word list" assumes "signed.lexordp xs ys"   and "signed.lexordp ys zs" shows "signed.lexordp xs zs"
bet_eq: shows "between (vertices f) u v = between (verticesFrom f v') u v"
common_first_summand: fixes x :: "i"   and y :: "i"   and z :: "i" shows "P x (x \<oplus> y) \<and> P x (x \<oplus> z)"
toTree_correct_1: fixes l :: "('a \<times> 'b) list" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.toTree l)"
cSup_eq: fixes X :: "'a set"   and c :: "'a" assumes "\<forall>x\<in>X. x \<le> c"   and "\<exists>x\<in>X. c \<le> x" shows "Sup X = c"
subst_impl: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and P :: "'b upred"   and Q :: "'b upred" shows "\<sigma> \<dagger> (P \<Rightarrow> Q) = (\<sigma> \<dagger> P \<Rightarrow> \<sigma> \<dagger> Q)"
nextl_init_state: fixes xs :: "'a list" shows "nextl (init M) xs \<in> states M"
mkinterp_eq: fixes I :: "interp"   and J :: "interp" shows "(Consts I = Consts J \<and> Funcs I = Funcs J \<and> Preds I = Preds J \<and> Games I = Games J) = (I = J)"
thetaSD_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis" shows "PL_Indis.thetaSD aval tval cval indis \<equiv> {(c, d) |c d. proper c \<and> proper d \<and> PL_Indis.discr aval tval cval indis c \<and> PL_Indis.discr aval tval cval indis d}"
locally_ringed_space_axioms_def: fixes 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 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" shows "locally_ringed_space_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<equiv> \<forall>x U. x \<in> U \<longrightarrow> is_open U \<longrightarrow> stalk.is_local is_open \<FF> \<rho> add_str mult_str zero_str one_str (topological_space.neighborhoods is_open x) x U"
S5: shows "nat_to_pr (sch_to_nat t) = sch_to_pr t"
Gid: fixes i :: "nat" assumes "i < length F" shows "dim_vec (F ! i) = n"
semiattractive_axioms: shows "semiattractive A (\<sim>)"
ix_def: fixes p :: "(nat \<times> int) list"   and d :: "nat" shows "ix p d = snd (p ! d)"
antisym_conv1: fixes x :: "'b word"   and y :: "'b word" assumes "\<not> x <s y" shows "(x \<le>s y) = (x = y)"
hom_vec_of_poly_n: fixes p :: "'a poly"   and n :: "nat" shows "vec\<^sub>h (vec_of_poly_n p n) = vec_of_poly_n (map_poly hom p) n"
diagonal_step_PQ_PAQ': fixes P :: "(('b, 'c) vec, 'c) vec"   and Q :: "(('b, 'd) vec, 'd) vec"   and A :: "(('b, 'd) vec, 'c) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and p :: "'b"   and q :: "'b"   and u :: "'b"   and v :: "'b"   and d :: "'b" assumes "(P, Q) = diagonal_step_PQ A i k bezout"   and "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)" shows "P ** A ** Q = mult_column (column_add (column_add (P ** A) (from_nat i) (from_nat k) q) (from_nat k) (from_nat i) u) (from_nat k) (- (1::'b))"
pab: shows "p = rat_of_int a / rat_of_int b"
continuous_within_eps_delta: fixes x :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous (at x within s) f = (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
\<tau>1'_m: shows "(r|labels \<gamma>1''|, {#fst \<beta>_step#}) \<in> mul r"
map_add_eq_empty_iff: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" shows "(f ++ g = Map.empty) = (f = Map.empty \<and> g = Map.empty)"
hn_refine_split_post: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and a :: "'b nres"   and \<Gamma>'' :: "assn" assumes "hn_refine \<Gamma> c \<Gamma>' R a" shows "hn_refine \<Gamma> c (\<Gamma>' \<or>\<^sub>A \<Gamma>'') R a"
properCh: fixes ch :: "'choice"   and s :: "'state" shows "0 \<le> cval ch s \<and> cval ch s \<le> 1"
w_1: shows "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"
Leaf_in_set_random_bst_iff: fixes A :: "'a set" assumes "finite A" shows "(\<langle>\<rangle> \<in> set_pmf (random_bst A)) = (A = {})"
deconstruct_rules_6: fixes x :: "'b \<times> 'c"   and P :: "assn"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd Heap"   and Q :: "'d \<Rightarrow> assn" assumes "\<And>a b. x = (a, b) \<Longrightarrow> <P> f a b <Q>" shows "<P> case x of (a, b) \<Rightarrow> f a b <Q>"
assms_2: shows "fst (cs (S t i) cid) \<noteq> []"
intro: fixes invar :: "'c \<Rightarrow> bool"   and reverse_iterateoi :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "\<And>m. invar m \<Longrightarrow> map_iterator_rev_linord (reverse_iterateoi m) (\<alpha> m)" shows "map_reverse_iterateoi_axioms \<alpha> invar reverse_iterateoi"
lin_indpt_list_length_le_n: fixes n :: "nat"   and xs :: "'b vec list" assumes "cof_vec_space.lin_indpt_list n xs" shows "length xs \<le> n"
config_rand: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init \<and> distinct (fst x) = distinct init \<and> length (fst x) = length init"
Call: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and bdy :: "('a, 'b, 'c) com"   and s :: "'a" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile>bdy \<down> Normal s" shows "\<Gamma>\<turnstile>Call p \<down> Normal s"
sympI: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
take_bit_push_bit: fixes m :: "nat"   and n :: "nat"   and a :: "'a" shows "take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)"
Gromov_boundary_def: shows "Gromov_boundary = UNIV - range to_Gromov_completion"
valuate_uminus: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "- p \<lbrace> v \<rbrace> = - p \<lbrace> v \<rbrace>"
treewidth_upper_bound_V: shows "treewidth \<le> card V - 1"
subset_imp_LeadsTo: fixes A :: "'c set"   and B :: "'c set"   and F :: "'c program" assumes "A \<subseteq> B" shows "F \<in> A \<longmapsto>w B"
boundedE: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> inf b c" obtains "a \<le> b" and "a \<le> c"
ctyping_preservation: fixes c :: "com"   and s :: "char list \<Rightarrow> val"   and c' :: "com"   and s' :: "char list \<Rightarrow> val"   and \<Gamma> :: "char list \<Rightarrow> ty" assumes "(c, s) \<rightarrow> (c', s')"   and "\<Gamma> \<turnstile> c" shows "\<Gamma> \<turnstile> c'"
trans_def: fixes r :: "('b \<times> 'b) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
relation_of_Schema: fixes sc :: "'d \<times> 'd \<Rightarrow> bool" shows "action.relation_of (Schema sc) = R (\<lambda>(A, A'). Pre sc (alpha_rp.more A) \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A')"
IH_2: assumes "poly q x\<^sub>0 \<noteq> 0"   and "poly (hd (q # r # ps)) x\<^sub>0 \<noteq> 0" shows "sign_changes' (q # r # ps) x\<^sub>0 = sign_changes (q # r # ps) x\<^sub>0"
fempty_iff: fixes c :: "'d" shows "(c |\<in>| {||}) = False"
l12: shows "\<not> (e\<^sup>2 + d\<^sup>2 = 0 \<and> 2 * e * b * d - 2 * a * e\<^sup>2 - 2 * d * f = 0 \<and> a\<^sup>2 * e\<^sup>2 + f\<^sup>2 - 2 * e * b * f + b\<^sup>2 * e\<^sup>2 - e\<^sup>2 * c = 0)"
minus_unique: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus"   and "plus a b = zero" shows "uminus a = b"
nth_equalityI: fixes xs :: "'f list"   and ys :: "'f list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
lpp_q': fixes k :: "nat" shows "lpp (q' k) = monomial k y + lpp q"
col: fixes ja :: "nat" assumes "ja \<le> n" shows "signof (permutation_insert n (n - ja) p) = (- (1::'a)) ^ (n + (n - ja)) * signof p"
wt_t_ge_wt_s1s2: shows "wt t \<ge>\<^sub>p wt (App s1 s2)"
o_def: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'd" shows "f \<circ> g = (\<lambda>x. f (g x))"
phis_in_\<alpha>n: fixes g :: "'g"   and n :: "'node"   and v :: "'a"   and vs :: "'a list" assumes "phis' g (n, v) = Some vs" shows "n \<in> set (\<alpha>n g)"
cvector_fraction_eq_iff: fixes u :: "complex"   and v :: "complex"   and a :: "'b"   and x :: "'b" shows "((u / v) *\<^sub>C a = x) = (if v = 0 then x = (0::'b) else u *\<^sub>C a = v *\<^sub>C x)"
assms_2: shows "local_ring_morphism g B addB multB zeroB oneB C addC multC zeroC oneC"
reachable_mems_def: fixes cms :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and mem :: "'Var\<^sub>A \<Rightarrow> 'Val" shows "abs.reachable_mems cms mem \<equiv> {mem'. \<exists>sched cms'. abs.meval_sched_abv (cms, mem) sched (cms', mem')}"
unique_sig_Ext_foo: fixes sig :: "sig" assumes "tprg\<turnstile> mdecl (sig, snd Ext_foo) declared_in Ext" shows "sig = foo_sig"