hcju/isabelleps huggingface dataset:
mulex_iff_mult: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" shows "mulex r M N = ((M, N) \<in> mult {(M, N). r M N})"
hd_butlast: fixes xs :: "'a list" assumes "1 < length xs" shows "hd (butlast xs) = hd xs"
complex_split_polar: fixes z :: "complex" shows "\<exists>r a. z = complex_of_real r * (complex_of_real (cos a) + \<i> * complex_of_real (sin a))"
simps_6: fixes x :: "'a rexp" shows "lang (Star x) = lang x\<^sup>\<star>"
prefixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le> ys" obtains zs :: "'b list" where "ys = xs @ zs"
set_image_mset: fixes f :: "'l \<Rightarrow> 'k"   and M :: "'l multiset" shows "set_mset (image_mset f M) = f ` set_mset M"
ty: shows "[] \<turnstile> v : (\<forall>X<:T\<^sub>1. T\<^sub>2)"
event_bound: fixes t :: "real"   and p :: "nat"   and q :: "nat" assumes "0 \<le> t"   and "correct p t"   and "correct q t" shows "\<exists>i. t < max (te p i) (te q i)"
ll_on_open_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" shows "ll_on_open T f X \<equiv> interval T \<and> ll_on_open_axioms T f X"
times_divide_eq_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "c \<noteq> \<zero>" shows "b \<oslash> c \<otimes> a = b \<otimes> a \<oslash> c"
cont_fun: fixes x :: "'a" shows "cont (\<lambda>f. f x)"
tendsto_min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "'b \<Rightarrow> 'a"   and x :: "'a"   and net :: "'b filter"   and Y :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "class.linorder_topology less_eq less open"   and "filterlim X (topological_space.nhds open x) net"   and "filterlim Y (topological_space.nhds open y) net" shows "LIM x net. ord.min less_eq (X x) (Y x) :> topological_space.nhds open (ord.min less_eq x y)"
Re_divide_of_real: fixes z :: "complex"   and r :: "real" shows "Re (z / complex_of_real r) = Re z / r"
is_prefix_cons: fixes x :: "'a"   and a :: "'a list"   and b :: "'a list" shows "is_prefix (x # a) b = (\<exists>c. b = x # c \<and> is_prefix a c)"
partial_ordering_axioms: shows "partial_ordering UNIV (\<le>) (<)"
mat_size_alt: shows "M_mat signs subsets \<in> carrier_mat (length subsets) (length subsets)"
no_return_to_zero: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and t :: "(String.literal \<times> value list) list" assumes "\<forall>((from, to), t)|\<in>|e. 0 < to" shows "\<forall>r n. \<not> visits 0 e (Suc n) r t"
left_neutral: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and a :: "'e" assumes "monoid f z" shows "f z a = a"
simps_3: fixes c :: "'b"   and first :: "indices"   and "next" :: "indices list"   and S :: "'b trm" shows "subterm (Const c) (first # next) S = False"
infinite_subset: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "A \<subseteq> B" shows "infinite B"
equals_minim_Under: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> B"   and "a \<in> local.Under B" shows "a = minim B"
lub_upper: fixes A :: "'f option set"   and x :: "'f option" assumes "Complete_Partial_Order.chain option_ord A"   and "x \<in> A" shows "option_ord x (flat_lub None A)"
eta_cfun: fixes f :: "'c \<rightarrow> 'd" shows "(\<Lambda> x. f\<cdot>x) = f"
cSup_subset_mono: fixes A :: "'c multiset set"   and B :: "'c multiset set" assumes "A \<noteq> {}"   and "subset_mset.bdd_above B"   and "A \<subseteq> B" shows "Sup A \<subseteq># Sup B"
single_add: fixes k :: "'d"   and a :: "'e"   and b :: "'e" shows "monomial (a + b) k = monomial a k + monomial b k"
sifum_lang_axioms: shows "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"
preserves_section_retraction: fixes e :: "'d"   and m :: "'d" assumes "D.ide (e \<cdot>\<^sub>D m)" shows "C.ide (G e \<cdot>\<^sub>C G m)"
count_a: assumes "count P a \<le> count M a" shows "\<exists>z>a. count M z < count P z"
transpose_mat_of_cols: fixes n :: "nat"   and vs :: "'a vec list" shows "mat_of_cols n vs\<^sup>t = mat_of_rows n vs"
hyps: fixes y :: "nat"   and u :: "nat" assumes "y < k"   and "y \<le> length Doc" shows "items_le y (\<J> y u) = Gen (paths_le y (\<P> y u)) \<and> \<T> y u = \<Z> y u \<and> items_le y (\<I> y) = Gen (paths_le y (\<Q> y))"
mat_eqI: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list" assumes "mat nr nc m1"   and "mat nr nc m2"   and "\<And>i j. \<lbrakk>i < nc; j < nr\<rbrakk> \<Longrightarrow> m1 ! i ! j = m2 ! i ! j" shows "m1 = m2"
finite_pred_imp_se_updatable: fixes c :: "('c, 'd) conf" assumes "finite (Bexp.vars (conjunct (pred c)))" shows "updatable c"
get_ancestors_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ptr |\<in>| object_ptr_kinds h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> ok Shadow_DOM.get_ancestors ptr"
vcard_vsingleton: fixes a :: "V" shows "vcard (set {a}) = 1\<^sub>\<nat>"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree" assumes "\<And>l a uu_ r. (l \<noteq> \<langle>\<rangle> \<Longrightarrow> P l) \<Longrightarrow> P \<langle>l, (a, uu_), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
unsorted_term_poly_list_rel_def: shows "unsorted_term_poly_list_rel = {(xs, ys). ys = mset xs \<and> distinct xs}"
AbruptProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> \<Rightarrow> Abrupt s"
collapse_2: fixes tllist :: "('a, 'b) tllist" assumes "\<not> is_TNil tllist" shows "TCons (thd tllist) (ttl tllist) = tllist"
one_zhmultiset_def: shows "1 \<equiv> \<omega>\<^sub>z^ 0"
CVdI: fixes u :: "nat" assumes "u \<in> C" shows "u \<in> Vd d"
has_integral_mult_right: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and i :: "'c set"   and c :: "'d" assumes "(f has_integral y) i" shows "((\<lambda>x. c * f x) has_integral c * y) i"
P_def: shows "P = (\<lambda>x. \<exists>p\<in>set (pders p). poly p x = 0)"
exhaustive_correct: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "linorder.sorted less_eq xs"   and "distinct xs" shows "proper_intrvl.exhaustive proper_interval xs = (set xs = UNIV)"
eval_subst_fm: fixes e :: "(name, hf) finfun"   and fm :: "fm"   and i :: "name"   and t :: "tm" shows "eval_fm e (fm(i::=t)) = eval_fm (finfun_update e i \<lbrakk>t\<rbrakk>e) fm"
quotient_map_restriction: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and V :: "'d set"   and U :: "'c set" assumes "quotient_map X Y f"   and "{x \<in> topspace X. f x \<in> V} = U"   and "openin Y V \<or> closedin Y V" shows "quotient_map (subtopology X U) (subtopology Y V) f"
path2_def: fixes g :: "'a"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" shows "gp2.path2 g n ns m \<equiv> gp2.path g ns \<and> n = hd ns \<and> m = last ns"
Fvars_pls: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "FvarsT (pls t1 t2) = FvarsT t1 \<union> FvarsT t2"
assms_3: shows "\<beta> \<in> elts (ordertype M r)"
prob_space_axioms_def: fixes M :: "'c measure" shows "prob_space_axioms M \<equiv> emeasure M (space M) = 1"
space_def: fixes M :: "'b measure" shows "space M = fst (Rep_measure M)"
inv_is_inverse: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.inverse_arrows f (CC.inv f)"
Setcompr_eq_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "{f x |x. x \<in> A} = f ` A"
simps_3: fixes d :: "int"   and i :: "int"   and ks :: "int list" shows "hd_coeff (NDvd d i ks) = (case ks of [] \<Rightarrow> 0 | k # x \<Rightarrow> k)"
assms: shows "bij_betw f {..<length as} {..<length as}"
elims: fixes x :: "ereal"   and y :: "ereal" assumes "sgn x = y" obtains r :: "real" where "x = ereal r" and "y = ereal (sgn r)"   | "x = \<infinity>" and "y = 1"   | "x = - \<infinity>" and "y = - 1"
test_bit_of_bl: fixes bl :: "bool list"   and n :: "nat" shows "of_bl bl !! n = (rev bl ! n \<and> n < LENGTH('b) \<and> n < length bl)"
vec_of_inj: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and sp' :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "spp_inv sp"   and "vec_of sp = vec_of sp'" shows "sp = sp'"
s2r_compl_hom: fixes A :: "'b set" shows "s2r (- A) = rel_n (s2r A)"
max_fun_diff_max: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "f \<noteq> g"   and "f \<in> rt.FINFUNC"   and "g \<in> rt.FINFUNC"   and "x \<in> {a \<in> Field t. f a \<noteq> g a}" shows "(x, t.max_fun_diff f g) \<in> t"
hd_concat: fixes xs :: "'a list list" assumes "xs \<noteq> []"   and "hd xs \<noteq> []" shows "hd (concat xs) = hd (hd xs)"
has_derivative_componentwise_within: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and a :: "'c"   and S :: "'c set" shows "(f has_derivative f') (at a within S) = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. f' x \<bullet> i)) (at a within S))"
suffix_appendI: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<le>s ys" shows "xs \<le>s zs \<cdot> ys"
self_finance_updated_suc_0: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market"   and asset :: "'b"   and v :: "real"   and w :: "'c" assumes "portfolio pf"   and "\<forall>n w. prices Mkt asset n w \<noteq> 0" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc 0) w = cls_val_process Mkt pf (Suc 0) w + prices Mkt asset (Suc 0) w / prices Mkt asset 0 w * (val_process Mkt (self_finance Mkt v pf asset) 0 w - val_process Mkt pf 0 w)"
abs_divide: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "abs (divide a b) = divide (abs a) (abs b)"
vsv_ex1_app1: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> xs" shows "(b = xs\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs)"
card_UNIV_length_enum: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "CARD('a) = length enum"
eucl_of_list_inner_le: fixes i :: "'a"   and xs :: "real list" assumes "i \<in> Basis"   and "DIM('a) \<le> length xs" shows "eucl_of_list xs \<bullet> i = xs ! index Basis_list i"
subspace_is_vs: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme"   and W :: "'i set" assumes "vectorspace K V"   and "subspace K W V" shows "vectorspace K (V\<lparr>carrier := W\<rparr>)"
psubst_eq_rawpsubst2: fixes \<phi> :: "fm"   and x1 :: "name"   and x2 :: "name"   and t1 :: "tm"   and t2 :: "tm" assumes "\<phi> \<in> UNIV"   and "x1 \<in> UNIV"   and "x2 \<in> UNIV"   and "t1 \<in> UNIV"   and "t2 \<in> UNIV"   and "x1 \<noteq> x2"   and "x2 \<notin> Fvars t1" shows "psubst \<phi> [(t1, x1), (t2, x2)] = rawpsubst \<phi> [(t1, x1), (t2, x2)]"
perm_2: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and x2 :: "form"   and x1 :: "form_list" shows "pi \<bullet> FCons x2 x1 = FCons (pi \<bullet> x2) (pi \<bullet> x1)"
append_Cons: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "append (Cons x xs) ys = Cons x (append xs ys)"
assms: fixes i :: "nat" shows "u !! i = v !! i"
ord_eq_le_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vrat_uminus" shows "-\<^sub>\<rat> a = b"
if_split: fixes P :: "'c \<Rightarrow> bool"   and Q :: "bool"   and x :: "'c"   and y :: "'c" shows "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
even_and_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (and a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
succ_rvar_valuation: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and x :: "nat" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "x \<in> rvars (\<T> s')" shows "\<langle>\<V> s'\<rangle> x = \<langle>\<V> s\<rangle> x \<or> \<langle>\<V> s'\<rangle> x = the (\<B>\<^sub>l s x) \<or> \<langle>\<V> s'\<rangle> x = the (\<B>\<^sub>u s x)"
intro: assumes "var_infinite undefined"   and "var_regular undefined"   and "varSort_lt_var undefined undefined" shows "FixVars TYPE('d) TYPE('e)"
rep_eq: fixes x :: "('c, 'd) fmap"   and xa :: "('c, 'd) fmap" shows "(x \<subseteq>\<^sub>f xa) = (($$) x \<subseteq>\<^sub>m ($$) xa)"
b_pos: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and b :: "real" assumes "akra_bazzi_params k as bs"   and "b \<in> set bs" shows "0 < b"
obj_is_equivalence_map: fixes a :: "'d" assumes "D.obj a" shows "D.equivalence_map a"
eqvtY: shows "eqvt (Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim)"
inter_plus_ideal_eq_carrier_arbitrary: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set"   and j :: "nat" assumes "\<And>i. i \<le> Suc n \<Longrightarrow> ideal (I i) R"   and "\<And>i j. \<lbrakk>i \<le> Suc n; j \<le> Suc n; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R"   and "j \<le> Suc n" shows "\<Inter> (I ` ({..Suc n} - {j})) <+> I j = carrier R"
vcard_vempty: fixes A :: "V" shows "(vcard A = []\<^sub>\<circ>) = (A = []\<^sub>\<circ>)"
commutes_with_unit: shows "\<phi>\<^sub>f V (zero_str\<^sub>Y V) = zero_im_sheaf V"
iso_iff_mono_and_retraction: fixes f :: "'d" shows "D.iso f = (D.mono f \<and> D.retraction f)"
mult_of_simps_4: fixes R :: "('c, 'd) ring_scheme" shows "\<one>\<^bsub>mult_of R\<^esub> = \<one>\<^bsub>R\<^esub>"
False: shows "\<not> (p = 0 \<or> r \<le> 0)"
b_join_c_eq_e: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "b_aux a b c \<squnion> c_aux a b c = e_aux a b c"
c: shows "rawpsubst (cnj \<phi>1 \<phi>2) (zip (map Var us) (map snd txs)) = cnj (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (rawpsubst \<phi>2 (zip (map Var us) (map snd txs)))"
cod_eqI': fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "a \<in> VoV.codomains f" shows "a = VoV.cod f"
Pd_shuffle_to_top_linear: fixes Y :: "'a set"   and ds' :: "'b set"   and d :: "'b" assumes "inj_on Xd Y" shows "Linear_order (TruePref.Pd_shuffle_to_top ds' Y d)"
mat_of_cblinfun_ell2_carrier: fixes a :: "'b ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c ell2" shows "mat_of_cblinfun a \<in> carrier_mat CARD('c) CARD('b)"
wf_inst_formula: fixes P :: "ast_problem"   and Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and f :: "variable \<Rightarrow> object"   and \<phi> :: "PDDL_STRIPS_Semantics.term atom formula" assumes "\<And>x T. ast_domain.is_of_type (domain P) Q x T \<Longrightarrow> ast_domain.is_of_type (domain P) (ast_problem.objT P) (f x) T"   and "ast_domain.wf_fmla (domain P) (ty_term Q (ast_domain.constT (domain P))) \<phi>" shows "ast_domain.wf_fmla (domain P) (ast_problem.objT P) ((map_formula \<circ> map_atom \<circ> subst_term) f \<phi>)"
eq: shows "set (add_pairs gs bs ps hs data) = {(p, q) |p q. (False, p, q) \<in> set ps1 \<or> (p, q) \<in> set ps \<and> \<not> ocrit data hs ps1 p q}"
simps_16: fixes b :: "char list exp"   and e :: "char list exp" shows "fv (while (b) e) = fv b \<union> fv e"
rep_eq: fixes x :: "('a, 'b) RBT.rbt"   and xa :: "('a, 'b) RBT.rbt" shows "impl_of (RBT.union x xa) = rbt_union (impl_of x) (impl_of xa)"
partial_summation_strong: fixes X :: "real set"   and y :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a" assumes "finite X"   and "0 \<le> y"   and "y < x"   and "\<And>z. z \<in> {y..x} - X \<Longrightarrow> (f has_vector_derivative f' z) (at z)"   and "continuous_on {y..x} f" shows "((\<lambda>t. sum_upto a t * f' t) has_integral sum_upto a x * f x - sum_upto a y * f y - (\<Sum>n\<in>real -` {y<..x}. a n * f (real n))) {y..x}"
without_def: fixes t :: "latin_type" shows "without t \<equiv> \<lambda>e. (e (next t), e (next (next t)))"
is_OK_def: fixes h :: "'l"   and p :: "('l, 'm, 'n) prog" shows "h \<turnstile> ok p = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr x \<Rightarrow> True)"
ctxt_eq: fixes C :: "('d, 'e) ctxt"   and s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" shows "(C\<langle>s\<rangle> = C\<langle>t\<rangle>) = (s = t)"
mtf2_forward_beforeq: fixes q :: "'c"   and xs :: "'c list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "i < index xs q" shows "index (mtf2 n q xs) (xs ! i) \<le> index xs q"
in_derived_set_of: fixes x :: "'b"   and X :: "'b topology"   and S :: "'b set" shows "(x \<in> X derived_set_of S) = (x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T)))"
len\<^sub>C: shows "length cms\<^sub>A = length cms"
arc_implies_awalk: fixes e :: "'b" assumes "e \<in> arcs G" shows "G.awalk (tail G e) [e] (head G e)"
not_eq_extremum: fixes a :: "'b word" shows "(a \<noteq> - 1) = (a < - 1)"
sint_int_min: shows "sint (- (2 ^ (LENGTH('b) - Suc 0))) = - (2 ^ (LENGTH('b) - Suc 0))"
assms_2: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (map snd D) \<inter> set X = {}"
induct: fixes x1 :: "TP"   and x2 :: "IMP"   and P :: "TP \<Rightarrow> IMP \<Rightarrow> bool" assumes "(x1, x2) \<in> VS_com"   and "\<And>pc. P pc Skip"   and "\<And>x pc e. CONTEXT x = high \<Longrightarrow> P pc (Assign x e)"   and "\<And>x pc e. \<lbrakk>CONTEXT x = low; pc = low; (e, low) \<in> VS_expr\<rbrakk> \<Longrightarrow> P pc (Assign x e)"   and "\<And>pc c1 c2. \<lbrakk>(pc, c1) \<in> VS_com; P pc c1; (pc, c2) \<in> VS_com; P pc c2\<rbrakk> \<Longrightarrow> P pc (Comp c1 c2)"   and "\<And>b pc c1 c2. \<lbrakk>(b, pc) \<in> VS_Bexpr; (pc, c1) \<in> VS_com; P pc c1; (pc, c2) \<in> VS_com; P pc c2\<rbrakk> \<Longrightarrow> P pc (Iff b c1 c2)"   and "\<And>b pc c. \<lbrakk>(b, pc) \<in> VS_Bexpr; (pc, c) \<in> VS_com; P pc c\<rbrakk> \<Longrightarrow> P pc (While b c)"   and "\<And>c. \<lbrakk>(high, c) \<in> VS_com; P high c\<rbrakk> \<Longrightarrow> P low c" shows "P x1 x2"
map: shows "(y + q - x) mod q = (y + q - ya) mod q"
fd_iso1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ad (ad x) \<le> ad (ad y)" shows " |z\<rangle> x \<le>  |z\<rangle> y"
simps_1: shows "RBT_Impl.entries rbt.Empty = []"
simps_1: fixes F :: "nat \<Rightarrow> nat"   and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and k :: "nat"   and x :: "real" shows "horner F G 0 i k x = 0"
except_min_Nil: fixes xs :: "('b, 'c) oalist_ntm"   and ko :: "'b nat_term_order" assumes "fst (list_of_oalist_ntm xs) = []" shows "fst (list_of_oalist_ntm (OAlist_except_min_ntm ko xs)) = []"
discrete_def: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "topological_space.discrete open S = (\<forall>x\<in>S. topological_space.isolate open x S)"
leq_pathSh: 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"   and D :: "char list" assumes "Subobjs P C Cs"   and "P \<turnstile> last Cs \<prec>\<^sub>S D" shows "P,C \<turnstile> Cs \<sqsubset>\<^sup>1 [D]"
paths_with_new_path: fixes xs :: "'a list" assumes "xs \<in> paths_with_new" shows "path xs"
closest_supchamberD_1: fixes F :: "'a set"   and D :: "'a set" assumes "F \<in> X"   and "domain.chamber D" shows "domain.chamber (domain.closest_supchamber F D)"
tendsto: fixes g :: "'d \<Rightarrow> 'a"   and a :: "'a"   and F :: "'d filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
SteamBoiler_def: fixes x :: "nat \<Rightarrow> bit list"   and s :: "nat \<Rightarrow> nat list"   and y :: "nat \<Rightarrow> nat list" shows "SteamBoiler x s y \<equiv> ts x \<longrightarrow> ts y \<and> ts s \<and> y = s \<and> s 0 = [500] \<and> (\<forall>j. \<exists>r>0. r \<le> 10 \<and> hd (s (Suc j)) = (if hd (x j) = Zero then hd (s j) - r else hd (s j) + r))"
convex_convex_hull: fixes S :: "'a vec set" assumes "convex S" shows "convex_hull S = S"
gt: shows "poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'a), PVar v] else PVar w)) p >p p"
compositionality: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and list :: "'d list" shows "map f (map g list) = map (f \<circ> g) list"
diff_diff_eq: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a - b - c = a - (b + c)"
stability: fixes j :: "nat" assumes "t j \<le> t1" shows "dist (grid_function (discrete_evolution (\<lambda>h t x. incr h t x + s h t x)) (x (t 0) + s0) t j) (grid_function (discrete_evolution incr) (x (t 0)) t j) \<le> B / L * (exp (L * (t1 - t 0) + 1) - 1) * max_stepsize j ^ p"
comp_red_basis_dgrad_p_set_le: fixes d :: "'a \<Rightarrow> nat"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "dickson_grading d" shows "dgrad_p_set_le d (set (comp_red_basis xs)) (set xs)"
closed_map_into_discrete_topology: fixes X :: "'c topology"   and U :: "'d set"   and f :: "'c \<Rightarrow> 'd" shows "closed_map X (discrete_topology U) f = (f ` topspace X \<subseteq> U)"
tendsto_rabs_zero_cancel: fixes f :: "'c \<Rightarrow> real"   and F :: "'c filter" assumes "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
add_assoc: fixes a :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and b :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and c :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "a \<in> \<O> U"   and "b \<in> \<O> U"   and "c \<in> \<O> U" shows "add_sheaf_spec U (add_sheaf_spec U a b) c = add_sheaf_spec U a (add_sheaf_spec U b c)"