hcju/isabelleps huggingface dataset:
finsum_Un_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier R"   and "g \<in> B \<rightarrow> carrier R" shows "finsum R g (A \<union> B) = finsum R g A \<oplus> finsum R g B"
foldli_snoc: fixes xs :: "'d list"   and x :: "'d"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma> :: "'c" shows "foldli (xs @ [x]) c f \<sigma> = (if c (foldli xs c f \<sigma>) then f x (foldli xs c f \<sigma>) else foldli xs c f \<sigma>)"
field_has_derivative_at: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a" shows "(f has_derivative (*) D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'a\<rightarrow> D"
redT_updT_None: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and ta :: "('c, 'a, 'd) new_thread_action"   and t :: "'c" assumes "redT_updT ts ta t = None" shows "ts t = None"
lang_PLUS: fixes n :: "nat"   and xs :: "'b rexp list" shows "lang n (PLUS xs) = \<Union> (lang n ` set xs)"
that: fixes t' :: "'state" assumes "(d, t) \<rightarrow>t t'"   and "s' \<approx> t'" shows "thesis"
sw_sym: fixes xs :: "'b"   and x :: "'a"   and y :: "'a"   and zs :: "'b"   and z :: "'a" shows "z @zs[x \<and> y]_xs = z @zs[y \<and> x]_xs"
residualFreshChain2_1: fixes xvec :: "name list"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" assumes "xvec \<sharp>* \<alpha>"   and "xvec \<sharp>* P" shows "xvec \<sharp>* (\<alpha> \<prec> P)"
in_set_inline1_lift_spmf1: fixes y :: "'d \<times> 'e + 'f \<times> ('h \<times> 'e, 'f, 'g) rpv \<times> ('d, 'i, 'h) rpv"   and p :: "'e \<Rightarrow> 'i \<Rightarrow> ('h \<times> 'e) spmf"   and gpv :: "('d, 'i, 'h) gpv"   and s :: "'e" assumes "y \<in> set_spmf (inline1 (\<lambda>s x. lift_spmf (p s x)) gpv s)" shows "\<exists>r s'. y = Inl (r, s')"
affine_dependent_translation: fixes S :: "'a set"   and a :: "'a" assumes "affine_dependent S" shows "affine_dependent ((+) a ` S)"
terminatesI: fixes g :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'd \<times> 'c)" assumes "\<And>s. s \<in> terminates_on g" shows "terminates g"
n_min: fixes i :: "nat" assumes "i < n" shows "P $ i \<in> A - W"
inv_greek_invol: fixes s :: "(accent \<times> 'a) list" shows "inv_greek (inv_greek s) = s"
upto_single: fixes i :: "int" shows "[i..i] = [i]"
assms_3: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "f \<in> F" shows "homogeneous f"
col_cop_orth_at__eq: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p"   and Y :: "'p" assumes "X OrthAt A B C U V"   and "Coplanar A B C Y"   and "Col U V Y" shows "X = Y"
recognises_possible_steps_not_empty: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and t :: "(String.literal \<times> value list) list" assumes "recognises_execution e s r (h # t)" shows "possible_steps e s r (fst h) (snd h) \<noteq> {||}"
sign_simps_33: fixes am :: "'c"   and bm :: "'c" shows "am * bm = bm * am"
match_list_False: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ms :: "'b match_expr list"   and a :: "action"   and p :: "'c" assumes "\<not> match_list \<gamma> ms a p" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) ms) Undecided = Undecided"
uneq: fixes u :: "complex"   and z :: "complex"   and n :: "nat" assumes "cmod u = 1"   and "z \<in> s" shows "u ^ n * (deriv ^^ n) f (u * z) = u * (deriv ^^ n) f z"
simps_1: fixes n :: "nat" shows "(0 \<le> n) = True"
with_bot_cases: fixes u :: "'a option"   and A :: "'a set" assumes "u \<in> A\<^sub>\<bottom>" obtains x :: "'a" where "x \<in> A" and "u = Some x"   | "u = None"
MkI_def: shows "MkI \<equiv> \<Lambda> int. Integer_abs\<cdot>(up\<cdot>int)"
plus_tree_def: fixes x :: "'b tree"   and y :: "'b tree" shows "x + y = pure (+) \<diamondop> x \<diamondop> y"
ND_Rules_Inst_def: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b" shows "ND_Rules_Inst freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP \<equiv> Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"
less_TC_iff: fixes x :: "V"   and y :: "V" shows "(x \<sqsubset> y) = (\<exists>z\<in>elts y. x \<sqsubseteq> z)"
stream_dec_enc_Inl: fixes n :: "nat"   and FO :: "nat set"   and w :: "'a list"   and I :: "(nat + nat set) list"   and i :: "nat"   and p' :: "nat"   and p :: "nat" assumes "stream_dec n FO (stream_enc (w, I)) ! i = Inl p'"   and "I ! i = Inl p"   and "i \<in> FO"   and "i < n"   and "length I = n" shows "p = p'"
join_bot: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<squnion> \<bottom> = x"
lhd_conv_lnth: fixes xs :: "'c llist" assumes "\<not> lnull xs" shows "lhd xs = xs $ 0"
divide_le_eq_1_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq (divide b a) one = less_eq b a"
Church_Rosser_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Church_Rosser R = (\<forall>x y. (sup R R\<inverse>\<inverse>)\<^sup>*\<^sup>* x y \<longrightarrow> (\<exists>z. R\<^sup>*\<^sup>* x z \<and> R\<^sup>*\<^sup>* y z))"
vec_matrix: fixes A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e" shows "vec_to_iarray (A $ i) = matrix_to_iarray A !! mod_type_class.to_nat i"
map_comp_Some_iff: fixes m1 :: "'d \<Rightarrow> 'c option"   and m2 :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and v :: "'c" shows "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
terminals_take: fixes n :: "nat"   and p :: "(symbol \<times> character list) list" shows "terminals (take n p) = take n (terminals p)"
convex_imp_path_connected: fixes S :: "'c set" assumes "convex S" shows "path_connected S"
sel_2: fixes x11 :: "char list"   and x12 :: "fterm list" shows "get_sub_terms (Fun x11 x12) = x12"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> kerbIV_gets" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
that: fixes x :: "'a"   and y :: "'b" assumes "fmlookup m x = Some y" shows "thesis"
multiplicity_cong: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<And>r. dvd.dvd times (power.power one times p r) a = dvd.dvd times (power.power one times p r) b" shows "factorial_semiring.multiplicity times one p a = factorial_semiring.multiplicity times one p b"
existence_ivl0_def: fixes x0 :: "'a" shows "rev.existence_ivl0 x0 = ll_on_open.existence_ivl UNIV (\<lambda>_. - f) X 0 x0"
sup_eq_bot_iff: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x \<union># y = {#}) = (x = {#} \<and> y = {#})"
QF_non_Nil: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j < Suc k" shows "list_of (QF k (j, i)) \<noteq> []"
mbind'_unit: fixes f :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p [] f = unit\<^sub>S\<^sub>E []"
sp_pl: shows "simple_path (p +++ linepath (a - complex_of_real d) (a + complex_of_real e))"
not_eq_extremum: fixes a :: "'b" shows "(a \<noteq> \<top>) = (a < \<top>)"
InitProcessing\<^sub>1: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "nat exp prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "nat exp"   and h :: "heap"   and l :: "val list"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "sh C = \<lfloor>(sfs, Processing)\<rfloor>"   and "P \<turnstile>\<^sub>1 \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
additive_def: fixes f :: "'b \<Rightarrow> 'c" shows "additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
Mapping_lookup_map_default: fixes k :: "'b"   and d :: "'a"   and f :: "'a \<Rightarrow> 'a"   and m :: "('b, 'a) mapping"   and k' :: "'b" shows "Mapping.lookup (Mapping.map_default k d f m) k' = (if k = k' then (Some \<circ> f) (case Mapping.lookup m k of None \<Rightarrow> d | Some x \<Rightarrow> x) else Mapping.lookup m k')"
blinding_of_blindable_simps_2: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d blindable\<^sub>h"   and y :: "('c, 'd) blindable\<^sub>m" shows "blinding_of_blindable h bo (Blinded x) y = (hash_blindable h y = x)"
homeomorphic_AR_iff_AR: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "AR S = AR T"
uniqDistinct: fixes l :: "'b list" shows "uniq l = distinct l"
sees_methods_sub_Obj: fixes P :: "'a prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "P \<turnstile> C sees_methods Mm" shows "P \<turnstile> C \<preceq>\<^sup>* Object"
fresh_2: fixes a :: "name" shows "a \<sharp> \<tau> = True"
is_lub_bigger2: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" shows "is_lub (r\<^sup>*) x y x = ((y, x) \<in> r\<^sup>*)"
check_poly_strict_mono_def: fixes pm :: "bool"   and p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "check_poly_strict_mono pm p v \<equiv> list_ex (\<lambda>(m, c). (1::'d) \<le> c \<and> check_monom_strict_mono pm m v) p"
Q: fixes S :: "'a set"   and S' :: "'b set" assumes "P S"   and "homeomorphism S S' f g" shows "Q S'"
generate_valid_stateful_policy_IFSACS_2_wf_stateful_policy: fixes G :: "'b graph"   and edgesList :: "('b \<times> 'b) list"   and M :: "'b SecurityInvariant_configured list" assumes "wf_graph G"   and "set edgesList = edges G" shows "wf_stateful_policy (generate_valid_stateful_policy_IFSACS_2 G M edgesList)"
get_child_nodes_def: shows "get_child_nodes = (\<lambda>ptr. invoke a_get_child_nodes_tups ptr ())"
ev_induct_strong: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and x :: "'c stream"   and P :: "'c stream \<Rightarrow> bool" assumes "ev \<phi> x"   and "\<And>xs. \<phi> xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>ev \<phi> (stl xs); \<not> \<phi> xs; P (stl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P x"
sublocale_sas_plus_finite_domain_representation_ii_1: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). \<R>\<^sub>+ \<Psi> v \<noteq> {}"
nxt_substitute: fixes t :: "'b ttree"   and x :: "'b"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" assumes "possible t x" shows "nxt (substitute f T t) x = substitute (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x)"
restrict_correct_aux1_1: fixes \<alpha>1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar1 :: "'a \<Rightarrow> bool"   and \<alpha>2 :: "'d \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar2 :: "'d \<Rightarrow> bool"   and restrict :: "('b \<times> 'c \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'd"   and m :: "'a"   and P :: "'b \<times> 'c \<Rightarrow> bool" assumes "map_restrict \<alpha>1 invar1 \<alpha>2 invar2 restrict"   and "invar1 m" shows "\<alpha>2 (restrict P m) = \<alpha>1 m |` {k. \<exists>v. \<alpha>1 m k = Some v \<and> P (k, v)}"
H_llist_hom_id: shows "list2H\<cdot>lnil = ID"
pr_valid_key_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and X :: "agent" assumes "(evs, S, A, U) \<in> protocol" shows "Key (pubSK X) \<in> analz (A \<union> spies evs)"
restrict_map_subset_eq: fixes m :: "'d \<Rightarrow> 'e option"   and R :: "'d set"   and m' :: "'d \<Rightarrow> 'e option"   and R' :: "'d set" assumes "m |` R = m'"   and "R' \<subseteq> R" shows "m |` R' = m' |` R'"
val_def: shows "f'.val \<equiv> sum f' (cf.outgoing s) - sum f' (cf.incoming s)"
aencrypt_def: fixes \<alpha> :: "'grp"   and msg :: "bool list" shows "aencrypt \<alpha> msg = lift_spmf (sample_uniform (order \<G>)) \<bind> (\<lambda>y. hash (\<alpha> [^] y) \<bind> (\<lambda>h. Generative_Probabilistic_Value.Done (\<^bold>g [^] y, h [\<oplus>] msg)))"
D_def: shows "D = max 1 ((1 / 2) powr (a - 1))"
waiting_error_update: shows "\<forall>s d a' a. current s \<noteq> d \<and> waiting s d a \<longrightarrow> waiting (set_error_code s a') d a"
abs_eq: fixes xa :: "int"   and x :: "nat" shows "setBit (Word.Word xa) x = Word.Word (ring_bit_operations_class.set_bit x xa)"
well_valued0_r: fixes a :: "'a tree" assumes "local.well_valued0 a"   and "a \<noteq> Leaf" shows "r a \<le> length as"
subst_atm_mset_def: fixes AA :: "'a multiset"   and \<sigma> :: "'s" shows "AA \<cdot>am \<sigma> = {#A \<cdot>a \<sigma>. A \<in># AA#}"
abs_eq: fixes x :: "('a \<times> 'b) list" shows "AList_Mapping.Mapping x = mapping.Mapping (map_of x)"
bijective_map_def: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" shows "bijective_map \<alpha> S T \<equiv> map \<alpha> S T \<and> bijective \<alpha> S T"
union_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
Assign: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> Suc 0 \<Down> s(x := aval a s)"
uint_power_lower: fixes n :: "nat" assumes "n < LENGTH('a)" shows "uint (2 ^ n) = 2 ^ n"
extremum_strict: fixes a :: "'c multiset" shows "\<not> a \<subset># {#}"
isin_set: fixes t :: "'b tree23"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
RootEx_Root: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "RootEx F G" shows "Root F G \<in> F"
thetaParWL2_converse_Wretr: shows "thetaParWL2\<inverse> \<subseteq> Wretr (thetaParWL2\<inverse> \<union> Wbis)"
valid_eval_list: fixes env :: "v sem_env"   and s :: "'a state"   and es :: "exp0 list" shows "evaluate_list True env s es (eval_list env es s)"
card_order_csum: fixes r1 :: "('p \<times> 'p) set"   and r2 :: "('q \<times> 'q) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 +c r2)"
B: shows "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
Zr_zero: shows "\<zero>\<^bsub>Zr\<^esub> = 0"
small_Un_elts: fixes x :: "V"   and y :: "V" shows "small (elts x \<union> elts y)"
completeI: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set" assumes "class.metric_space dist uniformity open"   and "\<And>f. \<lbrakk>\<forall>n. f n \<in> s; uniform_space.Cauchy uniformity f\<rbrakk> \<Longrightarrow> \<exists>l\<in>s. filterlim f (topological_space.nhds open l) sequentially" shows "uniform_space.complete open uniformity s"
point_in_vector_sup: fixes p :: "'a"   and v :: "'a"   and w :: "'a" assumes "point p"   and "vector v"   and "regular v"   and "p \<le> v \<squnion> w" shows "p \<le> v \<or> p \<le> w"
pair_to_complex_def: fixes ri :: "real_alg_3 \<times> real_alg_3" shows "pair_to_complex ri \<equiv> case ri of (r, i) \<Rightarrow> Complex (real_of_3 r) (real_of_3 i)"
assms: fixes A :: "'a set" assumes "A \<subseteq> set_pmf p" shows "measure_pmf.prob p A \<le> measure_pmf.prob q {y \<in> set_pmf q. \<exists>x\<in>A. R x y}"
lagrange_finite: fixes H :: "'a set" assumes "finite (carrier G)"   and "subgroup H G" shows "card (rcosets H) * card H = order G"
homotopy_dominated_contractibility: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "continuous_map X Y f"   and "continuous_map Y X g"   and "homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id"   and "contractible_space X" shows "contractible_space Y"
is_rcoset_replistD_distinct: fixes H :: "'g set"   and gs :: "'g list" assumes "is_rcoset_replist H gs" shows "distinct (map (\<lambda>g. rcoset_rel H `` {g}) gs)"
splitAt_dist_ram2: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(a @ ram1 # b, c) = splitAt ram2 vs"
finite_dimensional_vector_space_axioms_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and Basis :: "'d set" shows "finite_dimensional_vector_space_axioms scale Basis \<equiv> finite Basis \<and> \<not> module.dependent scale Basis \<and> module.span scale Basis = UNIV"
separation_t1: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t1_space open" shows "(x \<noteq> y) = (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
T_vrestr_meas_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M (T ^^ n) A \<in> sets M"
triangle_left: shows "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sup>*, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
theFTOne_length_1: fixes cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list"   and n :: "nat" assumes "length cl = length dl"   and "n \<in> theFTOne cl dl" shows "n < length cl"
shallower_pos: fixes t :: "('c, 'd) tm"   and p :: "dir list"   and q :: "dir list"   and dq :: "dir"   and dp :: "dir" assumes "position_of t (p @ q @ [dq])" shows "position_of t (p @ [dp])"
psimps: fixes i :: "int"   and j :: "int" assumes "opt_bst_wpl2_dom (i, j)" shows "opt_bst_wpl2 i j = (if j < i then (\<langle>\<rangle>, 0) else if i = j then (\<langle>\<langle>\<rangle>, i, \<langle>\<rangle>\<rangle>, w i i) else let l = root (fst (opt_bst_wpl2 i (j - 1))); r = root (fst (opt_bst_wpl2 (i + 1) j)) in argmin snd (map (\<lambda>k. let (tl, wl) = opt_bst_wpl2 i (k - 1); (tr, wr) = opt_bst_wpl2 (k + 1) j in (\<langle>tl, k, tr\<rangle>, wl + wr + w i j)) [l..r]))"
assms: shows "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')"
variation_mult_pos_1: fixes c :: "real"   and x :: "real"   and y :: "real" assumes "0 < c" shows "variation (c * x) y = variation x y"
Substitution: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "t \<in> SMP M"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "t \<cdot> \<delta> \<in> SMP M"
has_integral_spike_finite_eq: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "(f has_integral y) T = (g has_integral y) T"
pmf_equiv_intro1: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "\<And>e. e \<in> set_pmf p \<Longrightarrow> pmf p e = pmf q e" shows "p = q"
nEq_neq_conv: fixes x :: "'a"   and y :: "'a" shows "(cmp x y \<noteq> Eq) = (x \<noteq> y)"
add_spvec_empty1: fixes a :: "(nat \<times> 'a) list" shows "add_spvec [] a = a"
splitAt_ram: fixes ram :: "'a"   and vs :: "'a list" assumes "ram \<in> set vs" shows "vs = fst (splitAt ram vs) @ ram # snd (splitAt ram vs)"
set2fun_def: fixes X :: "'c set" shows "set2fun X = (\<lambda>x. if s2p X x then \<eta> x else {})"
f_Img_ex_set: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<I> \<otimes> f = null"
independent_span_bound: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and T :: "'e set"   and S :: "'e set" assumes "vector_space scale"   and "finite T"   and "\<not> module.dependent scale S"   and "S \<subseteq> module.span scale T" shows "finite S \<and> card S \<le> card T"
Explained_def: fixes x :: "e" shows "Explained x \<equiv> \<lambda>w. \<exists>xa. ((actualizedAt) xa \<^bold>\<and> xa explains x) w"
f_shrink_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'d message_af list" shows "xs \<up> (n * k) \<div> k = xs \<div> k \<up> n"
clo_hS: shows "closedin (Euclidean_space n) (h ` S)"
reflexive: fixes A :: "('c, 'd) nba"   and p :: "'d" shows "transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) p p"
gcd_unique: fixes d :: "'a"   and a :: "'a"   and b :: "'a" shows "(d dvd a \<and> d dvd b \<and> normalize d = d \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d)) = (d = gcd a b)"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VxVxV.cod f = (if arr (fst f) \<and> VxV.arr (snd f) then (cod (fst f), VxV.cod (snd f)) else (null, VxV.null))"
ccINF_lower: fixes A :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'a" assumes "countable A"   and "i \<in> A" shows "Inf (f ` A) \<le> f i"
enc_extend_interp: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list"   and x :: "nat + nat set" assumes "enc (w, I) = enc (w', I')" shows "enc (w, x # I) = enc (w', x # I')"
simp: shows "local.neville_aitken_main i j = (if i < j then smult (inverse (x j - x i)) (local.X i * local.neville_aitken_main (i + 1) j - local.X j * local.neville_aitken_main i (j - 1)) else [:f i:])"
comm: fixes a :: "'c"   and b :: "'c" shows "ev_eq a b = ev_eq b a"
DERIV_const_ratio_const2: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and k :: "real" assumes "a \<noteq> b"   and "\<And>x. (f has_real_derivative k) (at x)" shows "(f b - f a) / (b - a) = k"
unitr_ex: fixes x :: "'c \<times> 'd" shows "\<exists>e\<in>pm_prod_E. ps_prod_D x e \<and> x = ps_prod_times x e"
emp_pow: fixes n :: "nat" shows "\<epsilon> \<^sup>@ n = \<epsilon>"
chine_hcomp_in_hom: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "\<guillemotleft>chine_hcomp \<nu> \<mu> : Leg0 (Dom \<nu>) \<down>\<down> Leg1 (Dom \<mu>) \<rightarrow>\<^sub>C Leg0 (Cod \<nu>) \<down>\<down> Leg1 (Cod \<mu>)\<guillemotright>"
rep_eq: fixes x :: "'a fls" shows "fls_nth (- x) = (\<lambda>n. - fls_nth x n)"
a: shows "abc_steps_l (0, lm) ap stp = (as, am)"
plus_period: fixes x :: "'a" shows "f (x + (1::'a)) = f x"
exec_n_drop_Cons: fixes instr :: "instr"   and P' :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and k :: "nat"   and n :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "instr # P' \<turnstile> (i, s, stk) \<rightarrow>^k (n, s', stk')"   and "1 \<le> i"   and "exits P' \<subseteq> {0..}" shows "P' \<turnstile> (i - 1, s, stk) \<rightarrow>^k (n - 1, s', stk')"