hcju/isabelleps huggingface dataset:
finite_has_maximal: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq m b \<longrightarrow> m = b"
internalize_ordLess: fixes r' :: "('c \<times> 'c) set"   and r :: "('d \<times> 'd) set" shows "(r' <o r) = (\<exists>p. Field p \<subset> Field r \<and> r' =o p \<and> p <o r)"
inv_eq_1_iff: fixes x :: "'a" assumes "x \<in> carrier R" shows "(\<ominus> x = \<zero>) = (x = \<zero>)"
assms: shows "(f expands_to MS (MSLCons (c, e) xs) g) (b # basis)"
replicate_length_same: fixes xs :: "'k list"   and x :: "'k" assumes "\<forall>y\<in>set xs. y = x" shows "replicate (length xs) x = xs"
upcast_max: shows "sint (SCAST(32 \<rightarrow> 64) 0x80000001) = sint 0x80000001"
not_is_zero_iarray_eq_iff: fixes A :: "('e, 'd) vec" shows "(\<exists>b. A $ b \<noteq> (0::'e)) = (\<not> is_zero_iarray (vec_to_iarray A))"
stabilizer_def: fixes m :: "'c" shows "stabilizer m = {g \<in> carrier G. \<phi> g m = m}"
sg6: shows "i1 (Suc t) = inf_last_ti dt t"
outstanding_non_volatile_writes_unshared: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "outstanding_non_volatile_writes_unshared \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "non_volatile_writes_unshared \<S> sb"
rel_spmf_bind_reflI: fixes p :: "'b spmf"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd spmf"   and g :: "'b \<Rightarrow> 'e spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf P (f x) (g x)" shows "rel_spmf P (p \<bind> f) (p \<bind> g)"
lexLessBackjump: fixes p :: "('a \<times> bool) list"   and level :: "nat"   and a :: "('a \<times> bool) list"   and x :: "'a" assumes "p = prefixToLevel level a"   and "0 \<le> level"   and "level < currentLevel a" shows "(p @ [(x, False)], a) \<in> lexLess"
abc_neq: shows "a \<noteq> b \<and> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<and> c \<noteq> d"
Tree_wf_eqvt: fixes p :: "perm" shows "p \<bullet> Tree_wf = Tree_wf"
surjective: fixes r :: "'a qtname_scheme" shows "r = \<lparr>pid = pid r, tid = tid r, \<dots> = qtname.more r\<rparr>"
eNorm_sum: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "eNorm N (sum u {..<n}) \<le> (\<Sum>i<n. ennreal (defect N ^ Suc i) * eNorm N (u i))"
connected_component_Union: fixes S :: "'c set"   and x :: "'c" shows "connected_component_set S x = \<Union> {T. connected T \<and> x \<in> T \<and> T \<subseteq> S}"
minSpanningForest_def: fixes F :: "nat uprod set" shows "minSpanningForest F \<equiv> spanningForest F \<and> (\<forall>F'. spanningForest F' \<longrightarrow> sum (uGraph_from_list_\<alpha>_weight L) F \<le> sum (uGraph_from_list_\<alpha>_weight L) F')"
openin_delete: fixes u :: "'b set"   and s :: "'b set"   and a :: "'b" assumes "openin (top_of_set u) s" shows "openin (top_of_set u) (s - {a})"
assms_2: shows "homotopic_with (\<lambda>x. True) Y Z g g'"
option_bind_def: fixes m :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "option_bind m f = (case m of None \<Rightarrow> None | Some r \<Rightarrow> f r)"
SNF_S: shows "Smith_normal_form_mat (fst (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))))"
F_def: shows "F \<equiv> \<lambda>P fs x. if x \<in> P then fs else []"
2: shows "lfp (\<lambda>y. snd (F (lfp (\<lambda>x. fst (F (x, lfp (\<lambda>y. snd (F (x, y)))))), y))) \<le> y"
Irint_C: fixes i :: "int"   and vs :: "int list" shows "Irint (IC i) vs = i"
bind_commute_pmf: fixes A :: "'c pmf"   and B :: "'d pmf"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> 'b pmf" shows "A \<bind> (\<lambda>x. B \<bind> C x) = B \<bind> (\<lambda>y. A \<bind> (\<lambda>x. C x y))"
Collect_conj_eq: fixes P :: "'k \<Rightarrow> bool"   and Q :: "'k \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
domains_subsume_globallyI: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "\<And>r \<sigma> \<tau> e. s r = Some (\<sigma>, \<tau>, e) \<Longrightarrow> \<S> (\<sigma>, \<tau>, e)" shows "\<S>\<^sub>G s"
run_cont_def: fixes contT :: "('d, 'e) contT" shows "run_cont contT = (case contT of ContT x \<Rightarrow> x)"
ro_ifex_unique: fixes x :: "'c ifex"   and y :: "'c ifex" assumes "ro_ifex x"   and "ro_ifex y"   and "\<And>ass. val_ifex x ass = val_ifex y ass" shows "x = y"
lift_character_adjoined: fixes \<chi>z :: "('a \<Rightarrow> complex) \<times> complex" assumes "character (G\<lparr>carrier := {\<one>}\<rparr>) (fst \<chi>z)" shows "H.lift_character \<chi>z x = snd \<chi>z"
assms_1: shows "k < 2 ^ (Suc n + 1)"
mult_mat_vec: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'c vec"   and k :: "'c" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "A *\<^sub>v (k \<cdot>\<^sub>v v) = k \<cdot>\<^sub>v (A *\<^sub>v v)"
ilist_eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f = g) = (\<forall>x. f x = g x)"
that: fixes i :: "nat" assumes "i < l"   and "\<alpha> ! i \<le> \<beta>" shows "thesis"
arr_iff_in_hom: fixes f :: "('obj, 'obj) concrete_category.arr" shows "arr f = \<guillemotleft>f : local.dom f \<rightarrow> cod f\<guillemotright>"
f_im: fixes v :: "'c \<Rightarrow> 'a" assumes "v \<in> carrier (func_space A)" shows "lincomb v A \<in> carrier M"
lowner_le_def: fixes A :: "complex mat"   and B :: "complex mat" shows "(A \<le>\<^sub>L B) = (dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> positive (B - A))"
emeasure_count_space_infinite: fixes X :: "'a set"   and A :: "'a set" assumes "X \<subseteq> A"   and "infinite X" shows "emeasure (count_space A) X = \<infinity>"
le_eq_less_or_eq: fixes m :: "nat"   and n :: "nat" shows "(m \<le> n) = (m < n \<or> m = n)"
mult_numeral_left_semiring_numeral: fixes v :: "num"   and w :: "num"   and z :: "'k" shows "numeral v * (numeral w * z) = numeral (v * w) * z"
vdomain_def: fixes D :: "V" shows "vdomain D = (\<lambda>r\<in>\<^sub>\<circ>D. set {a. \<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r})"
le_imp_less_or_eq: fixes x :: "'a list"   and y :: "'a list" assumes "x \<le>lex y" shows "x <lex y \<or> x = y"
simps_2: fixes lt :: "dag"   and a :: "ref"   and rt :: "dag" shows "set_of (Node lt a rt) = {a} \<union> set_of lt \<union> set_of rt"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool" shows "list_all P []"
euler_fas_def: fixes X0 :: "floatarith list"   and h :: "floatarith"   and CX :: "floatarith list" shows "euler_fas X0 h CX = map (\<lambda>i. euler_incr_fas_nth X0 h X0 i + euler_err_fas_nth X0 h CX i) [0..<length X0] @ euler_err_fas X0 h CX"
raw_filtermap_def: fixes f :: "'e \<Rightarrow> 'h" shows "raw_filtermap f = (f ---> id) ---> id"
rev_cases: fixes xs :: "'k list" obtains "xs = []"   | ys :: "'k list" and y :: "'k" where "xs = ys @ [y]"
main: shows "w \<in> {2 ^ (LENGTH(32) - 1) + 1..2 ^ LENGTH(32) - 1}"
eq_up_to_sym: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'a" assumes "IFC prob"   and "IFC_def.eq_up_to \<pi> k \<pi>'" shows "IFC_def.eq_up_to \<pi>' k \<pi>"
trunc_err_pdevs_raw_def: fixes p :: "nat"   and x :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "trunc_err_pdevs_raw p x i = trunc_pdevs_raw p x i - x i"
pred_mapping_mono: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'a option"   and R :: "'a \<Rightarrow> bool" assumes "pred_mapping Q P"   and "Q \<le> R" shows "pred_mapping R P"
N0_2: shows "\<forall>n'\<ge>n0. n' < length trace \<longrightarrow> execMsg n' \<noteq> msg"
assms_6: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
integral_scaleR_right: fixes M :: "'e measure"   and c :: "real"   and f :: "'e \<Rightarrow> 'd" shows "LINT x|M. c *\<^sub>R f x = c *\<^sub>R integral\<^sup>L M f"
inverse_unique: fixes a :: "'a perm"   and b :: "'a perm" assumes "a * b = 1" shows "inverse a = b"
complex_cvec_eq_def: fixes z1 :: "complex \<times> complex"   and z2 :: "complex \<times> complex" shows "(z1 \<approx>\<^sub>v z2) = (\<exists>k. k \<noteq> 0 \<and> z2 = k *\<^sub>s\<^sub>v z1)"
assms: shows "x \<in>\<^sub>\<circ> vconverse A"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "List_Supplement.embed (x # xs) = [x] # List_Supplement.embed xs"
gamma2_cont: shows "continuous_on {0..1} \<gamma>2"
assms: fixes s :: "'state"   and s1 :: "'state"   and vl :: "'value list"   and vl1 :: "'value list" assumes "reachNT s"   and "reach s1"   and "\<Delta> s vl s1 vl1" shows "vl \<noteq> [] \<and> exit s (hd vl) \<or> iaction \<Delta> s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> reaction \<Delta> s vl s1 vl1"
sel_1: fixes x1 :: "'a list"   and x2 :: "nat" shows "body (Fundef x1 x2) = x1"
assms_3: shows "g \<in> mor\<^bsub>C\<^esub>"
implies_itself: fixes x :: "'a" shows "x \<leadsto> x = top"
inject_2: fixes x21 :: "'a"   and x22 :: "real"   and x23 :: "'a kdt"   and x24 :: "'a kdt"   and y21 :: "'a"   and y22 :: "real"   and y23 :: "'a kdt"   and y24 :: "'a kdt" shows "(Node x21 x22 x23 x24 = Node y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
abs_le_zero_iff: fixes abs :: "'b \<Rightarrow> '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 a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) zero = (a = zero)"
pick_one_gt: fixes xs :: "real list" assumes "list_neq xs (mean xs) \<noteq> []" shows "\<exists>e. e \<in> set (list_neq xs (mean xs)) \<and> mean xs < e"
map_val_cong: fixes xs :: "('b, 'c) oalist_ntm"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>k v. (k, v) \<in> set (fst (list_of_oalist_ntm xs)) \<Longrightarrow> f k v = g k v" shows "OAlist_map_val_ntm f xs = OAlist_map_val_ntm g xs"
Lcm_Gcd: 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 Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<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 set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm A = Gcd {b. \<forall>a\<in>A. dvd.dvd times a b}"
BC: shows "B \<inter> C \<in> null_sets lborel"
out_conga_out: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "B Out A C"   and "A B C CongA D E F" shows "E Out D F"
worecSL_succ: fixes L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"   and j :: "'a"   and S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "adm_woL L"   and "local.aboveS j \<noteq> {}" shows "worecSL S L (succ j) = S j (worecSL S L j)"
abs_eq: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "total_degree (MPoly x) = Max (insert 0 ((\<lambda>m. sum (poly_mapping.lookup m) (Poly_Mapping.keys m)) ` Poly_Mapping.keys x))"
subdag_parentdag_high: fixes not :: "dag"   and rt :: "dag"   and lt :: "dag"   and p :: "ref" assumes "not \<le> rt" shows "not \<le> Node lt p rt"
f0_def: shows "f0 \<equiv> \<lambda>A. THE x. \<exists>a b c d. A = {a, b, c, d} \<and> a < b \<and> b < c \<and> c < d \<and> x = f {(a, b), (c, d)}"
sel_2: fixes x1 :: "'h word"   and x2 :: "nat" shows "pfxm_length (PrefixMatch x1 x2) = x2"
elims: fixes x :: "'a list \<times> 'a list"   and y :: "'a option \<times> 'a list \<times> 'a list" assumes "safe_last_t x = y" obtains "x = ([], [])" and "y = (None, [], [])"   | fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = (fs__, l # ls)" and "y = (Some l, fs__, l # ls)"   | f :: "'a" and fs :: "'a list" where "x = (f # fs, [])" and "y = undefined"
below_eq_trans: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "below a b"   and "b = c" shows "below a c"
blinfun_apply_integral: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow>\<^sub>L 'd"   and s :: "'b set"   and x :: "'c" assumes "f integrable_on s" shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
Max_atm_of_set_mset_commute: fixes C :: "'a literal multiset" assumes "C \<noteq> {#}" shows "Max (atm_of ` set_mset C) = atm_of (Max_mset C)"
hyps_2: fixes b :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and ba :: "'d" assumes "Rs \<prec> C = b \<prec> P'"   and "bn b \<sharp>* subject b"   and "distinct (bn b)" shows "Prop ba \<Psi> (P \<parallel> !P) b P'"
label_wrt_def: fixes B :: "'c set"   and f :: "'a \<Rightarrow> 'c" shows "label_wrt B f \<equiv> \<forall>C\<in>\<C>. bij_betw f C B"
verit_comp_simplify1_3: fixes b' :: "'k"   and a' :: "'k" shows "(\<not> b' \<le> a') = (a' < b')"
pelims: fixes x :: "acom"   and xa :: "bexp"   and xb :: "char list \<Rightarrow> int"   and y :: "char list \<Rightarrow> int" assumes "postQs x xa xb = y"   and "postQs_dom (x, xa, xb)" obtains C :: "acom" and b :: "bexp" and s :: "char list \<Rightarrow> int" where "x = C" and "xa = b" and "xb = s" and "y = (if bval b s then postQs C b (postQ C s) else s)" and "postQs_dom (C, b, s)"
prems_6: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
assms: shows "((\<lambda>x. (f x - f x\<^sub>0 - (x - x\<^sub>0) *\<^sub>R f' t) /\<^sub>R (\<parallel>x - x\<^sub>0\<parallel>)) \<longlongrightarrow> 0) (at t within T)"
fls_X_inv_subdegree: shows "fls_subdegree fls_X_inv = - 1"
assms: shows "P\<^sup>+\<^sup>+ x z"
UN_simps_4: fixes A :: "'c \<Rightarrow> 'b set"   and B :: "'b set"   and C :: "'c set" shows "(\<Union>x\<in>C. A x \<inter> B) = \<Union> (A ` C) \<inter> B"
cols_dim: fixes A :: "'a mat" shows "set (cols A) \<subseteq> carrier_vec (dim_row A)"
rev_update: fixes k :: "nat"   and xs :: "'c list"   and y :: "'c" assumes "k < length xs" shows "rev (xs[k := y]) = (rev xs)[length xs - k - 1 := y]"
2: fixes v :: "V"   and u :: "V" assumes "v \<in> elts u"   and "\<exists>x\<in>elts x. u \<in> elts (TC x)" shows "v \<in> elts (TC x)"
fresh_fact: fixes z :: "name"   and N :: "lam"   and L :: "lam"   and y :: "name" assumes "z \<sharp> N"   and "z \<sharp> L" shows "z \<sharp> N[y::=L]"
bound_on_all_plans_bounds_problem_plan_bound_thesis': fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and k :: "nat" assumes "finite PROB"   and "\<forall>as s. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < k)" shows "problem_plan_bound PROB < k"
conj_cancel_right: fixes x :: "'b" shows "x AND NOT x = (0::'b)"
2: shows "a [^]\<^bsub>units_of G\<^esub> (n + m) = a [^]\<^bsub>units_of G\<^esub> n \<otimes>\<^bsub>units_of G\<^esub> a [^]\<^bsub>units_of G\<^esub> m"
domain_atlas_subset_carrier: fixes c :: "('b, 'f) chart" assumes "c \<in> dest.atlas" shows "domain c \<subseteq> dest.carrier"
visits_via_visits: fixes P :: "'c llist"   and A :: "'c set"   and W :: "'c set" assumes "G'.visits_via P A W" shows "lset P \<inter> W \<noteq> {}"
boundedE: fixes b :: "'a word"   and c :: "'a word"   and a :: "'a word" assumes "signed.max b c \<le>s a" obtains "b \<le>s a" and "c \<le>s a"
getTvars_def: fixes \<sigma>l :: "'tp list" shows "TE.getTvars \<sigma>l \<equiv> map Var (TE.getVars \<sigma>l)"
set_subset_Cons: fixes xs :: "'c list"   and x :: "'c" shows "set xs \<subseteq> set (x # xs)"
zin_options: fixes x :: "game"   and g :: "game" assumes "zin x (options g)" shows "(x, g) \<in> option_of"
concat_map_maps: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list" shows "concat (map f xs) = List.maps f xs"
norm_zero2: fixes j :: "nat" assumes "j < m" shows "\<parallel>gs2.gso j\<parallel>\<^sup>2 \<noteq> 0"
a0n_agreement_def: shows "a0n_agreement \<equiv> {s. \<forall>A B M. 0 < signals s (Commit A B M) \<longrightarrow> 0 < signals s (Running A B M)}"
rtranclp_into_tranclp2: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
insert_absorb: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "set_ins a A = A"
simul: shows "\<forall>i\<le>Suc k. if pcss i = [] then CSS (f i) = css \<and> CS (f i) = pcs i @ cs else CS (f i) = pcs i \<and> CSS (f i) = butlast (pcss i) @ [(fst (last (pcss i)) @ cs, snd (last (pcss i)) @ cs)] @ css"
left_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and a :: "'a" assumes "Groups.monoid f z" shows "f z a = a"
dlist_insert_not_empty: fixes x :: "'b"   and dl :: "'b dlist" shows "DList.insert x dl \<noteq> DList.empty"
fds_nth_remainder: fixes m :: "nat"   and f :: "'b fds" shows "fds_nth (fds_remainder m f) = (\<lambda>n. if m < n then fds_nth f n else (0::'b))"
round_unique: fixes x :: "'b"   and y :: "int" assumes "x - (1::'b) / (2::'b) < of_int y"   and "of_int y \<le> x + (1::'b) / (2::'b)" shows "round x = y"
simple_def: fixes t :: "tm" shows "simple t = (let (c, s) = split0 (simptm t) in if c = 0\<^sub>p then le s else Le (CNP 0 c s))"
UNION_Cinfinite_bound: fixes I :: "'d set"   and r :: "('e \<times> 'e) set"   and A :: "'d \<Rightarrow> 'f set" assumes "|I| \<le>o r"   and "\<forall>i\<in>I. |A i| \<le>o r"   and "Cinfinite r" shows "|\<Union> (A ` I)| \<le>o r"
sum_2: fixes u :: "nat \<Rightarrow> 'b" shows "sum u {1..2} = u 1 + u 2"
Ln_in_Reals: fixes z :: "complex" assumes "z \<in> \<real>"   and "0 < Re z" shows "Ln z \<in> \<real>"
p_def: shows "p = Poly (map (\<lambda>k. \<Sum>i\<le>min N k. cs ! (N - i) * f (k - i)) [0..<N + m])"
cos_inj_pi: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> pi"   and "0 \<le> y"   and "y \<le> pi"   and "cos x = cos y" shows "x = y"
card_of_Un1: fixes A :: "'a set"   and B :: "'a set" shows "|A| \<le>o |A \<union> B|"
divides_refl: fixes a :: "'a list" assumes "a \<in> carrier (K [X])" shows "a divides\<^bsub>K [X]\<^esub> a"
this_wall_betwI: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>"   and "D \<in> g \<turnstile> folding_g.\<C>"   and "C \<sim> D" shows "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}}) \<and> separated_by H C D) = {f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}"
assms_1: shows "\<exists>x\<in>X. k \<le> x"
simps_3: fixes p :: "('c, 'b) fm" shows "nominals (\<^bold>\<not> p) = nominals p"
mult_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and q :: "'a" shows "mult_row A a q = (\<chi>i j. if i = a then q * A $ a $ j else A $ i $ j)"
quotient_of_number_1: shows "quotient_of 0 = (0, 1)"
deriv: shows "chain (redundancy_criterion.derive \<Gamma>' Rf) Ns"
ind_eq_0_iff: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "(ind P n = (0::'a)) = (n = 0 \<or> \<not> P n)"
sin_diff: fixes x :: "'a"   and y :: "'a" shows "sin (x - y) = sin x * cos y - cos x * sin y"
Sum2: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> Rs"   and "guarded Q" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> Q \<longmapsto> Rs"
arr_dom_iff_arr: fixes f :: "'a" shows "arr (local.dom f) = arr f"
normalize_quotE: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" obtains d :: "'b" where "a = fst (normalize_quot (a, b)) * d" and "b = snd (normalize_quot (a, b)) * d" and "d dvd a" and "d dvd b" and "d \<noteq> (0::'b)"
le_max_iff_disj: fixes z :: "'a Word.word"   and x :: "'a Word.word"   and y :: "'a Word.word" shows "(z \<le>s signed.max x y) = (z \<le>s x \<or> z \<le>s y)"
strict_trans2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "lt a b"   and "le b c" shows "lt a c"
distr_singleton: fixes M :: "'a \<Rightarrow> 'b measure"   and i :: "'a" assumes "product_sigma_finite M" shows "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i"
irreducibleD': fixes a :: "'c"   and b :: "'c" assumes "irreducible a"   and "b dvd a" shows "a dvd b \<or> is_unit b"
SS: shows "X closure_of S \<subseteq> X interior_of S"
left_commute: fixes b :: "'a Word.word"   and a :: "'a Word.word"   and c :: "'a Word.word" shows "signed.max b (signed.max a c) = signed.max a (signed.max b c)"
sets_Pair: fixes x :: "'c"   and M1 :: "'c measure"   and y :: "'d"   and M2 :: "'d measure" assumes "{x} \<in> sets M1"   and "{y} \<in> sets M2" shows "{(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
continuous_map_id: fixes X :: "'c topology" shows "continuous_map X X id"
a_create_character_data_def: fixes set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list" shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_character_data set_val get_disconnected_nodes set_disconnected_nodes document_ptr text = Heap_Error_Monad.bind new_character_data (\<lambda>new_character_data_ptr. Heap_Error_Monad.bind (set_val new_character_data_ptr text) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes document_ptr (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr # disc_nodes)) (\<lambda>_. return new_character_data_ptr))))"
2: fixes x :: "int" assumes "x \<in> carrier (residue_ring (p ^ k)) \<and> x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = \<zero>\<^bsub>residue_ring (p ^ k)\<^esub> \<and> y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> x = \<zero>\<^bsub>residue_ring (p ^ k)\<^esub>" shows "x = \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> (y mod p ^ k)"
assms_2: shows "Q \<longmapsto> a<x> \<prec> Q'"
l_cancel_one': fixes x :: "'c \<Rightarrow> 'a"   and a :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "a \<in> carrier F" shows "(x = x \<oplus>\<^bsub>F\<^esub> a) = (a = \<zero>\<^bsub>F\<^esub>)"
asymp_equiv_sandwich_real: fixes l :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and u :: "'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> real" assumes "l \<sim>[F] g"   and "u \<sim>[F] g"   and "\<forall>\<^sub>F x in F. f x \<in> {l x..u x}" shows "f \<sim>[F] g"
complete_2: shows "wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v"
subst_of_def: fixes ss :: "('a \<times> ('b, 'a) Term.term) list" shows "subst_of ss = foldr (\<lambda>(x, t) \<sigma>. \<sigma> \<circ>\<^sub>s subst x t) ss Var"
der_f_def: shows "der_f \<equiv> (deriv ^^ (n - 1)) h z / fact (n - 1)"
basic_type_less_left_simps_6: fixes \<sigma> :: "'a basic_type" shows "(UnlimitedNatural < \<sigma>) = (\<sigma> = OclAny \<or> \<sigma> = Real \<or> \<sigma> = Integer)"