hcju/isabelleps huggingface dataset:
Urysohn_strong: fixes S :: "'c set"   and T :: "'c set"   and a :: "'d"   and b :: "'d" assumes "closed S"   and "closed T"   and "S \<inter> T = {}"   and "a \<noteq> b" obtains f :: "'c \<Rightarrow> 'd" where "continuous_on UNIV f" and "\<And>x. f x \<in> closed_segment a b" and "\<And>x. (f x = a) = (x \<in> S)" and "\<And>x. (f x = b) = (x \<in> T)"
lprefix_expand: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<Longrightarrow> \<not> lnull ys \<and> lhd xs = lhd ys \<and> ltl xs \<sqsubseteq> ltl ys" shows "xs \<sqsubseteq> ys"
ZObis_pres_discrCfL: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cf :: "('c, 'd, 'a) cmd \<times> 'b"   and df :: "('c, 'd, 'a) cmd \<times> 'b"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(fst cf, fst df) \<in> PL_Indis.ZObis aval tval cval indis"   and "(snd cf, snd df) \<in> indis"   and "PL_Indis.discrCf aval tval cval indis df" shows "PL_Indis.discrCf aval tval cval indis cf"
t2': shows "\<Gamma> \<turnstile>\<^sub>c \<langle>e2\<rangle>\<^sub>c : REAL"
fmlookup_restrict_set_id: fixes N :: "('c, 'd) fmap"   and A :: "'c set" assumes "fset (fmdom N) \<subseteq> A" shows "fmrestrict_set A N = N"
basis_reduction_step_def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "local.basis_reduction_step upw i fs = (if i = 0 then (True, Suc i, fs) else let fs' = local.basis_reduction_add_rows upw i fs in if \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs') (i - 1)\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs') i\<parallel>\<^sup>2 then (True, Suc i, fs') else local.basis_reduction_swap i fs')"
SD_pref_profile: fixes i :: "'agent"   and q :: "'alt pmf"   and p :: "'alt pmf" assumes "i \<in> agents" shows "SD (R i) q p = (p \<in> lotteries_on alts \<and> q \<in> lotteries_on alts \<and> (\<forall>x\<in>alts. measure_pmf.prob q (preferred_alts (R i) x) \<le> measure_pmf.prob p (preferred_alts (R i) x)))"
b: shows "prime (3 * 2 ^ (2 - 1) - 1)"
in_ass_Hermite_of_row: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and i :: "'b"   and A :: "(('a, 'c) vec, 'b) vec" assumes "ass_function ass"   and "res_function res"   and "\<not> is_zero_row i A" shows "Hermite_of_row_i ass res A i $ i $ (LEAST n. Hermite_of_row_i ass res A i $ i $ n \<noteq> (0::'a)) \<in> range ass"
monoD: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
double_orthogonal_complement_increasing: fixes M :: "'a set" shows "M \<subseteq> orthogonal_complement (orthogonal_complement M)"
distinct_53: fixes x6 :: "msg"   and x81 :: "nat"   and x82 :: "msg" shows "Hash x6 \<noteq> Crypt x81 x82"
getEnv_idEnv: fixes xs :: "'g"   and x :: "'h" shows "idEnv xs x = None"
order_one_closed: shows "(top \<le> top \<and> top \<sqinter> top \<le> top) \<and> top \<sqinter> id top \<le> top"
Abs_fnat_hom_add: fixes a :: "nat"   and b :: "nat" shows "of_nat a + of_nat b = of_nat (a + b)"
weak_ranking_eqD: fixes alts :: "'c set"   and R1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite_total_preorder_on alts R1"   and "finite_total_preorder_on alts R2"   and "weak_ranking R1 = weak_ranking R2" shows "R1 = R2"
case_1: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x1 :: "bool" shows "(case Det x1 of Det x \<Rightarrow> f1 x | Indet x \<Rightarrow> f2 x) = f1 x1"
64282: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - - ((x \<squnion> top) \<sqinter> y)) = - (x \<squnion> - - y)"
ospec: fixes A :: "'e option"   and P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<forall>x\<in>set_option A. P x"   and "A = Some x" shows "P x"
weak_unit_self_composable_1: fixes a :: "'a" assumes "weak_unit a" shows "ide a"
edom_fun_upd_subset: fixes h :: "'c \<Rightarrow> 'd"   and x :: "'c"   and v :: "'d" shows "edom (h(x := v)) \<subseteq> insert x (edom h)"
l_get_component_get_element_by_id\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_component_get_element_by_id\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
that: assumes "nop A (1\<^sub>\<nat>) f" shows "thesis"
sqrt_rat_code: fixes x :: "rat" shows "sqrt_rat x = (case quotient_of x of (z, n) \<Rightarrow> case sqrt_int n of [] \<Rightarrow> [] | sn # xs \<Rightarrow> map (\<lambda>sz. rat_of_int sz / rat_of_int sn) (sqrt_int z))"
last_psums: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "last (psums xs) = sum_list xs"
bij_betw_mod_nat_atLeastLessThan: fixes k :: "nat"   and d :: "nat" assumes "0 < k" shows "bij_betw (\<lambda>i. i mod k) {d..<d + k} {..<k}"
ln_one: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and ln :: "'a \<Rightarrow> 'a" assumes "class.ln dist uniformity open minus norm plus zero uminus scaleR sgn times one ln" shows "ln one = zero"
from_nat_into_surj: fixes A :: "'d set"   and a :: "'d" assumes "countable A"   and "a \<in> A" shows "\<exists>n. from_nat_into A n = a"
HPhase0Read_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv5_inner s q" shows "HInv5_inner s' q"
snd_snd_rw: shows "snd (snd g) = matrix_to_iarray (snd (snd f))"
linear: fixes x :: "'b"   and y :: "'b" shows "c1.le x y \<or> c1.le y x"
Lxy_empty: fixes S :: "'b set" shows "Lxy [] S = []"
wneg_lemma: fixes w :: "32 Word.word"   and r :: "real" assumes "w \<equiv>\<^sub>E r" shows "wneg w \<equiv>\<^sub>E - r"
ball_imageD: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>\<P> f A. P x" shows "\<forall>x\<in>A. P (f x)"
Cauchy_Schwarz_ineq2: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'d"   and y :: "'d" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
H_def: shows "H \<equiv> pre_digraph.del_arc (G.del_arc a') a"
nb2: shows "\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j)"
sasp_op_to_strips_injective: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op\<^sub>1 :: "('c, 'd) sas_plus_operator"   and op\<^sub>2 :: "('c, 'd) sas_plus_operator" assumes "\<phi>\<^sub>O \<Psi> op\<^sub>1 = \<phi>\<^sub>O \<Psi> op\<^sub>2" shows "op\<^sub>1 = op\<^sub>2"
reduced_homology_group_def: fixes p :: "int"   and X :: "'a topology" shows "reduced_homology_group p X \<equiv> subgroup_generated (homology_group p X) (kernel (homology_group p X) (homology_group p (discrete_topology {()})) (hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ())))"
nabla_noether: fixes x :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)" shows "Noetherian x"
monom_mult_zero_left: fixes t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "monom_mult (0::'d) t p = 0"
extreme_point_of_singleton: fixes x :: "'b"   and a :: "'b" shows "(x extreme_point_of {a}) = (x = a)"
inducts: fixes x :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "p2 x"   and "\<And>n. \<lbrakk>p2 n; P n\<rbrakk> \<Longrightarrow> P n" shows "P x"
gen_ideal_group: fixes x :: "'a" assumes "x \<in> R" shows "Group_Theory.group \<langle>x\<rangle> (+) \<zero>"
less_const_def: fixes x :: "const"   and y :: "const" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
measure_lex_prod_def': fixes f :: "'a \<Rightarrow> nat"   and r :: "('a \<times> 'a) set" shows "f <*mlex*> r \<equiv> {(x, y). (x, y) \<in> measure f \<or> f x = f y \<and> (x, y) \<in> r}"
that: fixes n :: "nat" assumes "(x, y) \<in> r ^^ n" shows "thesis"
is_category_with_finite_products: shows "category_with_finite_products (\<cdot>)"
remdups_adj_sorted_distinct: fixes xs :: "'a list" assumes "sorted xs" shows "distinct (remdups_adj xs)"
c_def: shows "c = exp (fds_nth f (Suc 0))"
that: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "r \<noteq> 0"   and "lt r = u"   and "r \<in> dgrad_sig_set' (length fs) d"   and "rep_list r = 0" shows "thesis"
\<Sigma>_\<Delta>: fixes f :: "'b \<Rightarrow> 'a"   and j :: "int"   and l :: "int" shows "\<Sigma> (\<Delta> f) j l = f (of_int l) - f (of_int j)"
infinite_Un: fixes S :: "'f set"   and T :: "'f set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
Polys_closed_normal_form: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "p \<in> P[X]" shows "normal_form F p \<in> P[X]"
rev_path_empty: fixes p :: "('b \<times> 'b) list" shows "(rev_path p = []) = (p = [])"
Const\<^sub>0_def: fixes c :: "'c" shows "Const\<^sub>0 c \<equiv> Poly_Mapping.single 0 c"
af_composition: fixes g :: "('f \<Rightarrow> 'd) af"   and f :: "('e \<Rightarrow> 'f) af"   and x :: "'e af" shows "Applicative.ap (Applicative.ap (Applicative.ap (pure (\<circ>)) g) f) x = Applicative.ap g (Applicative.ap f x)"
lookup_present_eq_key: fixes al :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "distinct (map fst al)" shows "((k, v) \<in> set al) = (lookup (\<lambda>x. x = k) al = Some v)"
leq_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
A_1: fixes i :: "nat" shows "A i \<subseteq> C"
extending_maps_Union_aux: fixes \<F> :: "'a set set"   and K :: "'a set"   and T :: "'b set"   and h :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"   and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"   and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)" shows "\<exists>g. continuous_on (\<Union> \<F>) g \<and> g ` \<Union> \<F> \<subseteq> T \<and> (\<forall>x\<in>\<Union> \<F> \<inter> K. g x = h x)"
adds_pp_alt: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_pp pair_of_term t v = (\<exists>u. v = term_powerprod.splus pair_of_term term_of_pair t u)"
lmeasurable_cbox: fixes a :: "'b"   and b :: "'b" shows "cbox a b \<in> lmeasurable"
drinks2_0_invalid: fixes aa :: "String.literal"   and b :: "value list" assumes "\<not> (aa = STR ''select'' \<and> length b = 1)" shows "possible_steps drinks2 0 <> aa b = {||}"
a: shows "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)"
qs_1: shows "(qs, a) \<in>. (ps, a) # branch"
preserves_cod: fixes f :: "J.Arr" assumes "J.arr f" shows "C.cod (\<chi> f) = local.map (J.cod f)"
mult_closed: fixes x :: "'a"   and a :: "real" assumes "x \<in> H" shows "a \<cdot> x \<in> H"
span_minimal: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "csubspace T" shows "cspan S \<subseteq> T"
subgroupI: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> inv a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" shows "subgroup H G"
poly_vars_list_def: fixes p :: "('c monom \<times> 'd) list" shows "poly_vars_list p = remdups (concat (map (monom_vars_list \<circ> fst) p))"
compact_continuous_image: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "compact s" shows "compact (f ` s)"
visits_empty: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r :: "nat \<Rightarrow>f value option" shows "visits s e s' r [] = (s = s')"
prems_1: fixes i' :: "nat"   and j' :: "nat"   and k :: "nat" assumes "i' < length (y # xs)"   and "j' < length (y # xs)"   and "(y # xs) ! i' = Some k" shows "((y # xs) ! j' = Some k) = (\<not> tr_lookup T (i' + jj) (j' + jj))"
sat_for_sols_post: fixes res :: "'res" assumes "\<tau> \<Turnstile> POST self a1 (\<lambda>_. res)" shows "\<tau> \<Turnstile> E (\<lambda>_. res)"
step_dom_mono_aux: fixes p :: "nat"   and \<tau> :: "nat list"   and a :: "nat"   and b :: "nat list"   and \<tau>' :: "nat list" assumes "Sorted_Less.sorted (rev (transf p \<tau>))"   and "(a, b) \<in> set (step p \<tau>)"   and "\<tau> \<in> A"   and "p < n"   and "\<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>'" shows "\<exists>\<tau>''. (a, \<tau>'') \<in> set (step p \<tau>') \<and> b \<sqsubseteq>\<^bsub>r\<^esub> \<tau>''"
prime_factorization_subset_iff_dvd: 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 x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "(factorial_semiring.prime_factorization zero times one normalize x \<subseteq># factorial_semiring.prime_factorization zero times one normalize y) = dvd.dvd times x y"
hyps_1: shows "(s, t) \<in> TS.trans T"
owned_def: fixes t :: "'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g \<times> 'b \<times> 'h" shows "owned t = (let (p, instrs, \<theta>, sb, \<D>, \<O>, \<R>) = t in \<O>)"
dvd_triv_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times a b)"
section_retraction_of_iso_2: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.iso f"   and "CC.ide (CC.comp f g)" shows "CC.inverse_arrows f g"
False_eqvt: fixes p :: "perm" shows "p \<bullet> False = False"
mcompos_surj_surj: fixes L :: "('i, 'b, 'j) Module_scheme"   and N :: "('k, 'b, 'l) Module_scheme"   and f :: "'i \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'k" assumes "R module L"   and "R module N"   and "surjec\<^bsub>L,M\<^esub> f"   and "surjec\<^bsub>M,N\<^esub> g"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N" shows "surjec\<^bsub>L,N\<^esub> compos L g f"
fdisjnt_swap: fixes M :: "'a fset"   and N :: "'a fset" assumes "fdisjnt M N" shows "fdisjnt N M"
intro: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "H \<subseteq> carrier G"   and "\<And>x y. \<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H"   and "\<one>\<^bsub>G\<^esub> \<in> H"   and "\<And>x. x \<in> H \<Longrightarrow> inv\<^bsub>G\<^esub> x \<in> H" shows "subgroup H G"
right_neutral_mow: fixes a :: "'ag" assumes "a \<in> U" shows "a +\<^sub>o\<^sub>w 0\<^sub>o\<^sub>w = a"
left_commute: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry"   and c :: "'b DBMEntry" shows "dbm_add b (dbm_add a c) = dbm_add a (dbm_add b c)"
list_decomp_1: fixes l :: "'c list" assumes "length l = 1" shows "\<exists>a. l = [a]"
assms_7: shows "list_all (\<lambda>p. prime p \<and> coprime (b ^ (q div p) mod n - 1) n) ps"
is_nnf_cnf: fixes F :: "'b formula" assumes "is_cnf F" shows "is_nnf F"
hf_of_option_def: shows "hf_of \<equiv> case_option 0 (\<lambda>x. \<lbrace>hf_of x\<rbrace>)"
mdqd: shows "(\<exists>x. Ifm (real_of_int x # bs) p) = (Ifm (real_of_int i # bs) (evaldjf (\<lambda>j. simpfm (subst0 (C j) (simpfm (minusinf (fst (chooset p)))))) [1..snd (snd (chooset p))]) \<or> Ifm (real_of_int i # bs) (evaldjf (stage (fst (chooset p)) (snd (snd (chooset p)))) (fst (snd (chooset p)))))"
order_subst1: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and b :: "'d"   and c :: "'d" assumes "a \<le> f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
open_lessThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.order_topology less_eq less open" shows "open (ord.lessThan less a)"
e': shows "xs @ rs = aform_vals (e(max (degrees (map snd XS @ map (snd \<circ> pdevs_of_real) rs)) (degree_aform_err Y) := err)) (XS @ map pdevs_of_real rs)"
block: shows "a \<in> A \<and> A \<in> P"
I_def: shows "I \<equiv> \<lambda>l s. (\<exists>k. s ''r'' - s ''l'' = 2 ^ k) \<and> 0 \<le> s ''l''"
dpd3: fixes x :: "'a" shows "d x \<le> (1::'a)"
cinner_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and r :: "complex"   and x :: "'a"   and y :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (scaleC r x) y = cnj r * cinner x y"
distinct_7: fixes x1 :: "'f"   and x51 :: "('d, 'f) ODE"   and x52 :: "('d, 'e, 'f) formula" shows "$\<alpha> x1 \<noteq> EvolveODE x51 x52"
class_add_field: fixes P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> C\<^sub>0 sees F,b:T in D"   and "\<not> is_class P C" shows "field P C\<^sub>0 F = field (class_add P (C, cdec)) C\<^sub>0 F"
image_subset_iff: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" shows "order.greater_eq B (f ` A) = (\<forall>x\<in>A. f x \<in> B)"
cringE_2: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
O_sets: fixes i :: "nat" shows "\<Union> (Q' ` {..i}) \<in> sets M"
dupe_monic_int_def: fixes m :: "int"   and D :: "int poly"   and H :: "int poly"   and S :: "int poly"   and T :: "int poly"   and U :: "int poly" shows "poly_mod.dupe_monic_int m D H S T U = (case pdivmod_monic (poly_mod.Mp m (T * U)) D of (q, r) \<Rightarrow> (poly_mod.Mp m (S * U + H * q), poly_mod.Mp m r))"
funsignature_abbrevE: fixes f :: "'a"   and S :: "('b, 'a) Signature"   and A :: "'b"   and B :: "'b" assumes "f \<in> Sig S : A \<rightarrow> B" obtains "f \<in> Fn\<^bsub>S\<^esub>" and "A \<in> Ty\<^bsub>S\<^esub>" and "B \<in> Ty\<^bsub>S\<^esub>" and "sDom\<^bsub>S\<^esub> f = A" and "sCod\<^bsub>S\<^esub> f = B" and "Signature S"
simps_3: fixes x :: "'a"   and y :: "'a" shows "(Some x \<le> Some y) = (x \<le> y)"
lift_Suc_mono_less_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" assumes "class.order less_eq less"   and "\<And>n. less (f n) (f (Suc n))" shows "less (f n) (f m) = (n < m)"
dnf_to_bool_set: fixes \<gamma> :: "'b \<Rightarrow> bool"   and dnf :: "'b negation_type list list" shows "dnf_to_bool \<gamma> dnf = (\<exists>d\<in>set dnf. cnf_to_bool \<gamma> d)"
fcard_fminus_fsingleton_if: fixes A :: "'b fset"   and x :: "'b" shows "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
sset_smerge: fixes ss :: "'a stream stream" shows "i.R (smerge ss) = \<Union> (i.R ` i.R ss)"
eq: shows "tensor_mat m1 (matrix_sum d2 f (Suc n)) = tensor_mat m1 (f n) + tensor_mat m1 (matrix_sum d2 f n)"
subsumption: fixes t1 :: "transition"   and t2 :: "transition"   and r :: "nat \<Rightarrow>f value option" assumes "Label t1 = Label t2 \<and> Arity t1 = Arity t2"   and "\<forall>i. can_take_transition t1 i r \<longrightarrow> can_take_transition t2 i r"   and "\<forall>i. can_take_transition t1 i r \<longrightarrow> evaluate_outputs t1 i r = evaluate_outputs t2 i r"   and "\<forall>p1 p2 i. posterior_separate (Arity t1) (Guards t1) (Updates t2) i r = Some p2 \<longrightarrow> posterior_separate (Arity t1) (Guards t1) (Updates t1) i r = Some p1 \<longrightarrow> (\<forall>P r'. p1 $ r' = None \<or> (P (p2 $ r') \<longrightarrow> P (p1 $ r')))" shows "subsumes t2 r t1"
not_ordLess_ordIso: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r <o r'" shows "(r, r') \<notin> ordIso"
fls_divide_base_factor_denom: fixes f :: "'a fls"   and g :: "'a fls" shows "f / fls_base_factor g = fls_shift (- fls_subdegree g) (f / g)"
bdia_def: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_antirange_kleene_algebra.bdia x y = rel_ar (rel_ar (y ; x))"
assms: shows "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"
Abs_N_inverse: fixes y :: "'b" assumes "y \<in> N" shows "Rep_N (Abs_N y) = y"
nths_all: fixes xs :: "'a list"   and I :: "nat set" assumes "\<forall>i<length xs. i \<in> I" shows "nths xs I = xs"
lNact_active: fixes n :: "nat"   and cid :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<forall>k<n. \<parallel>cid\<parallel>\<^bsub>t k\<^esub>" shows "\<langle>cid \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> = n"
oquality_increases_nsqn_fresh: fixes i :: "nat" shows "opaodv i \<Turnstile>\<^sub>A (otherwith quality_increases {i} (orecvmsg msg_fresh), other quality_increases {i} \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<sigma>, uu_), uu_, \<sigma>', uu_). \<forall>j. quality_increases (\<sigma> j) (\<sigma>' j))"
IH: shows "cbelow_on A ord c (\<leftrightarrow>) a b0"
Gamma_series_nonpos_Ints_LIMSEQ: fixes z :: "'a" assumes "z \<in> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma_series z \<longlonglongrightarrow> (0::'a)"
assms: fixes Q :: "pi" shows "(Q \<parallel> \<zero>, Q) \<in> Rel"
bernoulli_odd_eq_0: fixes n :: "nat" assumes "n \<noteq> 1"   and "odd n" shows "bernoulli n = 0"