hcju/isabelleps huggingface dataset:
comm_semiring_axioms_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.comm_semiring_axioms plus times \<equiv> \<forall>a b c. times (plus a b) c = plus (times a c) (times b c)"
eq: shows "LINT x|MX. indicat_real A x / Sigma_Algebra.measure MX A * log b (indicat_real A x / Sigma_Algebra.measure MX A) = LINT x|MX. - log b (Sigma_Algebra.measure MX A) / Sigma_Algebra.measure MX A * indicat_real A x"
term_subst_eq: fixes t :: "('e, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term"   and \<tau> :: "'d \<Rightarrow> ('e, 'f) Term.term" assumes "\<And>x. x \<in> vars_term t \<Longrightarrow> \<sigma> x = \<tau> x" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
IH: fixes b :: "'b \<Rightarrow> 'a option" assumes "b x = Some y \<or> (\<exists>m\<in>set ms. m x = Some y)"   and "\<forall>m'\<in>set ms. m' x = Some y \<or> m' x = None" shows "foldl (++) b ms x = Some y"
st_part_Ex: fixes x :: "real star" assumes "x \<in> HFinite" shows "\<exists>t\<in>\<real>. x \<approx> t"
maybe_abs_def: shows "maybe_abs \<equiv> PRJ('c\<cdot>maybe) oo EMB(one \<oplus> 'c\<^sub>\<bottom>)"
map_fmap_rel_def: shows "map_fmap_rel = br map_of_fmap (\<lambda>a. finite (dom a))"
span_def: fixes S :: "'c set" shows "span S = {lincomb a A |a A. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R}"
4: shows "D ! ladder_n L 0 = (ladder_i L 1, rule)"
antipar_1: shows "trg (trg \<mu>) = src (trg \<mu>)"
enum_all_bool_def: fixes P :: "bool \<Rightarrow> bool" shows "enum_all P = (P False \<and> P True)"
d_indistinguishable_def: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and e1 :: "'a"   and e2 :: "'a" shows "Strongly_Secure_Programs.d_indistinguishable E DA d e1 e2 \<equiv> \<forall>m m'. Strong_Security.d_equal DA d m m' \<longrightarrow> E e1 m = E e2 m'"
3: shows "(cmod (nu / du - nv / dv))\<^sup>2 / ((1 - (cmod nu / cmod du)\<^sup>2) * (1 - (cmod nv / cmod dv)\<^sup>2)) = (cmod d)\<^sup>2 / ((1 - (cmod u')\<^sup>2) * (1 - (cmod v')\<^sup>2))"
mgu_vars_bounded: fixes M :: "('b, 'a) Term.term"   and N :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu M N = Some \<sigma>" shows "subst_domain \<sigma> \<union> range_vars \<sigma> \<subseteq> fv M \<union> fv N"
sum_nonzero_representation_eq: fixes basis :: "'a set"   and v :: "'a" assumes "rational_vector.independent basis"   and "v \<in> rational_vector.span basis" shows "(\<Sum>b | rational_vector.representation basis v b \<noteq> 0. rational_vector.representation basis v b *R b) = v"
ide_in_hom: fixes a :: "'a" shows "S.ide a = S.in_hom a a a"
ex_Maxf: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>m\<in>S. \<forall>s\<in>S. f s \<le> f m"
dom_fmlookup: fixes m :: "('c, 'd) fmap" shows "dom (fmlookup m) = fmdom' m"
hash_tree_def: fixes h :: "'c \<Rightarrow> 'd" shows "hash_tree h = hash_rt_tree' \<circ> map_rose_tree\<^sub>m h id"
rank_conjugate_le: fixes A :: "'a mat"   and d :: "nat" assumes "A \<in> carrier_mat n d" shows "rank (conjugate A) \<le> rank A"
b: shows "g \<in> mor\<^bsub>C\<^esub>"
fconverseE: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>" obtains a :: "V" and b :: "V" where "x = [b, a]\<^sub>\<circ>" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
prems_8: shows "ms'' \<noteq> [] \<longrightarrow> (\<exists>mx'. call_of_return_node (last ms'') mx' \<and> mx' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>)"
l'_def: fixes F' :: "'c filter"   and g' :: "'c \<Rightarrow> 'b" shows "l' F' g' = {f. \<forall>c>0. \<forall>\<^sub>F x in F'. R (norm (f x)) (c * norm (g' x))}"
abs_minus_le_zero: fixes abs :: "'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus (abs a)) zero"
sees_method_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
assms_1: shows "wf_matchP_aux \<sigma> V n R I r aux ne"
simps_1: fixes sprFst :: "('b, 'c, 'd) BEState"   and sprLst :: "('b, 'c, 'd) BEState"   and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more :: "'e" shows "sprFst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprFst"
preserves_reflects_arr: fixes f :: "'a" shows "\<tau>.B.arr (\<nu> f) = \<tau>.A.arr f"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(VNat x1 = VNat y1) = (x1 = y1)"
0: shows "\<forall>x\<in>one_chain1. case x of (k, \<gamma>) \<Rightarrow> real_of_int k * line_integral F basis \<gamma> = (case f x of (k, \<gamma>) \<Rightarrow> real_of_int k * line_integral F basis \<gamma>) \<and> line_integral_exists F basis \<gamma>"
subset_closure: fixes s :: "'a set" shows "s \<subseteq> closure s"
cases: fixes a1 :: "'b com option list"   and a2 :: "'b"   and a3 :: "'b com option list"   and a4 :: "'b" assumes "par_ctranp a1 a2 a3 a4" obtains i :: "nat" and Ps :: "'b com option list" and s :: "'b" and r :: "'b com option" and t :: "'b" where "a1 = Ps" and "a2 = s" and "a3 = Ps[i := r]" and "a4 = t" and "i < length Ps" and "ctranp (Ps ! i, s) (r, t)"
notin_set_remove1: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "x \<notin> set xs" shows "x \<notin> set (remove1 y xs)"
indep_vars_compose2: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and I :: "'b set"   and Y :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'b \<Rightarrow> 'd measure" assumes "indep_vars M' X I"   and "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> M' i \<rightarrow>\<^sub>M N i" shows "indep_vars N (\<lambda>i x. Y i (X i x)) I"
divide_le_cancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a c) (divide b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
strict_trans2: fixes b :: "nat list"   and a :: "nat list"   and c :: "nat list" assumes "b <\<^sub>v a"   and "c \<le>\<^sub>v b" shows "c <\<^sub>v a"
fmadd_empty_1: fixes m :: "('c, 'd) fmap" shows "{$$} ++\<^sub>f m = m"
inj_on_chamber: fixes C :: "'a set" assumes "folding.chamber C" shows "inj_on f C"
subst_atmss_def: fixes AAA :: "'a set set"   and \<sigma> :: "'s" shows "AAA \<cdot>ass \<sigma> = (\<lambda>AA. AA \<cdot>as \<sigma>) ` AAA"
B1_2: fixes a :: "'n"   and b :: "'n" assumes "ball (0::'n) B1 \<subseteq> cbox a b" shows "\<exists>z. ((\<lambda>x. if x \<in> i then f x else (0::'a)) has_integral z) (cbox a b) \<and> norm (z - k1) < norm (k1 - k2) / 2"
inner_axis_axis: fixes i :: "'b"   and x :: "'a"   and j :: "'b"   and y :: "'a" shows "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
root_set_def: fixes les :: "(real \<times> real \<times> real) set" shows "root_set les \<equiv> {x. \<exists>(a, b, c)\<in>les. a * x\<^sup>2 + b * x + c = 0}"
path_independent_on_symmetric: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "path_independent_on A f = (\<forall>B C. B \<subseteq> A \<and> C \<subseteq> A \<longrightarrow> f (B \<union> C) = f (f B \<union> f C))"
dterms_not_call: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "q \<in> dterms \<Gamma> p" shows "not_call q"
left_commute: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" shows "signed.max b (signed.max a c) = signed.max a (signed.max b c)"
case_prod_const: fixes c :: "'f" shows "(\<lambda>(i, j). c) = (\<lambda>_. c)"
count_mset_pos: fixes M :: "'a zmultiset"   and x :: "'a" shows "count (mset_pos M) x = nat (zcount M x)"
sums_SUP: fixes f :: "nat \<Rightarrow> 'a" shows "f sums (SUP n. sum f {..<n})"
castOp_stOf_Op: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp" shows "castOp (stOf delta) (Op delta inp binp) = (delta, inp, binp)"
comp: fixes Z :: "'c set set"   and g :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism Y Z g" shows "ChamberComplexMorphism X Z (g \<circ> f)"
strict_trans1: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "a \<subseteq># b"   and "b \<subset># c" shows "a \<subset># c"
a: shows "\<forall>j. \<exists>i. j < card ((\<tau> ^^ i) {})"
trans_equiv: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b = Equiv"   and "compare cmp b c = Equiv" shows "compare cmp a c = Equiv"
emeasure_eq_measure: fixes A :: "'b set" shows "emeasure MX A = ennreal (Sigma_Algebra.measure MX A)"
omega_coinduct: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<le> z + x \<cdot> y" shows "y \<le> x\<^sup>\<omega> + x\<^sup>\<star> \<cdot> z"
set_mult_def: fixes R :: "('d, 'e) Ring_scheme"   and A :: "'d set"   and B :: "'d set" shows "set_mult R A B = {z. \<exists>x\<in>A. \<exists>y\<in>B. x \<cdot>\<^sub>r\<^bsub>R\<^esub> y = z}"
iso_points_map: shows "\<pi> ` \<V> = \<V>'"
mru_opt_TS_def: shows "mru_opt_TS = \<lparr>init = opt_mru_init, trans = mru_opt_trans\<rparr>"
le: shows "radj f (y \<leftarrow> f x) \<le> radj f y \<leftarrow> x"
irreducible_altdef: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.idom times one plus zero minus uminus" shows "comm_semiring_1.irreducible times one zero x = (x \<noteq> zero \<and> \<not> dvd.dvd times x one \<and> (\<forall>b. dvd.dvd times b x \<longrightarrow> dvd.dvd times x b \<or> dvd.dvd times b one))"
D_ndet: fixes P :: "'c process"   and Q :: "'c process" shows "D (P |-| Q) = D P \<union> D Q"
bv_prepend_equal: fixes n :: "nat"   and b :: "'a"   and l :: "'a list" shows "bv_prepend (Suc n) b l = b # bv_prepend n b l"
ucast_eq: fixes w :: "'c word" shows "ucast w = word_of_int (uint w)"
\<phi>_def: fixes y :: "'d"   and f :: "'c" shows "F.\<phi> y f = (THE g. \<guillemotleft>g : y \<rightarrow>\<^sub>D F.G (C.cod f)\<guillemotright> \<and> F.\<psi> (C.cod f) g = f)"
simps_2: fixes vs :: "'b list"   and bs :: "'b list" shows "Ifm vs bs F = False"
less_eq_synlr_def: fixes x :: "('a, 'b) synlr"   and y :: "('a, 'b) synlr" shows "x \<le> y \<equiv> unsynlr x \<subseteq> unsynlr y"
get_child_nodes_pure: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "pure (get_child_nodes ptr) h"
set_replicate: fixes n :: "nat"   and x :: "'b" assumes "n \<noteq> 0" shows "set (replicate n x) = {x}"
coeff_map_poly_hom: fixes p :: "'c poly"   and i :: "nat" shows "coeff (map_poly (\<lambda>a. [:a:]) p) i = [:coeff p i:]"
simps: fixes \<C> :: "t_context"   and es :: "b_e list"   and ts :: "checker_type" shows "check \<C> es ts = (case es of [] \<Rightarrow> ts | e # es \<Rightarrow> case ts of Bot \<Rightarrow> Bot | _ \<Rightarrow> check \<C> es (check_single \<C> e ts))"
real_cond_exp_jensens_inequality_2: fixes X :: "'a \<Rightarrow> real"   and I :: "real set"   and a :: "real"   and b :: "real"   and q :: "real \<Rightarrow> real" assumes "integrable M X"   and "AE x in M. X x \<in> I"   and "I = {a<..<b} \<or> I = {a<..} \<or> I = {..<b} \<or> I = UNIV"   and "integrable M (\<lambda>x. q (X x))"   and "convex_on I q"   and "q \<in> borel_measurable borel" shows "AE x in M. q (real_cond_exp M F X x) \<le> real_cond_exp M F (\<lambda>x. q (X x)) x"
len': shows "\<forall>y\<in>set as. length y = q"
iT_Div_card_le: fixes k :: "nat"   and I :: "nat set" assumes "0 < k" shows "card (I \<oslash> k) \<le> card I"
finite_UNIV_expr'_def: shows "finite_UNIV = Phantom('a expr') False"
a_kernel_def': fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" shows "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
all_before_eq_1: fixes c :: "'b" shows "$#c \<equiv> #c"
inCC_eqvt: fixes \<pi> :: "perm"   and x :: "Vars.var"   and y :: "Vars.var"   and G :: "CoCalls" shows "\<pi> \<bullet> x--y\<in>G = (\<pi> \<bullet> x)--(\<pi> \<bullet> y)\<in>(\<pi> \<bullet> G)"
assms_1: shows "\<forall>x\<ge>1. g x = dirichlet_prod' a f x"
execn_Normal_elim_cases_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Skip,Normal s\<rangle> =n\<Rightarrow> t" obtains "t = Normal s"
t_decomp: shows "t = tm_comp prec ord I a (mid (compute_bound_tm prec a a tg)) tf tg"
entails_triv: fixes A :: "assn" shows "A \<Longrightarrow>\<^sub>A A"
l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root ShadowRootClass.type_wf get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "[] \<noteq> x21 \<Colon> x22"
ipv4range_UNIV_def: shows "ipv4range_UNIV \<equiv> wordinterval_UNIV"
subdegree_le_imp_dvd_left_ring1: fixes f :: "'a fps"   and g :: "'a fps" assumes "\<exists>y. fps_nth f (subdegree f) * y = (1::'a)"   and "subdegree f \<le> subdegree g" shows "\<exists>k. g = f * k"
lunit_simps_1: fixes f :: "'b" assumes "D.ide f" shows "D.arr \<l>\<^sub>D[f]"
comp_red_basis_aux_dgrad_p_set_le: fixes d :: "'a \<Rightarrow> nat"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "dickson_grading d" shows "dgrad_p_set_le d (set (comp_red_basis_aux xs ys)) (set xs \<union> set ys)"
frechet_derivative_compose: fixes g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and f :: "'d \<Rightarrow> 'e" assumes "g differentiable at x"   and "f differentiable at (g x)" shows "frechet_derivative (f \<circ> g) (at x) = frechet_derivative f (at (g x)) \<circ> frechet_derivative g (at x)"
gR_rel_defined: fixes x :: "'c"   and y :: "'c" assumes "x \<preceq>\<^sub>R y" shows "D x (rquot y x)"
fmran'I: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f"   and y :: "'e" assumes "fmlookup m x = Some y" shows "y \<in> fmran' m"
Plus_def: fixes A :: "'h set"   and B :: "'i set" shows "A <+> B = Inl ` A \<union> Inr ` B"
pre_between_symI: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "pre_between vs ram2 ram1"
that: fixes b :: "'a" assumes "b \<in> S"   and "x \<in> convex hull insert a (S - {b})" shows "thesis"
normal_form_id_iff: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]" shows "(normal_form F p = p) = (\<not> punit.is_red (punit.reduced_GB F) p)"
nn_integral_noteq_infinite: fixes g :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "g \<in> borel_measurable M"   and "integral\<^sup>N M g \<noteq> \<infinity>" shows "AE x in M. g x \<noteq> \<infinity>"
regular_comp: fixes f :: "'b set \<Rightarrow> 'b set"   and g :: "'b set \<Rightarrow> 'b set" assumes "regular f"   and "regular g" shows "regular (g \<circ> f)"
less_le_diff1: fixes n :: "nat"   and m :: "nat" assumes "n \<noteq> 0" shows "(m < n) = (m \<le> n - Suc 0)"
tllist_all2_tmap1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'a"   and g :: "'f \<Rightarrow> 'c"   and xs :: "('e, 'f) tllist"   and ys :: "('b, 'd) tllist" shows "tllist_all2 P Q (tmap f g xs) ys = tllist_all2 (\<lambda>x. P (f x)) (\<lambda>x. Q (g x)) xs ys"
comparable_cases: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" obtains "x \<sqsubseteq> y"   | "y \<sqsubseteq> x"
resPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and x :: "name" assumes "\<Psi> \<rhd> P \<leadsto>[Rel] Q"   and "eqvt Rel'"   and "x \<sharp> \<Psi>"   and "Rel \<subseteq> Rel'"   and "\<And>\<Psi>' R S y. \<lbrakk>(\<Psi>', R, S) \<in> Rel; y \<sharp> \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<leadsto>[Rel'] \<lparr>\<nu>x\<rparr>Q"
sinks_aux_less_single_event: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'c" shows "sinks_aux_less I D U [x] = (if \<exists>u\<in>U. (u, D x) \<in> I then {D x} else {})"
nn_integral_snd: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M x \<partial>M y = integral\<^sup>N (M x \<Otimes>\<^sub>M M y) f"
E_sup_add_left: fixes f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s"   and c :: "ennreal" assumes "f \<in> borel_measurable St" shows "E_sup s (\<lambda>x. f x + c) = E_sup s f + c"
fset_induct: fixes P :: "'b fset \<Rightarrow> bool"   and S :: "'b fset" assumes "P {||}"   and "\<And>x S. P S \<Longrightarrow> P (insert_fset x S)" shows "P S"
v0_V: shows "v0 \<in> V\<^bsub>remove_vertex v\<^esub>"
not_choice_simps_9: fixes l :: "'d"   and fmsg :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}receive(fmsg) . p)"
second_lesbegue_integral_eq: shows "LBINT x. LBINT y. f (x, y) = LBINT x. integral UNIV (\<lambda>y. f (x, y))"
nat_pow_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and n :: "nat" shows "a [^]\<^bsub>G\<^esub> n = rec_nat \<one>\<^bsub>G\<^esub> (\<lambda>u b. b \<otimes>\<^bsub>G\<^esub> a) n"
atLeast_Suc_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..<Suc n} = sum (g \<circ> Suc) {m..<n}"
hom_induced_carrier: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set"   and Y :: "'a topology"   and T :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set" shows "hom_induced p X S Y T f c \<in> carrier (relative_homology_group p Y T)"
tendsto_add_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> (0::'b)) F"   and "(g \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> (0::'b)) F"
assign_lem1: fixes I :: "('sf, 'a, 'sz) interp"   and i :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "dterm_sem I (if i = vid1 then Var vid1 else Const 0) (\<chi>y. if vid1 = y then Functions I fid1 (\<chi>i. dterm_sem I (local.empty i) \<nu>) else fst \<nu> $ y, snd \<nu>) = dterm_sem I (if i = vid1 then $f fid1 local.empty else Const 0) \<nu>"
\<rho>: fixes t :: "'s" assumes "(s, t) \<in> acc_on (- H)" shows "\<rho> t < \<infinity>"
simps_2: fixes p :: "dir list"   and d :: "dir"   and ps :: "(dir list \<times> dir) list"   and t :: "('c, 'd) tm" shows "emb_at ((p, d) # ps) t = emb_step_at p d (emb_at ps t)"
natLeq_ordLeq_cinfinite: fixes r :: "('d \<times> 'd) set" assumes "Cinfinite r" shows "natLeq \<le>o r"
brent_def: shows "brent \<equiv> find_lambda ;; find_mu"
pell_valuation_pos_neg: fixes x :: "int"   and y :: "int" assumes "0 < x"   and "y < 0"   and "solution (x, y)" shows "pell_valuation (x, y) \<in> {0<..<1}"
simps_1: fixes x :: "C list"   and ya :: "C list" shows "(C x = C ya) = (x = ya)"
induct: fixes x1 :: "ccs"   and x2 :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and P :: "ccs \<Rightarrow> ccs \<Rightarrow> bool" assumes "(x1, x2) \<in> bangRel Rel"   and "\<And>Pa Q. (Pa, Q) \<in> Rel \<Longrightarrow> P !Pa !Q"   and "\<And>R T Pa Q. \<lbrakk>(R, T) \<in> Rel; (Pa, Q) \<in> bangRel Rel; P Pa Q\<rbrakk> \<Longrightarrow> P (R \<parallel> Pa) (T \<parallel> Q)" shows "P x1 x2"
Collect_enabled_S: fixes x :: "'s" shows "Collect (enabled x) \<in> sets (stream_space (count_space UNIV))"
bigo_abs2: fixes f :: "'a \<Rightarrow> 'b" shows "f \<in> O(\<lambda>x. \<bar>f x\<bar>)"
simps_2: fixes \<gamma> :: "'b \<Rightarrow> 'a set"   and a :: "'b" shows "\<gamma>_option \<gamma> (Some a) = \<gamma> a"
nonprimitive_dchar_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" shows "nonprimitive_dchar n \<chi> \<equiv> dcharacter n \<chi> \<and> nonprimitive_dchar_axioms n \<chi>"
Cons: fixes xs :: "'a list"   and x2 :: "'a"   and x1 :: "'a" assumes "P xs"   and "P (x2 # xs)" shows "P (x1 # x2 # xs)"
d_nu: fixes x :: "'a"   and y :: "'a" shows "\<nu> (d x \<cdot> y) = d x \<cdot> \<nu> y"
assms_3: shows "\<forall>B\<in>M. B \<inter> L \<noteq> {}"
in_set_conv_decomp_last: fixes x :: "'a"   and xs :: "'a list" shows "x mem xs = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
L0: fixes lholed :: "Lholed"   and es :: "e list" assumes "lholed = LBase [] []" shows "Lfilled_exact 0 lholed es es"
simps_1: fixes i :: "'d" shows "poly_of (PNum i) = (if i = (0::'d) then [] else [(1, i)])"
sets_of_coll_def: fixes X :: "'a set" shows "sets_of_coll X = SPEC (\<lambda>XS. X = \<Union> XS)"