hcju/isabelleps huggingface dataset:
store_sops_issue_expr: fixes t :: "nat"   and e :: "expr" shows "store_sops (issue_expr t e) = {}"
antisym: fixes a :: "'b list"   and b :: "'b list" assumes "a \<le>p b"   and "b \<le>p a" shows "a = b"
add_set_single_nonempty1: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<exists>x. x \<in> add_set f {a} n"
wivs_index_def: shows "wivs_index \<equiv> dflt_size - 1"
awalk_def: fixes u :: "'c"   and p :: "'d list"   and v :: "'c" shows "H.awalk u p v \<equiv> u \<in> verts (app_iso hom G) \<and> set p \<subseteq> arcs (app_iso hom G) \<and> H.cas u p v"
CR_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>a b c. \<lbrakk>a \<in> A; (a, b) \<in> r\<^sup>*; (a, c) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (b, c) \<in> r\<^sup>\<down>" shows "CR_on r A"
closed_segment_commute: fixes a :: "'c"   and b :: "'c" shows "closed_segment a b = closed_segment b a"
sat_sub_by_sat: fixes c\<^sub>2 :: "('d, 'e) conf"   and c\<^sub>1 :: "('d, 'e) conf" assumes "Conf.sat c\<^sub>2"   and "c\<^sub>2 \<sqsubseteq> c\<^sub>1" shows "Conf.sat c\<^sub>1"
assms_3: shows "rep_list (spair p q) \<noteq> 0"
opefSU: shows "openin (top_of_set (f ` S)) U"
val_Zp_mult: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)" shows "padic_integers.val_Zp p (x \<otimes>\<^bsub>padic_int p\<^esub> y) = padic_integers.val_Zp p x + padic_integers.val_Zp p y"
cartesian_product: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and B :: "'e set"   and A :: "'d set" shows "(\<Sum>x\<in>A. sum (g x) B) = (\<Sum>(x, y)\<in>A \<times> B. g x y)"
strict_coboundedI1: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "c \<subset># a" shows "c \<subset># a \<union># b"
Least_equality: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y" shows "ord.Least less_eq P = x"
ext_irrefl_before_trans_axioms: shows "ext_irrefl_before_trans msetext_huet"
Abs_fresh_iff_3: fixes a :: "atom"   and cs :: "atom list"   and x :: "'a" shows "a \<sharp> [cs]lst. x = (a \<in> set cs \<or> a \<notin> set cs \<and> a \<sharp> x)"
not_eq_order_implies_strict: fixes a :: "complex"   and b :: "complex" assumes "a \<noteq> b"   and "b \<le>\<^sub>\<complex> a" shows "b <\<^sub>\<complex> a"
Gcd_eqI: 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"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "normalize a = a"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times a b"   and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times c b) \<Longrightarrow> dvd.dvd times c a" shows "Gcd A = a"
assms_3: fixes s1 :: "'s1"   and s2 :: "'s2"   and x :: "'a" assumes "X s1 s2"   and "x \<in> outs_\<I> \<I>" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"
minor: fixes x :: "'a" assumes "P x"   and "x \<in> S" shows "Q x"
vec_to_iarray_show_vec_real: fixes v :: "(real, 'a) vec" shows "vec_to_iarray (show_vec_real v) = show_vec_real_iarrays (vec_to_iarray v)"
W'_def: shows "W' \<equiv> corthogonal_inv W"
subset_spec_coll_def: fixes X :: "'c set"   and ivl :: "'c set" shows "subset_spec_coll X ivl = (sets_of_coll X ::: \<langle>\<langle>appr_rel\<rangle>list_wset_rel\<rangle>nres_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN True) (\<lambda>X. op_subset X ivl) (\<lambda>x y. RETURN (x \<and> y)))"
fdia_export_2: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "\<langle>x| z \<cdot> ar y = \<langle>x \<cdot> ar y| z"
addr_in_ipset_from_netmask_code: fixes addr :: "'a word"   and base :: "'a word"   and netmask :: "'a word" shows "(addr \<in> ipset_from_netmask base netmask) = (base && netmask \<le> addr \<and> addr \<le> base && netmask || ~~ netmask)"
finTrace_imp_MtransT: fixes tr :: "(('test, 'atom) com \<times> 'state) list \<times> 'state" assumes "finTrace tr" shows "iconfig tr \<rightarrow>*t fstate tr"
sum_norm_le: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (comm_monoid_add.sum plus zero f S) \<le> sum g S"
sumhr_eq_bounds: fixes n :: "nat star"   and f :: "nat \<Rightarrow> real" shows "sumhr (n, n, f) = 0"
preserves_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.dom (B.HoHV f) = B.HoHV (B.VVV.dom f)"
omega_coinduct_var1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> (1::'a) + x \<cdot> y" shows "y \<le> x\<^sup>\<omega> + x\<^sup>\<star>"
monad_fail_writerT': assumes "monad_fail Monad_Overloading.return Monad_Overloading.bind fail" shows "monad_fail Monad_Overloading.return Monad_Overloading.bind fail"
sel_1: fixes xs :: "('a, 'b) tllist" assumes "\<not> is_TNil xs" shows "lhd (llist_of_tllist xs) = (case xs of TCons x xs' \<Rightarrow> x)"
flip_at_base_simps_2: fixes a :: "'a"   and b :: "'a" assumes "sort_of (atom a) = sort_of (atom b)" shows "(a \<leftrightarrow> b) \<bullet> b = a"
norm_prod: fixes a :: "'a"   and b :: "'b" shows "norm (a ** b) \<le> norm a * norm b * K"
dvd_divide_Ints_iff: fixes b :: "int"   and a :: "int" shows "(b dvd a \<or> b = 0) = (of_int a / of_int b \<in> \<int>)"
is_meetI: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y"   and "\<And>a. \<lbrakk>a \<sqsubseteq> x; a \<sqsubseteq> y\<rbrakk> \<Longrightarrow> a \<sqsubseteq> z" shows "x \<sqinter> y = z"
dist_norm: fixes x :: "'c"   and y :: "'c" shows "dist x y = norm (x - y)"
valid_frac_def: fixes X :: "('a \<times> 'a) set" shows "cq.valid_frac X \<equiv> \<exists>r\<in>R. \<exists>s\<in>R\<setminus>I. cq.frac r s = X"
p_res_ring_0': fixes x :: "int" assumes "x \<in> carrier (residue_ring (p ^ 0))" shows "x = 0"
real_alg_2'_def: fixes ri :: "root_info"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "real_alg_2' ri p l r = (if degree p = 1 then Rational (Rat.Fract (- coeff p 0) (coeff p 1)) else real_alg_2_main ri (case tighten_poly_bounds_for_x p 0 l r (sgn (ipoly p r)) of (l', r', sr') \<Rightarrow> (p, l', r')))"
overrider_method_fun: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list"   and mthd' :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs'' :: "char list list" assumes "P \<turnstile> (C, Cs) has overrider M = mthd via Cs'"   and "P \<turnstile> (C, Cs) has overrider M = mthd' via Cs''" shows "mthd = mthd' \<and> Cs' = Cs''"
splitDigit_def: fixes p :: "'b \<Rightarrow> bool"   and i :: "'b"   and d :: "('a, 'b) Digit" shows "FingerTreeStruc.splitDigit p i d = FingerTreeStruc.splitNlist p i (FingerTreeStruc.digitToNlist d)"
IH: fixes c :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "c k \<noteq> (0::'a)"   and "k \<le> m" shows "finite {z. (\<Sum>i\<le>m. c i * z ^ i) = (0::'a)} \<and> card {z. (\<Sum>i\<le>m. c i * z ^ i) = (0::'a)} \<le> m"
set_eq_fold: fixes x :: "'a"   and xs :: "'a list" shows "Max (set (x # xs)) = fold max xs x"
prems_2: shows "\<I> \<turnstile>g gpv \<surd>"
index_unit_vec_3: fixes n :: "nat"   and i :: "nat" shows "dim_vec (unit\<^sub>v n i) = n"
has_integral_spike: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "(f has_integral y) T" shows "(g has_integral y) T"
lt_zero: shows "i.lpp 0 = i.punit.min_term"
le_iff_plus_unchanged: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = (x \<squnion>\<^bsub>f\<^esub> y = y)"
tensor_compose_distribution2: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "mat (row_length A1) (length A1) A1"   and "mat (row_length A2) (length A2) A2"   and "mat (row_length B1) (length B1) B1"   and "mat (row_length B2) (length B2) B2"   and "length A1 = row_length A2"   and "length B1 = row_length B2"   and "A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []" shows "mat (row_length A1 * row_length B1) (length A2 * length B2) ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2))"
swap_4: fixes k :: "nat" assumes "k < m" shows "\<parallel>fs'.gs.gso k\<parallel>\<^sup>2 = (if k = k - 1 then \<parallel>fs.gs.gso k\<parallel>\<^sup>2 + fs.gs.\<mu> k (k - 1) * fs.gs.\<mu> k (k - 1) * \<parallel>fs.gs.gso (k - 1)\<parallel>\<^sup>2 else if k = k then \<parallel>fs.gs.gso k\<parallel>\<^sup>2 * \<parallel>fs.gs.gso (k - 1)\<parallel>\<^sup>2 / \<parallel>fs'.gs.gso (k - 1)\<parallel>\<^sup>2 else \<parallel>fs.gs.gso k\<parallel>\<^sup>2)"
combine_is_incomplete: shows "incomplete_design \<V>\<^sup>+ \<B>\<^sup>+ \<k>"
treap_of_unique: fixes t :: "('a \<times> 'b) tree" assumes "treap t"   and "inj_on snd (set_tree t)" shows "treap_of (set_tree t) = t"
trimmed_neg_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "multiseries.trimmed_neg trimmed dominant_term F = (trimmed F \<and> fst (dominant_term F) < 0)"
add_basis_sorted_def: fixes rel :: "nat \<times> 'h \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ns :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" shows "add_basis_sorted rel gs bs ns data = merge_wrt (rel data) bs ns"
inf_compl_bot: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (uminus x) = bot"
rat_real_conversion: fixes cs :: "constraint set" assumes "finite cs" shows "(\<exists>v. v \<Turnstile>\<^sub>c\<^sub>s cs) = (\<exists>v. v \<Turnstile>\<^sub>r\<^sub>c\<^sub>s cs)"
isLim_def: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r" shows "isLim r i \<equiv> \<not> isSucc r i"
geod_mult: fixes x0 :: "'b"   and y0 :: "'b"   and n :: "nat"   and a :: "nat"   and k :: "nat" assumes "complete_space_with_middle TYPE('b)" shows "complete_space_with_middle.geod x0 y0 n a = complete_space_with_middle.geod x0 y0 (n + k) (a * 2 ^ k)"
matched_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''" shows "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns') n'"
write'': shows "justifying_ws (J (Suc n)) a'' \<in> write_actions (justifying_exec (J (Suc n)))"
fls_lr_inverse_X_inv_power'_2: fixes n :: "nat"   and y :: "'a" shows "fls_right_inverse (fls_X_inv ^ n) y = fls_const y * fls_X ^ n"
remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "Max A = (if A - {x} = {} then x else max x (Max (A - {x})))"
map_aux_AboveS_not_empty: fixes S :: "'a Bonk_Schramm_extension_unfolded set" assumes "map_aux ` S \<subseteq> Field wo_aux" shows "AboveS wo_aux (map_aux ` S) \<noteq> {}"
\<Phi>o_in_hom: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "\<guillemotleft>\<Phi>o yx : Hom_FopxC.map yx \<rightarrow>\<^sub>S Hom_DopxG.map yx\<guillemotright>"
UNION_empty_conv_2: fixes B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "(\<Union> (B ` A) = {}) = (\<forall>x\<in>A. B x = {})"
binrelchain_propcong_Cons_snoc: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "\<And>x y. \<lbrakk>Q x; P x y\<rbrakk> \<Longrightarrow> Q y"   and "Q x"   and "binrelchain P (x # xs @ [y])" shows "Q y"
nat_of_char_inject: fixes x :: "char"   and y :: "char" shows "(of_char x = of_char y) = (x = y)"
check_unsat_id: fixes s :: "('i, 'a) state" assumes "\<U> s" shows "check s = s"
s_nc_U: fixes x :: "'a" shows "1\<^sub>\<sigma> \<sqinter> x \<cdot> nc = 1\<^sub>\<sigma> \<sqinter> x \<cdot> U"
RN_deriv_finite: fixes N :: "'a measure" assumes "sigma_finite_measure N"   and "absolutely_continuous (density M (\<lambda>x. 1 + h x)) N"   and "sets N = sets (density M (\<lambda>x. 1 + h x))" shows "AE x in density M (\<lambda>x. 1 + h x). RN_deriv (density M (\<lambda>x. 1 + h x)) N x \<noteq> \<infinity>"
bij_\<phi>: shows "bij_betw \<phi> S.Univ S'.Univ"
rel_refl_strong: fixes x :: "'c stream"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> sset x \<Longrightarrow> Ra z z" shows "stream_all2 Ra x x"
HBL2_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}" shows "bprv (imp (PP \<langle>imp \<phi> \<chi>\<rangle>) (imp (PP \<langle>\<phi>\<rangle>) (PP \<langle>\<chi>\<rangle>)))"
qo_on_imp_reflp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "qo_on P A" shows "reflp_on P A"
h2: shows "\<forall>i<deg + 1. MPoly_Type.degree (g i) var = 0"
tuples'_def: shows "tuples' = Sigma Roots_ms tuples"
ppa_def: fixes R :: "'d"   and P :: "nat \<Rightarrow> 'c set"   and A :: "'c set"   and i :: "nat"   and l :: "nat" shows "ppa R P A i l = (SOME x. x \<in> A \<and> x \<in> P (skip i l) \<and> x \<notin> P i)"
gL_gH_rel: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pas times D" shows "partial_semigroup.gL_rel times D x y = partial_semigroup.gH_rel times D x y"
mult_minus_distrib_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat"   and C :: "'a mat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "C \<in> carrier_mat n nc" shows "A * (B - C) = A * B - A * C"
field_differentiable_within_csqrt: fixes z :: "complex"   and s :: "complex set" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "csqrt field_differentiable at z within s"
four_one_ind_half: fixes \<delta>S :: "real"   and i :: "nat"   and q :: "nat"   and p :: "nat" assumes "\<beta> \<le> rmin"   and "\<mu> \<le> \<delta>S"   and "\<gamma>1 \<delta>S \<le> \<delta>S"   and "okmaxsync i \<delta>S"   and "te q (i + 1) \<le> te p (i + 1)"   and "correct p (te p (i + 1))"   and "correct q (te p (i + 1))" shows "\<bar>IC p (i + 1) (te p (i + 1)) - IC q (i + 1) (te p (i + 1))\<bar> \<le> \<delta>S"
assms_4: shows "S1 \<subseteq> rel_frontier S2"
fst_image_mp: fixes A :: "('e \<times> 'f) set"   and B :: "'e set"   and x :: "'e"   and y :: "'f" assumes "fst ` A \<subseteq> B"   and "(x, y) \<in> A" shows "x \<in> B"
g_def: shows "g = (\<lambda>i. (f i, f i))"
fpairwiseD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a fset"   and m :: "'a"   and n :: "'a" assumes "fpairwise P M"   and "m |\<in>| M"   and "n |\<in>| M" shows "P m n"
abs_to_eq: fixes x :: "'c \<Rightarrow> 'c"   and y :: "'c \<Rightarrow> 'c" assumes "abs_endofun x = abs_endofun y" shows "x = y"
has_typ1_iff_typ_of1: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ" shows "Ts \<turnstile>\<^sub>\<tau> t : ty = (typ_of1 Ts t = Some ty)"
invI_1: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "x \<otimes> y = \<one>" shows "y = inv x"
simps_3: fixes v :: "('c \<times> ('d, 'e) rbt) list"   and vb :: "color"   and vc :: "('d, 'e) rbt"   and vd :: "'d"   and ve :: "'e"   and vf :: "('d, 'e) rbt" shows "rbt_has_next (v, Branch vb vc vd ve vf) = True"
pow_prop: fixes x :: "'a"   and n :: "nat" shows "p_power (1\<^sub>\<sigma> + x) (n + 1) = 1\<^sub>\<sigma> + x \<cdot> p_power (1\<^sub>\<sigma> + x) n"
neqE: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "semiconnex A less"   and "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" obtains "less x y"   | "less y x"
AsCondExpr1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com"   and a :: "'e set"   and r :: "'e set"   and a2 :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c2 :: "('e, 'd, 'f) com" assumes "assertionsR \<Gamma> \<Theta> Q A a1 c1 a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnBin r a1 a2) (Cond b c1 c2) a"
assms_1: shows "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) g"
inv2: shows "\<exists>hs. rev v @ rev u @ h = [x, x] @ hs"
mignotte_bound_def: fixes f :: "int poly"   and d :: "nat" shows "mignotte_bound f d = (let d' = d - 1; d2 = d' div 2; binom = d' choose d2 in mahler_approximation 2 binom f + int binom * \<bar>lead_coeff f\<bar>)"
Sink_f: fixes s1 :: "'state"   and s2 :: "'state" assumes "{s1, s2} \<subseteq> Sink" shows "f s1 s2"
set_lebesgue_integral_cong: fixes A :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "A \<in> sets M"   and "\<forall>x. x \<in> A \<longrightarrow> f x = g x" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A g"
a: shows "R' \<subseteq> upRules \<and> R = Ax \<union> R'"
differentiable_imp_continuous_within: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set" assumes "f differentiable at x within s" shows "continuous (at x within s) f"
distinct_143: fixes x71 :: "char list"   and x72 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "x71:=x72 \<noteq> Call x101 x102 x103 x104"
shift_def: fixes lab :: "'d list \<Rightarrow> 'e"   and k :: "'d" shows "BNF_Greatest_Fixpoint.shift lab k = (\<lambda>kl. lab (k # kl))"
ide_def: fixes a :: "'a" shows "C.ide a \<equiv> a \<cdot> a \<noteq> C.null \<and> (\<forall>f. (f \<cdot> a \<noteq> C.null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> C.null \<longrightarrow> a \<cdot> f = f))"
goodBinp_imp_qGoodBinp_pickBinp: fixes binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" assumes "goodBinp binp" shows "qGoodBinp (pickBinp binp)"
simps: fixes \<phi> :: "'b stream \<Rightarrow> 'a"   and xs :: "'b stream" shows "nxt \<phi> xs = \<phi> (stl xs)"
assms_2: shows "eint (int n) \<le> val_Zp x"
starfun_o2: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "(\<lambda>x. (*f* f) ((*f* g) x)) = *f* (\<lambda>x. f (g x))"
tdghm_dghm_comp_NTMap_app: fixes x :: "V"   and \<NN> :: "V" assumes "x \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<KK>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<NN>\<lparr>NTMap\<rparr>\<lparr>\<KK>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr>\<rparr>"
wt_compTP_compP2: fixes P :: "'a J1_prog" assumes "wf_J1_prog P" shows "wf_jvm_prog\<^bsub>compTP P\<^esub> (compP2 P)"
that: fixes n :: "nat" assumes "ntrancl_onp F r n a b" shows "thesis"
primes_dvd_imp_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and q :: "'c" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "dvd.dvd times p q" shows "p = q"