hcju/isabelleps huggingface dataset:
filter_filter_mset: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and M :: "'b multiset" shows "filter_mset P (filter_mset Q M) = {#x \<in># M. Q x \<and> P x#}"
assms_2: fixes T :: "'a set" assumes "T \<in> P" shows "T exposed_face_of S"
A'_2: fixes x :: "real" assumes "x \<in> {A<..<A'}" shows "norm (f x) \<le> c * norm (g x)"
assms_2: shows "\<exists>p. NPreflow.isAugmentingPath c s t f p"
ras_set_refine: fixes i' :: "nat"   and l :: "'c list"   and a :: "'d array \<times> nat"   and R :: "('d \<times> 'c) set"   and x :: "'d"   and x' :: "'c"   and i :: "nat" assumes "i' < length l"   and "(a, l) \<in> \<langle>R\<rangle>ras_rel"   and "(x, x') \<in> R"   and "(i, i') \<in> nat_rel" shows "(ras_set a i x, l[i' := x']) \<in> \<langle>R\<rangle>ras_rel"
leI: fixes x :: "'t"   and y :: "'t" assumes "\<not> x \<prec>\<^sub>t y" shows "y \<preceq>\<^sub>t x"
relcomp_mono: fixes r' :: "('a \<times> 'b) set"   and r :: "('a \<times> 'b) set"   and s' :: "('b \<times> 'c) set"   and s :: "('b \<times> 'c) set" assumes "r' \<subseteq> r"   and "s' \<subseteq> s" shows "r' O s' \<subseteq> r O s"
removeAt_def: fixes n :: "nat"   and ls :: "'b list" shows "removeAt n ls = take n ls @ drop (n + 1) ls"
concl_def: fixes "thm" :: "thm" shows "concl thm = (case thm of Thm x \<Rightarrow> x)"
AndL: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
simps_10: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list" shows "Cn x41 x42 x43 \<noteq> Z"
simps: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R" shows "ring.Span R K Us = foldr (ring.line_extension R K) Us {\<zero>\<^bsub>R\<^esub>}"
inv_comp_right_1: fixes g :: "'b"   and f :: "'b" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso f" shows "D.inv (g \<cdot>\<^sub>D f) = D.inv f \<cdot>\<^sub>D D.inv g"
abelian_groupE_5: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_group R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
le_less: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x \<subseteq># y) = (x \<subset># y \<or> x = y)"
DisjunctiveD: fixes x :: "'c \<Rightarrow> 'c"   and X :: "('c \<Rightarrow> 'c) set" assumes "x \<in> Comp.Disjunctive" shows "x \<circ> Sup X = Sup ((\<circ>) x ` X)"
eq_card_imp_inj_on: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "card (f ` A) = card A" shows "inj_on f A"
ratfps_nth_bigo: fixes R :: "real"   and q :: "complex poly"   and k :: "nat"   and p :: "complex poly" assumes "0 < R"   and "\<And>z. z \<in> ball 0 (1 / R) \<Longrightarrow> poly q z \<noteq> 0"   and "\<And>z. \<lbrakk>z \<in> sphere 0 (1 / R); poly q z = 0\<rbrakk> \<Longrightarrow> order z q \<le> Suc k" shows "($) (fps_of_poly p / fps_of_poly q) \<in> O(\<lambda>n. of_nat n ^ k * complex_of_real R ^ n)"
sum_list_subseteq_mset_is_ground_cls_list: fixes Cs :: "'a literal multiset list"   and Ds :: "'a literal multiset list" assumes "sum_list Cs \<subseteq># sum_list Ds"   and "is_ground_cls_list Ds" shows "is_ground_cls_list Cs"
if_P: fixes P :: "bool"   and x :: "'b"   and y :: "'b" assumes "P" shows "(if P then x else y) = x"
assms_3: fixes t :: "real" assumes "t \<in> T" shows "g t \<in> S"
borel_measurable_real_of_ereal: fixes f :: "'b \<Rightarrow> ereal"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
pmf_bernoulli_half: fixes x :: "bool" shows "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2"
filter_map: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
is_aligned_add_step_le: fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" assumes "is_aligned a n"   and "is_aligned b n"   and "a < b"   and "b \<le> a + mask n" shows "False"
knows3: fixes A :: "specID"   and e1 :: "Expression list"   and e2 :: "Expression list" shows "knows A e1 \<and> knows A e2 \<longrightarrow> knows A (e1 @ e2)"
exhaust: fixes y :: "vbl" obtains x :: "nat" where "y = X x"
simps_1: fixes e :: "'b" shows "prefixBeforeElement e [] = []"
simps_2: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat" shows "ir_runs2sigs runz (Commit [A, B] (Kab, Ts, Ta)) = card (ir_commit runz A B Kab Ts Ta)"
set_of_mul_inc: fixes A :: "'a interval"   and A' :: "'a interval"   and B :: "'a interval"   and B' :: "'a interval" assumes "set_of A \<subseteq> set_of A'"   and "set_of B \<subseteq> set_of B'" shows "set_of (A * B) \<subseteq> set_of (A' * B')"
that: fixes s2 :: "('a, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'b wait_set_status option) \<times> 't set" assumes "r2.actions_ok s2 t ta2" shows "thesis"
iT_Plus_neg_Plus_le_cut_eq: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b" shows "I \<oplus>- a \<oplus> b = I \<down>\<ge> a \<oplus> b - a"
SubstTermP_Eats: fixes v :: "tm"   and i :: "tm"   and t1 :: "tm"   and u1 :: "tm"   and t2 :: "tm"   and u2 :: "tm" shows "{SubstTermP v i t1 u1, SubstTermP v i t2 u2} \<turnstile> SubstTermP v i (Q_Eats t1 t2) (Q_Eats u1 u2)"
elementary_cartesian_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and trm :: "'b \<Rightarrow> 'b"   and dup :: "'b \<Rightarrow> 'b" shows "elementary_cartesian_monoidal_category C tensor unity lunit runit assoc trm dup \<equiv> elementary_monoidal_category C tensor unity lunit runit assoc \<and> elementary_cartesian_monoidal_category_axioms C tensor unity lunit runit trm dup"
numadd_nb: fixes t :: "Ferrack.num"   and s :: "Ferrack.num" assumes "numbound0 t"   and "numbound0 s" shows "numbound0 (numadd t s)"
mapp_2_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'd" shows "mapp_2 f p q = Abs_poly_mapping (\<lambda>k. f k (lookup p k) (lookup q k) when k \<in> keys p \<union> keys q)"
fls_times_conv_regpart: fixes f :: "'a fls"   and g :: "'a fls" assumes "0 \<le> fls_subdegree f"   and "0 \<le> fls_subdegree g" shows "fls_regpart (f * g) = fls_regpart f * fls_regpart g"
Lcm_Un: fixes A :: "'a set"   and B :: "'a set" shows "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
i: shows "i < dim_vec (A *\<^sub>v col B j)"
eq_refl: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.preorder (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(x::'b::type) = (y::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)"
simps: fixes v :: "'d" shows "\<lceil>\<lfloor>v\<rfloor>\<rceil> = v"
a: shows "\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> incomparable subseq x y"
1: shows "r !! k \<in> transition (complement A) (w !! k) (m !! k)"
card_of_ordLess: fixes A :: "'a set"   and B :: "'b set" shows "(\<nexists>f. inj_on f A \<and> f ` A \<subseteq> B) = (|B| <o |A|)"
ideD_3: fixes a :: "('obj, 'obj) concrete_category.arr" assumes "ide a" shows "cod a = a"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" obtains c :: "('Var, 'AExp, 'BExp) Stmt" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E__ cb__" and "(cb__, mem) \<leadsto>\<^sub>s (c'b__, mem')" and "c'a__ = cxt_to_stmt E__ c'b__" and "mds' = mds"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = c\<^sub>1__ ;; c\<^sub>2" and "c'a__ = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1__, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>w"   | c :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E (c@[mu])" and "c'a__ = cxt_to_stmt E c'" and "\<langle>c, update_modes mu mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"   | b :: "'BExp" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = Await b c\<^sub>1" and "eval\<^sub>B mem b" and "no_await c\<^sub>1" and "\<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w\<^sup>+ \<langle>c'a__, mds', mem'\<rangle>\<^sub>w" and "is_final c'a__"
rev_exhaust: fixes xs :: "'e list" obtains "xs = []"   | ys :: "'e list" and y :: "'e" where "xs = ys @ [y]"
c12_1: shows "(fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) \<in> conv ars"
normal_invE_1: fixes N :: "'a set set" assumes "N \<lhd> G Mod N" shows "subgroup N (G Mod N)"
parent_child_rel_parent_in_heap: fixes parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "(parent, child_ptr) \<in> Shadow_DOM.parent_child_rel h" shows "parent |\<in>| object_ptr_kinds h"
not_fresher_trans: fixes x :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and y :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and z :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "\<not> x \<sqsubseteq> y"   and "\<not> z \<sqsubseteq> x" shows "\<not> z \<sqsubseteq> y"
tensors_from_net_Conv_id: fixes m :: "real mat convnet"   and i :: "nat"   and nr :: "nat" assumes "valid_net' m"   and "i < nr" shows "tensors_from_net (Conv (id_matrix nr (output_size' m)) m) $ i = (if i < output_size' m then tensors_from_net m $ i else tensor0 (input_sizes m))"
to_bool_eq_0: fixes x :: "'b word" shows "(\<not> to_bool x) = (x = 0)"
that: fixes N :: "int" assumes "\<forall>n<N. f $$ n = (0::'a)" shows "thesis"
vrrestriction_atD_2: fixes Aa :: "V"   and A :: "V"   and a :: "V"   and b :: "V" assumes "((r \<restriction>\<^sub>\<circ> Aa) \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> (r \<restriction>\<^sub>\<circ> Aa) -`\<^sub>\<circ> A" shows "(r \<restriction>\<^sub>\<circ> Aa)\<lparr>a\<rparr> = b"
has_field_derivative_def: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and F :: "'a filter" shows "(f has_field_derivative D) F = (f has_derivative (*) D) F"
has_integral_affinity': fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b"   and m :: "real"   and c :: "'b" assumes "(f has_integral i) (cbox a b)"   and "0 < m" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral i /\<^sub>R m ^ DIM('b)) (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m))"
SUP_sup_distrib: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'c" shows "sup (Sup (f ` A)) (Sup (g ` A)) = (SUP a\<in>A. sup (f a) (g a))"
strict_sorted_list_of_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (linorder.sorted_list_of_set less_eq A)"
fmempty_def: shows "fmempty \<equiv> map_of_fmap Map.empty"
factor_dvd_f_0: fixes p :: "int"   and F :: "int poly"   and l :: "nat"   and u :: "int poly"   and f :: "int poly"   and n :: "nat"   and gs :: "int poly list"   and Degs :: "nat set"   and factor :: "int poly" assumes "prime p"   and "0 < degree F"   and "2 ^ (degree F)\<^sup>2 * B2_LLL F ^ (2 * degree F) \<le> (p ^ l)\<^sup>2"   and "0 < degree u"   and "monic u"   and "poly_mod.irreducible_m p u"   and "poly_mod.dvdm p u f"   and "f dvd F"   and "n \<equiv> degree f"   and "poly_mod.unique_factorization_m (p ^ l) f (lead_coeff f, mset gs)"   and "comm_monoid_mult_class.coprime (lead_coeff f) p"   and "poly_mod.square_free_m p f"   and "square_free f"   and "u \<in> set gs"   and "map (poly_mod.Mp (p ^ l)) gs = gs"   and "\<And>factor. \<lbrakk>factor dvd f; poly_mod.dvdm p u factor\<rbrakk> \<Longrightarrow> degree factor \<in> Degs"   and "factor dvd f" shows "poly_mod.Mp (p ^ l) factor \<noteq> 0"
prems_1: shows "P,E \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>es\<^sub>2,s\<^sub>2\<rangle>"
maximal_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "class.order less_eq less" shows "order.maximal less_eq f S = {x \<in> S. \<forall>y\<in>S. less_eq (f y) (f x)}"
\<ww>\<^sub>B: shows "max_lideal \<ww>\<^sub>B B addB multB zeroB oneB"
correct: fixes x0 :: "'g"   and x1 :: "'a"   and w0 :: "'h"   and w1 :: "'b"   and e :: "'e" assumes "((x0, x1), w0, w1) \<in> Rel_AND"   and "e \<in> challenge_space" shows "\<Sigma>_AND.completeness_game (x0, x1) (w0, w1) e = return_spmf True"
finite_mod_type: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "finite UNIV"
card_union_inter_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "card_fset xs + card_fset ys = card_fset (xs |\<union>| ys) + card_fset (xs |\<inter>| ys)"
is_slim_def: fixes V :: "'a set" shows "is_slim V \<equiv> (\<forall>v\<in>pverts G. v \<in> V \<or> in_degree (with_proj G) v \<le> 2 \<and> (\<exists>x p y. gen_iapath V x p y \<and> v \<in> set (pawalk_verts x p))) \<and> (\<forall>e\<in>parcs G. fst e \<noteq> snd e \<and> (\<exists>x p y. gen_iapath V x p y \<and> e \<in> set p)) \<and> (\<forall>u v p q. gen_iapath V u p v \<and> gen_iapath V u q v \<longrightarrow> p = q) \<and> V \<subseteq> pverts G"
times_complex_interval_tendsto: fixes A :: "nat \<Rightarrow> complex_interval"   and a :: "complex"   and B :: "nat \<Rightarrow> complex_interval"   and b :: "complex" assumes "A \<longlonglongrightarrow>\<^sub>c a"   and "B \<longlonglongrightarrow>\<^sub>c b" shows "(\<lambda>i. A i * B i) \<longlonglongrightarrow>\<^sub>c a * b"
abs_eq: fixes xa :: "'c ltl set"   and x :: "'c ltl" shows "xa \<up>\<Turnstile>\<^sub>P Abs x = xa \<Turnstile>\<^sub>P x"
vector_space_on_axioms: fixes p :: "'a" shows "vector_space_on (sub.tangent_space p) (*\<^sub>R)"
as: shows "\<not> (\<exists>z\<in>S. a \<bullet> z = b)"
grounding_of_grounding_of_cls: fixes E :: "'a literal multiset"   and D :: "'a literal multiset"   and C :: "'a literal multiset" assumes "E \<in> \<G>_F D"   and "D \<in> \<G>_F C" shows "E = D"
vsubsetE: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" obtains "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> x \<in>\<^sub>\<circ> B"
add: fixes u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal" assumes "u \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> u x < top"   and "P u"   and "v \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> v x < top"   and "\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0"   and "P v" shows "P (\<lambda>x. v x + u x)"
mult_gcd_left: 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 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 c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times c (gcd a b) = times (unit_factor c) (gcd (times c a) (times c b))"
fdom_upd: fixes f :: "('a, 'b) ffun"   and k :: "'a"   and v :: "'b" shows "fdom (f(k \<mapsto> v)\<^sub>f) = insert k (fdom f)"
bij_betw_inv_into_left: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A' :: "'d set"   and a :: "'c" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A f (f a) = a"
less_minus_one_simps_3: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less zero (uminus one)"
is_bintree_list_Nil: shows "is_bintree_list 0 []"
widen: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>T" shows "G\<turnstile>S\<preceq>? T"
cod_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.cod f = (if \<F>\<^sub>SC.arr f then cod f else null)"
intro: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "\<And>B N. \<lbrakk>B \<in> Bot; calculus.saturated Inf Red_I N; entails N {B}\<rbrakk> \<Longrightarrow> \<exists>B'\<in>Bot. B' \<in> N" shows "statically_complete_calculus_axioms Bot Inf entails Red_I"
assign: fixes E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and e :: "'b"   and m :: "'c \<Rightarrow> 'a"   and v :: "'a"   and x :: "'c"   and \<iota> :: "nat"   and BMap :: "'a \<Rightarrow> bool" assumes "E e m = v" shows "((x:=\<^bsub>\<iota>\<^esub> e, m), [], None, m(x := v)) \<in> MWLs_semantics.MWLsSteps_det E BMap"
TBtheorem4a_empty: fixes P :: "specID"   and E :: "Expression"   and Q :: "specID"   and PQ :: "specID" assumes "ine P E \<or> ine Q E"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "loc PQ = {}" shows "ine PQ E"
acc_run_run: fixes r :: "nat \<Rightarrow> 'a node" assumes "gba.is_acc_run r" shows "gba.is_run r"
samp_uni_plus_one_time_pad: fixes y :: "nat"   and q :: "nat" shows "map_spmf (\<lambda>b. (y + b) mod q) (sample_uniform q) = sample_uniform q"
intros_3: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Drop], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
gram: fixes k :: "nat" assumes "k \<le> length fs" shows "(\<Prod>j<k. \<parallel>gso j\<parallel>\<^sup>2) = Gramian_determinant fs k"
mat_addcol_def: fixes a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "addcol a k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then a * A $$ (i, l) + A $$ (i, j) else A $$ (i, j))"
abs_eq: fixes xa :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "MPoly xa + MPoly x = MPoly (xa + x)"
singular_relboundary_imp_chain: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S c" shows "singular_chain p X c"
nn_integral_count_space_prod_eq: fixes f :: "'b \<times> 'c \<Rightarrow> ennreal" shows "integral\<^sup>N (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = integral\<^sup>N (count_space UNIV) f"
set_intros_1: fixes x21 :: "'h"   and x22 :: "'h list" shows "x21 \<in> set (x21 # x22)"
preserves_adjoint_pair: fixes f :: "'b"   and g :: "'b" assumes "D.adjoint_pair f g" shows "C.adjoint_pair (F.right_map f) (F.right_map g)"
simps_4: fixes v :: "poscheckvariant"   and va :: "('a, 'b) Term.term"   and vb :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "dual\<^sub>s\<^sub>t (\<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S) = \<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # dual\<^sub>s\<^sub>t S"
wary_args: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "s \<in> set (args t)"   and "wary t" shows "wary s"
right_def: shows "right = (\<lambda>S. \<Sum>x\<in>R S. jumpF h (at_right x))"
map_poly_hom_smult: fixes c :: "rat"   and p :: "rat poly" shows "map_poly of_rat (smult c p) = smult (of_rat c) (map_poly of_rat p)"
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck_1: fixes X :: "'d list"   and F :: "(('e, 'd) Term.term \<times> ('e, 'd) Term.term) list"   and G :: "(('e, 'd) Term.term \<times> ('e, 'd) Term.term) list" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = X"
simps_3: fixes c :: "'b"   and first :: "indices"   and "next" :: "indices list"   and u :: "'b trm"   and S :: "'b trm" shows "replace_subterm (Const c) (first # next) u S = False"
sup_continuous_sup: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "sup_continuous f"   and "sup_continuous g" shows "sup_continuous (\<lambda>x. sup (f x) (g x))"
R: shows "\<nexists>E1 E2. closedin (top_of_set S) E1 \<and> closedin (top_of_set S) E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {}"
if_doesnt_modify: fixes e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and x :: "'Var" shows "doesnt_modify (Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos) x"
rep_eq: fixes x :: "real_int" shows "right x = snd (Rep_real_int x)"
simps_2: fixes st :: "'a"   and u :: "'b"   and c :: "'c" shows "local.getGMState (State st u c) = st"
partition_on_union: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a set set"   and Q :: "'a set set" assumes "A \<inter> B = {}"   and "partition_on A P"   and "partition_on B Q" shows "partition_on (A \<union> B) (P \<union> Q)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x + xa) = int_of_integer x + int_of_integer xa"
zAJ_def: shows "zAJ = restrict (A \<inter> J) zJ"
systemIN_def: fixes x :: "chanID"   and i :: "AbstrLevelsID" shows "systemIN x i \<equiv> (\<exists>C1\<in>AbstrLevel i. x \<in> IN C1) \<and> (\<forall>C2\<in>AbstrLevel i. x \<notin> OUT C2)"
degree_adjacent: fixes v :: "'v" assumes "finite E" shows "degree v G = card {n. adjacent v n}"
run_ask_env: fixes f :: "'e \<Rightarrow> ('e, 'd) envT"   and r :: "'e" shows "run_env (ask_env f) r = run_env (f r) r"
substT_subtype: fixes \<Delta> :: "binding list"   and Q :: "POPLmark.type"   and \<Gamma> :: "binding list"   and S :: "POPLmark.type"   and T :: "POPLmark.type"   and P :: "POPLmark.type" assumes "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> S <: T"   and "\<Gamma> \<turnstile> P <: Q" shows "\<Delta>[0 \<mapsto>\<^sub>\<tau> P]\<^sub>e @ \<Gamma> \<turnstile> S[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> P]\<^sub>\<tau> <: T[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> P]\<^sub>\<tau>"
minus_uexpr_def: fixes u :: "('c, 'd) uexpr"   and v :: "('c, 'd) uexpr" shows "u - v = bop (-) u v"