hcju/isabelleps huggingface dataset:
even_numeral: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (num.Bit0 n))"
AVar: fixes Env :: "env"   and B :: "lname set"   and e1 :: "expr"   and E1 :: "assigned"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1"   and "Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>e1.[e2]\<rangle>\<guillemotright> A"
static_def: fixes s :: "'a list"   and rs :: "'a list" shows "static s rs = (set rs \<subseteq> set s)"
surj_iff_all: fixes f :: "'c \<Rightarrow> 'b" shows "surj f = (\<forall>x. f (inv f x) = x)"
genby_S_reduced_word_for_arg_min: fixes A :: "'w set"   and a :: "'w" assumes "order.greater_eq S A"   and "a \<in> \<langle>A\<rangle>" shows "reduced_word_for A a (arg_min length (word_for A a))"
Sigma_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b set"   and D :: "'a \<Rightarrow> 'b set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
impl_of_distinct: fixes xs :: "('a, 'b) alist" shows "distinct (map fst (alist.impl_of xs))"
typeof_addr_loc_def: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and al :: "addr_loc" shows "typeof_addr_loc P h a al = (THE T. P,h \<turnstile> a@al : T)"
span_induct: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and S :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "module scale"   and "x \<in> module.span scale S"   and "module.subspace scale (Collect P)"   and "\<And>x. x \<in> S \<Longrightarrow> P x" shows "P x"
simps_3: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set" shows "min_gallery (x # xs @ [y]) = (x \<noteq> y \<and> is_arg_min length (\<lambda>zs. gallery (x # zs @ [y])) xs)"
r_result1_converg': fixes i :: "nat"   and x :: "nat"   and v :: "nat" assumes "eval r_phi [i, x] \<down>= v" shows "\<exists>t. (\<forall>t'\<ge>t. eval r_result1 [t', i, x] \<down>= Suc v) \<and> (\<forall>t'<t. eval r_result1 [t', i, x] \<down>= 0)"
assms: fixes h :: "heap" shows "execute x h = None \<or> execute x h = execute y h"
Red_I_def: fixes Inf :: "'a inference set"   and Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N :: "'a set" assumes "calculus_with_standard_redundancy Inf Bot entails" shows "calculus_with_standard_redundancy.Red_I Inf entails N = {\<iota> \<in> Inf. calculus_with_standard_redundancy.redundant_infer entails N \<iota>}"
poly_identity_mod_p: fixes v :: "'b mod_ring poly" shows "v ^ CARD('b) - v = (\<Prod>x\<in>UNIV. v - [:x:])"
minimal_basis_red_rtrancl_nonzero: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis B"   and "p \<in> B"   and "(red (B - {p}))\<^sup>*\<^sup>* p r" shows "r \<noteq> 0"
asym: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subset># a"   and "a \<subset># b" shows "False"
diff_union_single_conv: fixes a :: "'a"   and J :: "'a multiset"   and I :: "'a multiset" assumes "a \<in># J" shows "(I + J) \<ominus> a = I + J \<ominus> a"
assms_2: shows "f k (0::'b) = (0::'c)"
odd_iff_mod_2_eq_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one)"
io_vars_def: fixes p :: "prog" shows "io_vars p \<equiv> \<Gamma>\<^sub>0 \<subseteq> ctxt p"
awp_Join_eq: fixes F :: "'b program"   and G :: "'b program"   and B :: "'b set" shows "awp (F \<squnion> G) B = awp F B \<inter> awp G B"
list_eq_iff_zip_eq: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). x = y))"
valid_def2: fixes P :: "assn"   and c :: "stmt"   and Q :: "assn" shows "\<Turnstile> {P} c {Q} = (\<forall>n. \<Turnstile>n: (P, c, Q))"
R\<^sub>N_alt_def: fixes n :: "nat"   and u :: "nat"   and l :: "nat" shows "R\<^sub>N n u l = map_pmf (\<lambda>f. steps {..<n} f 0 u l) (SL\<^sub>N n)"
interp_eq: fixes f :: "'a \<Rightarrow> (real, 'b) vec \<Rightarrow> real"   and f' :: "'a \<Rightarrow> (real, 'b) vec \<Rightarrow> real"   and p :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> bool"   and p' :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> bool"   and c :: "'c \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set"   and c' :: "'c \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set"   and PP :: "'b \<Rightarrow> (((real, 'b) vec \<times> (real, 'b) vec) \<times> (real, 'b) vec \<times> (real, 'b) vec) set"   and PP' :: "'b \<Rightarrow> (((real, 'b) vec \<times> (real, 'b) vec) \<times> (real, 'b) vec \<times> (real, 'b) vec) set"   and ode :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> (real, 'b) vec"   and ode' :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> (real, 'b) vec"   and odebv :: "'b \<Rightarrow> 'b set"   and odebv' :: "'b \<Rightarrow> 'b set" assumes "f = f'"   and "p = p'"   and "c = c'"   and "PP = PP'"   and "ode = ode'"   and "odebv = odebv'" shows "\<lparr>Functions = f, Predicates = p, Contexts = c, Programs = PP, ODEs = ode, ODEBV = odebv\<rparr> = \<lparr>Functions = f', Predicates = p', Contexts = c', Programs = PP', ODEs = ode', ODEBV = odebv'\<rparr>"
satisfies_properties_sys1: shows "satisfies_properties_R p qs1 (get_subsets_R (calculate_data_R p qs1)) (get_signs_R (calculate_data_R p qs1)) (get_matrix_R (calculate_data_R p qs1))"
sylvester_index_mat2: fixes i :: "nat"   and p :: "'b poly"   and q :: "'b poly"   and j :: "nat" assumes "i < degree p + degree q"   and "j < degree p + degree q" shows "sylvester_mat p q $$ (i, j) = (if i < degree q then coeff (monom (1::'b) (degree q - i) * p) (degree p + degree q - j) else coeff (monom (1::'b) (degree p + degree q - i) * q) (degree p + degree q - j))"
is_ground_atm_mset_def: fixes AA :: "'a multiset" shows "is_ground_atm_mset AA = (\<forall>A. A \<in># AA \<longrightarrow> is_ground_atm A)"
changes_le_smods_def: fixes b :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_le_smods b p q = (let ps = smods p q in changes_poly_neg_inf ps - changes_poly_at ps b)"
curry_conv: fixes f :: "'d \<times> 'e \<Rightarrow> 'b"   and a :: "'d"   and b :: "'e" shows "curry f a b = f (a, b)"
topology_finer_closedin: fixes X :: "'c topology"   and Y :: "'c topology" assumes "topspace X = topspace Y" shows "(\<forall>S. openin Y S \<longrightarrow> openin X S) = (\<forall>S. closedin Y S \<longrightarrow> closedin X S)"
SkipI: fixes G :: "prog"   and s :: "state" shows "G\<turnstile>s \<midarrow>Skip\<rightarrow> s"
2: fixes theta :: "('a \<times> 'a) set" assumes "refl theta" shows "refl (Retr theta)"
lower_on_def: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and V :: "'a set" shows "lower_on \<sigma> \<eta> V = (\<forall>x\<in>V. lower_or_eq (Var x \<lhd> \<sigma>) (Var x \<lhd> \<eta>))"
in_hom_char: fixes f :: "('obj, 'obj) concrete_category.arr"   and a :: "('obj, 'obj) concrete_category.arr"   and b :: "('obj, 'obj) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> = (arr f \<and> f = a \<and> f = b)"
shift_map_keys_def: fixes t :: "'a"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('t, 'c) oalist_ntm" shows "shift_map_keys t f xs = OAlist_ntm (map_raw (\<lambda>kv. (t \<oplus> fst kv, f (snd kv))) (list_of_oalist_ntm xs))"
Spy_not_see_encrypted_key: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and nb :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"   and "Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Key K \<notin> analz (knows Spy evs)"
distinct_3: fixes x1 :: "nat" shows "ONat x1 \<noteq> OBad"
False: shows "mem_context_val_w32 9 (ucast (ptp OR ucast (ucast (va >> 24)))) (mem s1) \<noteq> None"
decomp_H'_H: fixes E :: "'c set"   and H :: "'c set"   and t :: "'c"   and x' :: "'c" assumes "vectorspace E"   and "Subspace.subspace H E"   and "t \<in> H"   and "x' \<notin> H"   and "x' \<in> E"   and "x' \<noteq> (0::'c)" shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
func_le_def: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "f \<preceq> g = (\<forall>x\<in>dom f. f x = g x)"
continuity_lem_gt0_expanded: fixes r :: "real"   and k :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "r < k"   and "0 < a * r\<^sup>2 + b * r + c" shows "\<exists>x\<in>{r<..k}. 0 < a * x\<^sup>2 + b * x + c"
hconf_upd_obj: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and fs' :: "char list \<times> char list \<Rightarrow> val option" assumes "P \<turnstile> h \<surd>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "P,h \<turnstile> (C, fs') \<surd>" shows "P \<turnstile> h(a \<mapsto> (C, fs')) \<surd>"
length_nat_mono: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "length_nat x \<le> length_nat y"
IH: shows "summable (\<lambda>n. f (n + m) * z ^ n) = summable (\<lambda>n. f n * z ^ n)"
magQ_one: shows "\<lbrakk>1\<rbrakk>\<^sub>Q = (1::'d)"
subspace_kernel: shows "m1.subspace {x. f x = (0::'c)}"
that: fixes N :: "nat set" assumes "N \<subseteq> M"   and "infinite N"   and "decides \<F> S N" shows "thesis"
Sup_llist_LCons: fixes X :: "'a set"   and Xs :: "'a set llist" shows "Sup_llist (LCons X Xs) = X \<union> Sup_llist Xs"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "a \<prec> b" shows "a \<preceq> b"
F_def: shows "F \<equiv> dirichlet_prod f moebius_mu"
fmapU_error_def: shows "fmapU \<equiv> error_map\<cdot>ID"
assms_3: shows "order.greater_eq {y. pseudominimal_in (P .\<^bold>\<le> x) y} a"
b_e_type_value: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and v :: "v" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = C v" shows "ts' = ts @ [typeof v]"
deleteMin_def: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.deleteMin bq \<equiv> let min = BinomialHeapStruc.getMinTree bq in BinomialHeapStruc.meld (rev (BinomialHeapStruc.children min)) (BinomialHeapStruc.remove1Prio (BinomialHeapStruc.prio min) bq)"
ireturn_eq: fixes x :: "'a" shows "RETURN x = SPEC ((=) x)"
Tw_equiv: fixes T :: "('e, 'f) Order_scheme"   and x :: "'a" assumes "Worder T"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier T. ord_equiv (Iod D (segment D a)) (Iod T (segment T b))"   and "x \<in> carrier D" shows "ord_equiv (Iod D (segment D x)) (Iod T (segment T ((Tw\<^bsub>D,T\<^esub>) x)))"
dist_le_cost: fixes p :: "'b list"   and v :: "'a" assumes "awalk s p v" shows "dist v \<le> ereal (awalk_cost c p)"
ord_isom_less: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec> b = f a \<prec>\<^bsub>E\<^esub> f b"
det_linear_rows_sum: fixes S :: "'a set"   and a :: "'a \<Rightarrow> 'a \<Rightarrow> ('b, 'a) vec" assumes "finite S" shows "det (\<chi>i. sum (a i) S) = (\<Sum>f | \<forall>i. f i \<in> S. det (\<chi>i. a i (f i)))"
f_Streams_Connected_nth_conv: fixes outS :: "'d message_af list"   and inS :: "'d message_af list" shows "f_Streams_Connected outS inS = (length inS = Suc (length outS) \<and> (\<forall>i<length inS. inS ! i = (case i of 0 \<Rightarrow> NoMsg | Suc k \<Rightarrow> outS ! k)))"
entails_\<G>_def: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile>\<inter>\<G> N2) = (\<forall>q\<in>Q. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2))"
bprules_3: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> S \<union> L}) = U"
map_of_SomeI: fixes l :: "('c \<times> 'd) list"   and k :: "'c"   and x :: "'d" assumes "unique l"   and "(k, x) \<in> set l" shows "table_of l k = Some x"
morphism_half_apartments: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ThinChamberComplexFolding X f" shows "ChamberComplexMorphism (ChamberComplexFolding.opp_half_apartment X f) (f \<turnstile> X) f"
count_space_singleton_conv_return: fixes x :: "'a" shows "count_space {x} = return (count_space {x}) x"
quot_of_fract_eq_iff: fixes x :: "'b fract"   and y :: "'b fract" shows "(quot_of_fract x = quot_of_fract y) = (x = y)"
dist_eq_dist_code_le: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int"   and p\<^sub>2 :: "int \<times> int"   and p\<^sub>3 :: "int \<times> int" shows "(dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>2 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>3 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) = (dist_code p\<^sub>0 p\<^sub>1 \<le> dist_code p\<^sub>2 p\<^sub>3)"
product_factor_tree: fixes t :: "'a factor_tree"   and p :: "int"   and u :: "int poly factor_tree"   and f :: "int poly" assumes "\<And>x. x \<in># factors_of_factor_tree t \<Longrightarrow> poly_mod.Mp p x = x"   and "u \<in> sub_trees (product_factor_tree p t)"   and "factor_node_info u = f" shows "poly_mod.Mp p f = f \<and> f = poly_mod.Mp p (\<Prod>\<^sub># (factors_of_factor_tree u)) \<and> factors_of_factor_tree (product_factor_tree p t) = factors_of_factor_tree t"
part_eqs_2: fixes X :: "'a set" assumes "part X" shows "I \<circ> K = K"
prefD: fixes A :: "'b \<Rightarrow> bool"   and w :: "'b" assumes "|~ A" shows "A w"
Mapping_lookup_combineE: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "('b, 'a) mapping"   and m' :: "('b, 'a) mapping"   and k :: "'b"   and v :: "'a" assumes "Mapping.lookup (Mapping.combine f m m') k = Some v" obtains "Mapping.lookup m k = Some v"   | "Mapping.lookup m' k = Some v"   | v' :: "'a" and v'' :: "'a" where "Mapping.lookup m k = Some v'" and "Mapping.lookup m' k = Some v''" and "f v' v'' = v"
comp_null_2: fixes f :: "'a1" shows "f \<cdot>\<^sub>1 C1.null = C1.null"
maxsimpD_simplex: fixes x :: "'a set" assumes "domain.chamber x" shows "x \<in> X"
L_def: shows "L = (\<Sum>n. \<chi> n * of_nat n powr - s)"
degree_two_windmill: assumes "\<And>v u. \<lbrakk>v \<in> V; u \<in> V; v \<noteq> u\<rbrakk> \<Longrightarrow> \<exists>!n. adjacent v n \<and> adjacent u n"   and "finite E"   and "2 \<le> card V" shows "(\<exists>v\<in>V. degree v G = 2) = (\<exists>v. \<forall>n\<in>V. n \<noteq> v \<longrightarrow> adjacent v n)"
ket_vec_index: fixes i :: "nat"   and v :: "complex vec" assumes "i < dim_vec v" shows "|v\<rangle> $$ (i, 0) = v $ i"
is_Ref_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "shadow_root_ptr.is_Ref shadow_root_ptr = (\<exists>x1. shadow_root_ptr = shadow_root_ptr.Ref x1)"
Suc_leD: fixes m :: "nat"   and n :: "nat" assumes "Suc (m::nat) \<le> (n::nat)" shows "(m::nat) \<le> (n::nat)"
open_unresolvable_not_ed_ou_card: fixes X :: "'a set" assumes "open_unresolvable X"   and "\<not> ed_ou X" shows "card {f. CK_nf_pos f} = 5"
pmfs: shows "\<forall>l\<in>lotteries_on outcomes. set_pmf l = {x}"
distinct_3: fixes x1 :: "val"   and x31 :: "expr"   and x32 :: "bop"   and x33 :: "expr" shows "Val x1 \<noteq> x31 \<guillemotleft>x32\<guillemotright> x33"
grdOf: fixes c' :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list"   and l' :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit"   and x :: "Preliminaries.var" assumes "c' \<in> gPB"   and "l' \<in> set c'"   and "x \<in> nv2L l'"   and "\<not> infTp (GE.tpOfV x)" shows "grdOf c' l' x \<in> set c' \<and> isGuard x (grdOf c' l' x)"
FAddSA_Union_ran: fixes S :: "'d"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> dom G"   and "States SA \<inter> dom G = {}" shows "\<Union> (ran (G [f+] (S, SA))) = insert SA (\<Union> (ran G))"
raw_has_prod_cases: fixes f :: "nat \<Rightarrow> 'b"   and M :: "nat"   and p :: "'b" assumes "raw_has_prod f M p" obtains i :: "nat" where "i < M" and "f i = (0::'b)"   | p :: "'b" where "raw_has_prod f 0 p"
p'_def: shows "p' = (atom sm \<rightleftharpoons> atom sm') + (atom sn \<rightleftharpoons> atom sn') + p"
parabolic_logarithmic_growth: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat"   and x :: "'b" assumes "parabolic_isometry f"   and "1 \<le> n" shows "dist x ((f ^^ n) x) \<le> dist x (f x) + real_of_int (\<lceil>log 2 (real n)\<rceil> * 16) * deltaG TYPE('b)"
A'_def: shows "A' = mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)"
m_rcancel: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "domain R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(b \<otimes>\<^bsub>R\<^esub> a = c \<otimes>\<^bsub>R\<^esub> a) = (b = c)"
distinct_3: fixes x1 :: "lname"   and x31 :: "expr"   and x32 :: "expr" shows "LVar x1 \<noteq> x31.[x32]"
\<gamma>: fixes \<D> :: "('a \<times> 'a set) set" assumes "\<D> tagged_division_of cbox a b"   and "\<gamma> fine \<D>" shows "norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R indicat_real S x) - Sigma_Algebra.measure lebesgue S) < e"
dense_accessible_frontier_points: fixes S :: "'b set"   and V :: "'b set" assumes "open S"   and "openin (top_of_set (frontier S)) V"   and "V \<noteq> {}" obtains g :: "real \<Rightarrow> 'b" where "arc g" and "g ` {0..<1} \<subseteq> S" and "pathstart g \<in> S" and "pathfinish g \<in> V"
viewL_aux_def: fixes t :: "('a, 'b) FingerTree" shows "FingerTree.viewL_aux t \<equiv> Abs_viewres (FingerTreeStruc.viewL (Rep_FingerTree t))"
notinset_notinbetween: fixes x :: "'a"   and xs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "x \<notin> set xs" shows "x \<notin> set (between xs a b)"
lgrid_def: fixes b :: "(nat \<times> int) list"   and ds :: "nat set"   and lm :: "nat" shows "lgrid b ds lm = {p \<in> grid b ds. level p < lm}"
invertible_mat: shows "invertible_mat (M_mat_R signs subsets)"
inner_zero_left: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x :: "'b" assumes "inner_product_space scale inner" shows "inner (0::'b) x = (0::'a)"
permute_Abs_res: fixes p :: "perm"   and as :: "atom set"   and x :: "'a" shows "p \<bullet> ([as]res. x) = [p \<bullet> as]res. p \<bullet> x"
sees_field_fun: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and b' :: "staticb"   and T' :: "ty"   and D' :: "char list" assumes "P \<turnstile> C sees F,b:T in D"   and "P \<turnstile> C sees F,b':T' in D'" shows "b = b' \<and> T' = T \<and> D' = D"
simps_2: fixes uv :: "'a"   and uw :: "real"   and l :: "'a kdt"   and r :: "'a kdt" shows "KD_Tree.size_kdt (Node uv uw l r) = KD_Tree.size_kdt l + KD_Tree.size_kdt r"
find_consistent_signs_at_roots_len1_R: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "length qs = 1" shows "set (find_consistent_signs_at_roots_R p qs) = set (characterize_consistent_signs_at_roots p qs)"
check_poly_weak_mono_all_pos: fixes p :: "('c monom \<times> 'd) list" assumes "check_poly_weak_mono_all p" shows "p \<ge>p zero_poly"
assume_SBE_def: fixes e :: "'b \<Rightarrow> bool" shows "assume\<^sub>S\<^sub>B\<^sub>E e = (\<lambda>\<sigma>. if e \<sigma> then Some {((), \<sigma>)} else Some {})"
1: shows "\<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot> (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b]) = \<d>[a \<otimes> b]"
runFw_no_interfaces: fixes rs :: "'a simple_rule list"   and s :: "'a word"   and d :: "'a word"   and c :: "'b parts_connection_scheme"   and y :: "char list"   and x :: "char list" assumes "simple_firewall_without_interfaces rs" shows "runFw s d c rs = runFw s d (c\<lparr>pc_iiface := x, pc_oiface := y\<rparr>) rs"
ty_REL: shows "TYREL (\<langle>Id\<rangle>dflt_rs_rel)"
cfun_cong: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "f = g"   and "x = y" shows "f\<cdot>x = g\<cdot>y"
exhaust: fixes y :: "'k option" obtains "y = None"   | x2 :: "'k" where "y = Some x2"
intro: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>"   and "primitive_dchar_axioms n \<chi>" shows "primitive_dchar n \<chi>"
induct: fixes x1 :: "'a literal set set"   and x2 :: "'a literal set"   and P :: "'a literal set set \<Rightarrow> 'a literal set \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>C S. C \<in> S \<Longrightarrow> P S C"   and "\<And>S C D k. \<lbrakk>S \<turnstile> C; P S C; S \<turnstile> D; P S D; k\<^sup>+ \<in> C; k\<inverse> \<in> D\<rbrakk> \<Longrightarrow> P S (C - {k\<^sup>+} \<union> (D - {k\<inverse>}))" shows "P x1 x2"
comparator_converse: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "comparator cmp" shows "comparator (\<lambda>x y. cmp y x)"
pred_entailment_mono_un: fixes P :: "'BExp set"   and P' :: "'BExp set"   and P'' :: "'BExp set" assumes "P \<turnstile> P'" shows "(P \<union> P'') \<turnstile> (P' \<union> P'')"
ms_aux_hd_exp_def: fixes xs :: "('a \<times> real) msllist" shows "ms_aux_hd_exp xs = (case xs of MSLNil \<Rightarrow> None | MSLCons (x, e) xa \<Rightarrow> Some e)"
segment_inc_psub: fixes x :: "'a"   and a :: "'a" assumes "x \<in> segment D a" shows "segment D x \<subset> segment D a"
LIMSEQ_unique: fixes "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.t2_space open"   and "filterlim X (topological_space.nhds open a) sequentially"   and "filterlim X (topological_space.nhds open b) sequentially" shows "a = b"
BINDINGn_def: fixes B :: "ty"   and X :: "ctrm set" shows "BINDINGn B X \<equiv> {x:M |x M. \<forall>a P. <a>:P \<in> X \<longrightarrow> SNa (M{x:=<a>.P})}"
cl_leq_one: shows "\<forall>c. |cl c| \<le> 1"
stake_run: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) stream"   and p :: "nat \<times> 'd" assumes "\<And>k. gpath A w V (stake k r) p" shows "grun A w V r p"
simps_5: fixes c :: "int"   and e :: "MIR.num" shows "plusinf (Lt (CN 0 c e)) = F"
dghm_ObjMap_vsv: shows "vsv (\<FF>\<lparr>ObjMap\<rparr>)"
union_single_eq_diff: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "add_mset x M = N" shows "M = N - {#x#}"
H_on_ket_one: shows "H * |Deutsch.one\<rangle> = Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, - 1 / sqrt 2]])"