hcju/isabelleps huggingface dataset:
intros_4: fixes s :: "s"   and i :: "nat"   and c :: "i32"   and cl :: "cl"   and j :: "nat"   and vs :: "v list" assumes "stab s i (nat_of_int c) = Some cl \<and> stypes s i j \<noteq> cl_type cl \<or> stab s i (nat_of_int c) = None" shows "\<lparr>s;vs;[$C ConstInt32 c, $Call_indirect j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
vy_def: shows "vy = (SOME ux. ux \<in> \<lfloor>V, y\<rfloor>)"
assms_2: shows "[\<^bold>\<not>\<lparr>A!,x\<rparr> in v]"
S: shows "S = S\<^sub>1 \<rightarrow> S\<^sub>2"
compseries_is_tW_cmpser: fixes r :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "compseries G r f" shows "tW_cmpser G r f"
Gcd_nat_def: fixes M :: "nat set" shows "Gcd M = Lcm {d. \<forall>m\<in>M. d dvd m}"
subdegrees: shows "fls_subdegree (f * g) = fls_subdegree f + fls_subdegree g"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and x1 :: "'c rule list"   and x2 :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> x1 \<Rightarrow>\<^sub>r x2"   and "P [] Undecided"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Accept # rs) (Decision FinalAllow)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Drop # rs) (Decision FinalDeny)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Reject # rs) (Decision FinalDeny)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Return # rs) Undecided"   and "\<And>rs t m. \<lbrakk>\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t; P rs t\<rbrakk> \<Longrightarrow> P (Rule m Log # rs) t"   and "\<And>rs t m. \<lbrakk>\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t; P rs t\<rbrakk> \<Longrightarrow> P (Rule m Empty # rs) t"   and "\<And>m rs t a. \<lbrakk>\<not> matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t; P rs t\<rbrakk> \<Longrightarrow> P (Rule m a # rs) t"   and "\<And>m chain rs X rrs. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r Decision X; P rs (Decision X)\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) (Decision X)"   and "\<And>chain rs rrs t m. \<lbrakk>\<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r Undecided; P rs Undecided; \<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>r t; P rrs t\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) t" shows "P x1 x2"
IdNatTrans'_def: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" shows "IdNatTrans' F \<equiv> \<lparr>NTDom = F, NTCod = F, NatTransMap = \<lambda>X. id\<^bsub>CatCod F\<^esub> (F @@ X)\<rparr>"
2: shows "inconsist i (f \<triangleright> n) \<down>= (LEAST x. x < Suc n \<and> \<phi> i x \<noteq> f x)"
P: fixes as :: "'a list"   and bs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "xs @ ys = as @ bs"   and "x \<in> set as"   and "y \<in> set bs" shows "x \<le> y"
subseteq_qle_refl: fixes M :: "'a set" shows "M \<subseteq>\<^sub>\<preceq> M"
echelon_form_upt_k_condition2: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and i :: "'f"   and j :: "'f" assumes "echelon_form_upt_k A k"   and "i < j"   and "\<not> is_zero_row_upt_k i k A"   and "\<not> is_zero_row_upt_k j k A" shows "(LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ j $ n \<noteq> (0::'d))"
finite_measure_mono: fixes A :: "'key set"   and B :: "'key set" assumes "A \<subseteq> B"   and "B \<in> K.events" shows "K.prob A \<le> K.prob B"
interpret_floatarith_diff: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Add a (Minus b)) vs = interpret_floatarith a vs - interpret_floatarith b vs"
eint_ord_code_1: fixes m :: "int"   and n :: "int" shows "(eint m \<le> eint n) = (m \<le> n)"
mod_add_left_eq: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a mod c + b) mod c = (a + b) mod c"
aux5c: fixes a :: "'b"   and b :: "'b"   and po :: "'c"   and p :: "('b, 'c) Combinators list" assumes "AllowPortFromTo a b po \<in> set p" shows "a \<in> set (net_list p)"
assert_top: shows "{\<cdot> \<top> } = (1::'b)"
vrat_inverse_transfer: shows "rel_fun cr_vrat cr_vrat vrat_inverse_app inverse"
right_mult_one_mat': fixes A :: "'c mat"   and n :: "nat" assumes "dim_col A = n" shows "A * 1\<^sub>m n = A"
measurable_snd: fixes M1 :: "'d measure"   and M2 :: "'e measure" shows "snd \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M2"
equal_length_sym: fixes l :: "tm list"   and l' :: "tm list" assumes "equal_length l l'" shows "equal_length l' l"
assms: fixes x :: "'a" assumes "x \<in> s" shows "f x = g x"
f3: shows "low_equal (snd (fst (write_reg w curr_win (rd AND 30) s1))) (snd (fst (write_reg w curr_win (rd AND 30) s2)))"
simps_5: fixes F :: "'b set \<Rightarrow> 'a"   and b :: "bexp"   and c :: "com"   and M :: "'b acom set" shows "lift F (WHILE b DO c) M = {F (invar ` M)} WHILE b DO lift F c (sub\<^sub>1 ` M) {F (post ` M)}"
locate_pred_def: fixes f :: "'b \<Rightarrow> nat"   and al :: "'b list"   and i :: "nat"   and n_j :: "nat \<times> nat" shows "locate_pred f al i n_j \<equiv> fst n_j < length al \<and> snd n_j < f (al ! fst n_j) \<and> i = lsum f (take (fst n_j) al) + snd n_j"
sol13: shows "(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid3 UNIV}"
seq_if_composable: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null" shows "src\<^sub>D \<nu> = trg\<^sub>D \<mu>"
f: shows "\<forall>j\<ge>i. \<forall>p. ps (S t j) p \<noteq> None"
div_exp_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (m + n))"
l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs"
hbase_evalC_mono: fixes n :: "C"   and b :: "nat"   and m :: "C" assumes "n \<in> hbase b"   and "m \<in> hbase b"   and "evalC b n < evalC b m" shows "evalC (Suc b) n < evalC (Suc b) m"
open_domain: fixes c :: "('c, 'd) chart" shows "open (domain c)"
lessLessTrace: shows "\<forall>i<length exec' - 1. i < length trace'"
invertible_matrix_iff_invertible_lf: fixes A :: "(('a, 'b) vec, 'b) vec" shows "invertible A = invertible_lf (*s) (*s) ((*v) A)"
CLS_chine: shows "\<lbrakk>\<lbrakk>TfTgh_TfHgh.chine\<rbrakk>\<rbrakk> = Maps.tuple \<lbrakk>\<lbrakk>f.chine \<star> TfTgh.p\<^sub>1\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>0 f\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 (g \<star> h)\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>Tgh_Hgh.chine \<star> TfTgh.p\<^sub>0\<rbrakk>\<rbrakk>"
iMOD_singleton_eq_conv: fixes r :: "nat"   and m :: "nat"   and a :: "nat" shows "([ r, mod m ] = {a}) = (r = a \<and> m = 0)"
ap_envir_apply: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'d \<Rightarrow> 'e"   and i :: "'d" shows "(f \<diamondop> x) i = f i (x i)"
bigthetaI': fixes c1 :: "real"   and c2 :: "real"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "0 < c1"   and "0 < c2"   and "\<forall>\<^sub>F x in F. c1 * norm (g x) \<le> norm (f x) \<and> norm (f x) \<le> c2 * norm (g x)" shows "f \<in> \<Theta>[F](g)"
of_nat_eq: fixes k :: "nat" shows "of_nat k = Abs (int k mod n)"
cSum2: assumes "Q \<longmapsto> Rs" shows "Prop"
of_nat_less_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
mpoly_of_poly_0: fixes i :: "nat" shows "mpoly_of_poly i 0 = 0"
walk_decomp_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "walk (xs @ ys)" shows "walk xs"
nfv_simps_3: fixes I :: "\<I>"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.nfv (formula.Prev I \<phi>) = MFOTL.nfv \<phi>"
path_decomp_2: fixes xs :: "'a list"   and ys :: "'a list" assumes "Q.path (xs @ ys)" shows "Q.path ys"
no_grey_refs_def: shows "no_grey_refs = (\<lambda>s. \<forall>x. \<not> grey x s)"
between_distinct: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs" shows "distinct (between vs ram1 ram2)"
inf_compl_bot_left1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) (inf x y) = bot"
S\<^sub>0_def: shows "S\<^sub>0 = \<lbrakk>\<lbrakk>tab\<^sub>0 s\<rbrakk>\<rbrakk>"
poly_roots_power: fixes p :: "'a poly"   and n :: "nat" shows "poly_roots (p ^ n) = repeat_mset n (poly_roots p)"
par_comp_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Secrets :: "('fun, 'var) Term.term set" shows "par_comp \<A> Secrets \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l1 \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l2 \<A>)) Secrets) \<and> (\<forall>s\<in>Secrets. \<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Secrets) \<and> ground Secrets"
qfree_qe_pres_o: fixes as :: "atom list" shows "qfree ((qe_pres\<^sub>1 \<circ> hd_coeffs1) as)"
norm: fixes x :: "'a" shows "norm x = sqrt (Re (cinner x x))"
poly_subst_plus: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_subst f (p + q) = poly_subst f p + poly_subst f q"
allAllocationsFinite: fixes N :: "'b set"   and G :: "'c set" assumes "finite N"   and "finite G" shows "finite (allAllocations N G)"
gronwall_general_left: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and C :: "real"   and K :: "real"   and t :: "real" assumes "\<And>t. t \<in> {a..b} \<Longrightarrow> g t \<le> C + K * integral {t..b} g"   and "continuous_on {a..b} g"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> 0 \<le> g t"   and "0 < C"   and "0 < K"   and "t \<in> {a..b}" shows "g t \<le> C * exp (- K * (t - b))"
Pg_plus_def: fixes G :: "Pg"   and H :: "Pg" shows "G + H = the_elem (\<Union>g\<in>Rep_Pg G. \<Union>h\<in>Rep_Pg H. {Abs_Pg (eq_game_rel `` {plus_game g h})})"
coboundedI2: fixes c :: "'b"   and b :: "'b"   and a :: "'b" assumes "order.greater_eq b c" shows "order.greater_eq (sup a b) c"
abs_majorities_intersect: fixes E :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "card E < card S + card T"   and "S \<subseteq> E"   and "T \<subseteq> E"   and "finite E" shows "S \<inter> T \<noteq> {}"
FinGroup_axioms_def: fixes G :: "'a set" shows "FinGroup_axioms G \<equiv> finite G"
add_diff_le_ennreal: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "a + b - c \<le> a + (b - c)"
A: fixes x :: "val"   and y :: "val" assumes "x \<in> space N'"   and "y \<in> space (stock_measure t2)" shows "indicator (fst' -` X \<inter> space N) <|x, y|> = indicator X x"
less_list_def: fixes s :: "'c list"   and t :: "'c list" shows "(s < t) = (s \<le> t \<and> s \<noteq> t)"
in_homE: fixes f :: "'t"   and a :: "'t"   and b :: "'t" assumes "\<guillemotleft>f : a \<rightarrow>' b\<guillemotright>" obtains "S'.arr f" and "S'.dom f = a" and "S'.cod f = b"
reachable_nexts: fixes g :: "(node \<times> node) list"   and x :: "node" shows "reachable g (nexts g x) \<subseteq> reachable g [x]"
ThrowThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
set_intros_2: fixes x22 :: "'b"   and x21 :: "'b tree"   and x23 :: "'b tree" shows "x22 \<in> set_tree \<langle>x21, x22, x23\<rangle>"
InterpExprWellDefined: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t"   and i :: "('o, 'm) IType" assumes "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> i" shows "Sig iS \<triangleright> Vx : A \<turnstile> e : B"
set_intros_3: fixes x32 :: "'c"   and x31 :: "'d" shows "x32 \<in> set2_step (Yield x31 x32)"
hyps_4: shows "\<exists>x. x \<in> F \<and> (\<forall>y\<in>F. f x \<le> f y)"
finite_product_sigma_finite_def: fixes M :: "'c \<Rightarrow> 'd measure"   and I :: "'c set" shows "finite_product_sigma_finite M I \<equiv> product_sigma_finite M \<and> finite_product_sigma_finite_axioms I"
abs_inverse: fixes a :: "'a" shows "\<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
IDC_def: fixes d :: "'d"   and c :: "('exp, 'id) MWLsCom"   and H :: "('d \<times> 'exp) set" shows "IDC d c H \<equiv> (\<exists>m m'. m =\<^bsub>d\<^esub> m' \<and> \<not> \<lbrakk>c\<rbrakk>(m) =\<^bsub>d\<^esub> \<lbrakk>c\<rbrakk>(m')) \<and> (\<forall>m m'. m \<sim>\<^bsub>d,H\<^esub> m' \<longrightarrow> \<lbrakk>c\<rbrakk>(m) =\<^bsub>d\<^esub> \<lbrakk>c\<rbrakk>(m'))"
cases: fixes a1 :: "('c, 'd) conf"   and a2 :: "('c, 'd) label list"   and a3 :: "('c, 'd) conf" assumes "se_star a1 a2 a3" obtains c :: "('c, 'd) conf" where "a1 = c" and "a2 = []" and "a3 = c"   | c1 :: "('c, 'd) conf" and l :: "('c, 'd) label" and c2 :: "('c, 'd) conf" and ls :: "('c, 'd) label list" and c3 :: "('c, 'd) conf" where "a1 = c1" and "a2 = l # ls" and "a3 = c3" and "se c1 l c2" and "se_star c2 ls c3"
index_locale_axioms: shows "index_locale index_class.less_eq index_class.less index_class.idx index_class.size"
smult_sum_list_distrib: fixes r :: "('f, 'g) aezfun"   and ms :: "'v list" assumes "r \<in> FG"   and "set ms \<subseteq> V" shows "r \<cdot> sum_list ms = sum_list (map ((\<cdot>) r) ms)"
eval_false: fixes L :: "real list" shows "\<not> eval FalseF L"
finite_reduced_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "finite (punit.reduced_GB F)"
bigtheta: shows "\<psi> \<in> \<Theta>(\<lambda>x. x)"
op_slg_succs_def: fixes E :: "('b \<times> 'a) set"   and v :: "'b" shows "op_slg_succs E v \<equiv> E `` {v}"
qe_FM\<^sub>1_def: fixes as :: "atom list" shows "qe_FM\<^sub>1 as = list_conj (concat (map (\<lambda>p. map (\<lambda>q. Atom (combine p q)) (ubounds as)) (lbounds as)))"
init_reach: fixes T :: "('a, 'b) TS_scheme" shows "init T \<subseteq> reach T"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" obtains "a = a \<cdot> b"
f_recurrence: fixes k :: "nat" assumes "0 < k"   and "k \<le> n" shows "f k = - of_nat k * (- 1) ^ k * e' k - (\<Sum>i = 1..<k. (- 1) ^ i * e' i * f (k - i))"
module_ow_axioms: shows "module_ow U\<^sub>M\<^sub>_\<^sub>2 (+\<^sub>M\<^sub>_\<^sub>2) 0\<^sub>M\<^sub>_\<^sub>2 (-\<^sub>M\<^sub>_\<^sub>2) uminus\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>M\<^sub>_\<^sub>2)"
wellformed1_sorted_auxQ: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "wellformed_policy1 p" shows "wellformed_policy1 (qsort p l)"
arr_Rel_ArrVal_vrange: shows "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrCod\<rparr>"
finite_Plus_UNIV_iff: shows "finite UNIV = (finite UNIV \<and> finite UNIV)"
add_zero_right: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "x + (0::'b) = x"
set_category_data_axioms: shows "set_category_data (\<cdot>)"
simps_33: fixes x3 :: "'c" shows "set2_sum3 (In3 x3) = {}"
cdep_SDG_path_cc_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n cd-ns\<rightarrow>\<^sub>d* n'" shows "n cc-ns\<rightarrow>\<^sub>d* n'"
ereal_one_not_less_zero_ereal: shows "\<not> 1 < 0"
alpha_sym: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "X #= X'" shows "X' #= X"
ResPres': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>', S, T) \<in> Rel'"   and "xvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel'"
I_and: fixes aneg :: "'a \<Rightarrow> 'a fm"   and anormal :: "'a \<Rightarrow> bool"   and I\<^sub>a :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and depends\<^sub>0 :: "'a \<Rightarrow> bool"   and decr :: "'a \<Rightarrow> 'a"   and \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm"   and xs :: "'b list" assumes "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr" shows "interpret I\<^sub>a (and \<phi>\<^sub>1 \<phi>\<^sub>2) xs = interpret I\<^sub>a (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
empty: 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 Empty], Undecided\<rangle> \<Rightarrow> Undecided"
simps_3: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "asym.implAuth A B M = Sign \<langle>Messages.Agent B, M\<rangle> (priK A)"
arr_def: fixes f :: "'b" shows "D.arr f \<equiv> D.domains f \<noteq> {} \<or> D.codomains f \<noteq> {}"
sum_of_prideals: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>l\<le>n. f l \<in> carrier R" shows "ideal R (sum_pr_ideals R f n)"
xf: fixes n :: "nat" shows "x n \<in> nnfis (f n) M"
konigDown_ipath: fixes t :: "('sequent \<times> 'rule) tree"   and S :: "'structure" assumes "local.wf t"   and "S \<in> structure"   and "\<not> sat S (fst (tree.root t))" shows "ipath t (smap fst (konigDown t S))"
intpow_mult_distrib: fixes x :: "'c"   and y :: "'c"   and m :: "int" shows "(x \<cdot> y) ^\<^sub>Z m = x ^\<^sub>Z m \<cdot> y ^\<^sub>Z m"
proj2_rep_abs: fixes v :: "(real, 3) vec" assumes "v \<in> non_zero_vectors" shows "(v, proj2_rep (proj2_abs v)) \<in> proportionality"
inner: shows "qtable (b + n) (fv \<phi>) (mem_restr (lift_envs' b R)) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>) rel"
assms_2: shows "\<Gamma> \<turnstile>\<^sub>c e2 : dist_result_type dst"
is_unit_triv: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "is_unit [:a:]"
step_3: shows "\<forall>b\<in>insert x basis. line_integral_exists F {b} \<gamma>"
swap_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set" shows "X #[x \<and> y]_zs = X #[y \<and> x]_zs"
bottom_meet: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<bottom> \<sqinter> x .= \<bottom>"
incomplete_alt_size: fixes bl :: "'a set" assumes "card bl < card \<V>\<^sup>+ \<and> bl \<in># \<B>\<^sup>+" shows "int (card bl) < int (card \<V>\<^sup>+)"
sum_clauses_1: fixes f :: "'e \<Rightarrow> 'd" shows "sum f {} = (0::'d)"
fps_left_inverse_eq_fps_right_inverse: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a" assumes "x * f $ 0 = (1::'a)"   and "f $ 0 * y = (1::'a)" shows "fps_left_inverse f x = fps_right_inverse f y"
cinner_mult_left: fixes m :: "complex"   and a :: "'c"   and b :: "'c" shows "cinner (of_complex m * a) b = cnj m * cinner a b"
refl_onP_empty: shows "refl_onP {} (\<lambda>x y. False)"
dom_def_aux_var: fixes x :: "'a" shows "d x = 1' \<cdot> x ; x\<^sup>T"
SAEvents_def: fixes SA :: "('e, 'd, 'f) seqauto" shows "SAEvents SA = (\<Union>l\<in>Label (Delta SA). fst (action l) \<union> ExprEvents (expr l))"
SubstTermP_Var_diff: fixes v :: "tm"   and w :: "tm"   and t :: "tm" shows "{VarP v, VarP w, Neg (v EQ w), TermP t} \<turnstile> SubstTermP v t w w"
Un: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a \<union> b \<in> fmeasurable M"
F'_start_le_F'_iter: shows "(XD_smallest ds X, dual (XH_largest ds X)) \<le> F'_iter"
sum_le_included: fixes s :: "'a set"   and t :: "'b set"   and g :: "'b \<Rightarrow> 'c multiset"   and i :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c multiset" assumes "finite s"   and "finite t"   and "\<forall>y\<in>t. {#} \<subseteq># g y"   and "\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<subseteq># g y" shows "comm_monoid_add.sum (+) {#} f s \<subseteq># comm_monoid_add.sum (+) {#} g t"
fst_foldl_Gauss_Jordan_column_k_less: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec" assumes "k < ncols A" shows "fst (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]) \<le> nrows A"
numeral_Bit0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "Num.num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
least_prefixpoint_same: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "is_least_prefixpoint f x" shows "x = p\<mu> f"
timesll: shows "wtimes l\<^sub>1 l\<^sub>2 \<equiv>\<^sub>E rl1 * rl2"
add_minus_cancel: fixes a :: "'b"   and b :: "'b" shows "a + (- a + b) = b"
get_IFS_def: fixes M :: "'b SecurityInvariant_configured list" shows "get_IFS M \<equiv> filter c_isIFS M"
hom_induced1: shows "\<exists>hom_relmap. (\<forall>p X S Y T f. continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<longrightarrow> hom_relmap p X S Y T f \<in> hom (relative_homology_group (int p) X S) (relative_homology_group (int p) Y T)) \<and> (\<forall>p X S Y T f c. continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> singular_relcycle p X S c \<longrightarrow> hom_relmap p X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c))"
proj_Cons_5: fixes l :: "'a strand_label"   and n :: "'a"   and a :: "'b"   and A :: "('a strand_label \<times> 'b) list" assumes "l = \<star>" shows "proj n ((l, a) # A) = (l, a) # proj n A"
ordLeq_ordLess_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r \<le>o r'"   and "r' <o r''" shows "r <o r''"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright> = (S.arr a \<and> S.arr b \<and> S.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"