hcju/isabelleps huggingface dataset:
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_trans ext" shows "ext ext"
validE: fixes X :: "msg"   and implem :: "chan \<Rightarrow> msg" assumes "X \<in> basic_implem.valid implem" obtains x :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "X = implem (Chan x A B M)" and "M \<in> payload"
card_of_Un2: fixes A :: "'b set"   and B :: "'b set" shows "|A| \<le>o |B \<union> A|"
project_act_Id: fixes h :: "'c \<times> 'd \<Rightarrow> 'e" assumes "Extend h" shows "project_act h Id = Id"
ListSum_compl_2: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and f :: "'a \<Rightarrow> nat" shows "ListSum (filter P xs) f + ListSum (filter (\<lambda>x. \<not> P x) xs) f = ListSum xs f"
interior_of_subset: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S \<subseteq> S"
indmhom1: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i" assumes "R module M"   and "submodule R M H"   and "R module N"   and "f \<in> mHom R M N"   and "H \<subseteq> ker\<^bsub>M,N\<^esub> f" shows "\<exists>!g. g \<in> mHom R (M /\<^sub>m H) N \<and> compos M g (mpj M H) = f"
cases: fixes x :: "('a, 'b) chain" obtains uu_ :: "'a" where "x = \<lbrace> uu_ \<rbrace>"   | uv_ :: "'a" and uw_ :: "'b" and \<Pi> :: "('a, 'b) chain" where "x = \<lbrace> uv_ \<rbrace> \<cdot> uw_ \<cdot> \<Pi>"
decisive2_witness: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and Is :: "'b set"   and P :: "'b \<Rightarrow> ('a \<times> 'a) set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "profile A Is P"   and "j \<in> Is" obtains P' :: "'b \<Rightarrow> ('a \<times> 'a) set" where "profile A Is P'" and "z \<^bsub>(P' j)\<^esub>\<prec> x \<and> x \<^bsub>(P' j)\<^esub>\<prec> y" and "\<And>i. i \<noteq> j \<Longrightarrow> z \<^bsub>(P' i)\<^esub>\<prec> x \<and> y \<^bsub>(P' i)\<^esub>\<prec> x \<and> (y \<^bsub>(P' i)\<^esub>\<preceq> z) = (y \<^bsub>(P i)\<^esub>\<preceq> z) \<and> (z \<^bsub>(P' i)\<^esub>\<preceq> y) = (z \<^bsub>(P i)\<^esub>\<preceq> y)"
igSwapClsSTR_imp_igSwapCls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igSwapClsSTR MOD" shows "FixSyn.igSwapCls Delta MOD"
fvu: assumes "r \<noteq> u" shows "between (vertices f) v u = between (vertices f) v r @ r # between (vertices f) r u"
foldl_spmf_Cons: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and p :: "'c spmf"   and x :: "'d"   and xs :: "'d list" shows "foldl_spmf f p (x # xs) = foldl_spmf f (p \<bind> (\<lambda>a. f a x)) xs"
n_eq: fixes N :: "'a stream measure"   and M :: "'a stream measure" assumes "R N M" shows "N = measure_pmf (p N M) \<bind> (\<lambda>y. distr (n N M y) (stream_space (count_space UNIV)) ((##) y))"
simps_1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c" shows "pre_digraph.awalk_verts G u [] = [u]"
m_inv_consistent: fixes H :: "'a set"   and x :: "'a" assumes "subgroup H (G\<lparr>carrier := P\<rparr>)"   and "x \<in> H" shows "inv\<^bsub>G\<lparr>carrier := P, carrier := H\<rparr>\<^esub> x = inv\<^bsub>G\<lparr>carrier := P\<rparr>\<^esub> x"
assms_3: shows "(a, b) \<notin>\<^sub>p set ps"
conc_fun_RETURN: fixes R :: "('a \<times> 'b) set"   and x :: "'b" shows "\<Down> R (RETURN x) = SPEC (\<lambda>y. (y, x) \<in> R)"
field_differentiable_minus: fixes f :: "'c \<Rightarrow> 'c"   and f' :: "'c"   and F :: "'c filter" assumes "(f has_field_derivative f') F" shows "((\<lambda>z. - f z) has_field_derivative - f') F"
preserved_code: fixes f :: "('aq, 'ar, 'as) prog"   and h :: "'aq"   and h' :: "'aq" shows "preserved f h h' = (h \<turnstile> ok f \<and> h' \<turnstile> ok f \<and> |h \<turnstile> f|\<^sub>r = |h' \<turnstile> f|\<^sub>r \<or> \<not> h \<turnstile> ok f \<and> \<not> h' \<turnstile> ok f)"
inv_solve_right': fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c = a) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
left_neutral: fixes a :: "'a" shows "\<bottom> \<squnion> a = a"
hyps: assumes "u \<le>p r \<cdot> u" shows "u \<le>p r \<^sup>@ k \<cdot> u"
balR_cl: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "cl_inv l"   and "cl_inv' r"   and "cl l = B" shows "cl_inv (balR l k v r)"
P_ge_0: fixes x :: "real" assumes "0 < x" shows "0 \<le> P x"
simCases: fixes Q :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<And>a y Q'. \<lbrakk>Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q'; y \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' Q' a y Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>[Rel] Q"
sound_isTrue: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv \<phi>" shows "isTrue \<phi>"
in_RSpan_Cons_obtain_same_length_coeffs: fixes R :: "'a set"   and n :: "'b"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b"   and ms :: "'b list" assumes "R_scalar_mult R"   and "n \<in> R_scalar_mult.RSpan R smult (m # ms)" shows "\<exists>r rs. set (r # rs) \<subseteq> R \<and> length rs = length ms \<and> n = smult r m + scalar_mult.lincomb smult rs ms"
singleton_simplex: fixes X :: "'b set set"   and v :: "'b" assumes "SimplicialComplex X"   and "v \<in> \<Union> X" shows "{v} \<in> X"
simps_2: fixes T :: "typ"   and body :: "Core.term"   and lev :: "nat"   and args :: "Core.term list" shows "subst_bvs1' (Abs T body) lev args = Abs T (subst_bvs1' body (lev + 1) (map (\<lambda>t. lift t 0) args))"
polytope_imp_compact: fixes S :: "'b set" assumes "polytope S" shows "compact S"
simps_7: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n X\<^sub>n \<phi> = suffix 1 \<xi> \<Turnstile>\<^sub>n \<phi>"
Dag_Ref: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "p \<noteq> Null" shows "Dag p l r t = (\<exists>lt rt. t = Node lt p rt \<and> Dag (l p) l r lt \<and> Dag (r p) l r rt)"
M_Signature_def: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool" shows "M_Signature TYPE('a) wtFsym wtPsym \<equiv> Sig.Signature TYPE('a) wtFsym wtPsym"
ideal_quot_set_ideal: fixes B :: "'a set"   and x :: "'a" shows "ideal (ideal B \<div> x) = ideal B \<div> x"
q_def: fixes t :: "real" shows "ego2.q t = ego.q \<delta> + v\<^sub>e * t"
Q: shows "Q = (\<lambda>((s, l, r), ss, x). dec_inv_2 ly x e (as, lm) (s, l, r) ires)"
join_sub': fixes R :: "nat set"   and L :: "nat set"   and n :: "nat"   and t1 :: "'b option list set"   and t2 :: "'b option list set" assumes "R \<subseteq> L"   and "table n L t1"   and "table n R t2" shows "join t2 True t1 = {as \<in> t1. proj_tuple_in_join True (join_mask n R) as t2}"
assms: shows "set_integrable (M1 \<Otimes>\<^sub>M M2) (A \<times> B) f"
trg_cod: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "trg\<^sub>D (D.cod \<mu>) = trg\<^sub>D \<mu>"
finite_dimensional_vector_space_def: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and Basis :: "'c set" shows "finite_dimensional_vector_space scale Basis \<equiv> vector_space scale \<and> finite_dimensional_vector_space_axioms scale Basis"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(X, Y) \<in> qSwapped" shows "\<forall>X' X''. Y #= X' \<and> X' #= X'' \<longrightarrow> Y #= X''"
fTo_conditional_inc_C: fixes C :: "'a set" assumes "C \<in> carrier fTo D" shows "C \<in> carrier (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S})"
positively_invariant_eq_rev: fixes M :: "'a set" shows "rev.positively_invariant M = negatively_invariant M"
id_left: fixes f :: "'a" assumes "f \<in> Ar" shows "Id (Cod f) \<bullet> f = f"
sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<sqinter> z) = x \<squnion> y \<sqinter> x \<squnion> z"
map_bind_pmf: fixes f :: "'d \<Rightarrow> 'b"   and M :: "'e pmf"   and g :: "'e \<Rightarrow> 'd pmf" shows "map_pmf f (M \<bind> g) = M \<bind> (\<lambda>x. map_pmf f (g x))"
swap_id_eq: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
foldr_plus_zero_le: fixes xs :: "'sum list"   and a :: "'sum" shows "sum_weights xs \<le> foldr followed_by xs a"
neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" obtains "x \<sqsubset> y"   | "y \<sqsubset> x"
\<eta>_p_lit: shows "\<forall>i<Suc n. \<forall>L. L \<in># (DA0' # CAs0') ! i \<longrightarrow> L \<cdot>l (\<eta>0' # \<eta>s0') ! i = L \<cdot>l \<eta>"
wffp_NE: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl" shows "sl \<noteq> []"
atLeastatMost_empty_iff: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b = {}) = (\<not> less_eq a b)"
Icarr: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and i :: "'c" assumes "ideal I R"   and "i \<in> I" shows "i \<in> carrier R"
of_int_less_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) zero = (z < 0)"
maximalD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and c :: "real" assumes "maximal t"   and "0 \<le> c" shows "t (\<lambda>_. c) = (\<lambda>_. c)"
hyps_3: shows "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A"
assms: shows "\<tau>trsys.\<tau>diverge (init_fin t) init_fin_\<tau>move ((status, x), m)"
distinct_30: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Stmt.If x51 x52 x53 \<noteq> x31@[x32]"
f: shows "f \<in> restrict_space M (- X) \<rightarrow>\<^sub>M N"
hom_add: fixes x :: "'a poly"   and y :: "'a poly" shows "pCons (0::'a) (x + y) = pCons (0::'a) x + pCons (0::'a) y"
subtype_x_Set: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> Set \<sigma>" obtains \<tau>' :: "'a OCL_Types.type" where "\<tau> = Set \<tau>'" and "\<tau>' \<sqsubset> \<sigma>"
y_def: shows "y = (apply_chart c3 \<circ> inv_chart c1) w'"
proj_in_K: fixes n :: "nat"   and m :: "nat"   and t :: "nat" assumes "n \<le> m"   and "t \<in> domain (f n)" shows "(f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
ENF_offending_not_empty_imp_ENF_offending_subseteq_rhs: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P"   and "set_offending_flows G nP \<noteq> {}" shows "{{(e1, e2). (e1, e2) \<in> edges G \<and> \<not> P (nP e1) (nP e2)}} \<subseteq> set_offending_flows G nP"
simps_3: fixes x21 :: "'a"   and x22 :: "'a msllist" shows "MSLCons x21 x22 \<noteq> MSLNil"
subdesign_image_is_design: fixes V :: "'a set"   and B :: "'a set multiset" assumes "sub_set_system V B \<V> \<B>"   and "design V B" shows "design (\<pi> ` V) (blocks_image B \<pi>)"
p_props: shows "(\<Sum>i<k. as ! i * bs ! i powr p) = 1"
rep_prm_def: shows "rep_prm \<equiv> quot_type.rep Rep_prm"
intensional_rews_6: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and w :: "'d" shows "(\<exists>x. A x) w \<equiv> \<exists>x. A x w"
setsum_one_less: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "\<forall>i<n. f i \<le> 1" shows "sum f {..<n} \<le> n"
inj_on_to_singleton: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and b :: "'c" assumes "inj_on f A"   and "f ` A = {b}" shows "\<exists>a. A = {a}"
simps_2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
hop4: shows "prf'.game_1 \<A> = map_spmf fst game_A"
double: fixes n :: "nat" assumes "2 \<le> n"   and "P (n div 2)" shows "P n"
foldr_strict_1: fixes f :: "'c \<rightarrow> 'b \<rightarrow> 'b"   and d :: "'b" shows "foldr\<cdot>f\<cdot>d\<cdot>\<bottom> = \<bottom>"
Indiscriminate_def: shows "Indiscriminate \<equiv> \<lambda>F. \<^bold>\<box>((\<^bold>\<exists>x. \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> (\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr>))"
degree_add_le: fixes p :: "'c poly"   and n :: "nat"   and q :: "'c poly" assumes "degree p \<le> n"   and "degree q \<le> n" shows "degree (p + q) \<le> n"
parts_insert_subset: fixes X :: "msg"   and H :: "msg set" shows "insert X (parts H) \<subseteq> parts (insert X H)"
delete_point_index_eq: fixes ps :: "'a set"   and p :: "'a" assumes "ps \<subseteq> del_point p" shows "del_point_blocks p index ps = \<B> index ps"
that: fixes p :: "(name \<times> name) list" assumes "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "yvec = p \<bullet> xvec"   and "N = p \<bullet> M"   and "Q = p \<bullet> P"   and "xvec \<sharp>* N"   and "xvec \<sharp>* Q"   and "(p \<bullet> xvec) \<sharp>* M"   and "(p \<bullet> xvec) \<sharp>* P" shows "thesis"
minf_5: fixes t :: "'a" shows "\<exists>z. \<forall>x<z. (x < t) = True"
unop_def: fixes A :: "V"   and f :: "V" shows "unop A f \<equiv> nop A (1\<^sub>\<nat>) f"
hyp1: shows "fst (Gauss_Jordan_upt_k_det_P A k) = fst (Gauss_Jordan_upt_k_det_P_iarrays (matrix_to_iarray A) k)"
pmf_bind: fixes N :: "'f pmf"   and f :: "'f \<Rightarrow> 'e pmf"   and i :: "'e" shows "pmf (N \<bind> f) i = measure_pmf.expectation N (\<lambda>x. pmf (f x) i)"
permutes_in_funpow_image: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b"   and n :: "nat" assumes "f permutes S"   and "x \<in> S" shows "(f ^^ n) x \<in> S"
vec_cbox_1_eq: fixes u :: "real"   and v :: "real" shows "vec ` cbox u v = cbox (vec u) (vec v)"
dbl_simps_3: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus one = numeral.numeral one plus (num.Bit0 num.One)"
open_Collect_conj: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}"   and "open {x. Q x}" shows "open {x. P x \<and> Q x}"
subst_dnf'_list_dnf: fixes \<phi> :: "'b ltln"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_dnf' (list_dnf \<phi>) m = list_dnf (subst \<phi> m)"
baIANow: fixes \<phi> :: "'BB \<Rightarrow> bool"   and c :: "'bid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'BB" assumes "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "pb.eval c t t' n [\<phi>]\<^sub>b"
arg_pi_iff: fixes z :: "complex" shows "(arg z = pi) = (is_real z \<and> Re z < 0)"
Kmult_def: shows "Kmult = (L + 4 * delta) / (L - 13 * delta) * (4 * exp (1 / 2 * ln 2) * lambda * exp (- (1 - alpha) * D * ln 2 / (5 * delta)) / K)"
standard_decompD: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "standard_decomp k ps"   and "(h, U) \<in> set (ps\<^sub>+)" shows "k \<le> poly_deg h"
A2: shows "Esigma (3\<^sup>2 * 7 * 13 * 5 * 17) = Esigma (3\<^sup>2 * 7 * 13) * Esigma (5 * 17)"
simps_1: shows "del_lo \<langle>\<rangle> = \<langle>\<rangle>"
g'_def: fixes y :: "nat" shows "g' y = (if y \<in> to_nat_on B ` B then enn2real (g (from_nat_into B y)) else 0)"
V_iff_Nil: fixes \<phi> :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and f :: "('b, 'c, 'd) trans \<Rightarrow> 'a"   and trn :: "('b, 'c, 'd) trans"   and tr :: "('b, 'c, 'd) trans list" shows "(BD_Security.V \<phi> f (trn # tr) = []) = (\<not> \<phi> trn \<and> BD_Security.V \<phi> f tr = [])"
bij_betw_nth_root_unity: fixes c :: "complex"   and n :: "nat" assumes "c \<noteq> 0"   and "0 < n" shows "bij_betw ((*) (complex_of_real (root n (cmod c)) * cis (arg c / real n))) {z. z ^ n = 1} {z. z ^ n = c}"
is_RB_uptD1: fixes d :: "'a \<Rightarrow> nat"   and canon :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" assumes "is_RB_upt d canon G u" shows "G \<subseteq> dgrad_sig_set' (length fs) d"
fst_INF: fixes f :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set" shows "fst (Inf (f ` A)) = (INF x\<in>A. fst (f x))"
tree_recurse_simps_1: fixes l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "root (tree_recurse l r x) = x"
precS_def: shows "precS = {w \<in> space M. stake n w = stake n x}"
topo_mono: fixes A :: "'b set set"   and B :: "'b set set" assumes "A \<subseteq> B" shows "topo A \<subseteq> topo B"
arr_dom: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.arr (VoV.dom f)"
b2_def: shows "b2 \<equiv> compute_bound_tm prec I a t2"
sel_rews_1: shows "runWriterT\<cdot>\<bottom> = \<bottom>"
intros: fixes c :: "nat"   and rcm :: "('a, 'b) ta_rule \<Rightarrow> nat option"   and r :: "('a, 'b) ta_rule"   and \<Sigma> :: "('a \<Rightarrow> 'b tree option) \<times> 'a set \<times> (('a, 'b) ta_rule \<Rightarrow> nat option)"   and Q :: "'a \<Rightarrow> 'b tree option"   and W :: "'a set"   and \<Sigma>' :: "('a \<Rightarrow> 'b tree option) \<times> 'a set \<times> (('a, 'b) ta_rule \<Rightarrow> nat option)"   and Q' :: "'a \<Rightarrow> 'b tree option"   and W' :: "'a set"   and rcm' :: "('a, 'b) ta_rule \<Rightarrow> nat option" assumes "c = the (rcm r)"   and "\<Sigma> = (Q, W, rcm)"   and "\<Sigma>' = (Q', W', rcm')"   and "if c \<le> 1 \<and> lhs r \<notin> dom Q then Q' = Q(lhs r \<mapsto> construct_witness Q r) else Q' = Q"   and "if c \<le> 1 \<and> lhs r \<notin> dom Q then W' = insert (lhs r) W else W' = W"   and "rcm' = rcm(r \<mapsto> c - 1)" shows "(\<Sigma>, \<Sigma>') \<in> brw_inner_step r"
inf_compl_bot_right: fixes x :: "'b upred"   and y :: "'b upred" shows "(x \<and> y \<and> \<not> x) = false"
negligible_UNIV: fixes S :: "'a set" shows "negligible S = (indicat_real S has_integral 0) UNIV"
simps_1: fixes G :: "('c, 'd) graph"   and v :: "'c"   and v' :: "'c" shows "is_path_undir G v [] v' = (v = v' \<and> v' \<in> nodes G)"
map_return_pmf: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
P_def: shows "P \<equiv> rec_nat {} (\<lambda>n. st_succ A (w !! n) (g (Suc n)))"
A: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
inverse_mult_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "inverse (times a b) = times (inverse a) (inverse b)"
integrable_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
absolutely_integrable_zero: fixes S :: "'c set" shows "(\<lambda>x. 0::'d) absolutely_integrable_on S"
ex_t: shows "\<exists>t. local.prefixes t j \<down> \<and> x < e_length (the (local.prefixes t j))"
upd_surj: shows "upd ` {..<n} = {..<n}"
IfTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "bval b s" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<rightarrow> (c\<^sub>1, s)"
ccNeighbors_bot: fixes x :: "Vars.var" shows "ccNeighbors x \<bottom> = {}"
lspasl_empl: fixes Gamma :: "bool"   and h :: "'a"   and Delta :: "bool" assumes "Gamma \<and> h = (0::'a) \<longrightarrow> Delta" shows "Gamma \<and> \<box> h \<longrightarrow> Delta"
zeta_zero_reflect: assumes "Re (1 - s) \<in> {0<..<1}"   and "zeta (1 - s) = 0" shows "zeta (1 - (1 - s)) = 0"
dvd_mult_unit_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (times c b) = dvd.dvd times a c"
ResB: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "y \<sharp> a"   and "y \<noteq> x" shows "<\<nu>y>P \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> <\<nu>y>P'"
mon: shows "monic (\<Prod>e\<leftarrow>es. [:- e, 1:])"
subdegree_le_imp_dvd_right_ring1: fixes f :: "'a fps"   and g :: "'a fps" assumes "\<exists>x. x * fps_nth f (subdegree f) = (1::'a)"   and "subdegree f \<le> subdegree g" shows "\<exists>k. g = k * f"
aM: fixes k :: "nat" shows "a k \<lless> M (Suc k)"
keys_code: shows "RBT_Impl.keys = RBT_Impl.gen_keys []"
Henstock_lemma: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and e :: "real" assumes "f integrable_on cbox a b"   and "0 < e" obtains \<gamma> :: "'a \<Rightarrow> 'a set" where "gauge \<gamma>" and "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk> \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e"
map_cong: fixes x :: "'b fset"   and ya :: "'b fset"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> fset ya \<Longrightarrow> f z = g z" shows "f |`| x = g |`| ya"
zero_pos_def: shows "0 \<equiv> Abs_pos (0::'a)"
det_col: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A = (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>j<n. A $$ (p j, j)))"
opC: fixes T :: "'a set" assumes "T \<in> \<C>" shows "open T"
bounded_lattice_bot_ow_axioms: shows "bounded_lattice_bot_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 bot\<^sub>1"
bis_incl_1: shows "Sbis \<subseteq> ZObisT"
gval_gAnd_True: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "(gval (gAnd g1 g2) s = true) = (gval g1 s = true \<and> gval g2 s = true)"
seqsim_suffix_suc: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and t :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "s |\<^sub>s n \<approx> t |\<^sub>s m" shows "\<exists>m'. s |\<^sub>s n \<approx> t |\<^sub>s m' \<and> (s |\<^sub>s Suc n \<approx> t |\<^sub>s Suc m' \<or> s |\<^sub>s Suc n \<approx> t |\<^sub>s m')"
simps: fixes xs :: "cell list" shows "bl2wc xs = bl2nat xs 0"
h'h: fixes y :: "'b" shows "h' (h y) = y"
sxalloc: shows "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1"
rejects_def: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set" shows "rejects \<F> S M \<equiv> comparables S M \<inter> \<F> = {}"
fmap_writer_simps_1: fixes f :: "'f \<rightarrow> 'd" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"