hcju/isabelleps huggingface dataset:
assms: shows "f ` set (x # xs) \<subseteq> Field r"
nofail: shows "nofail (gds_init gds \<bind> WHILE gen_cond gen_step)"
set_selsort: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "set (linorder_list0.selsort le xs) = set xs"
finite_Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "finite (A - (a \<triangleright> B)) = finite (A - B)"
distinct_1: fixes x21 :: "char list"   and x22 :: "aexp" shows "com.SKIP \<noteq> x21 ::= x22"
simps_2: fixes atom_ordering :: "('a \<times> 'a) set"   and I :: "'a set"   and A :: "'a" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.validate_literal I (Neg A) = (A \<notin> I)"
1: shows "forest_components (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e) = (f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T)\<^sup>\<star>"
qmsg_send_from_queue: shows "qmsg \<TTurnstile>\<^sub>A (\<lambda>((msgs, q), a, uu_). sendmsg (\<lambda>m. m \<in> set msgs) a)"
infinite_remove: fixes S :: "'c set"   and a :: "'c" assumes "infinite S" shows "infinite (S - {a})"
power_mod: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (power.power one times (modulo a b) n) b = modulo (power.power one times a n) b"
conjugate_dist_add: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (plus a b) = plus (conjugate a) (conjugate b)"
sylow_central_def: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set"   and M :: "'a set set" shows "sylow_central G p a m M1 M \<equiv> sylow G p a m \<and> sylow_central_axioms G p a m M1 M"
adjacentset_conv_facetchambersets: fixes C :: "'a set" assumes "X \<noteq> {{}}"   and "chamber C" shows "adjacentset C = (\<Union>v\<in>C. {D \<in> X. C - {v} \<lhd> D})"
ext_inject: fixes init :: "'d set"   and trans :: "('d \<times> 'e \<times> 'd) set"   and more :: "'f"   and init' :: "'d set"   and trans' :: "('d \<times> 'e \<times> 'd) set"   and more' :: "'f" shows "(\<lparr>init = init, trans = trans, \<dots> = more\<rparr> = \<lparr>init = init', trans = trans', \<dots> = more'\<rparr>) = (init = init' \<and> trans = trans' \<and> more = more')"
P_conforms: shows "path_conforms_with_strategy p P (\<sigma>(v := w))"
mono_Seq: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq> Q" shows "(P `;` S) \<sqsubseteq> (Q `;` S)"
VLambdaD: fixes a :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" assumes "\<langle>a, f a\<rangle> \<in>\<^sub>\<circ> VLambda A f" shows "a \<in>\<^sub>\<circ> A"
deriv_no_occurrence: fixes x :: "'a"   and r :: "'a rexp" assumes "x \<notin> atoms r" shows "nderiv x r = Zero"
mul_ext_SN_order_pair: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "SN_order_pair S NS" shows "SN_order_pair (s_mul_ext NS S) (ns_mul_ext NS S)"
aux0_def: shows "aux0 = concat (map (\<lambda>(t, rel). if enat (\<tau> \<sigma> ne - t) \<le> right I then [(t, join rel pos rel1)] else []) aux)"
aux_lemma2: fixes q :: "'a list"   and r :: "'a list" assumes "q \<in> carrier (K [X])"   and "q \<noteq> []"   and "r \<in> carrier (K [X])"   and "r \<noteq> []"   and "[a, b] = q \<otimes>\<^bsub>K [X]\<^esub> r"   and "degree q = 1"   and "degree r = 0" shows "r \<in> Units (K [X])"
group_l_invI: assumes "\<And>x. x \<in> carrier (Z n) \<Longrightarrow> \<exists>y\<in>carrier (Z n). y \<otimes>\<^bsub>Z n\<^esub> x = \<one>\<^bsub>Z n\<^esub>" shows "Group.group (Z n)"
charslength_of_prefix: fixes a :: "(symbol \<times> character list) list"   and b :: "(symbol \<times> character list) list" assumes "is_prefix a b" shows "charslength a \<le> charslength b"
weak_transition_def: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle> P' \<equiv> if \<alpha> = \<tau> then P \<Rightarrow> P' else P \<Rightarrow>{\<alpha>} P'"
invar_uniqify: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.queue_invar q" shows "SkewBinomialHeapStruc.queue_invar (SkewBinomialHeapStruc.uniqify q)"
trancl_mono: fixes p :: "'b \<times> 'b"   and r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "p \<in> r\<^sup>+"   and "r \<subseteq> s" shows "p \<in> s\<^sup>+"
assoc_simps_5: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.cod \<a>\<^sub>D[f, g, h] = D.cod f \<star>\<^sub>D D.cod g \<star>\<^sub>D D.cod h"
simps_1: fixes x :: "rat" shows "\<lbrace>Const x\<rbrace> = real_of_rat x"
fresh_HPair: fixes x :: "atom"   and a :: "tm"   and b :: "tm" shows "x \<sharp> HPair a b = (x \<sharp> a \<and> x \<sharp> b)"
istopology_subtopology: fixes V :: "'c set"   and U :: "'c topology" shows "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
hd_def: fixes list :: "'c list" shows "hd list = (case list of x21 # x22 \<Rightarrow> x21)"
val_Zp0_imp_unit0': fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "val_Zp x = 0"   and "x \<in> carrier Zp"   and "0 < n" shows "x n \<in> Units (residue_ring (p ^ n))"
inv_is_inverse: fixes f :: "'a" assumes "B.iso f" shows "B.inverse_arrows f (B.inv f)"
vs_def: shows "vs = prefix (\<psi> j) (z - 1)"
Unit_def: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" shows "Unit R a = (a \<in> carrier R \<and> (\<exists>b\<in>carrier R. a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = 1\<^sub>r\<^bsub>R\<^esub>))"
edges_are_in_V_1: fixes v :: "nat"   and w :: "nat" assumes "v \<rightarrow> w" shows "v \<in> V"
Diff_insert_absorb: fixes x :: "'b"   and A :: "'b set" assumes "x \<notin> A" shows "insert x A - {x} = A"
b: shows "length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) = roundup (length (nat_to_bv (p * q)) - Suc 0) 8 * 8 - 168"
Scan_subset_Gen: fixes I :: "item set"   and k :: "nat"   and u :: "nat"   and T :: "(symbol \<times> character list) set" assumes "I \<subseteq> Gen (\<P> k u)"   and "k \<le> length Doc"   and "T \<subseteq> \<Z> k u" shows "Scan T k I \<subseteq> Gen (\<P> k u)"
space_stream_space: fixes M :: "'a measure" shows "space (stream_space M) = streams (space M)"
apply_cltn2_inv: fixes b :: "proj2"   and g :: "cltn2" shows "apply_cltn2 (apply_cltn2 b g) (cltn2_inverse g) = b"
produces_imp_atms_leq: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset"   and A :: "'a"   and B :: "'a" assumes "production N C = {A}"   and "B \<in> atms_of C" shows "B \<le> A"
prefix_of_empty_is_empty: fixes q :: "'a list" assumes "is_prefix q []" shows "q = []"
Chains_restrict_relD: fixes M :: "'a set"   and A :: "'a set"   and leq :: "('a \<times> 'a) set" assumes "M \<in> Chains (restrict_rel A leq)" shows "M \<in> Chains leq"
wp_under_le_trans: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "wp_under_wlp a" shows "le_utrans (wp a) (wlp a)"
simps_3: fixes s1 :: "String.literal"   and s2 :: "String.literal" shows "(Str s1 < Str s2) = (s1 < s2)"
Spec: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and r :: "('f \<times> 'f) set"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. (\<forall>t. (s, t) \<in> r \<longrightarrow> t \<in> Q) \<and> (\<exists>t. (s, t) \<in> r)} Spec r Q,A"
lemma_3_5_ii_1: fixes a :: "'a" shows "d1 a a = (1::'a)"
a0i_inv1_iagree_def: shows "a0i_inv1_iagree \<equiv> {s. \<forall>h d. set h \<subseteq> good \<longrightarrow> d \<notin> corrupted s \<longrightarrow> signals s (Commit h d) \<le> signals s (Running h d)}"
inj_idom_hom_pcompose: fixes p :: "'b poly" assumes "degree p \<noteq> 0" shows "inj_idom_hom (\<lambda>q. q \<circ>\<^sub>p p)"
atLeastAtMost_borel: fixes a :: "'c"   and b :: "'c" shows "{a..b} \<in> sets borel"
set_resolve_eq_UNION_set_resolve_on: fixes C :: "'a literal list"   and D :: "'a literal list" shows "set (resolve C D) = (\<Union>L\<in>set D. case L of Pos x \<Rightarrow> {} | Neg A \<Rightarrow> if maximal_wrt A (mset D) then set (resolve_on A (remove1 L D) C) else {})"
ucast_of_bl_up: fixes bl :: "bool list" assumes "length bl \<le> size (of_bl bl)" shows "ucast (of_bl bl) = of_bl bl"
components_eq_sing_iff: fixes s :: "'a set" shows "(components s = {s}) = (connected s \<and> s \<noteq> {})"
homf_preserves_dom: fixes f :: "'a"   and A :: "'o" assumes "f \<in> Ar" shows "Hom(A,_) \<^bsub>\<o>\<^esub> (Dom f) = Dom\<^bsub>Set\<^esub> Hom(A,_) \<^bsub>\<a>\<^esub> f"
prems_1: shows "ins3 n d x \<langle>l, y, r\<rangle> = Unbal2 t' n' h'"
type_less_eq_x_Bag: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> Bag \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Bag \<rho>" and "\<rho> \<le> \<sigma>"
cert_stepsize_def: fixes ode_ops :: "ode_ops"   and m :: "'e set \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'e set \<Rightarrow> ('e set \<times> 'f) option nres" shows "cert_stepsize ode_ops m \<equiv> \<lambda>uu uua uuaa uuaaa. rec_nat (\<lambda>X0 h n. let _ = trace_set (ST ''cert_stepsize failed'') (Some X0) in SUCCEED) (\<lambda>i ia X0 h n. op_ivl_rep_of_set X0 \<bind> (\<lambda>(l, u). ASSERT (0 \<le> h) \<bind> (\<lambda>_. ASSERT (l \<le> u) \<bind> (\<lambda>_. mk_safe ode_ops {l..u} \<bind> (\<lambda>ivl. ASSERT (ivl \<noteq> {}) \<bind> (\<lambda>_. P_iter ode_ops X0 h n ivl \<bind> (\<lambda>X'. case X' of None \<Rightarrow> ia X0 (h / 2) n | Some X' \<Rightarrow> m X0 h h X' \<bind> (\<lambda>r1. m X0 0 h X' \<bind> (\<lambda>r2. case (r1, r2) of (None, b) \<Rightarrow> let _ = trace_set (ST ''cert_stepsize method failed'') (Some X') in ia X0 (h / 2) n | (Some (res, err), None) \<Rightarrow> let _ = trace_set (ST ''cert_stepsize method failed'') (Some X') in ia X0 (h / 2) n | (Some (res, err), Some (res_ivl, x)) \<Rightarrow> mk_safe ode_ops res \<bind> (\<lambda>_. mk_safe ode_ops res_ivl \<bind> (\<lambda>_. RETURN (h, res, res_ivl, err)))))))))))) uuaaa uu uua uuaa"
nat_pow_Suc2: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> Suc n = x \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> n"
n_1: fixes n' :: "nat" assumes "n' \<in> set ns" shows "n < n'"
notIrr: assumes "\<not> IrrFinGroupRepresentation G (\<cdot>) V" shows "\<exists>U. (RModule FG (\<cdot>) U \<and> U \<subseteq> V) \<and> U \<noteq> 0 \<and> U \<noteq> V"
mset_insort_key: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and xs :: "'c list" shows "mset (insort_key f x xs) = {#x#} + mset xs"
assert_bound_tableau_id: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<T> (assert_bound a s) = \<T> s"
size_simp1: fixes t :: "'d"   and ts :: "'d list" assumes "t \<in> set ts" shows "size t < Suc (size_list size ts)"
braun_del_min: fixes t :: "'b tree" assumes "braun t" shows "braun (del_min t)"
greaterThanLessThan_subseteq_atLeastLessThan_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b \<subseteq> ord.atLeastLessThan less_eq less c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
order_tendstoD_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and y :: "'c"   and F :: "'d filter"   and a :: "'c" assumes "class.order_topology less_eq less open"   and "filterlim f (topological_space.nhds open y) F"   and "less y a" shows "\<forall>\<^sub>F x in F. less (f x) a"
adj_map: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D"   and "C \<sim> D" shows "f ` C \<sim> f ` D"
refine: fixes ai :: "'b list"   and a :: "'c set"   and sctni :: "real list sctn"   and sctn :: "'c sctn" assumes "(ai, a) \<in> appr_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(RETURN (sbelow_sctn_impl ai sctni), sbelow_sctn $ a $ sctn) \<in> \<langle>bool_rel\<rangle>nres_rel"
AE_False: fixes s :: "'s" shows "(AE x in T s. False) = False"
Abs: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X"   and "x \<notin> vars xs"   and "\<And>s' Y. Y \<in> terms s' \<Longrightarrow> fresh xs x Y"   and "\<And>xs' s' A. A \<in> abs (xs', s') \<Longrightarrow> freshAbs xs x A"   and "\<And>rho. rho \<in> envs \<Longrightarrow> freshEnv xs x rho"   and "\<And>Y. (X, Y) \<in> rel s \<Longrightarrow> phi s Y" shows "phiAbs (xs, s) (Abs xs x X)"
fundamental_theorem_of_algebra: fixes a :: "nat \<Rightarrow> complex"   and n :: "nat" assumes "a 0 = 0 \<or> (\<exists>i\<in>{1..n}. a i \<noteq> 0)" obtains z :: "complex" where "(\<Sum>i\<le>n. a i * z ^ i) = 0"
dual_category_axioms: shows "dual_category (\<cdot>\<^sub>D)"
Liouville_polynomial: fixes f :: "complex \<Rightarrow> complex"   and A :: "real"   and B :: "real"   and n :: "nat"   and \<xi> :: "complex" assumes "f holomorphic_on UNIV"   and "\<And>z. A \<le> cmod z \<Longrightarrow> cmod (f z) \<le> B * cmod z ^ n" shows "f \<xi> = (\<Sum>k\<le>n. (deriv ^^ k) f 0 / fact k * \<xi> ^ k)"
sources_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "weak_composition V H" shows "weak_composition.sources V H \<mu> \<equiv> {a. weak_composition.weak_unit V H a \<and> H \<mu> a \<noteq> partial_magma.null V}"
join_raw_Nil: fixes f :: "'b \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c"   and xs :: "('b \<times> 'c) list" shows "join_raw f xs [] = xs"
subdiv_step: shows "subdivision_step I rev_I H rev_H (u, v, w) (uv, uw, vw)"
compute_is_float_pos: fixes m :: "int"   and e :: "int" shows "is_float_pos (Float m e) = (0 < m)"
equiv_refl_valid: fixes \<phi> :: "fml" shows "valid (\<phi> \<leftrightarrow> \<phi>)"
permutes_subset: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and T :: "'a set" assumes "p permutes S"   and "S \<subseteq> T" shows "p permutes T"
induct: fixes x :: "'c"   and F :: "'c program"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> reachable F"   and "\<And>s. s \<in> Init F \<Longrightarrow> P s"   and "\<And>act s s'. \<lbrakk>act \<in> Acts F; s \<in> reachable F; P s; (s, s') \<in> act\<rbrakk> \<Longrightarrow> P s'" shows "P x"
dist_ge_\<mu>: fixes v :: "'a" assumes "v \<in> verts G"   and "num v \<noteq> \<infinity>"   and "dist v \<noteq> - \<infinity>"   and "\<mu> c s s = ereal 0"   and "dist s = 0"   and "\<And>u. \<lbrakk>u \<in> verts G; u \<noteq> s; num u \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> num u \<noteq> enat 0" shows "\<mu> c s v \<le> dist v"
inputChainFresh: fixes x :: "name"   and xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "x \<sharp> inputChain xvec N P = (x mem xvec \<or> x \<sharp> N \<and> x \<sharp> P)"
list_collect_set_simps_1: fixes f :: "'b \<Rightarrow> 'a set" shows "list_collect_set f [] = {}"
lesspoll_trans2: fixes X :: "'a set"   and Y :: "'b set"   and Z :: "'c set" assumes "X \<prec> Y"   and "Y \<lesssim> Z" shows "X \<prec> Z"
2: fixes p' :: "('a, bool) fmap list" assumes "valid_path Pi p'"   and "distinct p'" shows "length p' - 1 \<le> length p - 1"
sign_changes_rat: fixes ps :: "rat poly list"   and x :: "rat" shows "sign_changes (map real_of_rat_poly ps) (real_of_rat x) = sign_changes_rat ps x"
weak_nominal_ts_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c" shows "weak_nominal_ts satisfies transition \<tau> \<equiv> nominal_ts satisfies transition \<and> weak_nominal_ts_axioms \<tau>"
card_Diff_subset_Int: fixes A :: "'d set"   and B :: "'d set" assumes "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
edges_complete: fixes m :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'val" assumes "m \<in> local.nodes" shows "(m, suc m \<sigma>) \<in> local.edges"
simps_2: fixes \<sigma> :: "'a basic_type" shows "OclAny \<squnion> \<sigma> = (if OclAny \<le> \<sigma> then \<sigma> else if \<sigma> \<le> OclAny then OclAny else OclAny)"
retraction_map_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "retraction_map X Y f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "retraction_map X Y g"
B_SMP_repr: shows "finite_SMP_representation arity Ana \<Gamma> (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)"
norm_cauchy_schwarz: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x y \<le> norm x * norm y"
degree_add_less: fixes p :: "'b poly"   and n :: "nat"   and q :: "'b poly" assumes "degree p < n"   and "degree q < n" shows "degree (p + q) < n"
measure_eq_AE: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M A = Sigma_Algebra.measure M B"
i_Exec_Stream_expand_aggregate_map_nth: fixes k :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd"   and n :: "nat" assumes "0 < k" shows "i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag n = ag (map f (f_Exec_Comp_Stream trans_fun (input n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (input \<Down> n \<odot> k) c)))"
changes_diff_def: shows "changes_diff \<equiv> \<lambda>x. changes_alt_poly_at (p # q # - (p mod q) # ps) x - changes_alt_poly_at (q # - (p mod q) # ps) x"
dma\<^sub>C_eq: shows "dma\<^sub>C mem\<^sub>1\<^sub>C = dma\<^sub>C mem\<^sub>2\<^sub>C"
d_OUT_current_of_bipartite: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal"   and x :: "'v" assumes "current (bipartite_web_of \<Gamma>) f" shows "d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)"
lfp_induct_pointwise: fixes pre :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and post :: "'f \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "('d \<Rightarrow> 'e) \<Rightarrow> 'd \<Rightarrow> 'e"   and a :: "'f"   and x :: "'d" assumes "\<And>a x. chain_admissible (\<lambda>b. \<forall>a x. pre a x \<longrightarrow> post a x (b x))"   and "\<And>a x. pre a x \<longrightarrow> post a x bot"   and "mono B"   and "pre a x"   and "\<And>f a x. \<lbrakk>\<And>a' x'. pre a' x' \<Longrightarrow> post a' x' (f x'); pre a x; f \<le> lfp B\<rbrakk> \<Longrightarrow> post a x (B f x)" shows "post a x (lfp B x)"
assms: shows "0\<^sub>h \<in> disc_compl H"
find_map_filter_None: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'a \<Rightarrow> bool"   and as :: "'b list" assumes "find_map_filter f p as = None" shows "\<forall>b\<in>f ` set as. \<not> p b"
set_append: fixes xs :: "'d list"   and ys :: "'d list" shows "set (xs @ ys) = set xs \<union> set ys"
simps_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'b"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x (y # ys) = (if less_eq (f x) (f y) then x # y # ys else y # linorder.insort_key less_eq f x ys)"
Abs_map_CONS: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set"   and g :: "('b, unit) Old_Datatype.node set \<Rightarrow> 'c" assumes "M \<in> sexp"   and "N \<in> sexp" shows "Abs_map g (CONS M N) = g M # Abs_map g N"
diff_intersect_left_idem: fixes M :: "'c multiset"   and N :: "'c multiset" shows "M - M \<inter># N = M - N"
eexp_minus_infty: shows "eexp (- \<infinity>) = 0"
mono_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
inj_permute: fixes p :: "perm" shows "inj (permute p)"
Cf_1: shows "prob {x \<in> space M. Cf \<le> \<bar>f x\<bar>} < alpha"
cancel_at_def: fixes i :: "nat"   and l :: "(bool \<times> 'e) list" shows "cancel_at i l = take i l @ drop (2 + i) l"
irreducible_prod_lI: fixes b :: "'a"   and a :: "'a" assumes "irreducible G b"   and "a \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "irreducible G (a \<otimes> b)"
component_le_norm_cart: fixes x :: "(real, 'a) vec"   and i :: "'a" shows "\<bar>x $ i\<bar> \<le> norm x"
L_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "O[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
is_sig_red_cong': fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'e) set"   and p :: "'c \<Rightarrow>\<^sub>0 'e"   and q :: "'c \<Rightarrow>\<^sub>0 'e" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail F p"   and "ordered_term.lt term_of_pair ord_term p = ordered_term.lt term_of_pair ord_term q"   and "qpm_inf_term.rep_list term_of_pair fs p = qpm_inf_term.rep_list term_of_pair fs q" shows "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail F q"
sup_top_right: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and x :: "'a" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup x top = top"
l_set_child_nodes_get_tag_name_axioms: shows "l_set_child_nodes_get_tag_name ShadowRootClass.type_wf Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs get_tag_name get_tag_name_locs"
LCons_2: shows "\<forall>(x, y)\<in>Y. llist_all2 P x y"