hcju/isabelleps huggingface dataset:
dist_not_less_zero: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "\<not> dist x y < 0"
runit_char_2: fixes f :: "'a" assumes "B.ide f" shows "B.R (B.runit f) = (f \<star>\<^sub>B \<i>[src\<^sub>B f]) \<cdot>\<^sub>B \<a>\<^sub>B[f, src\<^sub>B f, src\<^sub>B f]"
metric_eq_thm: fixes x :: "'b"   and s :: "'b set"   and y :: "'b" assumes "x \<in> s"   and "y \<in> s" shows "x = y \<equiv> \<forall>a\<in>s. dist x a = dist y a"
h_plus_outside: fixes e :: "'v \<times> 'v"   and i :: "nat" assumes "e \<notin> \<^bold>E" shows "h_plus i e = 0"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "cost (Node w t\<^sub>1 t\<^sub>2) = weight t\<^sub>1 + cost t\<^sub>1 + weight t\<^sub>2 + cost t\<^sub>2"
sleq_def: shows "sleq = (\<lambda>x y. smap (insert x (\<eta> y)) = y)"
one_bit_shiftl: fixes n :: "nat" shows "set_bit_class.set_bit 0 n True = 1 << n"
partition_onD2: fixes A :: "'a set"   and P :: "'a set set" assumes "partition_on A P" shows "disjoint P"
generate_in_carrier: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and h :: "'c" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "h \<in> generate G H" shows "h \<in> carrier G"
assms_7: shows "p = 2 ^ (k - l) * f - 1"
valid_SDG_CFG_node: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n" shows "CFG.valid_node sourcenode targetnode valid_edge (parent_node n)"
bij_swap_compose_bij: fixes p :: "'a \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "bij p" shows "bij (Fun.swap a b id \<circ> p)"
fun_type_param_idx: fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and g :: "'fun"   and S :: "('fun, 'atom) Term.term list"   and i :: "nat" assumes "\<Gamma> (Fun f T) = Fun g S"   and "i < length T" shows "\<Gamma> (T ! i) = S ! i"
real_of_rat_ineq: fixes x :: "rat"   and z :: "real" assumes "real_of_rat x \<le> z" shows "x \<le> rat_of_int \<lceil>z\<rceil>"
gd_ord_\<Gamma>_ngd_ord_\<Gamma>: shows "gr.ord_\<Gamma> \<subseteq> ground_sound_\<Gamma>"
simps_2: fixes c :: "'b" shows "vars_of (Const c) = {}"
coprime_iff_gcd_eq_1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a b = (gcd a b = one)"
eval_Methd: fixes G :: "prog"   and s :: "state"   and C :: "qtname"   and sig :: "sig"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>In1l (body G C sig)\<succ>\<rightarrow> (w, s')" shows "G\<turnstile>s \<midarrow>In1l (Methd C sig)\<succ>\<rightarrow> (w, s')"
le_trans_trans: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and y :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and z :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "le_trans x y"   and "le_trans y z" shows "le_trans x z"
add_le_same_cancel1: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b + a \<subseteq># b) = (a \<subseteq># \<Empt>)"
iftrue: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and \<iota> :: "nat"   and c1 :: "('b, 'c) MWLsCom"   and c2 :: "('b, 'c) MWLsCom" assumes "BMap (E b m) = True" shows "((if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi, m), [], Some c1, m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
is_natural_transformation: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "vertical_composite A B F G H \<sigma> \<tau>" shows "natural_transformation A B F H (vertical_composite.map A B \<sigma> \<tau>)"
length_Lev1: fixes xa :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and x :: "'d" assumes "xa \<in> Lev1 s1 s2 n x" shows "length xa = n"
nqfree_nnf: fixes \<phi> :: "atom fm" assumes "qfree \<phi>" shows "nqfree (DLO.nnf \<phi>)"
runFw_scheme: fixes s :: "'a word"   and d :: "'a word"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list"   and p :: "('a, 'c) simple_packet_scheme" shows "runFw s d c rs = runFw_scheme s d c p rs"
bdd_below_image_mono: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "mono f"   and "bdd_below A" shows "bdd_below (f ` A)"
Link_Diagram_Equivalence_def: fixes T1 :: "Link_Diagram"   and T2 :: "Link_Diagram" shows "T1 ~L T2 \<equiv> Rep_Link_Diagram T1 ~ Rep_Link_Diagram T2"
gen_mono_on_def: fixes f :: "'b \<Rightarrow> 'c"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'b set" shows "gen_mono_on f R S = (\<forall>i\<in>S. \<forall>j\<in>S. i \<le> j \<longrightarrow> R (f i) (f j))"
\<F>_semantics': fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<F> \<phi> w = subformulas\<^sub>\<mu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n F\<^sub>n \<psi>}"
bind_bind_resumption: fixes r :: "('j, 'g, 'h) Resumption.resumption"   and f :: "'j \<Rightarrow> ('i, 'g, 'h) Resumption.resumption"   and g :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "r \<bind> f \<bind> g = r \<bind> (\<lambda>x. f x \<bind> g)"
assms_1: fixes l :: "'b" shows "\<alpha> (fl l) \<le> Fl l"
fds_abs_converges_def: fixes f :: "'a fds"   and s :: "'a" shows "fds_abs_converges f s = summable (\<lambda>n. norm (fds_nth f n / nat_power n s))"
size_fsubset: fixes x2 :: "'b fset"   and x1 :: "'b fset" assumes "x2 |\<subset>| x1" shows "size x2 < size x1"
assms_4: shows "polynomial (\<lambda>x. real (q' x))"
INF_identity_eq: fixes Inf :: "'c set \<Rightarrow> 'c"   and A :: "'c set" shows "Inf ((\<lambda>x. x) ` A) = Inf A"
continuous_on_vec_lambda: fixes S :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<And>i. continuous_on S (f i)" shows "continuous_on S (\<lambda>x. \<chi>i. f i x)"
simps_27: fixes va :: "'b formula"   and vb :: "\<I>"   and vc :: "'b formula" shows "safe_formula (Neg (Since va vb vc)) = False"
zero_spaceN_subset_spaceN: fixes N :: "'c quasinorm" shows "zero_space\<^sub>N N \<subseteq> space\<^sub>N N"
ds_dom_llp_strong: fixes x :: "'a"   and y :: "'a" shows "(x = ans_d y \<cdot> x) = domain_semigroup.ds_ord ans_d (\<cdot>) (ans_d x) (ans_d y)"
cring0_sheaf_spec_empty: shows "cring0.sheaf_spec {} = {\<lambda>x. undefined}"
sum_encode_def: fixes x :: "nat + nat" shows "sum_encode x = (case x of Inl a \<Rightarrow> 2 * a | Inr b \<Rightarrow> Suc (2 * b))"
real_cond_exp_F_meas: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M f"   and "f \<in> borel_measurable F" shows "AE x in M. real_cond_exp M F f x = f x"
cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes node_ptr :: "(_) node_ptr" shows "cast node_ptr = (case node_ptr of node_ptr.Ext (Inl element_ptr) \<Rightarrow> Some element_ptr | node_ptr.Ext (Inr x) \<Rightarrow> Map.empty x)"
sum_nonzero_representation_eq: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and basis :: "'e set"   and v :: "'e" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis" shows "(\<Sum>b | module.representation scale basis v b \<noteq> (0::'d). scale (module.representation scale basis v b) b) = v"
assms: shows "\<I> \<turnstile>g gpv \<surd>"
continuousI: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"   and "\<And>m. m open\<^bsub>T\<^esub> \<Longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>" shows "continuous f S T"
prod_unfold_prod_mset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "prod f A = \<Prod>\<^sub># (image_mset f (mset_set A))"
eventually_conj: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F" shows "\<forall>\<^sub>F x in F. P x \<and> Q x"
bprules_1: shows "disjoint (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})"
m2zExt: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "mor2ZF C \<in> extensional (mor\<^bsub>C\<^esub>)"
assoc'_in_hom'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "B.arr \<tau>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<tau>" shows "\<guillemotleft>B.\<a>' \<mu> \<nu> \<tau> : B.dom \<mu> \<star>\<^sub>B B.dom \<nu> \<star>\<^sub>B B.dom \<tau> \<rightarrow>\<^sub>B (B.cod \<mu> \<star>\<^sub>B B.cod \<nu>) \<star>\<^sub>B B.cod \<tau>\<guillemotright>"
induct: fixes P :: "('t + 'n) list \<Rightarrow> 't \<Rightarrow> bool"   and a0 :: "('t + 'n) list"   and a1 :: "'t" assumes "\<And>a. P [] a"   and "\<And>b \<alpha> a. P (Inl b # \<alpha>) a"   and "\<And>N \<alpha> a. ([] |\<in>| cfg.prod G N \<Longrightarrow> P \<alpha> a) \<Longrightarrow> P (Inr N # \<alpha>) a" shows "P a0 a1"
d_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "d x \<le> d y"
indRelRTPO_to_TRel_2: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q" shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, TQ) \<in> (TRel \<union> _Collect (T1, T2) (\<exists>S. T1 = \<lbrakk>S\<rbrakk> \<and> T2 = \<lbrakk>S\<rbrakk>))\<^sup>+"
H_3: fixes i :: "real"   and j :: "real" assumes "i \<in> I"   and "j \<in> I" shows "sheaf_spec_morphisms (V i) (V i \<inter> V j) (s i) = sheaf_spec_morphisms (V j) (V i \<inter> V j) (s j)"
subst_apply_term_empty: fixes t :: "('a, 'b) Term.term" shows "t \<cdot> Var = t"
unique_factorization_m_alt_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "p.unique_factorization_m f cfs = (p.factorization_m f cfs \<and> (\<forall>dgs. p.factorization_m f dgs \<longrightarrow> p.Mf dgs = p.Mf cfs))"
rel_Grp: fixes A1 :: "'f set"   and f1 :: "'f \<Rightarrow> 'i"   and A2 :: "'g set"   and f2 :: "'g \<Rightarrow> 'j"   and A3 :: "'h set"   and f3 :: "'h \<Rightarrow> 'k" shows "F2rel (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) (BNF_Def.Grp A3 f3) = BNF_Def.Grp (F2in A1 A2 A3) (F2map f1 f2 f3)"
good_item_simps_53: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X" shows "X #[xa \<and> xa]_xsa = X"
assms: shows "\<forall>q\<in>qs. \<exists>c\<in>set cs. Prime q \<in> set c"
sum_diff_nat: fixes B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> nat" assumes "finite B"   and "B \<subseteq> A" shows "sum f (A - B) = sum f A - sum f B"
spmf: shows "lossless_spmf (the_gpv gpv)"
l2_init_def: shows "l2_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, signalsInit2 = \<lambda>x. 0, signalsResp2 = \<lambda>x. 0, chan = {}, bad = bad_init\<rparr>}"
ab_diff_conv_add_uminus: fixes a :: "'a"   and b :: "'a" shows "a - b = a + - b"
ball_cong: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> P x = Q x" shows "(\<forall>x\<in>A. P x) = (\<forall>x\<in>B. Q x)"
grover_state_sig_axioms: shows "grover_state_sig n f dims R K"
term_subst_eq_rev: fixes t :: "('d, 'f) Term.term"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term"   and \<tau> :: "'f \<Rightarrow> ('d, 'e) Term.term" assumes "t \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<forall>x\<in>vars_term t. \<sigma> x = \<tau> x"
assms_2: shows "threshold \<phi> (suffix i w) \<G> = Some k'"
induct: fixes P :: "'a literal \<Rightarrow> bool"   and literal :: "'a literal" assumes "\<And>x. P (Pos x)"   and "\<And>x. P (Neg x)" shows "P literal"
xor_left_self: fixes x :: "'c"   and y :: "'c" shows "x XOR x XOR y = y"
prems_1: fixes u :: "'a list"   and a :: "'a" assumes "w = u @- sconst a" shows "thesis"
ring_distribs_1: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a \<cdot> (b + c) = a \<cdot> b + a \<cdot> c"
CD_on_range': fixes c :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "c \<in> ManiPref'.CD_on ds A" shows "c \<in> A \<inter> (\<Union>d\<in>ds. Field (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d))"
prefix_remdups_adj: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys" shows "prefix (remdups_adj xs) (remdups_adj ys)"
open_chart_image: fixes X :: "'c set"   and c :: "('c, 'd) chart" assumes "open X"   and "X \<subseteq> domain c" shows "open (apply_chart c ` X)"
Un_def: fixes A :: "'e set"   and B :: "'e set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
msort_set: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "set (linorder.msort less xs) = set xs"
fun_upd_other: fixes z :: "'e"   and x :: "'e"   and f :: "'e \<Rightarrow> 'f"   and y :: "'f" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
simps_1: fixes carrier :: "'d set"   and more :: "'e" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
dim_vec_of_list: fixes as :: "'b list" shows "dim_vec (vec_of_list as) = length as"
pseudominimal_in_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b" assumes "ordering less_eq less" shows "ordering.pseudominimal_in less_eq less P x \<equiv> ordering.minimal_in less (P - {ordering.bottom less_eq P}) x"
is_unit_content_iff: fixes p :: "'a poly" shows "is_unit (content p) = (content p = (1::'a))"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes document_ptr :: "(_) document_ptr"   and document :: "('c, 'a, 'b, 'j option) RDocument_scheme" shows "put document_ptr document = put (cast document_ptr) (cast document)"
f5: fixes A :: "'a set"   and Aa :: "'a set"   and Ab :: "'a set" shows "A - - Aa - - Ab = A - - (Aa - - Ab)"
sumComposeSet_def: shows "sumComposeSet \<equiv> {(xa, x). sumComposeSetp xa x}"
I2e: fixes s :: "state"   and s' :: "state" assumes "HNext s s'"   and "HInv1 s \<and> HInv2 s \<and> HInv2 s' \<and> HInv3 s \<and> HInv4 s \<and> HInv5 s" shows "HInv5 s'"
n_star_sim_left: fixes y :: "'a"   and x :: "'a" assumes "!y \<cdot> x = x \<cdot> !y" shows "!y \<cdot> x\<^sup>\<star> = !y \<cdot> (x \<cdot> !y)\<^sup>\<star>"
simps_2: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ImZ (x + y) = ImZ x + ImZ y"
assms: shows "linear_inhomogenous_recurrence f (eval_polyexp g) cs fs"
le_Suc_ex: fixes k :: "nat"   and l :: "nat" assumes "order.greater_eq l k" shows "\<exists>n. l = k + n"
card_of_Pow_mono: fixes A :: "'a set"   and B :: "'b set" assumes "|A| \<le>o |B|" shows "|Pow A| \<le>o |Pow B|"
min_rvar_incdec_eq_Some_rvars: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and eq :: "nat \<times> linear_poly"   and x\<^sub>j :: "nat" assumes "min_rvar_incdec_eq dir s eq = Inr x\<^sub>j" shows "x\<^sub>j \<in> rvars_eq eq"
essential_leI: fixes \<Gamma> :: "('a, 'b) web_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> g"   and "wave \<Gamma> g"   and "\<And>e. f e \<le> g e"   and "x \<in> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g)" shows "essential \<Gamma> (B \<Gamma>) (TER\<^bsub>\<Gamma>\<^esub> f) x"
rel_predI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set" assumes "(R ===> (=)) (\<lambda>x. x \<in> A) (\<lambda>y. y \<in> B)" shows "rel_pred R A B"
D_eq_1: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list" assumes "trans r"   and "irrefl r"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<tau>'| -s r \<down>l \<sigma>, r|\<tau>|) \<in> mul_eq r"
addSubsetSetE: fixes A :: "'b set"   and s :: "'b set"   and ts :: "'b set set" assumes "A \<in> addSubsetSet s ts" obtains "A = s - \<Union> ts"   | T :: "'b set" where "T__ \<in> ts" and "A = s \<inter> T__"   | T :: "'b set" where "T \<in> ts" and "A = T - s"
set_dom_\<Phi>: fixes f :: "'t" assumes "S'.arr f" shows "S.set (S.dom (\<Psi> f)) = \<psi> ` S'.set (S'.dom f)"
sum_incoming_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "sum g (incoming u) = (\<Sum>v\<in>E\<inverse> `` {u}. g (v, u))"
ball_def: fixes x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla" shows "ball x t \<phi> \<equiv> all x (imp (LLq (Var x) t) \<phi>)"
bidel_arc_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc G M a \<equiv> digraph_map G M \<and> bidel_arc_axioms G a"
rtranclp_unfold: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "r\<^sup>*\<^sup>* a b \<equiv> a = b \<or> r\<^sup>+\<^sup>+ a b"
bound_ind: shows "f 0 = a \<and> f (card X - 1) = c"
assms: shows "A \<in> {uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'}"
infdist_nonneg: fixes x :: "'b"   and A :: "'b set" shows "0 \<le> infdist x A"
returns_to_cong: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and x :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x" shows "auto_ll_on_open.returns_to g Y A x = Y.returns_to A x"
of_int_0: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
rank_Some_card: fixes x :: "nat"   and n :: "nat"   and j :: "nat" assumes "rank x n = Some j" shows "card (older_seniors x n) = j"
groupWIs_groupWIs1_equi: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "groupWIs1 c rs = groupWIs c rs"
subspace_null_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec.subspace (null_space A)"
observable_transition_stepI: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" assumes "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>" shows "P \<Rightarrow>{\<alpha>} P'"
deriv_g: fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "(g has_real_derivative g' x) (at x)"
alt3: shows "\<exists>a b c. distinct [a, b, c]"
elims: fixes x :: "nat"   and xa :: "char list"   and y :: "nat" assumes "string_to_nat_aux x xa = y" obtains n :: "nat" where "x = n__" and "xa = []" and "y = n__"   | n :: "nat" and d :: "char" and ds :: "char list" where "x = n" and "xa = d # ds" and "y = string_to_nat_aux (n * 10 + char_to_digit d) ds"
equal_bit0_def: fixes x :: "'a bit0"   and y :: "'a bit0" shows "equal_class.equal x y = (Rep_bit0 x = Rep_bit0 y)"
1: shows "\<forall>ys. length ys < length xs \<longrightarrow> snd (sort_and_count_inversions ys) = inversion_number ys"
dg_is_arrD_4: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"
least_power_minimal: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "(p ^^ n) a = a" shows "least_power p a dvd n"
multiplicative_function_divide_nat_power: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "multiplicative_function f" shows "multiplicative_function (\<lambda>n. f n / nat_power n s)"
gauss_lemma: shows "Legendre 2 (int (nat p)) = (- 1) ^ card E"
simps: fixes \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl" shows "ltl_to_generalized_rabin_af\<^sub>\<UU> \<Sigma> \<phi> = ltl_to_rabin_base_def.ltl_to_generalized_rabin \<up>af\<^sub>\<UU> \<up>af\<^sub>G\<^sub>\<UU> (Abs \<circ> Unf) (Abs \<circ> Unf\<^sub>G) M\<^sub>\<UU>_fin \<Sigma> \<phi>"
induct: fixes P :: "pexpr \<Rightarrow> pexpr \<Rightarrow> bool"   and a0 :: "pexpr"   and a1 :: "pexpr" assumes "\<And>x y. (\<And>x2 x21 x22 x2a x21a x22a. \<lbrakk>x = PExpr2 x2; x2 = PPow x21 x22; y = PExpr2 x2a; x2a = PPow x21a x22a; x22 = x22a\<rbrakk> \<Longrightarrow> P x21 x21a) \<Longrightarrow> P x y" shows "P a0 a1"
find_index_first: fixes i :: "nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "i < find_index P xs" shows "\<not> P (xs ! i)"
set_val_get_disconnected_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" assumes "l_set_val_get_disconnected_nodes type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs" shows "\<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
inj_on_add_mult: fixes x :: "nat"   and q :: "nat"   and y :: "nat" assumes "coprime x q" shows "inj_on (\<lambda>b. (y + x * b) mod q) {..<q}"
cscalar_prod_is_scalar_prod: fixes x :: "'b vec"   and y :: "'b vec" shows "x \<bullet>c y = x \<bullet> y"
simps_1: fixes I :: "'d \<Rightarrow> ('c, 'd) Term.term"   and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t [] I D = D"
ide_trg: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "ide (trg \<mu>)"
right_neutral: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<odot> mone = a"
concurrent_store_expunge_independent_technical: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e :: "'b"   and mo :: "'a"   and r :: "'a" assumes "xs prefix of j"   and "(i, Store e mo i) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e i r) \<in> set (node_deliver_messages xs)" shows "hb (i, Store e mo i) (r, Expunge e i r)"
assms: shows "\<forall>i\<in>{0, 1}. dim_vec u = dim_vec (v i)"