hcju/isabelleps huggingface dataset:
dagger_unfoldr_eq: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<dagger> \<cdot> x = x\<^sup>\<dagger>"
inj_f: shows "inj_on f {i. F i \<noteq> {}}"
to_F_in: shows "to_F \<iota> \<in> Inf_F"
distinct_map_default: fixes xs :: "('b \<times> 'c) list"   and k :: "'b"   and v :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "distinct (map fst xs)" shows "distinct (map fst (map_default k v f xs))"
the_adj_chamber_facet: fixes C :: "'a set"   and z :: "'a set" assumes "folding.chamber C"   and "z \<lhd> C" shows "z \<lhd> the_adj_chamber C z"
pair_measure_density: fixes f :: "'c \<Rightarrow> ennreal"   and M1 :: "'c measure"   and g :: "'d \<Rightarrow> ennreal"   and M2 :: "'d measure" assumes "f \<in> borel_measurable M1"   and "g \<in> borel_measurable M2"   and "sigma_finite_measure M2"   and "sigma_finite_measure (density M2 g)" shows "density M1 f \<Otimes>\<^sub>M density M2 g = density (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x * g y)"
nat_of_name_Abs_eq: fixes n :: "nat" shows "nat_of_name (Abs_name (Atom (Sort ''SyntaxN.name'' []) n)) = n"
arrE: fixes f :: "'a" assumes "hhom\<^sub>C.arr f" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright>"
monad_commute_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind"   and "monad_discard return bind" shows "monad_commute (return_option return) (bind_option return bind)"
option_rel_Some1: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c option" shows "rel_option A (Some x) y = (\<exists>y'. y = Some y' \<and> A x y')"
assms_1: shows "RPREM (hn_refine P' c Q' R (RCALL $ a $ b))"
uniform_int_def: shows "uniform_int = map_int_pair (\<lambda>l u. density (stock_measure INTEG) (uniform_int_density (l, u) \<circ> extract_int)) (\<lambda>_. undefined)"
assms_5: shows "x\<^sub>1 = x\<^sub>2"
assoc_set: fixes ps :: "('b \<times> 'a) list"   and x :: "'b"   and y :: "'a" assumes "ps\<langle>x\<rangle>\<^sub>? = \<lfloor>y\<rfloor>" shows "(x, y) \<in> set ps"
glb_def: fixes S :: "'b set"   and po :: "'b potype" shows "glb S po = greatest (\<lambda>x. \<forall>y\<in>S. (x, y) \<in> order po) po"
has_vector_derivative_diff_const: fixes g :: "real \<Rightarrow> 'a"   and z :: "'a"   and f' :: "'a"   and net :: "real filter" shows "((\<lambda>t. g t - z) has_vector_derivative f') net = (g has_vector_derivative f') net"
seqI: fixes f :: "('obj, 'obj) concrete_category.arr"   and g :: "('obj, 'obj) concrete_category.arr" assumes "arr f"   and "arr g"   and "local.dom g = cod f" shows "seq g f"
nat_term_compare_inv_def: fixes to :: "'b nat_term_order" shows "nat_term_compare_inv to = (\<lambda>x y. nat_term_compare to y x)"
typ_of1_drop_extra_bounds: fixes t :: "Core.term"   and Ts :: "typ list"   and rest :: "typ list" assumes "\<not> loose_bvar t (length Ts)" shows "typ_of1 (Ts @ rest) t = typ_of1 Ts t"
basis': fixes a :: "'a"   and b :: "'b" assumes "a \<in> basisA'"   and "b \<in> basisB" shows "\<exists>F\<in>basis. \<forall>a'\<in>basisA'. F *\<^sub>V a' = (if a' = a then b else (0::'b))"
stocks: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b" assumes "CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk" shows "stocks Mkt = {stk, risk_free_asset}"
intro: fixes A :: "bool" assumes "A" shows "my_locale1 A"
cong_mat: fixes nr :: "nat"   and nr' :: "nat"   and nc :: "nat"   and nc' :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'c"   and f' :: "nat \<times> nat \<Rightarrow> 'c" assumes "nr = nr'"   and "nc = nc'"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "mat nr nc f = mat nr' nc' f'"
prems_1: shows "(\<lambda>[\<langle>\<langle>p\<^sub>1, p\<^sub>2\<rangle>\<rangle>]. t) = (\<lambda>[q]. u)"
factor_preserving_hom_comp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "factor_preserving_hom f"   and "factor_preserving_hom g" shows "factor_preserving_hom (f \<circ> g)"
fbox_shunt: fixes p :: "'a"   and q :: "'a"   and x :: "'a"   and t :: "'a" shows "(d p \<cdot> d q \<le>  |x] t) = (d p \<le> ad q +  |x] t)"
UnionE: fixes A :: "'c"   and C :: "'c set set" assumes "A \<in> \<Union> C" obtains X :: "'c set" where "A \<in> X" and "X \<in> C"
iso_lunit: fixes f :: "'a" assumes "Left.ide f" shows "Left.iso (Left.lunit f)"
regions_part_ex_reset_1: fixes u :: "'c \<Rightarrow> real"   and r :: "'c list"   and d :: "nat" assumes "u \<in> V" shows "[r\<rightarrow>real d]u \<in> [[r\<rightarrow>real d]u]\<^sub>\<R>"
abc_exec_null: fixes sam :: "nat \<times> nat list"   and n :: "nat" shows "abc_steps_l sam [] n = sam"
space_T: fixes s :: "'s \<times> 't" shows "space (T s) = space (stream_space (count_space UNIV))"
gcd_int_poly_code: shows "gcd = gcd_int_poly"
eSwap_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and eX :: "'c withERR"   and Y :: "'c" assumes "eSwap MOD zs z1 z2 eX = OK Y" shows "\<exists>s X. eX = OK X \<and> Y = igSwap MOD zs z1 z2 X \<and> igWls MOD s X"
one_sheaf_spec_in_sheaf_spec: fixes U :: "('a \<times> 'a) set set set" assumes "local.local.is_zariski_open U" shows "local.local.one_sheaf_spec U \<in> local.local.sheaf_spec U"
measurable_compose: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'b \<Rightarrow> 'd"   and L :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M L"
lunit_char_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.in_hom V (bicategory.lunit V H \<a> \<i> src trg f) (if partial_magma.arr V f then H (trg f) f else partial_magma.null V) f"
map_ident: fixes t :: "'k list" shows "map (\<lambda>x. x) t = t"
check_mult_l_mult_err_impl_def: fixes p :: "'a"   and q :: "'b"   and pq :: "'c"   and r :: "'d" shows "check_mult_l_mult_err_impl p q pq r = ''Multiplying '' @ show p @ '' by '' @ show q @ '' gives '' @ show pq @ '' and not '' @ show r"
multiplicity_unit_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times p one" shows "factorial_semiring.multiplicity times one p x = 0"
carr_prodag_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "carr_prodag I A = {f \<in> extensional I. f \<in> I \<rightarrow> Un_carrier I A \<and> (\<forall>i\<in>I. f i \<in> carrier (A i))}"
list_all2_same: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_all2 P xs xs = (\<forall>x\<in>set xs. P x x)"
InverseUnique: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "cinv f g"   and "cinv f h" shows "g = h"
character_data_ptr_kinds_eq_h: shows "character_data_ptr_kinds h2 = character_data_ptr_kinds h"
merges_preserves_type: fixes xs :: "nat list list"   and ps :: "(nat \<times> nat list) list" assumes "xs \<in> list n A"   and "\<forall>(p, x)\<in>set ps. p < n \<and> x \<in> A" shows "merges f ps xs \<in> list n A"
simps: fixes i :: "nat" shows "gso i = fs ! i + M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<i])"
cone_hull_eq: fixes S :: "'a set" shows "(cone hull S = S) = cone S"
val_type_eq_REAL: fixes x :: "val" shows "(val_type x = REAL) = (x \<in> range RealVal)"
obsf_resource_eq_\<I>_cong: fixes A :: "'y set"   and res1 :: "('y, 'z) resource"   and res2 :: "('y, 'z) resource" assumes "A \<turnstile>\<^sub>R res1 \<sim> res2" shows "A \<turnstile>\<^sub>R obsf_resource res1 \<sim> obsf_resource res2"
power2_norm_eq_cinner: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(complex_of_real (norm x))\<^sup>2 = cinner x x"
elims: fixes x :: "pseqp"   and y :: "(unit, unit, pseqp, pseqp label) seqp" assumes "\<Gamma>\<^sub>T\<^sub>O\<^sub>Y_skeleton x = y" obtains "x = PToy" and "y = seqp_skeleton (\<Gamma>\<^sub>T\<^sub>O\<^sub>Y PToy)"
mapCollectI: fixes m :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and v :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "m k = Some v" shows "f k v \<in> mapCollect f m"
assms_4: shows "pirreducible\<^bsub>L\<^esub> (carrier L) q"
Nil_is_append_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>n. P (Bound n)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>c a. P a \<Longrightarrow> P (Mul c a)"   and "\<And>a. P a \<Longrightarrow> P (Floor a)"   and "\<And>n c a. P a \<Longrightarrow> P (CN n c a)"   and "\<And>c a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (CF c a b)"   and "\<And>v. P (C v)" shows "P a0"
intro: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2" shows "subresultant_prs_gcd F n \<delta> f k \<beta> G1 G2"
ccSp_Append_call: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n cc-ns\<rightarrow>\<^sub>d* n''"   and "n'' -p\<rightarrow>\<^bsub>call\<^esub> n'" shows "n cc-ns @ [n'']\<rightarrow>\<^sub>d* n'"
count_derangements: fixes n :: "nat" shows "real (count_derangements n) = fact n * (\<Sum>k = 0..n. (- 1) ^ k / fact k)"
consistentI2: fixes \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and store :: "'c \<Rightarrow> nat" shows "consistent (\<lambda>v. \<sigma>\<^sub>s\<^sub>y\<^sub>m (symvar v store)) \<sigma>\<^sub>s\<^sub>y\<^sub>m store"
odif: fixes n :: "nat" shows "open (F (a n) - \<Union> {closure (G (a m)) |m. m < n})"
mult_diag: fixes A :: "('b, 'c) sq_matrix"   and x :: "'b" shows "A * diag x = x *\<^sub>S A"
hm_it_next_rule: fixes m' :: "'a \<Rightarrow> 'b option"   and m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" assumes "m' \<noteq> Map.empty" shows "<hm_is_it m ht m' it> hm_it_next it <\<lambda>((k, v), it'). hm_is_it m ht (m' |` (- {k})) it' * \<up> (m' k = Some v)>"
plus_prod_def: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "x + y = (fst x + fst y, snd x + snd y)"
args_to_set_subset_Times: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)" shows "args_to_set (gs, bs, ps) = fst ` set gs \<union> fst ` set bs"
SubstFormP: fixes i :: "name"   and t :: "tm"   and A :: "fm" shows "{} \<turnstile> SubstFormP \<guillemotleft>Var i\<guillemotright> \<guillemotleft>t\<guillemotright> \<guillemotleft>A\<guillemotright> \<guillemotleft>A(i::=t)\<guillemotright>"
rel_conversep: fixes R :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (=) R\<inverse>\<inverse> = (rel_fun (=) R)\<inverse>\<inverse>"
diag_lookup: fixes t :: "('b, 'c) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.diag t) = (\<lambda>(k, k'). if k = k' then RBT_Mapping2.lookup t k else None)"
strip_all_single_var_is_all: fixes t :: "Core.term" shows "(strip_all_single_var t \<noteq> None) = is_all t"
path_split: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c"   and t1 :: "'d list"   and t2 :: "'d list"   and sn :: "'c" assumes "s1 (t1 @ t2)\<Longrightarrow>\<^bsub>SES\<^esub> sn" shows "\<exists>s2. s1 t1\<Longrightarrow>\<^bsub>SES\<^esub> s2 \<and> s2 t2\<Longrightarrow>\<^bsub>SES\<^esub> sn"
left_neutral: fixes a :: "'c" shows "(0::'c) OR a = a"
strict_implies_order: fixes b :: "'b list"   and a :: "'b list" assumes "strict_prefix b a" shows "Sublist.prefix b a"
replicate_empty: fixes n :: "nat"   and x :: "'c" shows "(replicate n x = []) = (n = 0)"
test_pres2: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q"   and "pres x q" shows "pres (p \<cdot> x) q"
ksimplex: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "ksimplex p n s"
app: fixes u :: "'a"   and v :: "'a" assumes "t = app u v" shows "thesis"
ltake_all: fixes xs :: "'c llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ltake m xs = xs"
multiplicity_prime_elem_times_other: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "prime_elem p"   and "\<not> p dvd q" shows "multiplicity p (q * x) = multiplicity p x"
len_def: fixes a :: "'a array" shows "Array.len a = tap (\<lambda>h. Array.length h a)"
hyperUn_def: fixes A :: "'f set option"   and B :: "'f set option" shows "A \<squnion> B \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> case B of None \<Rightarrow> None | \<lfloor>B\<rfloor> \<Rightarrow> \<lfloor>A \<union> B\<rfloor>"
\<alpha>_limit_set_def: fixes x :: "'a" shows "\<alpha>_limit_set x = {p. \<alpha>_limit_point x p}"
pth_3: fixes x :: "'b" shows "- x \<equiv> - 1 *\<^sub>R x"
refine: fixes g :: "'G"   and v0 :: "'V" assumes "g.invar g" shows "RETURN (idijkstra g v0) \<le> cdijkstra g v0"
linear_simps_2: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "bounded_linear f" shows "f (a - b) = f a - f b"
lead_coeff_add_le: fixes p :: "'b poly"   and q :: "'b poly" assumes "degree p < degree q" shows "lead_coeff (p + q) = lead_coeff q"
H_def: shows "H = (\<lambda>x. sum_upto (\<lambda>m. g (m\<^sup>2) * (real_of_int \<lfloor>x / real (m\<^sup>2)\<rfloor> - x / real (m\<^sup>2))) (sqrt x))"
assn_imp_def2: fixes P :: "'b assn"   and Q :: "'b assn" shows "(P \<Rightarrow> Q) = (\<forall>Y s Z. P Y s Z \<longrightarrow> Q Y s Z)"
respection_of_binary_predicates_and_closures_5: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_respects_binary_pred Rel Pred" shows "rel_respects_binary_pred (Rel\<^sup>*) Pred"
Cod_dom: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "local.Cod (local.dom f) = local.Dom f"
pratt_triple_def: fixes p :: "nat"   and a :: "nat"   and x :: "nat" shows "pratt_triple p a x = (0 < x \<and> (\<forall>q\<in>#prime_factorization x. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)))"
mono_Field: fixes r :: "('d \<times> 'd) set"   and s :: "('d \<times> 'd) set" assumes "r \<subseteq> s" shows "Field r \<subseteq> Field s"
q2: fixes b :: "'b" shows "\<exists>!b'. q b' = b"
assms: shows "P \<equiv> \<lambda>n. prime_factors n \<subseteq> {p}"
"2_1": shows "(it_to_list ((map_iterator_dom \<circ>\<circ>\<circ> (\<circ>)) foldli it) f, xs) \<in> \<langle>Rk\<rangle>list_rel"
Bseq_eventually_mono: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'c" assumes "\<forall>\<^sub>F n in sequentially. norm (f n) \<le> norm (g n)"   and "Bseq g" shows "Bseq f"
ground_fatoms_def: shows "ground_fatoms \<equiv> {a. ground\<^sub>t\<^sub>s (snd a)}"
a4: shows "set pi \<subseteq> Xs \<times> Ys"
netsDistinct_def: fixes a :: "'a set set"   and b :: "'a set set" shows "netsDistinct a b = (\<nexists>x. x \<sqsubset> a \<and> x \<sqsubset> b)"
hyps_3: shows "cancels_to_1\<^sup>*\<^sup>* (map (map_prod f g) l) (map (map_prod f g) x)"
rev_path_append: fixes p :: "('b \<times> 'b) list"   and q :: "('b \<times> 'b) list" shows "rev_path (p @ q) = rev_path q @ rev_path p"
constr_def: shows "constr = map (\<lambda>G. (i, \<forall>X\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t)) (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' (map snd (dbproj i D)))"
objectsNonEmpty: fixes M :: "model" shows "objects M \<noteq> {}"
vector_meet_comp': fixes v :: "'a" assumes "is_vector v" shows "v ; v\<^sup>T = v \<cdot> v\<^sup>T"
t_ne_outside: shows "t_ne \<notin> {t0--t_max}"
A: shows "sinks (con_comp_pol I) (con_comp_map D E p q) u (filter (\<lambda>x. x \<in> range p \<union> range q) xs) = sinks (con_comp_pol I) (con_comp_map D E p q) u xs \<inter> range Some"
Null_not_in_Obj: fixes Obj :: "'b set"   and Null :: "'b" assumes "discrete_category Obj Null" shows "Null \<notin> Obj"
infinite_arbitrarily_large: fixes A :: "'a set"   and n :: "nat" assumes "infinite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
psubstInp_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option"   and inp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" shows "inp %[rho] \<equiv> lift (psubst rho) inp"
1: assumes "0 \<le> e" shows "real (nat_of_integer e) = real_of_int (int_of_integer e)"
ND_F_dir2': fixes s :: "'b event list"   and P :: "'b process"   and S :: "'b process"   and Q :: "'b process" assumes "s \<notin> D P"   and "s \<in> T P"   and "P \<sqsubseteq> S"   and "Q \<sqsubseteq> S" shows "s \<in> T Q"
meval_sched_det: fixes ns :: "nat list"   and c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and c'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" assumes "c \<rightarrow>\<^bsub>ns\<^esub> c'"   and "c \<rightarrow>\<^bsub>ns\<^esub> c''" shows "c' = c''"
devBC_def: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat" shows "devBC t n nid n' \<equiv> if \<exists>nid'. (n', nid') \<in> his t n nid then Some (THE nid'. (n', nid') \<in> his t n nid) else None"
assms_3: shows "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"
simps: fixes a :: "real"   and c :: "real"   and n :: "nat"   and l :: "nat" assumes "table_insert a c" shows "table_insert.pins a c (n, l) = a * real n - table_insert.b c * real l"
mod_ring_uminus32: shows "(mod_ring_rel32 ===> mod_ring_rel32) (uminus_p32 pp) uminus"
fix_cprod: fixes F :: "'a \<times> 'b \<rightarrow> 'a \<times> 'b" shows "fix\<cdot>F = (\<mu> x. fst (F\<cdot>(x, \<mu> xa. snd (F\<cdot>(x, xa)))), \<mu> x. snd (F\<cdot>(\<mu> x. fst (F\<cdot>(x, \<mu> xa. snd (F\<cdot>(x, xa)))), x)))"
ord_spmf_bindI: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'c spmf"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd spmf"   and g :: "'c \<Rightarrow> 'd spmf" assumes "ord_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> ord_spmf P (f x) (g y)" shows "ord_spmf P (p \<bind> f) (q \<bind> g)"
norm_bound_Basis_le: fixes b :: "'b"   and x :: "'b"   and e :: "real" assumes "b \<in> Basis"   and "norm x \<le> e" shows "\<bar>x \<bullet> b\<bar> \<le> e"
mst_reachable_inv: fixes w :: "'a"   and v :: "'a"   and e :: "'a"   and r :: "'a"   and t :: "'a" assumes "regular (w \<sqinter> - id v \<sqinter> (top \<sqinter> e \<sqinter> top))"   and "r \<sqinter> top = r"   and "e \<le> v \<sqinter> - id v"   and "v \<sqinter> top = v"   and "id v = id r \<sqinter> top"   and "t \<le> w"   and "t \<le> v \<sqinter> id v"   and "w \<sqinter> v \<le> v" shows "id r \<sqinter> top \<le> id r \<sqinter> top"
strict_order_iff: fixes a :: "'t"   and b :: "'t" shows "(a \<prec>\<^sub>t b) = (a = ord_term_lin.min a b \<and> a \<noteq> b)"
row_length_eq: fixes a :: "'a list"   and b :: "'a list"   and N :: "'a list list" shows "mat (row_length (a # b # N)) (length (a # b # N)) (a # b # N) \<longrightarrow> row_length (a # b # N) = row_length (b # N)"
power_mono: fixes y :: "int"   and x :: "int"   and n :: "nat" assumes "True"   and "y < x"   and "0 \<le> y"   and "1 \<le> n" shows "y ^ n < x ^ n"
Int_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
sylvester_mat_sub_index: fixes i :: "nat"   and m :: "nat"   and n :: "nat"   and j :: "nat"   and p :: "'c poly"   and q :: "'c poly" assumes "i < m + n"   and "j < m + n" shows "sylvester_mat_sub m n p q $$ (i, j) = (if i < n then if i \<le> j \<and> j - i \<le> m then coeff p (m + i - j) else (0::'c) else if i - n \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'c))"
induct: fixes x :: "fm"   and P :: "fm \<Rightarrow> bool" assumes "x \<in> special_axioms"   and "\<And>A i x. P (A(i::=x) IMP SyntaxN.Ex i A)" shows "P x"
multiset_inter_single: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "(a, {#}) \<inter># (b, {#}) = {#}"
parPresAux: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and R :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and T :: "ccs"   and Rel'' :: "(ccs \<times> ccs) set" assumes "P \<leadsto>[Rel] Q"   and "(P, Q) \<in> Rel"   and "R \<leadsto>[Rel'] T"   and "(R, T) \<in> Rel'"   and "\<And>P' Q' R' T'. \<lbrakk>(P', Q') \<in> Rel; (R', T') \<in> Rel'\<rbrakk> \<Longrightarrow> (P' \<parallel> R', Q' \<parallel> T') \<in> Rel''" shows "P \<parallel> R \<leadsto>[Rel''] Q \<parallel> T"
list_ops_def: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "list_ops ops \<equiv> spec_ops ops list_op_deps"
execute_of_list: fixes xs :: "'a list"   and h :: "heap" shows "execute (Array.of_list xs) h = Some (Array.alloc xs h)"
sep_add_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "plus x y = plus y x"
r_find_k_recfn: shows "recfn 1 r_find_k"
is_singletonI: fixes x :: "'c" shows "is_singleton {x}"
ChamberSubcomplexD_complex: fixes Y :: "'a set set" assumes "folding_g.ChamberSubcomplex Y" shows "ChamberComplex Y"
is_weakly_scdom_component_safe_impl: shows "is_weakly_scdom_component_safe = local.a_is_weakly_scdom_component_safe"
tail_app_iso: fixes hom :: "('e, 'f, 'd, 'c) digraph_isomorphism"   and G :: "('e, 'f) pre_digraph" shows "tail (app_iso hom G) = iso_tail hom"
inj_on_mk_doubleton: fixes X :: "V" shows "inj_on (mk_doubleton X) (elts X)"
strongest_rhs_def: fixes F :: "'b program"   and A :: "'b set" shows "strongest_rhs F A \<equiv> \<Inter> {B. F \<in> A co B}"
textbook_invalid: fixes \<tau> :: "'b state \<times> 'b state" shows "I\<lbrakk>invalid\<rbrakk> \<tau> = UML_Types.bot_class.bot"
tabl: shows "\<forall>v. v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t t"
Bjb: shows "B $ j $ b \<noteq> (0::'a)"