hcju/isabelleps huggingface dataset:
INF_insert: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and a :: "'c::type"   and A :: "'c::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` insert (a::'c::type) (A::'c::type set)) = (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (f a) (Inf (f ` A))"
ccw'_sortedP_implies_nonaligned: fixes x :: "real \<times> real"   and qs :: "(real \<times> real) list"   and y :: "real \<times> real"   and z :: "real \<times> real" assumes "ccw'.sortedP x qs"   and "y \<in> set qs"   and "z \<in> set qs"   and "y \<noteq> z" shows "det3 x y z \<noteq> 0"
not_eq_extremum: fixes a :: "nat" shows "(a \<noteq> 0) = (a dvd 0 \<and> a \<noteq> 0)"
inverse_positive_iff_positive: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (inverse a) = less zero a"
comm_inv_char: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x = y"
Int_atLeastAtMostL1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b \<inter> ord.atMost less_eq d = ord.atLeastAtMost less_eq a (ord.min less_eq b d)"
tdropn_Suc_TCons: fixes n :: "nat"   and x :: "'a"   and xs :: "('a, 'b) tllist" shows "tdropn (Suc n) (TCons x xs) = tdropn n xs"
wf_greek_less: fixes r :: "('a \<times> 'a) set" assumes "wf r"   and "trans r" shows "wf (greek_less r)"
cases: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and mgu :: "'b set set \<Rightarrow> 'c option"   and less_atm :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and a1 :: "'b literal multiset list"   and a2 :: "'b literal multiset"   and a3 :: "'b multiset list"   and a4 :: "'b list"   and a5 :: "'c"   and a6 :: "'b literal multiset" assumes "FO_resolution subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm"   and "FO_resolution.ord_resolve_rename subst_atm renamings_apart mgu less_atm S a1 a2 a3 a4 a5 a6" obtains CAs :: "'b literal multiset list" and n :: "nat" and AAs :: "'b multiset list" and As :: "'b list" and DA :: "'b literal multiset" and \<rho> :: "'c" and \<rho>s :: "'c list" and \<sigma> :: "'c" and E :: "'b literal multiset" where "a1 = CAs" and "a2 = DA" and "a3 = AAs" and "a4 = As" and "a5 = \<sigma>" and "a6 = E" and "length CAs = n" and "length AAs = n" and "length As = n" and "\<forall>i<n. poss (AAs ! i) \<subseteq># CAs ! i" and "negs (mset As) \<subseteq># DA" and "\<rho> = hd (renamings_apart (DA # CAs))" and "\<rho>s = tl (renamings_apart (DA # CAs))" and "FO_resolution.ord_resolve subst_atm mgu less_atm S (substitution_ops.subst_cls_lists subst_atm CAs \<rho>s) (substitution_ops.subst_cls subst_atm DA \<rho>) (substitution_ops.subst_atm_mset_lists subst_atm AAs \<rho>s) (substitution_ops.subst_atm_list subst_atm As \<rho>) \<sigma> E"
exhaust: fixes y :: "role_t" obtains "y = Init"   | "y = Resp"
block_complement_subset_points: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and ps :: "'b set"   and bl :: "'b set" assumes "incidence_system point_set block_collection"   and "ps \<subseteq> incidence_system.block_complement point_set bl" shows "ps \<subseteq> point_set"
rQuot_empty: fixes a :: "'b" shows "rQuot a {} = {}"
arr_char: fixes f :: "'c local.term set" shows "arr f = (f \<noteq> {} \<and> (\<forall>t. t \<in> f \<longrightarrow> f = mkarr t))"
lm103: fixes Y :: "'a set"   and X :: "'a set" shows "sum (X <| Y) (X \<union> Y) = sum (chi X) (X \<union> Y)"
good_qXrho: shows "qGood qVar xs x #[[pickE rho]]"
case_10: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x101 :: "'b ltln"   and x102 :: "'b ltln" shows "(case x101 W\<^sub>n x102 of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f10 x101 x102"
closed_GSgame_axioms: shows "GaleStewartDeterminedGames.closed_GSgame A N"
one: fixes G :: "('f, 'g) monoid_scheme"   and H :: "'f set" shows "\<one>\<^bsub>G\<^esub> \<in> generate G H"
iadd_Suc_right: fixes m :: "eint"   and n :: "eint" shows "m + (n + 1) = m + n + 1"
stc_SComplex: fixes x :: "complex star" assumes "x \<in> HFinite" shows "stc x \<in> SComplex"
transT_not_mustT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "(c, s) \<rightarrow>t s'" shows "mustT c s"
arr_char: fixes f :: "'a" shows "Left_a.arr f = left a f"
cod_cod: fixes f :: "'a" assumes "A.arr f" shows "A.cod (A.cod f) = A.cod f"
of_rat_minus: fixes a :: "rat" shows "of_rat (- a) = - of_rat a"
simps_1: fixes k :: "'a" shows "split \<langle>\<rangle> k = (\<langle>\<rangle>, False, \<langle>\<rangle>)"
mset_subset_eq_add_left: fixes A :: "'c multiset"   and B :: "'c multiset" shows "A \<subseteq># A + B"
sum_list_sum_nth: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "sum_list xs = sum ((!) xs) {0..<length xs}"
rep_eq: fixes x :: "\<nu>" shows "eval\<kappa> (x\<^sup>P) = Some x"
sup_dual_def: fixes x :: "'a"   and y :: "'a" shows "\<partial> (x \<squnion> y) = \<partial> x \<sqinter> \<partial> y"
zn2: shows "(\<lambda>z. z ^ n) ` (- {0}) = - {0}"
intrinsic_border_def: fixes ls :: "'a list" shows "intrinsic_border ls \<equiv> ARG_MAX length b. strict_border b ls"
set_zip_length_map: fixes x1 :: "'a"   and y1 :: "'b"   and xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'c" assumes "(x1, y1) \<in> set (zip xs ys)"   and "length xs = length ys" shows "(f x1, y1) \<in> set (zip (map f xs) ys)"
dropWhile_append2: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and ys :: "'g list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "dropWhile P (xs @ ys) = dropWhile P ys"
ovalidNF_combine: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "ovalid P f Q"   and "no_ofail P f" shows "ovalidNF P f Q"
AND_NOT_mask_plus_AND_mask_eq: fixes w :: "'b word"   and n :: "nat" shows "(w AND NOT (mask n)) + (w AND mask n) = w"
composites_are_arrows_2: shows "hseq (cod \<mu> \<star> cod \<nu>) (cod \<pi>)"
id_maps: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "name" shows "Lam_ml.id \<Gamma> maps a to Var a"
psubst_fv: fixes \<rho> :: "(nat \<times> exp) list"   and e :: "exp" assumes "closed_env \<rho>" shows "FV (psubst \<rho> e) = FV e - sdom \<rho>"
iso_map_dual: fixes f :: "'a \<Rightarrow> 'b" assumes "mono f" shows "mono (\<partial>\<^sub>F f)"
cong_mult_self_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (times a b) zero a"
sstart_eq: fixes x :: "'c stream"   and S :: "'c set"   and xs :: "'c list" assumes "x \<in> streams S" shows "(x \<in> sstart S xs) = (\<forall>i<length xs. x !! i = xs ! i)"
elims_2: fixes x :: "'b list"   and xa :: "'b list" assumes "subset_sorted x xa" obtains l2 :: "'b list" where "x = []" and "xa = l2__"   | x1 :: "'b" and l1 :: "'b list" and x2 :: "'b" and l2 :: "'b list" where "x = x1 # l1" and "xa = x2 # l2" and "if x1 < x2 then False else if x1 = x2 then subset_sorted l1 l2 else subset_sorted (x1 # l1) l2"
qSwap_preserves_qGood1: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and zs :: "'e"   and x :: "'d"   and y :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X" shows "FixVars.qGood (X #[[x \<and> y]]_zs)"
imply_head: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list" shows "p # ps \<turnstile> p"
winning_strategyI: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "\<And>P. vmc_path G' P v0 p \<sigma> \<Longrightarrow> G'.winning_path p P" shows "G'.winning_strategy p \<sigma> v0"
D''_int: shows "dbm_int (abstr g (\<lambda>i j. \<infinity>) v) n"
tfr_subset_2: fixes B :: "('fun, 'var) Term.term set"   and A :: "('fun, 'var) Term.term set" assumes "tfr\<^sub>s\<^sub>e\<^sub>t B"   and "A \<subseteq> B" shows "tfr\<^sub>s\<^sub>e\<^sub>t A"
simps_1: fixes a :: "atomUni"   and x :: "real" shows "evalUni (AtomUni a) x = aEvalUni a x"
bin_split_rule: fixes xs :: "('b \<times> 'a) list"   and c :: "nat"   and a :: "('b \<times> 'a) array"   and n :: "nat"   and p :: "'a" assumes "sorted_less (separators xs)" shows "<is_pfa c xs (a, n)> bin_split (a, n) p <\<lambda>r. is_pfa c xs (a, n) * \<up> (r \<le> n \<and> (\<forall>j<r. snd (xs ! j) < p) \<and> (r < n \<longrightarrow> p \<le> snd (xs ! r)))>\<^sub>t"
face_cycle_sets_da0: shows "H.face_cycle_sets = G.face_cycle_sets - {G.face_cycle_set a, G.face_cycle_set a'} \<union> H.face_cycle_set ` (G.face_cycle_set a \<union> G.face_cycle_set a' - {a, a'})"
fgw: shows "\<forall>w. v \<noteq> w \<longrightarrow> f w = g w"
sig_red_single_monom_mult: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a"   and c :: "'b"   and s :: "'a" assumes "sig_red_single sing_reg top_tail p q f t"   and "c \<noteq> (0::'b)" shows "sig_red_single sing_reg top_tail (monom_mult c s p) (monom_mult c s q) f (s + t)"
composition_series_triv_group: shows "(carrier G = {\<one>}) = (\<GG> = [{\<one>}])"
freshly_shared_owned: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "dom (share sb \<S>) - dom \<S> \<subseteq> \<O>"
sg_subset: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "H \<subseteq> carrier G"
ma_inverse: fixes r :: "mini_alg" shows "inverse (real_of r) = real_of (ma_inverse r)"
indep_vars_def: fixes i :: "'i"   and M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and I :: "'d set" shows "Y.indep_vars i M' X I = ((\<forall>i\<in>I. Y.random_variable i (M' i) (X i)) \<and> Y.indep_sets i (\<lambda>i. sigma_sets (space (distr M borel (Y i))) {X i -` A \<inter> space (distr M borel (Y i)) |A. A \<in> sets (M' i)}) I)"
f: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "a \<in> local.underS b" shows "f a \<noteq> f b"
map_def: shows "\<alpha>' = transformation_by_components.map VVV.comp (\<cdot>) HoVH (\<lambda>a. local.inv \<a>[fst a, fst (snd a), snd (snd a)])"
lookup_rep_nat_pp_pp: fixes t :: "('b, 'c) pp" shows "lookup_pp (rep_nat_pp t) = (\<lambda>n. rep_nat (lookup_pp t (abs_nat n)))"
dvd_0_right: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times a zero"
empty_neutral_1: fixes x :: "'b multiset" shows "\<Empt> + x = x"
right_minus_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(minus a b = zero) = (a = b)"
h_def: shows "h \<equiv> \<lambda>x. gfinprod G (\<lambda>a. f a [^] poly_mapping.lookup x a) S"
meet_shunt: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "(inf x y = bot) = less_eq x (uminus y)"
path_from_given_point_3: fixes x :: "'a"   and p :: "'a" assumes "path x"   and "point p" shows "end_points (x\<^sup>T\<^sup>\<star> ; p \<cdot> x) \<le> end_points x"
continuous_map_snd_of: fixes Z :: "'b topology"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "continuous_map Z (prod_topology X Y) f" shows "continuous_map Z Y (snd \<circ> f)"
assms_3: shows "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> ipd (\<pi> k)"
simple_bochner_integrable_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<Omega> \<inter> space M \<in> sets M" shows "simple_bochner_integrable (restrict_space M \<Omega>) f = simple_bochner_integrable M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x)"
embedding_map_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "embedding_map X Y f \<equiv> homeomorphic_map X (subtopology Y (f ` topspace X)) f"
Nadd_def: shows "(+\<^sub>N) = (\<lambda>(a, b) (a', b'). if a = 0 \<or> b = 0 then normNum (a', b') else if a' = 0 \<or> b' = 0 then normNum (a, b) else normNum (a * b' + b * a', b * b'))"
dummy: shows "decr 0 s = subst_bv2 s 0 dummy"
colossless_gpvI: fixes gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "lossless_spmf (the_gpv gpv)"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> colossless_gpv \<I> (c input)" shows "colossless_gpv \<I> gpv"
nsphere: fixes n :: "nat" shows "nsphere n = subtopology (powertop_real UNIV) {x. (\<Sum>i\<le>n. (x i)\<^sup>2) = 1 \<and> (\<forall>i>n. x i = 0)}"
mod_diff_left_eq: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus (modulo a c) b) c = modulo (minus a b) c"
times_N_def: fixes a :: "'a N"   and b :: "'a N" shows "a * b \<equiv> Abs_N (Rep_N a + Rep_N b)"
f_1_lemma_2: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)"   and "x \<in> S1 p" shows "f_1 p a (f_1 p a x) = x"
hd_Cons_tl: fixes xs :: "'f list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"
sum_weights_append: fixes ys :: "'lbl list"   and xs :: "'lbl list" shows "sum_weights (ys @ xs) = sum_weights ys + sum_weights xs"
PSuf_snoc: fixes s :: "'a list"   and c :: "'a" shows "PSuf (s @ [c]) = PSuf s @@ {[c]} \<union> {[c]}"
hyp2_axiom4: shows "\<forall>q a b c. \<exists>x. B\<^sub>K q a x \<and> a x \<congruent>\<^sub>K b c"
exhaust: fixes y :: "('a, 'b, 'c, 'd) itnode" obtains x11 :: "('a, 'b) graph_node" and x12 :: "'a" and x13 :: "nat" and x14 :: "'c" where "y = I x11 x12 x13 x14"   | x21 :: "nat" and x22 :: "'c" where "y = H x21 x22"
assms_2: shows "b = aa \<and> [Oc] = ba"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* C" shows "Prop \<alpha> (P \<parallel> Q')"
simps: fixes a :: "dag"   and nodes :: "ref set"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" shows "(a \<in> Dags nodes low high) = (\<exists>t p. a = t \<and> set_of t \<subseteq> nodes \<and> Dag p low high t \<and> t \<noteq> Tip)"
correct_state_Throwing_ex: fixes P :: "jvm_prog"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and xp :: "nat option"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and Cs :: "char list list"   and a :: "nat" assumes "P,\<Phi> |- (xp, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]"   and "ics = Throwing Cs a" shows "\<exists>obj. h a = \<lfloor>obj\<rfloor>"
eq_\<I>_converterD: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and conv' :: "('a, 'b, 'c, 'd) converter"   and q :: "'a" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "q \<in> outs_\<I> \<I>" shows "eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r. r \<in> responses_\<I> \<I> q)) (eq_\<I>_converter \<I> \<I>')) \<I>' (run_converter conv q) (run_converter conv' q)"
4: fixes x :: "'b"   and out :: "'b list"   and nexts :: "(nat \<times> 'b) list" assumes "x = f (tail !! j) v" shows "next_subseqs2_spec out nexts prevs v js (next_subseqs2 out nexts prevs v js)"
prems_1: shows "(C, i) \<in># wP_of_wstate (N, P + {#(C' + {#L#}, i')#}, Q, n)"
lunit_coherence1: fixes a :: "'c" assumes "C.ide a" shows "monoidal_category.lunit (\<cdot>\<^sub>D) D'.T D'.\<alpha> F'.\<iota>\<^sub>1 (F a) \<cdot>\<^sub>D D'.T (F'.\<psi>, F a) = D'.lunit (F a)"
injects: fixes a :: "'a"   and Stream :: "'a Stream"   and a' :: "'a"   and Stream' :: "'a Stream" shows "(a && Stream = a' && Stream') = (a = a' \<and> Stream = Stream')"
w4: fixes x :: "'a"   and y :: "'a" assumes "x \<in> {x. f x = (0::'b)}"   and "y \<in> {x. f x = (0::'b)}" shows "x + y \<in> {x. f x = (0::'b)}"
mat_O_mult_psi'_l: fixes l :: "nat" shows "mat_O *\<^sub>v psi'_l l = psi_l l"
tensor_mkarr: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u" shows "mkarr t \<otimes> mkarr u = mkarr (t \<^bold>\<otimes> u)"
intros_1: fixes p :: "'a pfm" assumes "ptautology p" shows "\<turnstile>\<^sub>! p"
zeta_reflect: assumes "1 - s \<noteq> 0"   and "1 - s \<noteq> 1" shows "rGamma (1 - s) * zeta (1 - (1 - s)) = 2 * complex_of_real (2 * pi) powr - (1 - s) * cos ((1 - s) * complex_of_real pi / 2) * zeta (1 - s)"
assms_2: shows "alw (\<lambda>xs. \<phi> xs \<longrightarrow> ev \<psi> xs) xs"
ring_n_mprod_mprodRTr: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" assumes "distinct_pds K n P" shows "(\<forall>j\<le>m. e j \<in> Zset) \<and> (\<forall>j\<le>m. 0 \<le> e j) \<and> (\<forall>j\<le>m. f j \<in> carrier (O\<^bsub>K P n\<^esub>) - {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}) \<longrightarrow> mprod_exp K e f m = mprod_expR (O\<^bsub>K P n\<^esub>) (nat \<circ> e) f m"
eventually_conj: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F" shows "\<forall>\<^sub>F x in F. P x \<and> Q x"
mantissa_pos_iff: fixes x :: "Float.float" shows "(0 < mantissa x) = (0 < x)"
max_wp_PC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "maximal (wp a)"   and "maximal (wp b)" shows "maximal (wp (a \<^bsub>P\<^esub>\<oplus> b))"
assms_5: shows "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
unitor_coincidence_1: shows "\<l>[\<I>] = \<iota>\<^sub>F\<^sub>M\<^sub>C"
axioms_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MonotModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
F'set1_natural: fixes f3 :: "'o \<Rightarrow> 'r"   and f4 :: "'p \<Rightarrow> 's"   and f1 :: "'t \<Rightarrow> 'q" shows "F'set2 \<circ> Fmap f3 f4 = (`) f1 \<circ> F'set2"
simple_restrict_none: fixes i :: "nat"   and X :: "'a option list"   and A :: "nat set" assumes "i < length X"   and "i \<notin> A" shows "restrict A X ! i = None"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U"   and "b <\<^sub>o\<^sub>w a" shows "b \<le>\<^sub>o\<^sub>w a"
fps_one_def: shows "1 = Abs_fps (\<lambda>n. if n = 0 then 1::'b else (0::'b))"
assms_1: shows "\<turnstile>_ i s;vs;es : ts"