hcju/isabelleps huggingface dataset:
map2_val_pair_ko_def: fixes ko :: "'a key_order" shows "map2_val_pair_ko ko \<equiv> comparator.map2_val_pair (key_compare ko)"
beta_preserves_typ_of1: fixes Ts :: "typ list"   and r :: "Core.term"   and T :: "typ"   and s :: "Core.term" assumes "typ_of1 Ts r = Some T"   and "r \<rightarrow>\<^sub>\<beta> s" shows "typ_of1 Ts s = Some T"
top_meet: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<top> \<sqinter> x = x"
cI: fixes x :: "'a vec" assumes "x \<in> C" shows "c x \<in> \<int>"
bM: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j \<le> k" shows "b k (j, i) \<lless> M (Suc k)"
dominates_trans: fixes g :: "'g"   and n :: "'node"   and n' :: "'node"   and n'' :: "'node" assumes "invar g"   and "dominates g n n'"   and "dominates g n' n''" shows "dominates g n n''"
frameResChainFresh: fixes x :: "name"   and xvec :: "name list"   and F :: "'d frame" shows "x \<sharp> \<lparr>\<nu>*xvec\<rparr>F = (x mem xvec \<or> x \<sharp> F)"
FullModel_axioms: shows "FFullModel InfModel.intTF local.intFF local.intPF"
induct_step: fixes P :: "'b hrel" assumes "\<lbrace>p\<rbrace> P \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> F P \<lbrace>q\<rbrace>\<^sub>u"
cases_scheme: fixes r :: "('b, 'c) node_scheme" obtains name :: "nat" and incoming :: "nat set" and new :: "'b ltlr set" and old :: "'b ltlr set" and "next" :: "'b ltlr set" and more :: "'c" where "r = \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr>"
not_FVar_le_Fun: fixes n :: "nat"   and sch1 :: "type_scheme"   and sch2 :: "type_scheme" shows "\<not> FVar n \<le> sch1 =-> sch2"
potential: shows "enat (p1 + p2) + P3 t2 \<le> P1 s"
integral_change_of_variables_linear: fixes g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "linear g"   and "f absolutely_integrable_on g ` S \<or> f \<circ> g absolutely_integrable_on S" shows "integral (g ` S) f = \<bar>det (matrix g)\<bar> *\<^sub>R integral S (f \<circ> g)"
rep_eq: fixes x :: "('d, 'c) fmap"   and xa :: "'d fset" shows "fset (fmimage x xa) = {uu_. \<exists>a b. uu_ = b \<and> fmlookup x a = Some b \<and> a \<in> fset xa}"
hyps_8: shows "y\<^sub>4 = \<ominus> y\<^sub>1 \<ominus> l \<otimes> (x\<^sub>4 \<ominus> x\<^sub>1)"
linear_simps_5: fixes f :: "'d \<Rightarrow> 'e"   and s :: "real"   and v :: "'d" assumes "bounded_linear f" shows "f (s *\<^sub>R v) = s *\<^sub>R f v"
length_pivot_positions_dim_row: fixes A :: "'b mat" assumes "row_echelon_form A" shows "length (pivot_positions A) \<le> dim_row A"
indep_vars_sum: fixes I :: "'c set"   and i :: "'c"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> real" assumes "finite I"   and "i \<notin> I"   and "indep_vars (\<lambda>_. borel) X (insert i I)" shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Sum>i\<in>I. X i \<omega>)"
finite_char_def: fixes C :: "'a set set" shows "finite_char C = (\<forall>S. (S \<in> C) = (\<forall>S'. finite S' \<longrightarrow> S' \<subseteq> S \<longrightarrow> S' \<in> C))"
printList_def: fixes f :: "'b \<Rightarrow> char list"   and xs :: "'b list"   and l :: "char list"   and r :: "char list"   and sep :: "char list" shows "printList f xs l r sep = (let f' = \<lambda>str x. if str = [] then f x else str @ sep @ f x in l @ foldl f' [] xs @ r)"
bind_cong: fixes x :: "'c option"   and y :: "'c option"   and f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "x \<bind> f = y \<bind> g"
assms: shows "eval cid t t' n (\<exists>\<^sub>bp. \<gamma> p)"
equivI: fixes c :: "com"   and c' :: "com" assumes "\<And>s t \<pi>. \<pi>: (c, s) \<Rightarrow> t \<Longrightarrow> \<pi>: (c', s) \<Rightarrow> t"   and "\<And>s t \<pi>. \<pi>: (c', s) \<Rightarrow> t \<Longrightarrow> \<pi>: (c, s) \<Rightarrow> t" shows "c \<sim> c'"
ideD_1: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "DopxC.arr a"
H'_eq: fixes x :: "'s"   and m :: "nat" shows "T.prob x (H' x (Suc m)) = U x x ^ Suc m"
assms_6: shows "{x. f x = (0::'d)} \<subseteq> local.span B"
hyps: assumes "a \<in> set xs"   and "b \<in> set xs" shows "(flag, a, b) \<in> set (pairs aps flag xs) \<or> (flag, b, a) \<in> set (pairs aps flag xs)"
assms_3: shows "\<forall>i<n. a ! i = b ! i"
False: shows "odd (nat \<lfloor>x\<rfloor>)"
strip_while_Cons: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "\<not> P x" shows "strip_while P (x # xs) = x # strip_while P xs"
i_2: shows "i \<in> existence_ivl0 (flow0 x0 h)"
2: fixes p' :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and q' :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "p' \<in> set gs \<union> set bs"   and "q' \<in> set gs \<union> set bs"   and "fst p' \<noteq> 0"   and "fst q' \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"
Hilbert_fun_eq_Hilbert_poly: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and k :: "nat"   and d :: "nat" assumes "X \<noteq> {}"   and "valid_decomp X ps"   and "hom_decomp ps"   and "cone_decomp T ps"   and "standard_decomp k ps"   and "exact_decomp X 0 ps"   and "\<b> ps 0 \<le> d" shows "int (Hilbert_fun T d) = Hilbert_poly (\<b> ps) (int d)"
6: shows "(dom a \<cong> dom a \<^bold>\<leftarrow> E (a \<cdot> (dom a))) \<^bold>\<and> (E (a \<cdot> (dom a)) \<^bold>\<leftarrow> dom a \<cong> dom a)"
r: fixes q :: "nat"   and e1 :: "nat"   and y :: "nat"   and b :: "nat"   and s1 :: "nat"   and x :: "nat"   and r :: "nat" assumes "secure_mult q"   and "e1 = (y + (q - b)) mod q"   and "s1 = (x * ((y + (q - b)) mod q) + (q - r)) mod q"   and "b < q"   and "x < q"   and "y < q"   and "r < q" shows "r = (x * e1 + (q - s1)) mod q"
Seq: fixes A\<^sub>1 :: "char list set"   and c\<^sub>1 :: "com"   and A\<^sub>2 :: "char list set"   and c\<^sub>2 :: "com"   and A\<^sub>3 :: "char list set" assumes "D A\<^sub>1 c\<^sub>1 A\<^sub>2"   and "D A\<^sub>2 c\<^sub>2 A\<^sub>3" shows "D A\<^sub>1 (c\<^sub>1;; c\<^sub>2) A\<^sub>3"
is_maxpref_def: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "is_maxpref P xs ys = (prefix xs ys \<and> (xs = [] \<or> P xs) \<and> (\<forall>zs. prefix zs ys \<and> P zs \<longrightarrow> prefix zs xs))"
isomorphic_implies_ide_1: fixes f :: "'a"   and g :: "'a" assumes "f \<cong> g" shows "ide f"
w: fixes t :: "real" assumes "c - w < t"   and "t < c" shows "norm (f c) * norm (c - t) < e / 3"
image_cong: fixes M :: "'ap set"   and N :: "'ap set"   and f :: "'ap \<Rightarrow> 'aq"   and g :: "'ap \<Rightarrow> 'aq" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
r_2: fixes x :: "nat" shows "eval r [x] = f x"
OclAnd_false1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(false and X) = false"
ParAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<equiv>\<^sub>s P \<parallel> Q \<parallel> R"
a_set_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes document_ptr disc_nodes = put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update disc_nodes"
LST_def: fixes s :: "('i, 'j, 'g, 'k, 'h) system_state" shows "s\<down> = cLST \<circ> GST s"
set_iterator_image_filter_foldli_conv: fixes g :: "'c \<Rightarrow> 'b option"   and xs :: "'c list" shows "set_iterator_image_filter g (foldli xs) = foldli (List.map_filter g xs)"
bij_lens_vwb: fixes x :: "'b \<Longrightarrow> 'c" assumes "bij_lens x" shows "vwb_lens x"
assms: shows "n \<in> phiDefNodes_aux g v un m"
left_neutral: fixes a :: "'a" shows "\<top> \<sqinter> a = a"
lnth_ltl: fixes xs :: "'a llist"   and n :: "nat" assumes "\<not> lnull xs" shows "lnth (ltl xs) n = lnth xs (Suc n)"
minus_unique: fixes G :: "('c, 'd) ring_scheme"   and y :: "'c"   and x :: "'c"   and y' :: "'c" assumes "abelian_monoid G"   and "y \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"   and "x \<oplus>\<^bsub>G\<^esub> y' = \<zero>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "y = y'"
flat_pm_of_poly_def: fixes x :: "'a" shows "flat_pm_of_poly x = flatten \<circ> pm_of_poly x"
permutes_less_1: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p permutes {0..<n}"   and "i < n" shows "p i < n"
lookup_tensors_from_net: fixes m :: "real mat convnet"   and "is" :: "nat list"   and j :: "nat" assumes "valid_net' m"   and "is \<lhd> input_sizes m"   and "j < output_size' m" shows "lookup (tensors_from_net m $ j) is = evaluate_net m (base_input m is) $ j"
d'_2: shows "\<forall>t\<in>S_Min_Lt. t < d'"
finite_SigmaI: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)"
that: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)"   and "x\<^sub>j \<in> rvars (\<T> s)"   and "lvars (\<T> s') = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"   and "rvars (\<T> s') = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i}" shows "thesis"
berlekamp_mat_i_def: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and u :: "'a list" shows "berlekamp_mat_i p ff_ops u = (let n = degree_i u; ze = arith_ops_record.zero ff_ops; on = arith_ops_record.one ff_ops; mul_p = power_poly_f_mod_i ff_ops (\<lambda>v. mod_field_poly_i ff_ops v u) [ze, on] (nat p); xks = power_polys_i ff_ops mul_p u [on] n in mat_of_rows_list n (map (\<lambda>cs. cs @ replicate (n - length cs) ze) xks))"
less_add_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less a (plus a one)"
consec_un_not_elem2: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<^bold>\<notin> i \<squnion> j \<longrightarrow> n \<^bold>\<notin> j"
summable: shows "summable (\<lambda>n. norm (fds_nth f n / nat_power n s))"
I_nnf: fixes \<phi> :: "atom fm"   and xs :: "int list" shows "Z.I (Z.nnf \<phi>) xs = Z.I \<phi> xs"
uncountable_convex: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "convex S"   and "a \<in> S"   and "b \<in> S"   and "a \<noteq> b" shows "uncountable S"
poly_list_rel_with0_def: fixes R :: "('a \<times> char list multiset) set" shows "poly_list_rel_with0 R = {(xs, ys). (xs, ys) \<in> \<langle>R \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel}"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) ((f \<star> g) \<star> h)"
rbt_foldi_fold_disj: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt"   and val :: "bool" shows "foldi (\<lambda>s. s = False) (\<lambda>k v s. s \<or> P k) t val = RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t val"
r_swap_def: fixes f :: "recf" shows "r_swap f \<equiv> Cn 2 f [recf.Id 2 1, recf.Id 2 0]"
g1: shows "\<forall>b\<in>B. \<exists>c\<in>C. map g b \<cong> c"
measurable_lborel1: fixes M :: "'c measure" shows "M \<rightarrow>\<^sub>M lborel = borel_measurable M"
assms_2: fixes i :: "nat" assumes "i \<in> set ls" shows "i < length s"
poly_rel_unit_factor: shows "(poly_rel ===> poly_rel) (unit_factor_poly_i ops) unit_factor_class.unit_factor"
sets_PiF_single: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "finite I"   and "I \<noteq> {}" shows "sets (Pi\<^sub>F {I} M) = sigma_sets (\<Pi>' i\<in>I. space (M i)) {{f \<in> \<Pi>' i\<in>I. space (M i). (f)\<^sub>F i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
q2: shows "q = Fract (a div gcd a b) (b div gcd a b)"
\<tau>J_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<psi> :: "'c" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv enc F encF N ssap" shows "Jeroslow_Diagonalization.\<tau>J var subst enc encF ssap \<psi> \<equiv> Generic_Syntax.instInp var subst (encF (ssap \<psi>)) (enc (encF (ssap \<psi>)))"
abscissa: shows "abs_conv_abscissa f \<le> ereal c"
type_definition_odlist: shows "type_definition toList odlist_Abs (rel_ext (\<lambda>x. sorted x \<and> distinct x))"
cospan_\<mu>\<nu>: shows "Maps.cospan SPN_fgh.\<mu>.leg0 SPN_fgh.\<nu>.leg1"
simps_2: fixes v :: "'a"   and va :: "'a list"   and u :: "'a \<Rightarrow> 'b" shows "set_clocks (v # va) [] u = u"
bdd_above_norm: fixes X :: "'c set" shows "bdd_above (norm ` X) = bounded X"
card_Diff1_le: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "card (A - {x}) \<le> card A"
bspec: fixes A :: "'k set"   and P :: "'k \<Rightarrow> bool"   and x :: "'k" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
eucl_le: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less_eq x y = (\<forall>i\<in>Basis. inner x i \<le> inner y i)"
set_drop_subset_set_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" assumes "n \<le> m" shows "set (drop m xs) \<subseteq> set (drop n xs)"
distinct_1: fixes x1 :: "'d"   and x21 :: "'d rexp"   and x22 :: "'d rexp" shows "Atom x1 \<noteq> Alt x21 x22"
ub: fixes n :: "nat" shows "t n + p (n + 1) - p n \<le> u n"
listrel_reflcl_if_listrel1: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" assumes "(xs, ys) \<in> listrel1 r" shows "(xs, ys) \<in> listrel (r\<^sup>*)"
binop_onto_def: fixes A :: "V"   and f :: "V" shows "binop_onto A f \<equiv> nop_onto A (2\<^sub>\<nat>) f"
lim_stream: shows "K.lim_stream \<in> S \<rightarrow>\<^sub>M prob_algebra (stream_space S)"
image_set_plus: fixes U :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "linear U" shows "U ` (A + B) = U ` A + U ` B"
Ker0_imp_inj_on: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T"   and "ker T \<inter> V \<subseteq> 0" shows "inj_on T V"
abs_eq: fixes x :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>"   and xa :: "'c" assumes "eq_onp Formula.hereditarily_fs x x" shows "Formula.Act xa (Formula.formula.Abs_formula x) = Formula.formula.Abs_formula (Formula.Act\<^sub>\<alpha> xa x)"
compP\<^sub>2_def: shows "compP\<^sub>2 \<equiv> compP compMb\<^sub>2"
mult_1_left: fixes a :: "'c" shows "bounded_distrib_lattice_class.comp_inf.covector a"
subset'_code_2: fixes ys :: "'b list"   and B :: "'b set" shows "Cardinality.subset' (set ys) B = (\<forall>y\<in>set ys. y \<in> B)"
intros_17: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile>c \<down> Abrupt s"
power2_less_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less (power.power one times a 2) zero"
VSagree_def: fixes \<nu> :: "(real, 'a) vec"   and \<nu>' :: "(real, 'a) vec"   and V :: "'a set" shows "VSagree \<nu> \<nu>' V = (\<forall>i\<in>V. \<nu> $ i = \<nu>' $ i)"
preserves_hseq: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "arr \<mu> \<and> arr \<nu> \<and> src \<mu> = trg \<nu>" shows "S.arr (S.UP \<mu>) \<and> S.arr (S.UP \<nu>) \<and> S.src (S.UP \<mu>) = S.trg (S.UP \<nu>)"
sbintrunc_bintrunc: fixes n :: "nat"   and w :: "int" shows "signed_take_bit n (take_bit (Suc n) w) = signed_take_bit n w"
rotate_before_vFrom: fixes f :: "face"   and r :: "nat"   and u :: "nat"   and v :: "nat" assumes "distinct (vertices f)"   and "r \<in> \<V> f"   and "r \<noteq> u"   and "before (verticesFrom f r) u v" shows "before (verticesFrom f v) r u"
implmt: fixes G :: "prog"   and C :: "qtname"   and I :: "qtname" assumes "G\<turnstile>C\<leadsto>I" shows "G\<turnstile>Class C\<preceq>Iface I"
lepi: shows "\<bar>\<theta> - real_of_int k * (2 * pi)\<bar> \<le> pi"
abstr_id1: fixes c :: "'c"   and cc :: "('c, 'd) cconstraint"   and v :: "'c \<Rightarrow> nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "c \<notin> collect_clks cc"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" shows "abstr cc M v 0 (v c) = M 0 (v c)"
LN_4: fixes k :: "nat"   and \<C> :: "t_context"   and ts :: "t list" assumes "j + 1 + ka__ = j + k"   and "\<S>\<bullet>\<C> \<turnstile> lfilledk : [] _> ts"   and "label \<C> ! k = tvs" shows "\<exists>lholed' vs \<C>'. Lfilled j lholed' (vs @ [$Br (j + k)]) lfilledk \<and> \<S>\<bullet>\<C>' \<turnstile> vs : [] _> tvs \<and> const_list vs"
rec_rng_of_frac_def: shows "rec_rng_of_frac \<equiv> \<lparr>carrier = set_class_of\<^bsub>rel\<^esub>, mult = mult_rng_of_frac, one = \<one> |\<^bsub>rel\<^esub> \<one>, zero = \<zero> |\<^bsub>rel\<^esub> \<one>, add = add_rng_of_frac\<rparr>"
AllN_1: fixes z :: "nat"   and L :: "real list" assumes "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 ((drop z L)[var := B / C])) b = B"   and "insertion (nth_default 0 ((drop z L)[var := B / C])) c = C"   and "var + z < length L" shows "eval \<phi> (L[var + z := B / C]) = eval (linear_substitution_fm_helper var b c \<phi> z) (L[var + z := v])"
currentLevelNonMarked: fixes M :: "('a \<times> bool) list"   and l :: "'a" shows "currentLevel M = currentLevel (M @ [(l, False)])"
map_cong: fixes x :: "'b list"   and ya :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> set ya \<Longrightarrow> f z = g z" shows "map f x = map g ya"
ang_vec_opposite1: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0" shows "\<angle> (- z1) z2 = \<downharpoonright>\<angle> z1 z2 - pi\<downharpoonleft>"
cond_bind_pmf_nonneg: fixes x :: "'a" shows "0 \<le> F x"
elims_2: fixes x :: "'a word \<times> nat"   and xa :: "'a word" assumes "simple_match_ip x xa" obtains base :: "'a word" and len :: "nat" and p_ip :: "'a word" where "x = (base, len)" and "xa = p_ip" and "p_ip \<in> ipset_from_cidr base len"
cSUP_upper2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and 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 f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'b"   and u :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq (f ` A)"   and "x \<in> A"   and "less_eq u (f x)" shows "less_eq u (Sup (f ` A))"
removeAll_id: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "removeAll x xs = xs"
conv_complement: fixes x :: "'a" shows "(- x)\<^sup>T = - x\<^sup>T"
not_lnull_conv: fixes xs :: "'b llist" shows "(\<not> lnull xs) = (\<exists>x xs'. xs = LCons x xs')"
assms_2: shows "\<forall>as s. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < k)"
PS\<^sub>R_def: shows "PS\<^sub>R = ps\<^sub>R - {p\<^sub>0}"
not_Carmichael_number_imp_card_fermat_witness_bound_3: fixes n :: "nat" assumes "\<not> prime n"   and "\<not> Carmichael_number n"   and "odd n"   and "1 < n" shows "real (card {a. 2 \<le> a \<and> a < n \<and> fermat_liar a n}) < real (n - 2) / 2"
map\<^sub>0_def: fixes a :: "'b" shows "GoH.map\<^sub>0 a \<equiv> src\<^sub>D (GoH.map a)"
sequence_flatten_normalize: fixes c :: "('d, 'e, 'f) com" shows "sequence Seq (flatten (Language.normalize c)) = Language.normalize c"
case_2: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> ('e, 'd) Term.term list \<Rightarrow> 'c"   and x21 :: "'e"   and x22 :: "('e, 'd) Term.term list" shows "(case Fun x21 x22 of Var x \<Rightarrow> f1 x | Fun x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
inject_1: fixes x11 :: "nat"   and x12 :: "'a"   and y11 :: "nat"   and y12 :: "'a" shows "(Leq x11 x12 = Leq y11 y12) = (x11 = y11 \<and> x12 = y12)"
keys_uses'_codem': fixes g :: "'g"   and nodes_of_uses :: "('val, 'node set) mapping"   and "next" :: "'node \<times> 'val" assumes "uninst_code.phi_equiv_mapping (p_g g s1_phis) g nodes_of_uses (uninst_code.ssa.useNodes_of (u_g g s1_uses) g)"   and "next \<in> Mapping.keys (p_g g s1_phis g)" shows "Mapping.keys (uninst_code.uses'_codem (u_g g s1_uses) g next (uninst_code.substitution_code (p_g g s1_phis) g next) nodes_of_uses) = Mapping.keys (u_g g s1_uses g)"
hyps: assumes "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<leadsto><Rel> \<lparr>\<nu>*xvec\<rparr>Q"
simps_2: fixes p :: "('a, 'b) form"   and ps :: "('a, 'b) form list" shows "conjoin (p # ps) = And p (conjoin ps)"
select_convs_2: fixes edge_rev :: "'b \<Rightarrow> 'b"   and edge_succ :: "'b \<Rightarrow> 'b"   and more :: "'c" shows "edge_succ \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_succ"
uncross_negative_flip_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "uncross_negative_flip w1 w2" shows "kauff_mat w1 = kauff_mat w2"
in_set_polychain_ofD: fixes ab :: "'a \<times> 'a"   and x :: "'a"   and xs :: "'a list" assumes "ab \<in> set (polychain_of x xs)" shows "snd ab - fst ab \<in> set xs"