hcju/isabelleps huggingface dataset:
x: fixes x :: "nat" assumes "x < Suc i" shows "x - (i - degree g) \<le> degree g"
inorder_inv_update: fixes t :: "'t"   and a :: "'a"   and b :: "'b" assumes "inv t \<and> sorted1 (inorder t)" shows "inv (update a b t)"
sees_method_decl_above: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
interchange: fixes g :: "'a"   and f :: "'a"   and g' :: "'a"   and f' :: "'a" assumes "seq g f"   and "seq g' f'" shows "(g \<otimes> g') \<cdot> (f \<otimes> f') = g \<cdot> f \<otimes> g' \<cdot> f'"
of_hypreal_hyperpow: fixes x :: "real star"   and n :: "nat star" shows "of_hypreal (x pow n) = of_hypreal x pow n"
sorted_append: fixes xs :: "'a list list"   and ys :: "'a list list" shows "linorder.sorted (\<lambda>x y. y \<le>lex x) (xs \<cdot> ys) = (linorder.sorted (\<lambda>x y. y \<le>lex x) xs \<and> linorder.sorted (\<lambda>x y. y \<le>lex x) ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. y \<le>lex x))"
minus_matrix_def: fixes A :: "'a matrix" shows "- A = apply_matrix uminus A"
\<delta>_2: shows "ball (complex_of_real c) \<delta> \<subseteq> A"
same_postrec_suffix: shows "\<forall>k\<ge>j + 1. postrecording_event t k = postrecording_event (swap_events i j t) k"
assms_2: shows "x\<^sub>i \<in> lvars t"
rel_witness_spmf1: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf" assumes "rel_spmf A p q" shows "rel_spmf (\<lambda>a (a', b). a = a' \<and> A a' b) p (rel_witness_spmf A (p, q))"
map_of_map_Pair_key: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and x :: "'c" shows "map_of (map (\<lambda>k. (k, f k)) xs) x = (if x \<in> set xs then Some (f x) else None)"
assms_2: shows "integral\<^sup>N M f = ennreal c * emeasure M (space M)"
leg_simps_4: shows "C.dom \<mu>\<nu>_\<pi>.leg1 = \<mu>\<nu>_\<pi>.apex"
f2: shows "length (get_subsets_R (take_rows_from_matrix (mat_of_cols (dim_row (M_mat_R signs subsets)) (map ((!) (cols (M_mat_R signs subsets))) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))) (rows_to_keep (mat_of_cols (dim_row (M_mat_R signs subsets)) (map ((!) (cols (M_mat_R signs subsets))) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))), map ((!) subsets) (rows_to_keep (mat_of_cols (dim_row (M_mat_R signs subsets)) (map ((!) (cols (M_mat_R signs subsets))) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))), map ((!) signs) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))) = length (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
bit_word_rotr_iff: fixes m :: "nat"   and w :: "'b word"   and n :: "nat" shows "bit (word_rotr m w) n = (n < LENGTH('b) \<and> bit w ((n + m) mod LENGTH('b)))"
vec_lambda_inject: fixes x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(vec_lambda x = vec_lambda y) = (x = y)"
wfp_on_ord_strict: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat" assumes "dickson_grading d" shows "wfp_on (\<prec>) {x. d x \<le> n}"
function_smult_eval: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set"   and b :: "'c" assumes "ring_functions R"   and "a \<in> carrier R"   and "f \<in> carrier (function_ring S R)"   and "b \<in> S" shows "(a \<odot>\<^bsub>function_ring S R\<^esub> f) b = a \<otimes>\<^bsub>R\<^esub> f b"
lbutlast_simps_2: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "lbutlast (a ## r) = (if r = LNil then LNil else a ## lbutlast r)"
orthogonal_self: fixes x :: "'a" shows "orthogonal x x = (x = (0::'a))"
join_ge1: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.SL_top le join Top" shows "le x (join x y)"
fls_X_nth: fixes n :: "int" shows "fls_nth fls_X n = (if n = 1 then 1::'a else (0::'a))"
update_by_fun_pair_gr: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv_raw xs"   and "xs = [] \<or> comp k (fst (last xs)) = Gt" shows "update_by_fun_pair k f xs = xs @ (if f (0::'c) = (0::'c) then [] else [(k, f (0::'c))])"
\<Delta>_naturality: shows "(\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> = (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>"
igSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igSwapCls MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbs MOD \<and> igSwapIGOp MOD"
mod_div_decomp: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" obtains q :: "'a" and r :: "'a" where "q = divide a b" and "r = modulo a b" and "a = plus (times q b) r"
dW0k: fixes k :: "nat" shows "denote_while_n M0 M1 DS k \<rho> \<in> carrier_mat d d"
fac_pow_pref_conjug: fixes u :: "'b list"   and t :: "'b list"   and k :: "nat" assumes "u \<le>f t \<^sup>@ k" obtains t' :: "'b list" where "t \<sim> t'" and "u \<le>p t' \<^sup>@ k"
union_multisetsD: fixes M :: "'b multiset"   and N :: "'b multiset"   and A :: "'b set" assumes "M + N \<in> multisets A" shows "M \<in> multisets A \<and> N \<in> multisets A"
weighted_majority_refl: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "weighted_majority R x x = 0"
insert_index_inj_on: fixes i :: "nat"   and S :: "nat set" shows "inj_on (insert_index i) S"
hom_mult: fixes x :: "'b"   and y :: "'b" shows "to_fract (x * y) = to_fract x * to_fract y"
mult_isol_var: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "'c"   and x :: "'c"   and v :: "'c"   and y :: "'c" assumes "class.pre_dioid plus times less_eq less"   and "less_eq u x"   and "less_eq v y" shows "less_eq (times u v) (times x y)"
rep_eq: fixes x :: "('a, 'b) tllist" shows "tnth x = lnth (fst (llist_of_tllist x, terminal x))"
lgrid_sum: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and lm :: "nat"   and F :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < length b"   and "level b < lm" shows "sum F (lgrid b {d} lm) = sum F (lgrid (child b left d) {d} lm) + sum F (lgrid (child b right d) {d} lm) + F b"
element_ptr_kinds_def: fixes heap :: "(_) heap" shows "element_ptr_kinds heap = the |`| 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 |`| ffilter is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (node_ptr_kinds heap)"
isin: shows "g \<in> isin (s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}), roomk := (roomk s)(r1 := k2), safe := (safe s)(r1 := owns s r1 = \<lfloor>g1\<rfloor> \<and> isin s r1 = {} \<and> k2 = currk s r1 \<or> safe s r1)\<rparr>) r"
res_at_most_two: fixes c :: "cars" shows "\<forall>ts v. \<nexists>va u. v=va--u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (res ts) c = lan va \<and> |lan va| = 1) \<and> (\<exists>v ua. u=v--ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext ua\<parallel> \<and> len ua ts c = ext ua \<and> restrict ua (res ts) c = lan ua \<and> |lan ua| = 1)"
beh_immediate_prefix_closed: fixes s :: "'a"   and b :: "'a list"   and T :: "('a, 'b) TS_scheme" assumes "s # b \<in> beh T" shows "b \<in> beh T"
set_mtf2: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "set (mtf2 n x xs) = set xs"
span_eq_iff: fixes s :: "('a, 'b) vec set" shows "(vec.span s = s) = vec.subspace s"
d1: fixes x :: "real"   and y :: "real" assumes "g x = g y"   and "u \<le> x"   and "x \<le> v"   and "u \<le> y"   and "y \<le> v" shows "x = y"
max_add_distrib_right: 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus x (ord.max less_eq y z) = ord.max less_eq (plus x y) (plus x z)"
prems_2: shows "\<mu>_small_row i fs (Suc j)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" obtains "A.arr f" and "A.dom f = a" and "A.cod f = b"
real_sqrt_lt_0_iff: fixes x :: "real" shows "(sqrt x < 0) = (x < 0)"
isomorphic_free_Abelian_groups: fixes S :: "'b set"   and T :: "'c set" shows "free_Abelian_group S \<cong> free_Abelian_group T = (S \<approx> T)"
simps_2: fixes f :: "'a"   and Tl :: "'a trm list" shows "gT (Fn f Tl) = Fn (Oldf f) (map gT Tl)"
get_alloc: fixes x :: "'a"   and h :: "heap"   and x' :: "'a" shows "Ref.get (snd (Ref.alloc x h)) (fst (Ref.alloc x' h)) = x"
fun_star_inductr: fixes h :: "'b nd_fun"   and g :: "'b nd_fun"   and f :: "'b nd_fun" assumes "h \<squnion> g \<cdot> f \<le> g" shows "h \<cdot> qstar f \<le> g"
soundnessFCut: fixes C :: "formula"   and Gamma :: "formula list"   and Delta :: "formula list" assumes "validS (C # Gamma)"   and "validS (FNot C # Delta)" shows "validS (Gamma @ Delta)"
Units_l_inv_ex: fixes x :: "'a" assumes "x \<in> Units (units_group K)" shows "\<exists>y\<in>carrier (units_group K). y \<otimes>\<^bsub>units_group K\<^esub> x = \<one>\<^bsub>units_group K\<^esub>"
gamma3_def: shows "gamma3 = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
tangent_space_def: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and p :: "'c" assumes "c_manifold charts k" shows "c_manifold.tangent_space charts k p = {X. c_manifold.is_derivation charts k X p \<and> extensional0 (c_manifold.diff_fun_space charts k) X}"
one_le_numeral: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one (numeral.numeral one plus n)"
Rep_proper_base: fixes x :: "proper_base" shows "Rep_proper_base x \<in> {x. 2 \<le> x}"
create_initial_object_simps: fixes h :: "'heap"   and ads :: "'addr list"   and b :: "bool"   and C :: "String.literal" shows "create_initial_object (h, ads, b) C = (if b then let HA = allocate h (Class_type C) in if HA = {} then (h, ads, False) else let (h', a'') = SOME ha. ha \<in> HA in (h', ads @ [a''], True) else (h, ads, False))"
bot_eq_sup_iff: fixes x :: "'b multiset"   and y :: "'b multiset" shows "({#} = x \<union># y) = (x = {#} \<and> y = {#})"
lincomb_def: fixes a :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "lincomb a A = (\<Oplus>\<^bsub>M\<^esub>v\<in>A. a v \<odot>\<^bsub>M\<^esub> v)"
snd_sym_preproc_dgrad_set_le: fixes d :: "'a \<Rightarrow> nat"   and gs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and fs :: "('a \<Rightarrow>\<^sub>0 'd) list" assumes "dickson_grading d" shows "dgrad_set_le d (local.punit.pp_of_term ` Keys (set (snd (punit.sym_preproc gs fs)))) (local.punit.pp_of_term ` Keys (set gs \<union> set fs))"
map_vec_def: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d vec" shows "map_vec f v \<equiv> vec (dim_vec v) (\<lambda>i. f (v $ i))"
as_needed_asses_submap_exec_ix: fixes f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" shows "(f \<subseteq>\<^sub>f g) = (\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v)"
simps_1: fixes x :: "real" shows "sqrtu x 0 = (x + 1) / 2"
seg_set_impl_correct: fixes i :: "nat" assumes "i < length B" shows "seg_set_impl (seg_start i) (seg_end i) \<le> SPEC (\<lambda>r. r = p_\<alpha> ! i)"
globally_consistent_loweq: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and A :: "'b \<Rightarrow> ('c \<times> 'c) option"   and mds :: "Mode \<Rightarrow> 'b set"   and x :: "'b"   and v :: "'c"   and v' :: "'c" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.globally_consistent dma A mds"   and "A x = Some (v, v')"   and "dma x = Low" shows "v = v'"
hom_induced_hom: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set"   and Y :: "'c topology"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c" shows "hom_induced p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)"
l2_inv7_def: shows "l2_inv7 \<equiv> {s. secret s \<subseteq> {Exp (Exp Gen (NonceF (R $ N))) (NonceF (R' $ N')) |N N' R R'. R = test \<and> R' \<in> partners \<and> (N = nx \<or> N = ny) \<and> (N' = nx \<or> N' = ny)}}"
rec_scan_def: shows "rec_scan = Cn 1 rec_mod [recf.id 1 0, constn 2]"
measurable_Pair1': fixes x :: "'c"   and M1 :: "'c measure"   and M2 :: "'d measure" assumes "x \<in> space M1" shows "Pair x \<in> M2 \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
flow_usolves_ode: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t0 \<in> T"   and "x0 \<in> X" shows "(ll_on_open.flow T f X t0 x0 usolves_ode f from t0) (ll_on_open.existence_ivl T f X t0 x0) X"
mset_merge_all: fixes xss :: "'c list list" shows "mset (merge_all xss) = mset_mset xss"
MakeCat: fixes C :: "('a, 'b, 'c) Category_scheme" assumes "Category_axioms C" shows "Category (MakeCat C)"
ext_model: fixes S :: "'b formula set"   and F :: "'b formula" assumes "fin_sat S" shows "(\<lambda>k. Atom k \<in> extended S) \<Turnstile> F = (F \<in> extended S)"
valid_prog_inv: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and p' :: "'a"   and is' :: "instr list" assumes "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')"   and "valid_prog p" shows "valid_prog p'"
face_cycle_pred_succ: fixes a :: "'b" shows "H.face_cycle_pred (H.face_cycle_succ a) = a"
rref_suc_if_zero_below_greatest: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)"   and "\<forall>a>GREATEST m. \<not> is_zero_row_upt_k m k A. is_zero_row_upt_k a (Suc k) A" shows "reduced_row_echelon_form_upt_k A (Suc k)"
median': shows "is_median x (map med (chop (Suc (2 * d)) xs))"
OAlist_tc_update_by_fun_gr_def: fixes k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_update_by_fun_gr k f xs = OAlist_tc (update_by_fun_gr_pair_tc k f (list_of_oalist_tc xs))"
summable_norm_cancel: fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>n. norm (f n))" shows "summable f"
prev_in_A: shows "prev_in A u \<in> A \<inter> {..<u}"
assms_2: shows "S \<subseteq> \<langle>G\<rangle>"
b_in_ys: shows "b \<in> set (sorted_list_of_set J)"
AMin_def: fixes A :: "ant set" shows "AMin A = (THE m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x))"
beukers_aux_by_parts: fixes x :: "real"   and z :: "real" assumes "x \<in> {0<..<1}"   and "z \<in> {0<..<1}" shows "LBINT y=0..1. P y / (1 - (1 - x * y) * z) = LBINT y=0..1. (x * y * z) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * z) ^ (n + 1)"
abs_le_D1: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq a b"
support_flow_def: shows "support_flow \<equiv> \<lambda>f. {x. support_flowp f x}"
onll_ostep_invariantE: fixes A :: "('e \<times> ('f, 'g, 'h, 'i) seqp, 'j) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'j \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "('e \<times> 'i) \<times> 'j \<times> 'e \<times> 'i \<Rightarrow> bool"   and \<sigma> :: "'e"   and p :: "('f, 'g, 'h, 'i) seqp"   and a :: "'j"   and \<sigma>' :: "'e"   and p' :: "('f, 'g, 'h, 'i) seqp"   and l :: "'i"   and l' :: "'i" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a"   and "l \<in> labels \<Gamma> p"   and "l' \<in> labels \<Gamma> p'" shows "P ((\<sigma>, l), a, \<sigma>', l')"
a_def: fixes f :: "'a \<Rightarrow> real \<times> real" shows "a f = (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i)"
indep_sets_finite_index_sets: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set" shows "indep_sets F I = (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
complex_Im_numeral: fixes v :: "num" shows "is_real (numeral v)"
distinct_75: fixes x3 :: "val"   and x14 :: "'a exp" shows "Val x3 \<noteq> throw x14"
fgh1: fixes x :: "real" shows "(\<lambda>t. f (g x) (h t)) = (\<lambda>(y1, y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
algebraic_int_minus: fixes x :: "'b" assumes "algebraic_int x" shows "algebraic_int (- x)"
simps_2: fixes ze :: "'c"   and on :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and m :: "'c list list"   and i :: "nat" shows "mat_powI ze on pl ti n m (Suc i) = mat_multI ze pl ti n (mat_powI ze on pl ti n m i) m"
atLeast_int_atMost_int_shift: fixes g :: "int \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {int m..int n} = sum (g \<circ> int) {m..n}"
singular_cycle_singleton: fixes X :: "'c topology"   and a :: "'c"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {a}" shows "singular_relcycle p X {} c = (singular_chain p X c \<and> (p = 0 \<or> odd p \<or> c = 0))"
ord_eq_le_trans: fixes a :: "'b Word.word"   and b :: "'b Word.word"   and c :: "'b Word.word" assumes "a = b"   and "b \<le>s c" shows "a \<le>s c"
set_3: fixes x1 :: "('b, 'a) hd" shows "vars (Hd x1) = vars_hd x1"
characteriseAx: fixes r :: "'b list \<times> 'c sequent" assumes "r \<in> Ax" shows "r = ([],  \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) \<or> (\<exists>i. r = ([],  \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>))"
stable_pair_on_CH: fixes ds :: "'d set"   and XD_XH :: "'x set \<times> 'x set" assumes "stable_pair_on ds XD_XH" shows "match XD_XH = CH (snd XD_XH)"
supp_map_transform_step: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and t :: "'d \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'd\<^sub>\<bottom>" assumes "\<And>x e a. (x, e) \<in> set \<Gamma> \<Longrightarrow> supp (t a e) \<subseteq> supp e" shows "supp (map_transform t ae \<Gamma>) \<subseteq> supp \<Gamma>"
less_eqE: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'b" where "b = plus a c"
1092: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> x = bot"
binary_code_axioms: shows "binary_code u\<^sub>0 u\<^sub>1"
lr_POR_result_rel: shows "POR_result_rel \<notin> unlr POR.delta"
simps_3: fixes y :: "'b"   and ys :: "['b]" shows "compare\<cdot>[]\<cdot>(y : ys) = LT"
simps_2: fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "receive\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = receive\<langle>t \<cdot> \<theta>\<rangle>"
ln_upper_5_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_5 x"
mbisim_thrD1: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and bisim_wait :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and t :: "'a"   and x :: "'b"   and ln :: "'f \<Rightarrow>f nat" assumes "FWbisimulation_base.mbisim bisim bisim_wait s1 s2"   and "thr s1 t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>x'. thr s2 t = \<lfloor>(x', ln)\<rfloor> \<and> bisim t (x, shr s1) (x', shr s2) \<and> (wset s1 t = None \<or> bisim_wait x x')"
a_d_mult_closure: fixes x :: "'a"   and y :: "'a" shows "am_d (ad x \<cdot> ad y) = ad x \<cdot> ad y"
quot_of_fract_in_normalized_fracts: fixes x :: "'a fract" shows "quot_of_fract x \<in> normalized_fracts"
CFG_SSA_Transformed_notriv_linorder_code_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> ('b, 'e set) mapping"   and phis :: "'a \<Rightarrow> ('b \<times> 'e, 'e list) mapping"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and chooseNext_all :: "('b \<Rightarrow> 'e set) \<Rightarrow> ('b \<times> 'e, 'e list) mapping \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'e" shows "CFG_SSA_Transformed_notriv_linorder_code \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var chooseNext_all \<equiv> CFG_SSA_Transformed_code \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var \<and> CFG_SSA_Transformed_notriv_linorder \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) var (\<lambda>uses phis. chooseNext_all uses (Mapping phis)) \<and> CFG_SSA_Transformed_notriv_linorder_code_axioms \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs var chooseNext_all"
mId_mHom: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "R module M" shows "mId\<^bsub>M\<^esub>  \<in> mHom R M M"
fmlookup_restrict_set_id': fixes N :: "('c, 'd) fmap"   and A :: "'c set" assumes "set_mset (dom_m N) \<subseteq> A" shows "fmrestrict_set A N = N"
payer_def: fixes n :: "nat"   and dc :: "'a \<times> 'b" assumes "dining_cryptographers_space n" shows "payer dc = fst dc"
l_get_owner_document_wf_get_root_node_wf_axioms: shows "l_get_owner_document_wf_get_root_node_wf heap_is_wellformed DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_root_node get_owner_document"