hcju/isabelleps huggingface dataset:
set_child_nodes_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_child_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
smap_alt: fixes f :: "'b \<Rightarrow> 'a"   and s :: "'b stream"   and s' :: "'a stream" shows "(smap f s = s') = (\<forall>n. f (s !! n) = s' !! n)"
verticesFrom_is_nextElem: fixes v :: "nat"   and f :: "face"   and a :: "nat"   and b :: "nat" assumes "v \<in> \<V> f" shows "is_nextElem (vertices f) a b = is_nextElem (verticesFrom f v) a b"
hyp: shows "matrix_to_iarray (Gram_Schmidt_upt_k A k) = foldl Gram_Schmidt_column_k_iarrays_efficient (matrix_to_iarray A) [0..<Suc k]"
sup_mono: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (sup a b) (sup c d)"
chine_hcomp_SPN_SPN: shows "Span.chine_hcomp (SPN g) (SPN h) = Maps.MkIde (src g\<^sub>0h\<^sub>1.p\<^sub>0)"
rep: fixes x :: "('b, 'c) mapping" shows "rep x \<in> UNIV"
lookup_tensor_prod: fixes is1 :: "nat list"   and A :: "'a tensor"   and is2 :: "nat list"   and B :: "'a tensor" assumes "is1 \<lhd> dims A"   and "is2 \<lhd> dims B" shows "lookup (A \<otimes> B) (is1 @ is2) = lookup A is1 * lookup B is2"
exhaust_sel: fixes exception :: "'h exception" obtains "exception = Fault"   | "exception = OK (ok exception)"
le_fin_reflp: fixes w :: "'action list" shows "w \<preceq>\<^sub>F w"
119: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> x \<sqinter> - z) = y \<squnion> x"
imply_Cons: fixes ps :: "('a, 'b) fm list"   and q :: "('a, 'b) fm"   and p :: "('a, 'b) fm" assumes "ps \<turnstile> q" shows "p # ps \<turnstile> q"
simps_2: fixes x :: "name"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and u :: "Lambda_Z.term" shows "subst x s (App t u) = App (subst x s t) (subst x s u)"
a: shows "g -` s' = \<Union> {g -` {x} |x. x \<in> s'}"
returnOk_liftE: fixes x :: "'e" shows "returnOk x = liftE (return x)"
update_since_def: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b"   and args :: "Monitor.args"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and nt :: "nat"   and aux :: "'a" assumes "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux" shows "maux.update_since add_new_ts_msaux join_msaux add_new_table_msaux result_msaux args rel1 rel2 nt aux = (let aux0 = join_msaux args rel1 (add_new_ts_msaux args nt aux); aux' = add_new_table_msaux args rel2 aux0 in (result_msaux args aux', aux'))"
sinpoly_lb_imp_cospoly_ub: fixes x :: "real"   and k :: "nat" assumes "0 \<le> x"   and "0 < k"   and "\<And>x. 0 \<le> x \<Longrightarrow> sinpoly (k - 1) x \<le> sin x" shows "cos x \<le> cospoly k x"
coeff_smult: fixes a :: "'c"   and p :: "'c poly"   and n :: "nat" shows "coeff (smult a p) n = a * coeff p n"
trm_ord_subterm: shows "\<forall>x1 x2. (x1, x1 \<cdot> x2) \<in> trm_ord \<and> (x2, x1 \<cdot> x2) \<in> trm_ord"
dropWhile_cong: fixes l :: "'c list"   and k :: "'c list"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "dropWhile P l = dropWhile Q k"
ToCT_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "H.ToCT f \<equiv> if VxVxV.arr f then fst (fst f, fst (fst (snd f), snd (snd f)) \<star> snd (fst (snd f), snd (snd f))) \<star> snd (fst f, fst (fst (snd f), snd (snd f)) \<star> snd (fst (snd f), snd (snd f))) else null"
partitions_eq: shows "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = (\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}}"
writes_union_left_I: fixes S :: "('l, 'm, 'n) prog set"   and f :: "('l, 'm, 'o) prog"   and h :: "'l"   and h' :: "'l"   and S' :: "('l, 'm, 'n) prog set" assumes "writes S f h h'" shows "writes (S \<union> S') f h h'"
le_fininf_concat_start: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a list"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream" assumes "traces ind" shows "traces.le_fininf ind (w @ w\<^sub>1) (w @- w\<^sub>2) = traces.le_fininf ind w\<^sub>1 w\<^sub>2"
e2lenas_lenas: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2lenas (encode_config ((f, xs, ls) # ss, rv)) = length xs"
subtype_trans_1: fixes \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type"   and Q :: "POPLmarkRecord.type"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> S <: Q"   and "\<Gamma> \<turnstile> Q <: T" shows "\<Gamma> \<turnstile> S <: T"
infprod_in_sets: fixes E :: "nat \<Rightarrow> 'd set"   and M :: "nat \<Rightarrow> 'd measure" assumes "\<And>i. E i \<in> sets (M i)" shows "Pi UNIV E \<in> sets (Pi\<^sub>M UNIV M)"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> a2 = ((\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>S. S \<in>S a1 \<and> S \<in>S a2) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> R))"
widen_null_array: fixes P :: "'a prog"   and A :: "ty" shows "P \<turnstile> NT \<le> A\<lfloor>\<rceil>"
that: fixes \<delta> :: "fm" assumes "{} \<turnstile> \<delta> IFF Neg (PfP \<guillemotleft>\<delta>\<guillemotright>)"   and "\<not> {} \<turnstile> \<delta>"   and "\<not> {} \<turnstile> Neg \<delta>"   and "eval_fm e \<delta>"   and "ground_fm \<delta>" shows "thesis"
soundness_diagonalize_JNF': fixes diagonalize_JNF :: "'d mat \<Rightarrow> ('d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd) \<Rightarrow> 'd mat \<times> 'd mat \<times> 'd mat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "'d mat"   and m :: "nat"   and n :: "nat"   and P :: "'d mat"   and S :: "'d mat"   and Q :: "'d mat" assumes "Diagonalize.diagonalize diagonalize_JNF"   and "is_bezout_ext bezout"   and "A \<in> carrier_mat m n"   and "diagonalize_JNF A bezout = (P, S, Q)" shows "P \<in> carrier_mat m m \<and> Q \<in> carrier_mat n n \<and> S \<in> carrier_mat m n \<and> invertible_mat P \<and> invertible_mat Q \<and> isDiagonal_mat S \<and> S = P * A * Q"
path_connectedin_continuous_map_image: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "continuous_map X Y f"   and "path_connectedin X S" shows "path_connectedin Y (f ` S)"
t: shows "t = monomial (poly_deg p - deg_pm t') x + t'"
lat_dist3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y + z) = x \<sqinter> y + x \<sqinter> z"
lens_source_def: fixes X :: "'b \<Longrightarrow> 'a" shows "\<S>\<^bsub>X\<^esub> = {s. \<exists>v s'. s = put\<^bsub>X\<^esub> s' v}"
config'_append2: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "Partial_Cost_Model.config' A s (xs @ ys) = Partial_Cost_Model.config' A (Partial_Cost_Model.config' A s xs) ys"
prod_zero_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.semidom plus minus zero times one"   and "finite A" shows "(comm_monoid_mult.prod times one f A = zero) = (\<exists>a\<in>A. f a = zero)"
elims: fixes x :: "nat"   and xa :: "nat list"   and y :: "real mat convnet" assumes "witness' x xa = y" obtains Y :: "nat" where "x = Y__" and "xa = []" and "y = Input Y__"   | Y :: "nat" and r :: "nat" and rs :: "nat list" where "x = Y" and "xa = r # rs" and "y = Pool (witness Y r rs) (witness Y r rs)"
map_upd_def: fixes k :: "'c"   and v :: "'d"   and m :: "'c \<Rightarrow> 'd option" shows "map_upd k v m = m(k \<mapsto> v)"
myeq: shows "ennreal (LINT w|M. X w * indicat_real (Y -` (A \<inter> range Y) \<inter> space M) w) = ennreal (LINT w|M. expl_cond_expect M Y X w * indicat_real (Y -` (A \<inter> range Y) \<inter> space M) w)"
LT19: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and D :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and B :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> (A \<leadsto> D \<or> B) \<longrightarrow> (B \<leadsto> D) \<longrightarrow> A \<leadsto> D"
check_tableau: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and v :: "nat \<Rightarrow> 'b" assumes "Check check"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "let s' = check s in v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s' \<and> \<triangle> (\<T> s') \<and> \<nabla> s'"
match_list_False: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and ms :: "'c match_expr list"   and a :: "action"   and p :: "'d" assumes "\<not> match_list \<gamma> ms a p" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) ms) Undecided = Undecided"
simps: fixes b :: "natural"   and i :: "natural" shows "log b i = (if b \<le> 1 \<or> i < b then 1 else 1 + log b (i div b))"
assms_2: shows "\<guillemotleft>g : trg f \<rightarrow> src f\<guillemotright>"
has_method_def: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal" shows "P \<turnstile> C has M \<equiv> \<exists>Ts T m D. P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"
CFG_edge_no_Def_equal: fixes a :: "'edge"   and V :: "'var"   and s :: "'state" assumes "valid_edge a"   and "V \<notin> Def (sourcenode a)"   and "pred (kind a) s" shows "state_val (transfer (kind a) s) V = state_val s V"
wf_nfa_def: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" shows "wf_nfa A n = (list_all (bddh n) (fst A) \<and> list_all (bdd_all (nfa_is_node A)) (fst A) \<and> length (snd A) = length (fst A) \<and> 0 < length (fst A))"
supr_greater: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "local.Above B \<noteq> {}"   and "b \<in> B" shows "(b, supr B) \<in> r"
assms_3: fixes u :: "'t" assumes "d (pp_of_term u) \<le> dgrad_max d"   and "component_of_term u < length fs" shows "is_sig_GB_in d G' u"
insert_absorb2: fixes x :: "'e"   and A :: "'e set" shows "insert x (insert x A) = insert x A"
mono_bij_Inf: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "mono f"   and "bij f" shows "f (Inf A) = Inf (f ` A)"
redundant_infer_def: fixes N :: "'a literal multiset set"   and \<gamma> :: "'a inference" shows "redundant_infer N \<gamma> = (\<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD + side_prems_of \<gamma> \<longrightarrow> I \<Turnstile> concl_of \<gamma>) \<and> (\<forall>D. D \<in># DD \<longrightarrow> D < main_prem_of \<gamma>))"
make_affine_infinity: fixes x :: "'a"   and y :: "'a" shows "make_affine (x, y, \<zero>) = Infinity"
at: shows "local.at_step (\<lambda>i. bnd_get ((bnd_update ^^ i) init)) 0 init"
Quotient_alt_def5: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool" shows "Quotient R Abs Rep T = (T \<le> Grp UNIV Abs \<and> Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>)"
walk_rev: fixes xs :: "'a list" assumes "walk xs" shows "walk (rev xs)"
nearly_healthy_unitaryD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "nearly_healthy t"   and "unitary P" shows "unitary (t P)"
ranking_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ranking R = map the_elem (weak_ranking R)"
sum_list_append: fixes xs :: "'c list"   and ys :: "'c list" shows "sum_list (xs @ ys) = sum_list xs + sum_list ys"
sum_nat_le_single: fixes y :: "'a"   and S :: "'a set"   and x :: "nat"   and f :: "'a \<Rightarrow> nat" assumes "y \<in> S"   and "x \<le> f y"   and "finite S" shows "x \<le> sum f S"
longest_common_suffix: fixes xs :: "'a list"   and ys :: "'a list" shows "\<exists>ss xs' ys'. xs = xs' @ ss \<and> ys = ys' @ ss \<and> (xs' = [] \<or> ys' = [] \<or> last xs' \<noteq> last ys')"
assms_2: shows "InvariantReasonClauses (getF state) (getM state)"
conseq12: fixes G :: "'b triple set"   and P' :: "'b assn"   and c :: "com"   and Q' :: "'b assn"   and P :: "'b assn"   and Q :: "'b assn" assumes "G|-{P'}. c .{Q'}"   and "\<forall>Z s. P Z s \<longrightarrow> (\<forall>s'. (\<forall>Z'. P' Z' s \<longrightarrow> Q' Z' s') \<longrightarrow> Q Z s')" shows "G|-{P}. c .{Q}"
succ_valuation_satisfies: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s" shows "\<langle>\<V> s'\<rangle> \<Turnstile>\<^sub>t \<T> s'"
cases: fixes a1 :: "'opSym"   and a2 :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsBinp a1 a2" obtains delta :: "'opSym" and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" where "a1 = delta" and "a2 = binp" and "wlsOpS delta" and "\<And>i. barOf delta i = None \<and> binp i = None \<or> (\<exists>us s A. barOf delta i = Some (us, s) \<and> binp i = Some A \<and> wlsAbs (us, s) A)"
test_sum: fixes x :: "'a"   and y :: "'a" assumes "is_test x"   and "is_test y" shows "is_test (x + y)"
select_convs_2: fixes iso_verts :: "'e \<Rightarrow> 'f"   and iso_arcs :: "'c \<Rightarrow> 'd"   and iso_head :: "'d \<Rightarrow> 'f"   and iso_tail :: "'d \<Rightarrow> 'f"   and more :: "'g" shows "iso_arcs \<lparr>iso_verts = iso_verts, iso_arcs = iso_arcs, iso_head = iso_head, iso_tail = iso_tail, \<dots> = more\<rparr> = iso_arcs"
strict_iff_order: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subset># b) = (a \<subseteq># b \<and> a \<noteq> b)"
discounted_adapted: fixes F :: "nat \<Rightarrow> 'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and r :: "real" assumes "borel_adapt_stoch_proc F X" shows "borel_adapt_stoch_proc F (discounted_value r X)"
exhaust_sel: fixes trans :: "('a, 'b, 'c) trans" obtains "trans = Trans (srcOf trans) (actOf trans) (outOf trans) (tgtOf trans)"
f_closed: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "hensel p f a" shows "f \<in> carrier (UP (padic_int p))"
B: shows "(+) (- a) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (+) (- a) ` V \<and> independent B \<and> (+) (- a) ` V \<subseteq> span B"
eq_fold: fixes M :: "'a multiset" shows "F M = fold_mset (\<^bold>*) \<^bold>1 M"
norm_rot: fixes v :: "'a" shows "norm (rot v) = norm v"
zero_ext_g: fixes y :: "'a" assumes "y \<in> GH - supp g" shows "f (x - y) * g y = (0::'b)"
lincomb_list_snoc: fixes Vs :: "'b vec list"   and n :: "nat"   and x :: "'b vec"   and f :: "nat \<Rightarrow> 'b" assumes "set Vs \<subseteq> carrier_vec n"   and "x \<in> carrier_vec n" shows "module.lincomb_list (module_vec TYPE('b) n) f (Vs @ [x]) = module.lincomb_list (module_vec TYPE('b) n) f Vs + f (length Vs) \<cdot>\<^sub>v x"
sc_completion_shift: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, ('f, 'b) obs_event action) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> ('f, 'b) obs_event action list"   and P :: "'g prog"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and vs :: "'f \<times> addr_loc \<Rightarrow> ('f val \<times> bool) option"   and ttas :: "('b \<times> ('d, 'b, 'a, 'c, 'e, ('f, 'b) obs_event action) thread_action) list"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" assumes "jmm_multithreaded.sc_completion final r convert_RA P s vs"   and "multithreaded_base.RedT final r convert_RA s ttas s'"   and "ta_seq_consist P vs (lconcat (lmap (\<lambda>(t, ta). llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) (llist_of ttas)))" shows "jmm_multithreaded.sc_completion final r convert_RA P s' (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"
Rep_uint16_inverse: fixes x :: "uint16" shows "Abs_uint16 (Rep_uint16 x) = x"
lift_poly_fun_2_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and f :: "('b \<Rightarrow>\<^sub>0 'd) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 'd) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'd"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.lift_poly_fun_2 pair_of_term term_of_pair f p q = term_powerprod.atomize_poly pair_of_term (mapp_2 (\<lambda>_. f) (term_powerprod.vectorize_poly term_of_pair p) (term_powerprod.vectorize_poly term_of_pair q))"
coinduct: fixes X :: "('test, 'atom) com \<Rightarrow> bool"   and x :: "('test, 'atom) com" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>c. x = c \<and> (\<forall>x xa xb. (c, x) \<rightarrow>c (xa, xb) \<longrightarrow> x \<approx> xb \<and> (X xa \<or> discr xa)) \<and> (\<forall>x xa. (c, x) \<rightarrow>t xa \<longrightarrow> x \<approx> xa)" shows "discr x"
rebase_mult_eq: fixes a :: "'pq mod_ring"   and a' :: "'pq mod_ring" shows "(of_nat CARD('q) * a = of_nat CARD('q) * a') = (@a = @a')"
irreducible_mat_def: fixes A :: "'a mat" shows "irreducible_mat A = (let n = dim_row A in \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> (i, j) \<in> (graph_of_mat A)\<^sup>+)"
simps_6: fixes ux :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Es :: "char list set"   and SS :: "char list \<Rightarrow> char list set"   and b :: "bexp"   and C :: "acom"   and LV :: "char list set" shows "fune ({(ux, S, E, Es, SS)} WHILE b DO C) LV = \<Union> (SS ` LV)"
lnull_def: fixes llist :: "'a llist" shows "lnull llist = (llist = []\<^sub>l)"
vbrelation_vcomp_vid_on_right: fixes A :: "V" assumes "\<D>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> A" shows "F\<lparr>ArrVal\<rparr> \<circ>\<^sub>\<circ> vid_on A = F\<lparr>ArrVal\<rparr>"
right_def: shows "right = (\<lambda>f lb ub. {x. jumpF f (at_right x) \<noteq> 0 \<and> lb \<le> x \<and> x \<le> ub})"
ufd_axioms: shows "class.ufd (*) (1::'a) (+) (0::'a) (-) uminus"
G_def: shows "G = {x \<in> space M. upper_asymptotic_density {n. P x n} < a}"
intros_15: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<in> b" shows "\<Gamma>\<turnstile> (While b c # cs, css, Normal s) \<rightarrow> (c # While b c # cs, css, Normal s)"
the_\<nu>_in_hom_2: shows "\<guillemotleft>\<chi>.the_\<nu> : s\<^sub>1 \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine\<guillemotright>"
subst_Nil: fixes t :: "'b trm" shows "t \<lhd> [] = t"
star_below_circ: fixes x :: "'a" shows "x\<^sup>\<star> \<le> x\<^sup>\<circ>"
process_prop_3_def: fixes P :: "('a list \<times> 'a set) set \<times> 'a list set" shows "process_prop_3 P \<equiv> \<forall>xs X Y. (xs, Y) \<in> fst P \<and> X \<subseteq> Y \<longrightarrow> (xs, X) \<in> fst P"
\<Psi>_simp: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and \<psi> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and yx :: "'d \<times> 'c" assumes "meta_adjunction C D F G \<phi> \<psi>"   and "partial_magma.ide (product_category.comp (dual_category.comp D) C) yx" shows "inverse_transformation.map (product_category.comp (dual_category.comp D) C) replete_setcat.comp (hom_functor.map D replete_setcat.comp (\<lambda>_. meta_adjunction.inD) \<circ> product_functor.map (dual_category.comp D) C (dual_category.comp D) D (identity_functor.map (dual_category.comp D)) G) (transformation_by_components.map (product_category.comp (dual_category.comp D) C) replete_setcat.comp (hom_functor.map C replete_setcat.comp (\<lambda>_. meta_adjunction.inC) \<circ> product_functor.map (dual_category.comp D) C (dual_category.comp C) C (dual_functor.map F) (identity_functor.map C)) (meta_adjunction.\<Phi>o C D F G \<phi>)) yx = set_category_given_img.mkArr replete_setcat.comp (set_category.some_img replete_setcat.comp (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set D (\<lambda>_. meta_adjunction.inD) (fst yx, G (snd yx))) (hom_functor.set C (\<lambda>_. meta_adjunction.inC) (F (fst yx), snd yx)) (meta_adjunction.inC \<circ> \<psi> (snd yx) \<circ> hom_functor.\<psi> D (\<lambda>_. meta_adjunction.inD) (fst yx, G (snd yx)))"
roofed_mono: fixes S :: "'a set"   and S' :: "'a set"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set" assumes "S \<subseteq> S'" shows "roofed_gen G B S \<subseteq> roofed_gen G B S'"
assms_3: shows "\<not> dep_var_set PROB vs_1 vs_2"
c: shows "prime (3 * 2\<^sup>2 - 1)"