hcju/isabelleps huggingface dataset:
mdecls_visible: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and 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 D :: "char list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list" assumes "wf_prog wf_md P"   and "is_class P C"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" shows "\<exists>Mm. P \<turnstile> C sees_methods Mm \<and> (\<forall>(M, Ts, T, m)\<in>set ms. Mm M = \<lfloor>((Ts, T, m), C)\<rfloor>)"
imethds_rec': fixes I :: "qtname"   and i :: "iface" assumes "iface tprg I = Some i" shows "DeclConcepts.imethds tprg I = Un_tables (DeclConcepts.imethds tprg ` set (isuperIfs i)) \<oplus>\<oplus> (set_option \<circ> table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)))"
intros_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and s' :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Seq c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
lookup_lower_when: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and u :: "'a"   and v :: "'a" shows "lookup (punit.lower p u) v = (lookup p v when v \<prec> u)"
good_context_6: fixes s :: "'b sparc_state"   and s'' :: "'b sparc_state"   and v1 :: "word32"   and v2 :: "sparc_operation \<times> inst_operand list" assumes "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') = 0 \<and> get_addr (snd v2) s'' AND 3 \<noteq> 0" shows "False"
ln_fun_in_smallo_fun: fixes f :: "'a \<Rightarrow> real"   and p :: "real" assumes "filterlim f at_top at_top" shows "(\<lambda>x. ln (f x) powr p) \<in> o(f)"
image_UN: fixes f :: "'b \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "f ` \<Union> (B ` A) = (\<Union>x\<in>A. f ` B x)"
is_pole_cong: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" assumes "\<forall>\<^sub>F x in at a. f x = g x"   and "a = b" shows "is_pole f a = is_pole g b"
assms: fixes i :: "nat" assumes "d \<le> i" shows "Abstract_Linear_Poly.coeff x i = 0"
is_num_normalize_8: 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.neg_numeral minus plus zero uminus" shows "uminus (plus a b) = plus (uminus b) (uminus a)"
collapse_1: fixes operation :: "'a operation" assumes "is_INS operation" shows "INS (pos operation) (op_idx operation) (op_int operation) = operation"
simps_12: fixes t :: "'a local.term"   and u :: "'a local.term"   and a :: "'a" shows "(t \<^bold>\<star> u) \<^bold>\<Down> \<^bold>\<langle>a\<^bold>\<rangle>\<^sub>0 = \<^bold>\<r>\<^bold>[t \<^bold>\<star> u\<^bold>]"
inverse_less_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse x) one = (less_eq x zero \<or> less one x)"
Sbis_transT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>s d"   and "s \<approx> t"   and "(c, s) \<rightarrow>t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>t t'" and "s' \<approx> t'"
supports_def: fixes S :: "atom set"   and x :: "'a" shows "S supports x \<equiv> \<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
\<iota>: shows "\<iota> \<in> Inf_from (Liminf_llist Ns)"
refine: fixes ai :: "'b list list"   and a :: "'c set"   and sctni :: "real list sctn"   and sctn :: "'c sctn" assumes "(ai, a) \<in> clw_rel appr_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(RETURN (intersects_clw_impl ai sctni), intersects_clw $ a $ sctn) \<in> \<langle>bool_rel\<rangle>nres_rel"
fmadd_disjnt: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" assumes "fdisjnt (fmdom m) (fmdom n)" shows "m ++\<^sub>f n = n ++\<^sub>f m"
presOp_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)" shows "presOp (fromIMor h) (fromIMorAbs hA) MOD"
trans_FD: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q"   and "Q \<sqsubseteq>\<^sub>F\<^sub>D S" shows "P \<sqsubseteq>\<^sub>F\<^sub>D S"
tau: shows "r \<down>s set_mset r|\<tau>| \<subseteq> r \<down>s S"
simps_1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and r :: "String.literal"   and ts :: "Formula.trm list" shows "Formula.sat \<sigma> V v i (formula.Pred r ts) = (case V r of None \<Rightarrow> (r, map (Formula.eval_trm v) ts) \<in> \<Gamma> \<sigma> i | Some X \<Rightarrow> map (Formula.eval_trm v) ts \<in> X i)"
"2_3": shows "k' \<noteq> Suc i_e \<longrightarrow> 0 < Suc i_e \<longrightarrow> A $$ (Suc i_e - 1, Suc i_e) \<noteq> (1::'b)"
simps_2: fixes l :: "'d strand_label"   and t :: "('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, receive\<langle>t\<rangle>) = (l, send\<langle>t\<rangle>)"
dual_sup: fixes x :: "'c"   and y :: "'c"   and inf :: "'c" shows "is_sup (dual x) (dual y) (dual inf) = is_inf x y inf"
StdUprio_axioms: shows "StdUprio aluprioi_ops"
weak_reduction_bisimulation_is_correspondence_simulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_correspondence_simulation Rel Cal"
coeff_plus: fixes lp1 :: "linear_poly"   and lp2 :: "linear_poly"   and var :: "nat" shows "coeff (lp1 + lp2) var = coeff lp1 var + coeff lp2 var"
has_derivative_iff_norm: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
B: shows "B \<in> M \<rightarrow>\<^sub>M prob_algebra N"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'm" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
IH: fixes x :: "'s" shows "ennreal (p x y (n + m)) = \<integral>\<^sup>+ x. ennreal (p x x n * p x y m) \<partial>count_space UNIV"
interior_def: fixes f :: "'a \<Rightarrow> 'a" shows "interior f \<equiv> (\<forall>x. f x \<le> x) \<and> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y) \<and> (\<forall>x. f (f x) = f x)"
ogroupcastT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>i\<^sub>p\<^sub>s :: "'e \<Rightarrow> nat set"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p), groupcast (s\<^sub>i\<^sub>p\<^sub>s (\<sigma> i)) (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i)), \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
hyps: assumes "\<And>kv. kv \<in> set xs \<Longrightarrow> f kv = g kv" shows "map_pair f xs = map_pair g xs"
strand_substI_2: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" assumes "subst_domain \<theta> \<inter> vars\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<theta> = S"
Arg_eq_0_pi: fixes z :: "complex" shows "(Arg z = 0 \<or> Arg z = pi) = (z \<in> \<real>)"
scale_shift_ms_aux_MSLCons_eval: fixes c :: "'a"   and e :: "real"   and c' :: "'a"   and e' :: "real"   and xs :: "('a \<times> real) msllist" shows "scale_shift_ms_aux (c, e) (MSLCons (c', e') xs) = MSLCons (c * c', e + e') (scale_shift_ms_aux (c, e) xs)"
var_is_subterm: fixes x :: "'a"   and t :: "('b, 'a) Term.term" assumes "x \<in> fv t" shows "Var x \<sqsubseteq> t"
simps_2: fixes \<theta> :: "(W.tvar \<times> ty) list"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" shows "\<theta><T\<^sub>1\<rightarrow>T\<^sub>2> = \<theta><T\<^sub>1>\<rightarrow>\<theta><T\<^sub>2>"
Infinitesimal_add_st_less: fixes x :: "real star"   and y :: "real star"   and u :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite"   and "u \<in> Infinitesimal"   and "st x < st y" shows "st x + u < st y"
eq: shows "[a, 1::'a] *** p = a %* (c # cs) +++ ((0::'a) # (1::'a) %* (c # cs))"
image_filter: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> bool" shows "f ` {g a |a. P a} = {f (g a) |a. P a}"
def_hmemrec: fixes f :: "hf \<Rightarrow> 'a"   and G :: "(hf \<Rightarrow> 'a) \<Rightarrow> hf \<Rightarrow> 'a"   and a :: "hf" assumes "f \<equiv> hmemrec G" shows "f a = G (ecut (hmemrec G) a) a"
connectedin_sing: fixes X :: "'a topology"   and a :: "'a" shows "connectedin X {a} = (a \<in> topspace X)"
monotone_bind_spmf2: fixes ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> 'e spmf"   and p :: "'c spmf" assumes "\<And>x. monotone ord (ord_spmf (=)) (\<lambda>y. g y x)" shows "monotone ord (ord_spmf (=)) (\<lambda>y. p \<bind> g y)"
that: fixes p :: "(name \<times> name) list" assumes "set p \<subseteq> set xvec \<times> set yvec"   and "distinctPerm p"   and "\<Psi>' = p \<bullet> \<Psi>" shows "thesis"
le_fininf_range: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2" shows "set w\<^sub>1 \<subseteq> sset w\<^sub>2"
divisor_add_diff_mod_eq2: fixes b :: "nat"   and m :: "nat"   and a :: "nat" assumes "b mod m < a mod m" shows "(m + b mod m - a mod m) mod m = m + b mod m - a mod m"
squanderTarget_def: shows "squanderTarget \<equiv> 15410"
product_fold: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "A \<times> B = Finite_Set.fold (\<lambda>x z. Finite_Set.fold (\<lambda>y. insert (x, y)) z B) {} A"
inner_minus_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (uminus y) = - inner x y"
le_enumerate: fixes S :: "nat set"   and n :: "nat" assumes "infinite S" shows "n \<le> enum S n"
D_comm: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pas times D" shows "D x y = D y x"
taq_def: fixes s :: "'a set"   and q :: "'a poly" shows "taq s q \<equiv> \<Sum>x\<in>s. sign (poly q x)"
uminus_id_iff: fixes z :: "complex_homo" shows "(~\<^sub>h z = z) = (z = 0\<^sub>h \<or> z = \<infinity>\<^sub>h)"
Crypt_parts_imp_used: fixes K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt K X \<in> parts (knows Spy evs)"   and "K \<in> symKeys"   and "evs \<in> set_pur" shows "Key K \<in> used evs"
cases: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and a1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and a2 :: "nat" assumes "count_it \<pi> b c a1 a2" obtains s :: "char list \<Rightarrow> int \<Rightarrow> int" where "a1 = sa__" and "a2 = 0" and "\<not> bval b sa__"   | s :: "char list \<Rightarrow> int \<Rightarrow> int" and s' :: "char list \<Rightarrow> int \<Rightarrow> int" and n :: "nat" where "a1 = s" and "a2 = Suc n" and "bval b s" and "\<pi>: (c, s) \<Rightarrow> s'" and "count_it \<pi> b c s' n"
Eps_case_prod: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(SOME (x, y). P x y) = (SOME xy. P (fst xy) (snd xy))"
pbeta_beta: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option"   and l :: "Label"   and p :: "sterm"   and p' :: "sterm" assumes "Obj f T => Obj f' T"   and "l \<in> dom f"   and "p => p'"   and "lc (Obj f T)"   and "lc p" shows "Call (Obj f T) l p => (the (f' l)\<^bsup>[Obj f' T,p']\<^esup>)"
csubspace_space_as_set: fixes S :: "'d ccsubspace" shows "csubspace (space_as_set S)"
errMOD_igSwapIGVarSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igVarIPresIGWls Delta MOD"   and "igSwapIGVar MOD" shows "igSwapIGVar (FixSyn.errMOD Delta MOD)"
conth: shows "continuous_on (\<Union> \<G> - C) h"
assms_1: shows "m \<le>\<^sub>n SPEC Q"
idirprod_generate_ind: fixes gs :: "'a set"   and g :: "'a" assumes "finite gs"   and "gs \<subseteq> carrier G"   and "g \<in> carrier G"   and "is_idirprod (generate G gs) (\<lambda>g. generate G {g}) gs"   and "complementary (generate G {g}) (generate G gs)" shows "is_idirprod (generate G (gs \<union> {g})) (\<lambda>g. generate G {g}) (gs \<union> {g})"
ap_list_def: fixes fs :: "('d \<Rightarrow> 'c) list"   and xs :: "'d list" shows "fs \<diamondop> xs = List.bind fs (\<lambda>f. List.bind xs (\<lambda>x. [f x]))"
length_lesss_le': fixes i :: "nat"   and xs :: "'b list"   and A :: "'b set"   and R :: "('b \<times> 'b) set" assumes "i < length xs"   and "linorder_on A R"   and "set xs \<subseteq> A" shows "length (lesss R (insort_wrt R xs ! i) xs) \<le> i"
only_call_get_return_edges: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and a' :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>Q r p fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
semilat_Def: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "semilat (A, r, f) = (order r \<and> closed A f \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
oa_facts_4: fixes v :: "i"   and x :: "\<nu>" shows "[\<^bold>\<diamond>\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>A!,x\<^sup>P\<rparr> in v]"
Quotient_residual: shows "Quotient alpha_residual abs_residual rep_residual cr_residual"
fps_divide_unit: fixes g :: "'a fps"   and f :: "'a fps" assumes "fps_nth g 0 \<noteq> (0::'a)" shows "f / g = f * inverse g"
struct_Lbd: fixes T :: "Syntax.type"   and t :: "trm"   and j :: "nat"   and k :: "nat"   and E :: "ctxt" shows "(\<lambda>T:t)[j=k E]\<^sup>T = \<lambda>T:t[j=k liftL_ctxt E 0]\<^sup>T"
length_DataValue_PartNum: fixes D :: "'b data" shows "length (DataValue D) = PartNum (Data.DataSpace D)"
int_set_bit_Suc: fixes x :: "int"   and n :: "nat"   and b :: "bool" shows "set_bit_class.set_bit x (Suc n) b = of_bool (odd x) + 2 * set_bit_class.set_bit (x div 2) n b"
wf_measure: fixes f :: "'h \<Rightarrow> nat" shows "wf (Wellfounded.measure f)"
minus_add_uminus_vec: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "v - w = v + - w"
brc_det_while_algo: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" assumes "rqrm_invar rqrm"   and "rqrm_prop (ls.\<alpha> \<delta>) rqrm"   and "ls.invar \<delta>" shows "det_while_algo (brc_det_algo rqrm \<delta>)"
BINDING_imply_SNa_2: fixes x :: "name"   and M :: "trm"   and B :: "ty" assumes "x:M \<in> BINDINGn B (\<parallel><B>\<parallel>)" shows "SNa M"
of_nat_min: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "nat"   and y :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "semiring_1.of_nat one plus zero (min x y) = ord.min less_eq (semiring_1.of_nat one plus zero x) (semiring_1.of_nat one plus zero y)"
is_processT5_S1: fixes s :: "'c event list"   and X :: "'c event set"   and P :: "'c process"   and Y :: "'c event set" assumes "(s, X) \<in> F P"   and "(s, X \<union> Y) \<notin> F P" shows "\<exists>c. c \<in> Y \<and> (s @ [c], {}) \<in> F P"
pentagon': fixes f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "D.ide k"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h"   and "src\<^sub>D h = trg\<^sub>D k" shows "((\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] \<star>\<^sub>D k) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g \<star>\<^sub>D h, k]) \<cdot>\<^sub>D (f \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[g, h, k]) = \<a>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g, h, k] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h \<star>\<^sub>D k]"
elims: fixes x :: "indices list"   and y :: "indices list" assumes "left_app x = y" obtains xa :: "indices list" where "x = xa" and "y = indices.Left # xa"
CD_on_inj_on_Xd: fixes ds :: "'d set"   and X :: "'x set" shows "inj_on Xd (CD_on ds X)"
replicate_Suc: fixes n :: "nat"   and x :: "'b" shows "replicate (Suc n) x = x # replicate n x"
UNIV_I: fixes x :: "'e" shows "x \<in> UNIV"
Rectangle_for_One: fixes t :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes " t \<longrightarrow>\<^sub>1\<^sup>* t1"   and " t \<longrightarrow>\<^sub>1 t2" shows "\<exists>t3.  t1 \<longrightarrow>\<^sub>1 t3 \<and>  t2 \<longrightarrow>\<^sub>1\<^sup>* t3"
abs_pos: fixes abs :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.partial_abs_if abs uminus zero less_eq"   and "less_eq zero a" shows "abs a = a"
max_fun_diff_oprod: fixes f :: "'b \<times> 'c \<Rightarrow> 'a"   and g :: "'b \<times> 'c \<Rightarrow> 'a" assumes "Field r \<noteq> {}"   and "f \<noteq> g"   and "f \<in> FinFunc r (s *o t)"   and "g \<in> FinFunc r (s *o t)" shows "wo_rel.max_fun_diff (s *o t) f g = (wo_rel.max_fun_diff s (local.rev_curr f (wo_rel.max_fun_diff t (local.rev_curr f) (local.rev_curr g))) (local.rev_curr g (wo_rel.max_fun_diff t (local.rev_curr f) (local.rev_curr g))), wo_rel.max_fun_diff t (local.rev_curr f) (local.rev_curr g))"
nat_int_comparison_3: fixes a :: "nat"   and b :: "nat" shows "(a \<le> b) = (int a \<le> int b)"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "a \<le> dual_wellorder.Min A"
conj_disj_abs: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(P \<and> (P \<or> Q)) = P"
inverts: fixes a :: "'b"   and list :: "['b]"   and a' :: "'b"   and list' :: "['b]" shows "(a : list \<sqsubseteq> a' : list') = (a \<sqsubseteq> a' \<and> list \<sqsubseteq> list')"
mslfg: shows "matrix_seq.lowner_is_lub (\<lambda>n. f n + g n) (matrix_seq.lowner_lub (\<lambda>n. f n + g n))"
power2_eq_square: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> 2 = x \<otimes>\<^bsub>R\<^esub> x"
linear_add: fixes f :: "'d \<Rightarrow> 'e"   and b1 :: "'d"   and b2 :: "'d" assumes "clinear f" shows "f (b1 + b2) = f b1 + f b2"
ensures_imp_wens: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> A ensures B" shows "\<exists>act\<in>Acts F. A \<subseteq> wens F act B"
assms_2: shows "\<D>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
funion_assoc: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" shows "A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)"
prems_4: shows "Re (trace (complex_of_real c \<cdot>\<^sub>m \<rho>)) \<le> 1"
reds_preserves_hconf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "expr list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and Ts :: "ty list" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>"   and "P,E,h \<turnstile> es [:] Ts"   and "P \<turnstile> h \<surd>" shows "P \<turnstile> h' \<surd>"
T_def: shows "T = (+) a ` insert (0::'n) B"
pi_nodes_it: fixes S :: "'G" shows "proper_it (nodes_it S) (nodes_it S)"
inv_gorder_inv: fixes L :: "'d gorder" shows "inv_gorder (inv_gorder L) = L"
monom_is_UP_monom_1: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "is_UP_monom (monom P a n)"
rbb_def: shows "bb\<^sub>\<R> = bb\<^sub>\<F> \<circ> \<F>"
f24: shows "(r \<otimes> r', s \<otimes> s') \<in> carrier rel"
preimage_of_max_lideal: fixes \<ww>\<^sub>A :: "'b set"   and \<ww>\<^sub>B :: "'c set" assumes "max_lideal \<ww>\<^sub>A B addB multB zeroB oneB"   and "max_lideal \<ww>\<^sub>B C addC multC zeroC oneC" shows "g \<^sup>\<inverse> B \<ww>\<^sub>B = \<ww>\<^sub>A"