hcju/isabelleps huggingface dataset:
HomFtorContraMor_3: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>SET\<^esub> (HomC\<^bsub>C\<^esub>[f,X]) = Hom\<^bsub>C\<^esub> dom\<^bsub>C\<^esub> f X"
DOWN_UP: fixes \<AA> :: "('a \<times> 'a) set"   and x :: "'b" assumes "setcat \<AA>" shows "setcat.DOWN (setcat.UP x) = x"
bisim: fixes s1 :: "'b"   and s2 :: "'a"   and x :: "'c" assumes "X s1 s2" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"
x_1: fixes i :: "nat" assumes "i \<in> {1..k1}" shows "0 \<le> u1 i \<and> x1 i \<in> S"
max_spec_Base_foo: fixes S :: "qtname" shows "max_spec tprg S (ClassT Base) \<lparr>name = foo, parTs = [NT]\<rparr> = {((ClassT Base, \<lparr>access = Public, static = False, pars = [z], resT = Class Base\<rparr>), [Class Base])}"
valid_sops_stmt: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "valid_sops_stmt t s"
cmiov: shows "m\<inverse> O ov \<subseteq> ov\<inverse> \<union> d \<union> f"
mat_of_rows_carrier_1: fixes n :: "nat"   and vs :: "'a Matrix.vec list" shows "mat_of_rows n vs \<in> carrier_mat (length vs) n"
\<iota>_inff': shows "\<iota> \<in> Inf_from (Liminf_llist As)"
predict_stoch_proc_def: fixes F :: "nat \<Rightarrow> 'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure" shows "predict_stoch_proc F X N = (X 0 \<in> F 0 \<rightarrow>\<^sub>M N \<and> (\<forall>n. X (Suc n) \<in> F n \<rightarrow>\<^sub>M N))"
horiz_vector_helper_pos_ind_R2: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x"   and "list_constr I (length qs)" shows "(prod_list (map ((!) sign) I) = 1) = (0 < poly (prod_list (retrieve_polys qs I)) x)"
ChiRel_HAStates_Self: fixes s :: "'a"   and a :: "('a, 'b, 'c) hierauto" shows "(s, s) \<notin> ChiRel a"
upset_anti_iff: fixes dual :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.preorder_with_dual dual less_eq less" shows "(ord.upset less_eq y \<subseteq> ord.upset less_eq x) = less_eq x y"
NewInitDoneReds2: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and h :: "heap"   and P :: "(char list list \<times> char list exp) prog"   and l :: "sfields" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "new_Addr h = None"   and "is_class P C" shows "P \<turnstile> \<langle>new C,(h, l, sh),False\<rangle> \<rightarrow>* \<langle>THROW OutOfMemory,(h, l, sh),False\<rangle>"
real_divl_pos_less1_bound: fixes x :: "real"   and prec :: "nat" assumes "0 < x"   and "x \<le> 1" shows "1 \<le> real_divl prec 1 x"
Max_wrt_subset: shows "Max_wrt (Pareto R) \<subseteq> alts"
deg_lcoeff_cancel: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P"   and "deg R r \<noteq> 0"   and "deg R p \<le> deg R r"   and "deg R q \<le> deg R r"   and "coeff P p (deg R r) = \<ominus> coeff P q (deg R r)" shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"
continuous_on_mult: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x * g x)"
init_mapstate_Some_aodv_init: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree"   and i :: "nat"   and v :: "'c" assumes "s \<in> init (pnet np p)"   and "netmap s i = Some v" shows "v \<in> init (np i)"
monoD: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "mono f"   and "x \<le> y" shows "f x \<le> f y"
poly_degree: fixes x :: "'c"   and p :: "('d monom \<times> 'c) list" assumes "(1::'c) \<le> x" shows "eval_poly (\<lambda>_. x) p \<le> poly_coeff_sum p * x ^ poly_degree p"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and sl :: "name"   and y :: "tm" assumes "atom l \<sharp> (s, k, sl)"   and "atom sl \<sharp> s" shows "SeqWRP s k y = LstSeqP s k y AND HPair Zero Zero IN s AND All2 l k (SyntaxN.Ex sl (HPair (Var l) (Var sl) IN s AND HPair (SUCC (Var l)) (Q_Succ (Var sl)) IN s))"
null_rec_1: fixes x :: "'b"   and xs :: "'b list" shows "List.null (x # xs) = False"
normal_range_le: fixes F :: "ordinal \<Rightarrow> ordinal"   and G :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "normal F"   and "normal G"   and "range G \<subseteq> range F" shows "F x \<le> G x"
simps_167: fixes x7 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Rfork x7) = LID\<^sub>E x7"
tfilter_TNil: fixes b' :: "'d"   and P :: "'c \<Rightarrow> bool"   and b :: "'d" shows "tfilter b' P (TNil b) = TNil b"
NoDSourceNoSource: fixes S :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" assumes "S \<notin> Sources i C" shows "S \<notin> DSources i C"
hpsafe_DiffAssign_simps: fixes x :: "'c"   and e :: "('a, 'c) trm" shows "hpsafe (DiffAssign x e) = dsafe e"
zmset_of_less: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(zmset_of M < zmset_of N) = (M < N)"
simps_3: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and \<phi> :: "('e, 'f) Kform"   and \<psi> :: "('e, 'f) Kform" shows "M, w \<Turnstile> Kand \<phi> \<psi> = (M, w \<Turnstile> \<phi> \<and> M, w \<Turnstile> \<psi>)"
brc_inner_step_def: fixes r :: "('a, 'b) ta_rule" shows "brc_inner_step r \<equiv> \<lambda>(Q, W, rcm). let c = the (hm.lookup r rcm); rcm' = hm.update r (c - 1) rcm; Q' = if c \<le> 1 then hs.ins (lhs r) Q else Q; W' = if c \<le> 1 \<and> \<not> hs.memb (lhs r) Q then lhs r # W else W in (Q', W', rcm')"
continuous_on_components_open: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "open S"   and "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f" shows "continuous_on S f"
subst_fv_bounded_if_img_bounded: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and V :: "'a set" assumes "range_vars \<theta> \<subseteq> fv t \<union> V" shows "fv (t \<cdot> \<theta>) \<subseteq> fv t \<union> V"
prv_imp_scnj: fixes \<chi> :: "'fmla"   and F :: "'fmla set" assumes "\<chi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F"   and "\<And>\<phi>. \<phi> \<in> F \<Longrightarrow> prv (imp \<chi> \<phi>)" shows "prv (imp \<chi> (scnj F))"
d_inst_true: shows "I \<Turnstile> DA \<cdot> \<sigma> \<cdot> \<eta>"
big_subsetI: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)" shows "O[F](f) \<subseteq> O[F](g)"
apply_cltn2_line_inv: fixes b :: "proj2_line"   and g :: "cltn2" shows "apply_cltn2_line (apply_cltn2_line b g) (cltn2_inverse g) = b"
coinduct_strong: fixes R :: "'b llist \<Rightarrow> 'b llist \<Rightarrow> bool"   and llist :: "'b llist"   and llist' :: "'b llist" assumes "R llist llist'"   and "\<And>llist llist'. R llist llist' \<Longrightarrow> lnull llist = lnull llist' \<and> (\<not> lnull llist \<longrightarrow> \<not> lnull llist' \<longrightarrow> lhd llist = lhd llist' \<and> (R (ltl llist) (ltl llist') \<or> ltl llist = ltl llist'))" shows "llist = llist'"
SUP_empty: 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 bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (\<P> f {}) = bot"
Fax_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.Fax wtFsym arOf resOf prot protFw protCl \<equiv> {ProblemIkTpartG.fax arOf resOf f |f. wtFsym f \<and> \<not> ProblemIkTpart.unprot prot protFw (resOf f) \<and> \<not> protCl (resOf f)} \<union> {ProblemIkTpartG.faxCD arOf resOf f |f. wtFsym f \<and> protCl (resOf f)}"
simps_18: fixes e\<^sub>1 :: "char list exp"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" shows "fv (try e\<^sub>1 catch(C V) e\<^sub>2) = fv e\<^sub>1 \<union> (fv e\<^sub>2 - {V})"
enum_offs_num: fixes i :: "nat"   and n :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d" assumes "i < n" shows "enum xs index key n mi ma ! i = offs_num n xs index key mi ma i"
update_no_left: fixes y :: "nat"   and t :: "(nat \<times> linear_poly) list"   and s :: "('c, 'a) state"   and x :: "nat"   and v :: "'a" assumes "y \<notin> lvars t" shows "Mapping.lookup (\<V> s) y = Mapping.lookup (foldl (\<lambda>v' e. Mapping.update (lhs e) (rhs_eq_val (\<V> s) x v e) v') (\<V> s) t) y"
divide_nonneg_nonpos: fixes x :: "'b"   and y :: "'b" assumes "(0::'b) \<le> x"   and "y \<le> (0::'b)" shows "x / y \<le> (0::'b)"
eqn: shows "\<forall>A\<in>sets N. LINT w|M. max 0 (- X w) * indicat_real (Y -` A \<inter> space M) w = LINT w|M. expl_cond_expect M Y (\<lambda>w. max 0 (- X w)) w * indicat_real (Y -` A \<inter> space M) w"
WTrt_hext_mono: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "P,E,h \<turnstile> e : T"   and "h \<unlhd> h'" shows "P,E,h' \<turnstile> e : T"
locally_sound_def: fixes R :: "fml list \<times> fml" shows "locally_sound R \<equiv> \<forall>I. (\<forall>k\<ge>0. k < length (fst R) \<longrightarrow> valid_in I (fst R ! k)) \<longrightarrow> valid_in I (snd R)"
that: fixes \<gamma> :: "V" assumes "\<alpha> + \<gamma> = \<beta>"   and "\<gamma> \<le> \<beta>"   and "Ord \<gamma>" shows "thesis"
assms_7: fixes x :: "'a"   and y :: "'c" assumes "x \<in> carrier A"   and "y \<in> carrier B" shows "(L x \<sqsubseteq>\<^bsub>B\<^esub> y) = (x \<sqsubseteq>\<^bsub>A\<^esub> R y)"
below_env_deleteI: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and g :: "'b \<Rightarrow> 'a" assumes "f x = \<bottom>"   and "f \<sqsubseteq> g" shows "f \<sqsubseteq> env_delete x g"
mIH: fixes val :: "bool" shows "ifex_ordered (ifex_ite (restrict_top i a_ val) (restrict_top t a_ val) (restrict_top e a_ val))"
is_group: shows "Group.group \<F>\<^bsub>I\<^esub>"
trss_add_context: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and ce :: "'a list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)"   and "mon_w fg w \<inter> mon_c fg ce = {}" shows "((s, c + ce), w, s', c' + ce) \<in> trcl (trss fg)"
Crypt_Spy_analz_bad: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) X \<in> analz (knows Spy evs)"   and "A \<in> bad" shows "X \<in> analz (knows Spy evs)"
Out_def: shows "DFIG.Out = (\<lambda>(A, A'). list_select DFIG_alphabet.v A' = DFIG_alphabet.max A \<and> DFIG_alphabet.max A' = DFIG_alphabet.max A + 1 \<and> retidS A' = retidS A - {list_select DFIG_alphabet.v A'})"
ipd': shows "ipd (\<pi> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k))) \<noteq> ipd (\<pi> k)"
domain: shows "Domainp cr_ffun = (\<lambda>f. finite (pdom f))"
dg_Dom_app_in_Obj: fixes \<alpha> :: "V"   and \<CC> :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
simps: fixes G :: "prog"   and C :: "qtname"   and t :: "'b"   and f :: "qtname \<Rightarrow> class \<Rightarrow> 'b \<Rightarrow> 'b" shows "class_rec G C t f = (case class G C of Some c \<Rightarrow> if ws_prog G then f C c (if C = Object then t else class_rec G (super c) t f) else undefined)"
s_top_elem1: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a"   and b :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "a \<in> H"   and "b \<in> K" shows "a \<cdot> b \<in> H \<diamondop> K"
bullet_app_redsI: fixes s :: "CL"   and t :: "CL"   and u :: "CL" assumes "(s, ` t u) \<in> red\<^sup>*" shows "(s, bullet_app t u) \<in> red\<^sup>*"
var_rel_set_rel_iff: fixes \<V> :: "char list set"   and \<V>' :: "nat set"   and yb :: "char list"   and x2 :: "nat" assumes "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel"   and "(yb, x2) \<in> var_rel" shows "(yb \<in> \<V>) = (x2 \<in> \<V>')"
pseudoinverse': fixes n :: "nat" shows "\<forall>\<omega>\<in>{0<..<1}. \<forall>x. (\<omega> \<le> \<mu>.C n x) = (\<mu>.I n \<omega> \<le> x)"
s_top_commute: fixes H :: "'a set"   and K :: "'a set"   and u :: "'a"   and v :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "K \<diamondop> H = H \<diamondop> K"   and "u \<in> H \<diamondop> K"   and "v \<in> H \<diamondop> K" shows "u \<cdot> v \<in> H \<diamondop> K"
neg_thms_3: shows "neg\<cdot>\<bottom> = \<bottom>"
eq_reflection: fixes x :: "'e"   and y :: "'e" assumes "x = y" shows "x \<equiv> y"
assms_2: shows "g \<in> carrier P \<and> f = X_poly R \<otimes>\<^bsub>P\<^esub> g"
left_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "Groups.monoid f z" shows "f z a = a"
fin_inf_append_def: fixes us :: "'d list"   and s :: "nat \<Rightarrow> 'd" shows "fin_inf_append us s \<equiv> \<lambda>i. if i < length us then us ! i else s (i - length us)"
af_letter_const_congruent: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" assumes "\<phi> \<sim>\<^sub>C \<psi>" shows "af_letter \<phi> \<nu> \<sim>\<^sub>C af_letter \<psi> \<nu>"
determ: fixes act :: "('a \<times> 'a) set" assumes "act \<in> Acts F" shows "single_valued act"
c_types_agree_id: fixes ts :: "t list" shows "c_types_agree (Type ts) ts"
chain'_imp_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain' R xs" shows "chain R xs"
step: fixes p :: "('c, 'a, ('c, 'a, 'b) rpv) generat spmf" assumes "lossless_spmf p"   and "\<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> lossless_gpv \<I> gpv"   and "\<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> P gpv" shows "P (GPV p)"
fresh_swap_qFresh_qSwap: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" assumes "good X" shows "fresh xs x (X #[y1 \<and> y2]_ys) = qFresh xs x (pick X #[[y1 \<and> y2]]_ys)"
owhile_rule: fixes I :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'c"   and s :: "'d"   and M :: "('c \<times> 'c) set"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> 'c option"   and Q :: "'c option \<Rightarrow> bool" assumes "I r s"   and "wf M"   and "\<And>r r'. \<lbrakk>I r s; C r s; B r s = Some r'\<rbrakk> \<Longrightarrow> (r', r) \<in> M"   and "\<And>r r'. \<lbrakk>I r s; C r s; B r s = Some r'\<rbrakk> \<Longrightarrow> I r' s"   and "\<And>r r'. \<lbrakk>I r s; C r s; B r s = None\<rbrakk> \<Longrightarrow> Q None"   and "\<And>r. \<lbrakk>I r s; \<not> C r s\<rbrakk> \<Longrightarrow> Q (Some r)" shows "Q (owhile C B r s)"
ccw_half_segments_of_aform_all: fixes c :: "real \<times> real"   and d :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(c, d) \<in> set (half_segments_of_aform X)" shows "list_all (\<lambda>(xi, xj). ccw xi xj c \<and> ccw xi xj d) (half_segments_of_aform X)"
put_option_def: fixes put :: "'g \<Rightarrow> 'i \<Rightarrow> 'i" shows "put_option put \<equiv> \<lambda>uu uua. rec_optionT (\<lambda>m s. OptionT (put s m)) uua uu"
closed_glob_consistent_def: fixes \<R> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "closed_glob_consistent \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R> \<longrightarrow> (\<forall>x. (dma x = High \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v\<^sub>1 v\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v\<^sub>1)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v\<^sub>2)\<rangle>) \<in> \<R>)) \<and> (dma x = Low \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v)\<rangle>) \<in> \<R>))))"
inorder_balL: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and r :: "('c \<times> nat) tree" shows "Tree2.inorder (balL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
concat_rsp_pre: fixes x :: "'b list list"   and x' :: "'b list list"   and y' :: "'b list list"   and y :: "'b list list"   and xa :: "'b" assumes "list_all2 (\<approx>) x x'"   and "x' \<approx> y'"   and "list_all2 (\<approx>) y' y"   and "\<exists>x\<in>set x. xa \<in> set x" shows "\<exists>x\<in>set y. xa \<in> set x"
arr_cod_iff_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "partial_magma.arr C (partial_magma.cod C f) = partial_magma.arr C f"
progress_le_gen: fixes j :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula" assumes "pred_mapping (\<lambda>x. x \<le> j) Map.empty" shows "Monitor.progress \<sigma> Map.empty \<phi> j \<le> j"
sep_empty_def: shows "\<box> \<equiv> \<lambda>h. h = (0::'a)"
LSC_AndL_inv: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F \<^bold>\<and> G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F, G, \<Gamma> \<Rightarrow>\<^sub>n"
trivial_factors_imp_splitted: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "p \<in> carrier (poly_ring M)"   and "\<And>q. \<lbrakk>q \<in> carrier (poly_ring M); pirreducible\<^bsub>M\<^esub> (carrier M) q; q pdivides\<^bsub>M\<^esub> p\<rbrakk> \<Longrightarrow> degree q \<le> 1" shows "M.splitted p"
fresh_atm_2: fixes a :: "vrs"   and b :: "vrs" shows "a \<sharp> b = (a \<noteq> b)"
iNextWeak_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextWeak t0 I P = (iNext t0 I P \<or> I \<down>> t0 = {} \<or> t0 \<notin> I)"
assms: fixes x :: "'b"   and i :: "nat" assumes "x \<in> X" shows "poly.coeff (f x) i \<in> A"
satisfiable_clause_set_def: fixes S :: "'b literal set set" shows "satisfiable_clause_set S = (\<exists>I. fo_interpretation I \<and> validate_clause_set I S)"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "((0::'a) = a + b) = (a = (0::'a) \<and> b = (0::'a))"
fps_lr_inverse_gp_ring1_1: shows "fps_left_inverse (Abs_fps (\<lambda>n. 1::'a)) (1::'a) = Abs_fps (\<lambda>n. if n = 0 then 1::'a else if n = 1 then - (1::'a) else (0::'a))"
hyps_2: fixes thesis :: "bool" assumes "\<And>f. \<lbrakk>f \<in> F; f \<noteq> 0; lt f adds\<^sub>t t\<rbrakk> \<Longrightarrow> thesis"   and "t \<in> keys p" shows "thesis"
mv_box_max_plus_3_correct: fixes xs :: "nat list"   and n :: "nat"   and x :: "nat"   and fft :: "nat"   and gft :: "nat"   and anything :: "nat list" assumes "length xs = n" shows "{\<lambda>nl. nl = xs @ x # 0 \<up> (max (n + 3) (max fft gft) - n) @ anything} mv_box n (max (n + 3) (max fft gft)) {\<lambda>nl. nl = xs @ 0 \<up> (max (n + 3) (max fft gft) - n) @ x # anything}"
\<psi>: shows "homeomorphic_map (top_of_set V) (Euclidean_space (dim V)) \<psi>"
simps_11: fixes x1 :: "'b"   and x2 :: "'b" shows "\<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0 \<noteq> \<^bold>\<langle>x2\<^bold>\<rangle>"
inext_def: fixes n :: "nat"   and I :: "nat set" shows "inext n I \<equiv> if n \<in> I \<and> I \<down>> n \<noteq> {} then iMin (I \<down>> n) else n"
bin_cat_eqD1: fixes n :: "nat"   and b :: "int"   and a :: "int"   and d :: "int"   and c :: "int" assumes "concat_bit n b a = concat_bit n d c" shows "a = c"
Un_insert_right: fixes A :: "'h set"   and a :: "'h"   and B :: "'h set" shows "A \<union> insert a B = insert a (A \<union> B)"
induct: fixes x1 :: "dB"   and x2 :: "dB"   and P :: "dB \<Rightarrow> dB \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sub>\<eta> x2"   and "\<And>s dummy. \<not> free s 0 \<Longrightarrow> P (Abs (s \<degree> Var 0)) (s[dummy/0])"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<eta> t; P s t\<rbrakk> \<Longrightarrow> P (s \<degree> u) (t \<degree> u)"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<eta> t; P s t\<rbrakk> \<Longrightarrow> P (u \<degree> s) (u \<degree> t)"   and "\<And>s t. \<lbrakk>s \<rightarrow>\<^sub>\<eta> t; P s t\<rbrakk> \<Longrightarrow> P (Abs s) (Abs t)" shows "P x1 x2"
box_subset_cbox: fixes a :: "'c"   and b :: "'c" shows "box a b \<subseteq> cbox a b"
a_\<beta>_ex: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and l :: "int" assumes "iszlfm p (a # bs)"   and "d_\<beta> p l"   and "0 < l" shows "(\<exists>x. l dvd x \<and> Ifm (real_of_int x # bs) (a_\<beta> p l)) = (\<exists>x. Ifm (real_of_int x # bs) p)"
pw_SUP_nofail: fixes f :: "'c \<Rightarrow> 'b nres"   and X :: "'c set" shows "nofail (Sup (f ` X)) = (\<forall>x\<in>X. nofail (f x))"
cases: fixes x :: "parity" obtains "x = Even"   | "x = Odd"   | "x = Either"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "max b (max a c) = max a (max b c)"
IH: fixes s :: "state" assumes "\<forall>r\<in>set rs. \<exists>a. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow> a" shows "\<exists>a. \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> a"
NO_MATCH_def: fixes pat :: "'a"   and val :: "'b" shows "NO_MATCH pat val \<equiv> True"
fail_bind: fixes f :: "int \<Rightarrow> 'm" shows "bind fail f = fail"
results_gpv_inline: fixes x :: "'h"   and s' :: "('i, 'j, 'k, 'l) converter"   and \<I>' :: "('k, 'l) \<I>"   and gpv :: "('h, 'i, 'j) gpv"   and s :: "('i, 'j, 'k, 'l) converter"   and \<I> :: "('i, 'j) \<I>" assumes "(x, s') \<in> results_gpv \<I>' (inline run_converter gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "pfinite_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "x \<in> results_gpv \<I> gpv \<and> pfinite_converter \<I> \<I>' s' \<and> \<I>, \<I>' \<turnstile>\<^sub>C s' \<surd>"
inject_14: fixes x141 :: "'a exp"   and x142 :: "'a exp"   and y141 :: "'a exp"   and y142 :: "'a exp" shows "(x141;; x142 = y141;; y142) = (x141 = y141 \<and> x142 = y142)"
rs_4: shows "the_inv_into A f s < the_inv_into A f r"
fetch: shows "fetch (ci ly (start_of ly as) ins) (Suc s - start_of ly as) b = (ac, ns)"
upper_real_interval: fixes x :: "float interval" shows "upper (real_interval x) = real_of_float (upper x)"