hcju/isabelleps huggingface dataset:
parallel_fixp_induct_1_1: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and luba :: "'c set \<Rightarrow> 'c"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and F :: "('e \<Rightarrow> 'c) \<Rightarrow> 'e \<Rightarrow> 'c"   and G :: "('f \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and P :: "('e \<Rightarrow> 'c) \<Rightarrow> ('f \<Rightarrow> 'd) \<Rightarrow> bool" assumes "partial_function_definitions orda luba"   and "partial_function_definitions ordb lubb"   and "\<And>x. monotone (fun_ord orda) orda (\<lambda>f. F f x)"   and "\<And>y. monotone (fun_ord ordb) ordb (\<lambda>g. G g y)"   and "f \<equiv> ccpo.fixp (fun_lub luba) (fun_ord orda) F"   and "g \<equiv> ccpo.fixp (fun_lub lubb) (fun_ord ordb) G"   and "ccpo.admissible (prod_lub (fun_lub luba) (fun_lub lubb)) (rel_prod (fun_ord orda) (fun_ord ordb)) (\<lambda>x. P (fst x) (snd x))"   and "P (\<lambda>_. luba {}) (\<lambda>_. lubb {})"   and "\<And>f g. P f g \<Longrightarrow> P (F f) (G g)" shows "P f g"
vrestr_of_set: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "g \<in> M \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "qmpt.vimage_restr M g A = g -` A \<inter> space M"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<phi> i\<lparr>ArrMap\<rparr>)" shows "\<exists>a\<in>elts (\<CC>\<lparr>Arr\<rparr>). \<phi> i\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
bounded_linear_mult_left: fixes y :: "'a" shows "bounded_linear (\<lambda>x. x * y)"
silent_moves_slice_keeps_state: fixes S :: "'node SDG_node set"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,slice_kind S \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "s = s'"
blinfun_eqI: fixes x :: "'a \<Rightarrow>\<^sub>L 'c"   and y :: "'a \<Rightarrow>\<^sub>L 'c" assumes "\<And>i. blinfun_apply x i = blinfun_apply y i" shows "x = y"
distinct_7: fixes x1 :: "'a"   and x5 :: "'a" shows "broadcast x1 \<noteq> send x5"
mwb_lens_def: fixes x :: "'a \<Longrightarrow> 'b" shows "mwb_lens x \<equiv> weak_lens x \<and> mwb_lens_axioms x"
finite_set_mset_mset_set: fixes A :: "'a set" assumes "finite A" shows "set_mset (mset_set A) = A"
subterms_sterms_in_cterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp"   and pn :: "'e"   and q :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>"   and "p \<in> subterms (\<Gamma> pn)"   and "q \<in> sterms \<Gamma> p" shows "q \<in> cterms \<Gamma>"
inject_6: fixes x61 :: "msg"   and x62 :: "msg"   and y61 :: "msg"   and y62 :: "msg" shows "(\<lbrace>x61, x62\<rbrace> = \<lbrace>y61, y62\<rbrace>) = (x61 = y61 \<and> x62 = y62)"
comp_cod_arr: fixes f :: "'d"   and b :: "'d" assumes "D.arr f"   and "D.cod f = b" shows "b \<cdot>\<^sub>D f = f"
sharing_consistent_mono_owns: fixes \<O> :: "nat set"   and \<O>' :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list" assumes "\<O> \<subseteq> \<O>'"   and "sharing_consistent \<S> \<O> sb" shows "sharing_consistent \<S> \<O>' sb"
columns_rw: shows "columns (fst (QR_decomposition A)) = (\<lambda>x. column x (fst (QR_decomposition A))) ` (UNIV - {ia})"
deadlocked'_no_red: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "deadlocked' s" shows "False"
cfg_onD_cont: fixes cfg :: "'s cfg"   and s :: "'s"   and t :: "'s" assumes "cfg \<in> cfg_on s"   and "t \<in> set_pmf (action cfg)" shows "cont cfg t \<in> cfg_on t"
atLeast1_lessThan_eq_remove0: fixes n :: "nat" shows "{Suc 0..<n} = {..<n} - {0}"
body_def: fixes fundef :: "'a fundef" shows "body fundef = (case fundef of Fundef x1 x2 \<Rightarrow> x1)"
simps: fixes t\<^sub>1 :: "'b"   and t\<^sub>2 :: "'b" shows "\<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>1 + \<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>2 = \<tau>\<^sub>c\<^sub>s\<^sub>t (t\<^sub>1 + t\<^sub>2)"
fst_set_path_verties: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and v :: "'d"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "Port_Graph nodes inPorts outPorts vertices nodeOf edges"   and "Pre_Port_Graph.path edges v v' pth" shows "fst ` fst ` set pth \<subseteq> fset vertices"
abs_add_2_mult_bound: fixes x :: "'a"   and y :: "'a"   and i :: "int" assumes "\<bar>x\<bar> \<le> y" shows "\<bar>x\<bar> \<le> \<bar>x + (2::'a) * of_int i * y\<bar>"
Sup2_E: fixes A :: "('b \<Rightarrow> 'c \<Rightarrow> bool) set"   and a :: "'b"   and b :: "'c" assumes "(\<Squnion> A) a b" obtains r :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where "r \<in> A" and "r a b"
mem_ordinal_number_Worder: fixes D :: "'c Order"   and X :: "'c Order" assumes "Worder D"   and "X \<in> ordinal_number D" shows "Worder X"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)"
g_def: shows "g = (\<lambda>t. \<gamma> t - z)"
intro: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool" assumes "\<And>Q p p'. \<lbrakk>quorum_of p Q; p' \<in> Q\<rbrakk> \<Longrightarrow> quorum_of p' Q" shows "personal_quorums quorum_of"
set_intros_19: fixes yq :: "'b"   and x142 :: "'b ltlc"   and x141 :: "'b ltlc" assumes "yq \<in> atoms_ltlc x142" shows "yq \<in> atoms_ltlc (x141 M\<^sub>c x142)"
skl1_inv5_def: shows "skl1_inv5 \<equiv> {s. \<forall>gny A B Ra Nr. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<longrightarrow> in_progressS (progress s Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<longrightarrow> sklvl1.guessed_frame Ra xgny = Some gny \<longrightarrow> sklvl1.guessed_frame Ra xnr = Some Nr \<longrightarrow> can_signal s A B \<longrightarrow> 0 < signalsResp s (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>)}"
eq_Nil_appendI: fixes xs :: "'d list"   and ys :: "'d list" assumes "xs = ys" shows "xs = [] @ ys"
eventually_ge_at_top: fixes c :: "'d" shows "eventually ((\<le>) c) at_top"
msort_bu_def: fixes xs :: "'c list" shows "msort_bu xs = merge_all (map (\<lambda>x. [x]) xs)"
cases: fixes x :: "nat \<times> nat" obtains xa :: "nat" and y :: "nat" where "x = (xa, y)"
DynT_mheadsD: fixes G :: "prog"   and sm :: "mhead"   and e :: "expr"   and invC :: "qtname"   and statT :: "ref_ty"   and C :: "qtname"   and L :: "lenv"   and statDeclT :: "ref_ty"   and sig :: "sig"   and s :: "st"   and a' :: "val"   and declC :: "qtname" assumes "G\<turnstile>invmode sm e\<rightarrow>invC\<preceq>statT"   and "wf_prog G"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-RefT statT"   and "(statDeclT, sm) \<in> mheads G C statT sig"   and "invC = invocation_class (invmode sm e) s a' statT"   and "declC = invocation_declclass G (invmode sm e) s a' statT sig" shows "\<exists>dm. methd G declC sig = Some dm \<and> dynlookup G statT invC sig = Some dm \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy sm \<and> wf_mdecl G declC (sig, mthd dm) \<and> declC = declclass dm \<and> is_static dm = is_static sm \<and> is_class G invC \<and> is_class G declC \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and> (if invmode sm e = IntVir then \<forall>statC. statT = ClassT statC \<longrightarrow> G\<turnstile>invC\<preceq>\<^sub>C statC else ((\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC) \<or> (\<forall>statC. statT \<noteq> ClassT statC \<and> declC = Object)) \<and> statDeclT = ClassT (declclass dm))"
compactin_euclidean_iff: fixes S :: "'c set" shows "compactin euclidean S = compact S"
LBINT_I0i_exp_mscale: fixes u :: "real" assumes "0 < u" shows "LBINT x=0..\<infinity>. exp (- (x * u)) = 1 / u"
matches_rule_and_simp_help: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m' :: "'c match_expr"   and a' :: "action"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule (MatchAnd m m') a'], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m' a'], s\<rangle> \<Rightarrow> t"
bdd_aboveI: fixes A :: "'c list set"   and M :: "'c list" assumes "\<And>x. x \<in> A \<Longrightarrow> prefix x M" shows "prefix_order.bdd_above A"
up_scast_inj: fixes x :: "'c word"   and y :: "'c word" assumes "scast x = scast y"   and "size x \<le> LENGTH('b)" shows "x = y"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>a. P [a]"   and "\<And>a b cs. P cs \<Longrightarrow> P (a # b # cs)" shows "P a0"
smallomegaD_filterlim_at_top_norm: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<omega>[F](g)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "LIM x F. norm (f x / g x) :> at_top"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "Ivl xa x = Interval (min xa x, x)"
IH: fixes b :: "bool list" assumes "branch b (deeptree n)" shows "length b = n"
assms_7: shows "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"
hyps: fixes xs :: "'a llist" assumes "enat n < llength xs" shows "LCons (lnth xs n) (ldropn (Suc n) xs) = ldropn n xs"
L: shows "fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma ! j = Some x"
rel_inject_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and x22 :: "'c list"   and y21 :: "'d"   and y22 :: "'d list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
simps: fixes a1 :: "'a eclause"   and a2 :: "'a eclause set" shows "derivable_ecl a1 a2 = ((\<exists>C S. a1 = C \<and> a2 = S \<and> C \<in> S) \<or> (\<exists>C S D. a1 = D \<and> a2 = S \<and> derivable_ecl C S \<and> renaming_cl C D) \<or> (\<exists>P S C S' \<sigma> C'. a1 = C \<and> a2 = S \<and> (\<forall>x. x \<in> P \<longrightarrow> derivable_ecl x S) \<and> derivable C P S' \<sigma> FirstOrder C'))"
simps_2: fixes uu :: "'a list" shows "sorted_list_subset [] uu = None"
simps_1: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree"   and y21 :: "nat"   and y22 :: "nat"   and y23 :: "'b option"   and y24 :: "'b tree"   and y25 :: "'b tree" shows "(Node x21 x22 x23 x24 x25 = Node y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
mult_right_le_one_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times x y) x"
trivial_hom_iff: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "(h ` carrier G = {\<one>\<^bsub>H\<^esub>}) = (kernel G H h = carrier G)"
obs_neq_ret_contradicts: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and i :: "nat" assumes "path \<sigma> n = local.return"   and "obs \<sigma>' i \<noteq> obs \<sigma> i"   and "obs \<sigma>' i \<noteq> None" shows "\<exists>k k'. (\<sigma>', k') \<cc> (\<sigma>, k) \<and> path \<sigma> k \<in> dom local.att \<or> (\<sigma>, k) \<cc> (\<sigma>', k') \<and> path \<sigma>' k' \<in> dom local.att"
ord_recp_succ_iff: fixes k :: "hf"   and T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf"   and z :: "hf" assumes "Ord k" shows "ord_recp T G H (succ k) z = (\<exists>y. z = G y \<and> ord_recp T G H k y)"
hoare_pre: fixes R :: "'a"   and P :: "'a"   and S :: "'b \<Rightarrow> 'a"   and Q :: "'b" assumes "R \<sqsubseteq> P"   and "\<Turnstile> P{| S |}Q" shows "\<Turnstile> R{| S |}Q"
eq_fold: fixes A :: "'a set" shows "Lcm\<^sub>f\<^sub>i\<^sub>n A = (if finite A then Finite_Set.fold lcm (1::'a) A else (0::'a))"
simps_1: shows "\<^bold>\<And>[] = \<^bold>\<not> \<bottom>"
dim_row_transfer_rule: fixes A :: "'d mat"   and A' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "HMA_Connect.HMA_M A A'" shows "dim_row A = CARD('f)"
inf_K_next_behaviour: fixes a :: "'a"   and b :: "'a"   and s :: "'b" assumes "a \<le>\<^sub>\<K> b" shows "s \<circ> a \<le>\<^sub>\<K> s \<circ> b"
connected_component_refl_eq: fixes S :: "'c set"   and x :: "'c" shows "connected_component S x x = (x \<in> S)"
deadlockD2: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   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 t :: "'b"   and x :: "'a"   and ln :: "'d \<Rightarrow>f nat"   and l :: "'d" assumes "multithreaded_base.deadlock final r s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "0 < ln $ l"   and "\<not> waiting (wset s t)" obtains l' :: "'d" and t' :: "'b" where "0 < ln $ l'" and "t \<noteq> t'" and "thr s t' \<noteq> None" and "has_lock (locks s $ l') t'"
splus_pprod_def: shows "splus_pprod = pprod.splus"
xor_inverse: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<oplus> x = \<bottom>"
ImageE: fixes b :: "'e"   and r :: "('f \<times> 'e) set"   and A :: "'f set" assumes "b \<in> r `` A" obtains x :: "'f" where "(x, b) \<in> r" and "x \<in> A"
ordinal_0_div: fixes x :: "ordinal" shows "0 div x = 0"
rreq_nsqn_is_fresh: fixes \<sigma> :: "nat \<Rightarrow> state"   and sip :: "nat"   and hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and handled :: "bool" assumes "rreq_rrep_fresh (rt (\<sigma> sip)) (Rreq hops rreqid dip dsn dsk oip osn sip handled)"   and "rreq_rrep_sn (Rreq hops rreqid dip dsn dsk oip osn sip handled)" shows "msg_fresh \<sigma> (Rreq hops rreqid dip dsn dsk oip osn sip handled)"
cone_def: fixes s :: "'b set" shows "cone s = (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
assms: shows "length (xvec1 @ x # xvec2) = length yvec"
w: shows "wqo_on (\<lambda>a b. (a, b) \<in> r\<^sup>=) UNIV"
hyps_3: shows "(remove_rels ts'', m'', \<S>'') \<Rightarrow>\<^sub>v\<^sup>* (remove_rels ts', m', \<S>')"
anyP_is_any: fixes f :: "'form" shows "\<exists>s. subst s anyP = f"
the_adj_chamber_is_adjacent: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "C \<sim> the_adj_chamber C z"
cone: shows "D.cone (C.dom p0') (D.mkCone p0' p1')"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a iarray"   and b :: "'b iarray" assumes "rel_iarray R a b" obtains xa :: "'a list" and ya :: "'b list" where "a = IArray xa" and "b = IArray ya" and "list_all2 R xa ya"
prems_2: shows "check_no_overlap \<gamma> (a # ft)"
max_c'_lt_a: fixes A :: "'a" assumes "A \<in> set As"   and "C_of A \<noteq> {#}" shows "Max (atms_of (C_of A)) < A"
simps_1: fixes x1 :: "'a"   and x2 :: "'a"   and y1 :: "'a"   and y2 :: "'a" shows "(Interval x1 x2 = Interval y1 y2) = (x1 = y1 \<and> x2 = y2)"
Q_rec: shows "Q = (\<lambda>s. Mprefix (Tr\<^sub>2 s) (\<lambda>e. Q (Up\<^sub>2 s e)))"
rtrancl_unfold: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
less_not_le: fixes x :: "nat"   and y :: "nat" assumes "x < y"   and "y \<le> x" shows "False"
nil_gt_s: shows "extf f (>\<^sub>t) [] [s]"
FUpdate: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a"   and t :: "('b, 'a) Term.term" assumes "\<theta> \<in> fsubst"   and "v \<notin> subst_domain \<theta>"   and "t \<noteq> Var v" shows "\<theta>(v := t) \<in> fsubst"
simps_9: fixes v :: "cmsg"   and vb :: "cmsg" shows "catomic (cAenc v vb) = False"
DiffD2: fixes c :: "'q"   and A :: "'q set"   and B :: "'q set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
simps_2: fixes at :: "('fsym, 'psym) atm" shows "glitOfL (Neg at) = {grdLit x |x. x \<in> varsA at \<and> prot (GE.tpOfV x)}"
L_in_interval: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and b :: "'b"   and S :: "'b set"   and a :: "'b"   and L :: "'b" assumes "CLF cl f"   and "b \<in> pset cl"   and "S \<subseteq> interval (order cl) a b"   and "isLub S cl L"   and "S \<noteq> {}" shows "L \<in> interval (order cl) a b"
intros_2: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t' :: "'c"   and Ts :: "'c set"   and t :: "'c" assumes "final_thread.not_final_thread final s t'"   and "t' \<in> Ts" shows "final_thread.must_wait final s t (Inr (Inl t')) Ts"
HC_intros_5: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> F \<^bold>\<or> (G::'b formula)"
QF_subset_N: fixes k :: "nat"   and j :: "nat"   and i :: "nat" shows "QF k (j, i) \<subseteq> N"
dsj_cong: fixes H :: "fmla set"   and A :: "fmla"   and A' :: "fmla"   and B :: "fmla"   and B' :: "fmla" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> A OR B IFF A' OR B'"
pot_comp_monotone: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a"   and t :: "'a" assumes "cmp u v = Lt \<Longrightarrow> cmp (splus t u) (splus t v) = Lt"   and "pot_comp cmp u v = Lt" shows "pot_comp cmp (splus t u) (splus t v) = Lt"
pq: shows "CARD('pq) = CARD('p) * CARD('q)"
components_are_iso: fixes a :: "'a" assumes "hom.ide a" shows "hom.iso (hom.map a)"
Vr_tOp_f_tOp: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "y \<in> carrier (Vr K v)" shows "x \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> y = x \<cdot>\<^sub>r y"
not_eq_order_implies_strict: fixes a :: "'c"   and b :: "'c" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
has_density_def: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "has_density M N f = (f \<in> borel_measurable N \<and> space N \<noteq> {} \<and> M = density N f)"
0: shows "(x \<otimes>\<^bsub>F\<^esub> y \<otimes>\<^bsub>F\<^esub> z) a = x a \<otimes> y a \<otimes> z a"
load_word_mem_def: fixes state :: "'b sparc_state"   and va :: "word32"   and asi :: "word8" shows "load_word_mem state va asi \<equiv> let pair = virt_to_phys va (mmu state) (mem state) in case pair of None \<Rightarrow> None | Some pair \<Rightarrow> if mmu_readable (get_acc_flag (snd pair)) asi then mem_val_w32 asi (fst pair) state else None"
pref_in_at_least_as: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y" shows "x \<in> at_least_as_good y carrier relation"
upright_vec_add: fixes v1 :: "'c zmultiset"   and v2 :: "'c zmultiset" assumes "upright v1"   and "upright v2" shows "upright (v1 + v2)"
intro: fixes H :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "subgroup H G composition unit"   and "group G composition unit" shows "subgroup_of_group H G composition unit"
rel_eq_onp: fixes P :: "'d \<Rightarrow> bool" shows "rel_cset (eq_onp P) = eq_onp (pred_cset P)"
bprv_compl_isTrue_PP_enc: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (PP \<langle>\<phi>\<rangle>)" shows "bprv (PP \<langle>\<phi>\<rangle>)"
bipartite_web_of_sel_3: fixes \<Gamma> :: "('a, 'b) web_scheme"   and v :: "'a"   and uv :: "'a + 'a" shows "edge (bipartite_web_of \<Gamma>) (Inr v) uv = False"
bounded_clinear_eq_on: fixes A :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e"   and G :: "'d set"   and t :: "'d" assumes "bounded_clinear A"   and "bounded_clinear B"   and "\<And>x. x \<in> G \<Longrightarrow> A x = B x"   and "t \<in> closure (cspan G)" shows "A t = B t"
tllist_ord_terminalD: fixes xs :: "('d, 'b) tllist"   and ys :: "('d, 'b) tllist" assumes "local.tllist_ord xs ys"   and "is_TNil ys" shows "flat_ord b (terminal xs) (terminal ys)"
carrier_one_zero: shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
commute: fixes a :: "'a"   and b :: "'a" shows "min a b = min b a"
silent_moves_into_RedT_\<tau>_inv: fixes t :: "'t"   and x :: "'x1"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x' :: "'x1"   and m' :: "'m1" assumes "r1.silent_moves t (x, shr s) (x', m')"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "wset s t = None" shows "\<tau>mRed1 s (redT_upd_\<epsilon> s t x' m')"
rec_all3_def: fixes f :: "recf" shows "rec_all3 f = CN rec_sign [rec_accum3 f]"
r_e2frame_prim: shows "prim_recfn 1 r_e2frame"
dependent_def: fixes P :: "'b set" shows "vs1.dependent P = (\<exists>a\<in>P. a \<in> vs1.span (P - {a}))"
uniq_line: fixes P :: "Points"   and Q :: "Points"   and l :: "Lines"   and m :: "Lines" assumes "P \<noteq> Q"   and "l \<in> lines P Q"   and "m \<in> lines P Q" shows "l = m"
markedElementsPrefixAreMarkedElementsTrail: fixes p :: "('b \<times> bool) list"   and M :: "('b \<times> bool) list"   and m :: "'b" assumes "isPrefix p M"   and "m \<in> set (markedElements p)" shows "m \<in> set (markedElements M)"
cryptinverts: fixes p :: "nat"   and q :: "nat"   and n :: "nat"   and m :: "nat"   and e :: "nat"   and d :: "nat" assumes "prime p"   and "prime q"   and "p \<noteq> q"   and "n = p * q"   and "m < n"   and "e * d mod (pred p * pred q) = 1" shows "rsa_crypt (rsa_crypt m e n) d n = m"
Oops_range_spies2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "servK \<notin> range shrK \<and> servK \<in> symKeys"
lcm_proj2_if_dvd: fixes a :: "'a"   and b :: "'a" assumes "a dvd b" shows "lcm a b = normalize b"
rbt_generator_induct: fixes P :: "('c \<times> ('d, 'e) rbt) list \<times> ('d, 'e) rbt \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and ktst :: "('c \<times> ('d, 'e) rbt) list \<times> ('d, 'e) rbt" assumes "P ([], rbt.Empty)"   and "\<And>k t kts. P (kts, t) \<Longrightarrow> P ((k, t) # kts, rbt.Empty)"   and "\<And>kts c l k v r. P ((f k v, r) # kts, l) \<Longrightarrow> P (kts, Branch c l k v r)" shows "P ktst"
H_ia_least_A: shows "H $ ia $ (LEAST n. A $ ia $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
add: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>R (x + y) = a *\<^sub>R x + a *\<^sub>R y"