hcju/isabelleps huggingface dataset:
InvariantCnCharacterization_def: fixes Cn :: "nat"   and C :: "Literal list"   and M :: "(Literal \<times> bool) list" shows "InvariantCnCharacterization Cn C M \<equiv> Cn = length (filter (\<lambda>l. elementLevel (opposite l) M = currentLevel M) (remdups C))"
DERIV_abs_powr: fixes p :: "real"   and x :: "real" assumes "1 < p" shows "((\<lambda>x. \<bar>x\<bar> powr p) has_real_derivative p * sgn x * \<bar>x\<bar> powr (p - 1)) (at x)"
disjnt_sym: fixes A :: "'b set"   and B :: "'b set" assumes "disjnt A B" shows "disjnt B A"
set_nns: fixes n :: "nat"   and ps :: "(real, 'b) vec list"   and p :: "(real, 'b) vec"   and kdt :: "'b kdt" shows "set (nearest_nbors n ps p kdt) \<subseteq> set_kdt kdt \<union> set ps"
mmonitorable_exec_mmonitorable: fixes \<phi> :: "Formula.formula" assumes "mmonitorable_exec \<phi>" shows "mmonitorable \<phi>"
Fun_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "Fun (tuple f g) = (\<lambda>x\<^bold>\<in>Dom f. UP \<langle>DOWN (Fun f x), DOWN (Fun g x)\<rangle>)"
equal_conv_eq: fixes x :: "'a"   and y :: "'a" shows "equal x y = (x = y)"
Var: fixes name :: "name" shows "wf_dbtm (DBVar name)"
abs_triangle_ineq: fixes a :: "'a"   and b :: "'a" shows "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
part_witness_not_discrete: fixes X :: "part_witness set" shows "\<not> discrete X"
simps_19: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "char list"   and ve :: "aexp"   and vf :: "'a" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> vd ::= ve {vf}) = False"
discard_alt: shows "discard = filter (\<lambda>(t, X). right I < enat (nt - t)) (linearize data_in)"
pmf_pos: fixes p :: "'c pmf"   and x :: "'c" assumes "pmf p x \<noteq> 0" shows "0 < pmf p x"
integrable_on_affinity: fixes m :: "real"   and f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "m \<noteq> 0"   and "f integrable_on cbox a b" shows "(\<lambda>x. f (m *\<^sub>R x + c)) integrable_on (\<lambda>x. (1 / m) *\<^sub>R x - (1 / m) *\<^sub>R c) ` cbox a b"
bij_\<pi>: fixes k :: "nat" assumes "k < length \<alpha>s" shows "bij_betw (\<pi> k) (\<phi> ` elts (E k)) (elts (\<omega> \<up> \<alpha>s ! k))"
more_than_one: shows "\<nexists>m. obs (sourcenode a) (PDG_BS S) = {m} \<or> obs (sourcenode a) (PDG_BS S) = {}"
inverse_sgn: 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 a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "inverse (sgn a) = sgn a"
select_convs_1: fixes verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d" shows "V\<^bsub>\<lparr>Graph.verts = verts, arcs = arcs, \<dots> = more\<rparr>\<^esub> = verts"
tr_def: shows "target (w || tr) (qA, qB) = (pA, pB) \<and> path AB (w || tr) (qA, qB) \<and> length w = length tr"
ex_conj_contr_left: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred"   and Q :: "'b upred" assumes "x \<sharp> P" shows "(\<exists> x \<bullet> P \<and> Q) = (P \<and> (\<exists> x \<bullet> Q))"
bounded_set_imp_lmeasurable: fixes S :: "'c set" assumes "bounded S"   and "S \<in> sets lebesgue" shows "S \<in> lmeasurable"
SeqHRP_imp_LstSeqP: fixes x :: "tm"   and y :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqHRP x y s k} \<turnstile> LstSeqP s k (HPair x y)"
ih: fixes ab :: "('s, 'v) tm"   and ac :: "('s, 'v) tm"   and ba :: "('s, 'v) tm" assumes "hsize ab + hsize ac + hsize ba < hsize s' + hsize s + hsize t"   and "ab >\<^sub>t ac"   and "t' \<ge>\<^sub>t ba" shows "App ab t' >\<^sub>t App ac ba"
finite_sub_comp_iff_card_eq_mult: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "finite H"   and "finite J" shows "(card (H <#> J) = card J * card H) = complementary H J"
PartE: fixes a :: "'b"   and A :: "'b set"   and h :: "'c \<Rightarrow> 'b" assumes "a \<in> Part A h" obtains z :: "'c" where "a \<in> A" and "a = h z"
disj_dom_drest_fupdate_eq: fixes x :: "('a, 'b) fmap"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "disjnt (fmdom' x) vs" shows "fmrestrict_set vs s = fmrestrict_set vs (x ++ s)"
simps_1: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and S :: "char list \<Rightarrow> 'a"   and X :: "char list set" assumes "Measure1_fun m h" shows "Measure1_fun.m_o m h (Some S) X = Measure1_fun.m_s m S X"
inductive_set_def: fixes D :: "('h, 'i) Order_scheme" shows "inductive_set D = (\<forall>C. Chain D C \<longrightarrow> (\<exists>b. ub\<^bsub>D\<^esub> C b))"
pr_dup_2: fixes a :: "'a" assumes "ide a" shows "prj1' a a \<cdot> ECC.dup a = a"
freshEnv_def: fixes xs :: "'a"   and x :: "'b"   and rho :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm set option" shows "freshEnv xs x rho \<equiv> rho xs x = None \<and> (\<forall>ys. liftAll (fresh xs x) (rho ys))"
lcsTr6: fixes H :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "\<rho> a \<cdot> b \<in> H"   and "x \<in> a \<diamondsuit> H" shows "x \<in> b \<diamondsuit> H"
arg_div: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "arg (z1 / z2) = \<downharpoonright>arg z1 - arg z2\<downharpoonleft>"
Pbij_Rng_def: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_Rng \<beta> = {ll. \<exists>l. (l, ll) \<in> \<beta>}"
face_cycle_succ_closed: fixes a :: "'b" shows "(face_cycle_succ a \<in> arcs G) = (a \<in> arcs G)"
assms_6: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> \<le> r\<^sup>T * t\<^sup>\<star>"
bisim_simple_\<R>: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" shows "bisim_simple (A.\<R> \<Gamma> \<S> P)"
assms_1: shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"
index_mat_addrow_4: fixes ad :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and k :: "nat"   and l :: "nat"   and A :: "'c mat" shows "dim_row (mat_addrow_gen ad mul a k l A) = dim_row A"
run_V0: fixes r :: "nat \<Rightarrow> 'v" assumes "G'.is_run r" shows "r 0 \<in> G'.V0"
select_first_INF: fixes x :: "'a"   and p :: "'a \<Rightarrow> real"   and y :: "'a" assumes "select_first x p y" shows "\<Sqinter> (p ` I x) = p y"
finite_theFTOne: fixes cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list" shows "finite (theFTOne cl dl)"
assms_1: shows "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"
pcr_\<Pi>\<^sub>1_def: shows "pcr_\<Pi>\<^sub>1 \<equiv> rel_fun (=) (rel_fun (=) (rel_fun (=) (=))) OO cr_\<Pi>\<^sub>1"
lfp_def: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.lfp Inf less_eq f = Inf {u. less_eq (f u) u}"
language_def: shows "ST.language \<equiv> {xs. ST.nextl (dfa.init \<lparr>dfa.states = {\<langle>q1, q2\<rangle> |q1 q2. q1 \<in> dfa.states MS \<and> q2 \<in> dfa.states MT}, init = \<langle>dfa.init MS, dfa.init MT\<rangle>, final = {\<langle>q1, q2\<rangle> |q1 q2. q1 \<in> dfa.final MS \<and> q2 \<in> dfa.final MT}, nxt = \<lambda>\<langle>qs, qt\<rangle> x. \<langle>dfa.nxt MS qs x, dfa.nxt MT qt x\<rangle>\<rparr>) xs \<in> dfa.final \<lparr>dfa.states = {\<langle>q1, q2\<rangle> |q1 q2. q1 \<in> dfa.states MS \<and> q2 \<in> dfa.states MT}, init = \<langle>dfa.init MS, dfa.init MT\<rangle>, final = {\<langle>q1, q2\<rangle> |q1 q2. q1 \<in> dfa.final MS \<and> q2 \<in> dfa.final MT}, nxt = \<lambda>\<langle>qs, qt\<rangle> x. \<langle>dfa.nxt MS qs x, dfa.nxt MT qt x\<rangle>\<rparr>}"
crypts_comp: fixes X :: "msg"   and H :: "msg set"   and K :: "key" assumes "X \<in> crypts H" shows "Crypt K X \<in> crypts (Crypt K ` H)"
Above_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "Above r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. (a, b) \<in> r}"
spanning_subset_independent: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'c set"   and A :: "'c set" assumes "module scale"   and "B \<subseteq> A"   and "\<not> module.dependent scale A"   and "A \<subseteq> module.span scale B" shows "A = B"
know_composition_neg_ine_k: fixes P :: "specID"   and key :: "Keys"   and Q :: "specID"   and PQ :: "specID" assumes "\<not> know P (kKS key)"   and "\<not> know Q (kKS key)"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ" shows "\<not> ine PQ (kE key)"
graph_block_size: fixes bl :: "'a set" assumes "bl \<in># arcs_blocks" shows "card bl = 2"
n_not_Suc_n: fixes n :: "nat" shows "\<not> n = Suc n"
fcs_x_eq_x: fixes x :: "'b" shows "(H.face_cycle_succ x = x) = (H.face_cycle_set x = {x})"
times0_left: fixes A :: "(('a, 'b) vec, 'd) vec" shows "0 ** A = 0"
simps_160: fixes x8 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Rjoin x8) = RID\<^sub>E x8"
ty: shows "\<Gamma> \<turnstile> (\<lambda>x:T. t) : S"
indRelRT_def: fixes Enc :: "'b \<Rightarrow> 'a" shows "encoding.indRelRT Enc \<equiv> \<lambda>TRel. _Collect (xa, x) (encoding.indRelRTp Enc (\<lambda>x xa. (x, xa) \<in> TRel) xa x)"
alphaAbs_qAbs_iff: fixes A :: "('d, 'e, 'f, 'g, 'h) qAbs"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(A $= qAbs xs' x' X') = (\<exists>x y X. A = qAbs xs' x X \<and> y \<notin> {x, x'} \<and> qAFresh xs' y X \<and> qAFresh xs' y X' \<and> X #[[y \<and> x]]_xs' #= X' #[[y \<and> x']]_xs')"
B: shows "(xs @ ipurge_tr I D u ys @ zs, Z) \<in> failures P"
acomp_exits: fixes a :: "aexp" shows "exits (acomp a) = {size (acomp a)}"
nsqn\<^sub>r_zero: fixes dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "nsqn\<^sub>r (0, dsk, flag, hops, nhip, pre) = 0"
distinct_181: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x111 :: "char list"   and x112 :: "ty"   and x113 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> {x111:x112; x113}"
omit_id: fixes ps :: "('c, 'd) fm list" shows "omit {} ps = ps"
buffon_altdef': shows "buffon l d = distr (uniform_measure lborel ({- d / 2..d / 2} \<times> {- pi..pi})) (count_space UNIV) (\<lambda>z. z \<in> buffon_set l d)"
\<mu>\<nu>: shows "B.VV.arr \<mu>\<nu>"
minus_minus: fixes x :: "'b" assumes "x \<in> UNIV" shows "- (- x) = x"
prime_factorization_prime_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize (power.power one times p n) = replicate_mset n p"
Pm: fixes m :: "M" shows "Preorder (Pm m) \<and> Total (Pm m)"
induces_equivalence: shows "\<exists>F \<eta> \<epsilon>. equivalence_of_categories (\<cdot>) (\<cdot>) F (\<lambda>f. f \<otimes> \<I>) \<eta> \<epsilon>"
Zfun_minus: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "Zfun f F" shows "Zfun (\<lambda>x. - f x) F"
countable_Un: fixes A :: "'f set"   and B :: "'f set" assumes "countable A"   and "countable B" shows "countable (A \<union> B)"
lowner_le_trans: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "A \<le>\<^sub>L B"   and "B \<le>\<^sub>L C" shows "A \<le>\<^sub>L C"
gen_new_keys_not_used: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_cr" shows "Key K \<notin> used evs \<longrightarrow> K \<in> symKeys \<longrightarrow> K \<notin> keysFor (parts (Key ` KK \<union> knows Spy evs))"
assn_supd_def2: fixes P :: "'b assn"   and f :: "state \<Rightarrow> state"   and Y :: "vals"   and s' :: "state"   and Z :: "'b" shows "(P ;. f) Y s' Z = (\<exists>s. P Y s Z \<and> s' = f s)"
simplex_map: fixes x :: "'a set" assumes "x \<in> X" shows "g ` x \<in> X"
discrete: fixes discrete :: "bool"   and y :: "int"   and x :: "int" assumes "discrete"   and "y \<le> x" shows "\<exists>k. x = ((+) 1 ^^ k) y"
neq_Leaf2_iff: fixes t :: "('b \<times> 'c) tree" shows "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a n r. t = \<langle>l, (a, n), r\<rangle>)"
jump_poly_not_root: fixes p :: "real poly"   and x :: "real"   and q :: "real poly" assumes "poly p x \<noteq> 0" shows "jump_poly q p x = 0"
srtc_in_pmdl: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "relation.srtc (red F) p q" shows "p - q \<in> pmdl F"
K_def: fixes n :: "nat" shows "K n = fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
p_def: shows "path M p q1 \<and> target p q1 = q2"
sinits_bottom_iff: fixes xs :: "[:'a:]" shows "(sinits\<cdot>xs = \<bottom>) = (xs = \<bottom>)"
simps: fixes a1 :: "'a list"   and a2 :: "sequent" shows "((a1, a2) \<in> upRules) = (\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> NominalSequents.mset c = \<LM> Cpd0 R Fs  \<RM> \<and> ps \<noteq> [])"
restr_eqE: fixes F :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and F' :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "restrict F A = restrict F' A"   and "x \<in> A" shows "F x = F' x"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>y x gt. y \<noteq> x \<Longrightarrow> ext gt [y] [x] = gt y x" shows "ext_singleton_axioms ext"
M2_axioms_def: shows "M2_axioms \<equiv> (\<forall>x y. PP x y = (P x y \<and> x \<noteq> y)) \<and> (\<forall>x y. PP x y = (P x y \<and> \<not> P y x))"
continuous_diff: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x - g x)"
pathfinish_shiftpath: fixes a :: "real"   and g :: "real \<Rightarrow> 'a" assumes "0 \<le> a"   and "pathfinish g = pathstart g" shows "pathfinish (shiftpath a g) = g a"
subword_split: fixes i :: "nat"   and j :: "nat"   and w :: "'c list"   and k :: "nat" assumes "i + j \<le> length w"   and "0 < k"   and "k < j" shows "subword w i j = subword w i k \<cdot> subword w (i + k) (j - k)"
carrier_eq: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le" shows "carrier = {x. le x x}"
mwb_lens_axioms_def: fixes x :: "'a \<Longrightarrow> 'b" shows "mwb_lens_axioms x \<equiv> \<forall>\<sigma> v u. put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
assms_2: shows "g = \<lbrace>u\<rbrace>"
moebius_circline: fixes M :: "moebius"   and H :: "circline" shows "{z. on_circline (moebius_circline M H) z} = moebius_pt M ` {z. on_circline H z}"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "ordering less_eq less"   and "ordering less_eq' less'"   and "OrderingSetMap_axioms less_eq less_eq' P f" shows "OrderingSetMap less_eq less less_eq' less' P f"
fin: shows "finite {Pr \<guillemotleft>s\<guillemotright> |s. toplevel_summands s \<subseteq> (\<Union>xs. toplevel_summands \<guillemotleft>lderivs xs r\<guillemotright>)}"
simps_2: fixes x :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" shows "height (MKT x l r h) = max (height l) (height r) + 1"
sees_methods_decl_above: fixes P :: "'a prog"   and C :: "String.literal"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option"   and M :: "String.literal"   and m :: "ty list \<times> ty \<times> 'a option"   and D :: "String.literal" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "P \<turnstile> C \<preceq>\<^sup>* D"
dom_map_upds: fixes m :: "'a \<Rightarrow> 'b option"   and xs :: "'a list"   and ys :: "'b list" shows "dom (m(xs [\<mapsto>] ys)) = set (take (length ys) xs) \<union> dom m"
adopt_node_writes: fixes h :: "(_) heap"   and node :: "(_) node_ptr"   and document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" shows "writes (adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M |h \<turnstile> get_parent node|\<^sub>r |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node)|\<^sub>r document_ptr) (adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M document_ptr node) h h'"
dfa_trans_def: fixes A :: "nat bdd list \<times> bool list"   and q :: "nat"   and bs :: "bool list" shows "dfa_trans A q bs \<equiv> bdd_lookup (fst A ! q) bs"
ACI_norm_alt_Plus: fixes r :: "'b rexp"   and s :: "'b rexp"   and t :: "'b rexp" assumes "ACI_norm_alt r = Plus s t" shows "\<exists>s t. r = Plus s t"
product_over_sum_right: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CEM"   and "O x y \<and> O x z" shows "P (x \<otimes> y \<oplus> x \<otimes> z) (x \<otimes> (y \<oplus> z))"
init_sfields_def: fixes FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" shows "init_sfields FDTs \<equiv> (map_of \<circ> map (\<lambda>((F, D), b, T). (F, default_val T))) FDTs"
lift_f_trivial_P: fixes ps :: "('a \<times> bool) list" assumes "ps \<in> P" shows "lift_f (Abs_freeword ps) = (0::'b)"
linorder_neqE_nat: fixes x :: "nat"   and y :: "nat" assumes "\<not> x = y" obtains "x < y"   | "y < x"
seqE: fixes g :: "'b"   and f :: "'b" assumes "J.seq g f" obtains "J.arr f" and "J.arr g" and "J.dom g = J.cod f"
bigtheta_refl: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "f \<in> \<Theta>[F](f)"
cexpr_typing_subst_val: fixes t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t' :: "pdf_type"   and v :: "val" assumes "case_nat t \<Gamma> \<turnstile>\<^sub>c e : t'"   and "val_type v = t" shows "\<Gamma> \<turnstile>\<^sub>c cexpr_subst_val e v : t'"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('u \<Rightarrow> 'u) \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme"   and v :: "'u" assumes "l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
assms_9: shows "e\<^sup>T \<le> v\<^sup>\<star>"
abs_mult_self_eq: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "times (abs a) (abs a) = times a a"
cls_val_process_def: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "cls_val_process Mkt p = (if \<not> portfolio p then \<lambda>n w. 0 else up_cl_proc Mkt p)"
n_less_n_mult_m: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "Suc 0 < m" shows "n < n * m"
ground_sub_of_denotl: fixes l :: "fterm literal"   and E :: "char list \<Rightarrow> hterm" shows "ground\<^sub>l (l \<cdot>\<^sub>l sub_of_denot E)"
Domainp_Domain_eq: fixes r :: "('b \<times> 'c) set" shows "Domainp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Domain r)"
CondTrue: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and t :: "('a, 'c) xstate"   and c\<^sub>2 :: "('a, 'b, 'c) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
induced_automorphism_chamber_list_image_fg: fixes Cs :: "'a set list" assumes "order.greater_eq (f \<turnstile> folding_g.\<C>) (set Cs)" shows "\
 \<Turnstile> Cs = g \<Turnstile> Cs"
ordering_axioms: shows "ordering (\<preceq>\<^sub>t) (\<prec>\<^sub>t)"
IH1: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "Vagree \<nu> \<omega> (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF x2)))"   and "fsafe x2" shows "fml_sem (local.adjoint I \<sigma> \<nu>) x2 = fml_sem (local.adjoint I \<sigma> \<omega>) x2"
idle_squareI: fixes v :: "state \<Rightarrow> 'a"   and s :: "state"   and t :: "state"   and A :: "state \<times> state \<Rightarrow> bool" assumes "(s, t) \<Turnstile> unchanged v" shows "(s, t) \<Turnstile> [A]_v"
snd_of_exps: fixes x :: "'a"   and i :: "nat"   and a :: "nat"   and b :: "nat" assumes "snd (rep_nat_term x) = i" shows "snd (rep_nat_term (Term_Order.of_exps a b i)) = i"
rtranclp_imp_eq_image: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "\<forall>x y. R x y \<longrightarrow> f x = f y"   and "R\<^sup>*\<^sup>* x y" shows "f x = f y"
div_poly_less: fixes x :: "'c poly"   and y :: "'c poly" assumes "degree x < degree y" shows "x div y = 0"
icrit_specI: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" assumes "\<And>d m data gs bs hs p q. \<lbrakk>dickson_grading d; fst ` (set gs \<union> set bs \<union> set hs) \<subseteq> dgrad_p_set d m; unique_idx (gs @ bs @ hs) data; is_Groebner_basis (fst ` set gs); p \<in> set hs; q \<in> set gs \<union> set bs \<union> set hs; fst p \<noteq> 0; fst q \<noteq> 0; crit data gs bs hs p q\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs \<union> set hs)) (fst p) (fst q)" shows "icrit_spec crit"
swapVal_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'g" shows "val ^[x \<and> y]_zs = val ^[y \<and> x]_zs"
show_l_zero: fixes a :: "'a"   and x :: "'a" shows "(a + x = x) = (a = (0::'a))"
lta_os__ts: fixes A :: "'p"   and O1 :: "'p"   and P :: "'p"   and B :: "'p" assumes "A O1 P LtA A O1 B"   and "O1 A OS B P" shows "O1 P TS A B"