hcju/isabelleps huggingface dataset:
SatP: fixes b :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>@ a p) at b in' set ((ps, a) # branch)"   and "new p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
eventually_ge_at_top: fixes c :: "'a" shows "eventually ((\<le>) c) at_top"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> zg" shows "\<exists>A. Says A B X \<in> set evs"
cong_eval_def: fixes bdt\<^sub>1 :: "bdt"   and bdt\<^sub>2 :: "bdt" shows "bdt\<^sub>1 \<sim> bdt\<^sub>2 = (eval bdt\<^sub>1 = eval bdt\<^sub>2)"
assms: shows "learn_lim \<psi> U (\<phi> i)"
is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D GF.map GF.cmp"
weak_equality_l_weak_equality_spec: shows "(uncurry weak_equality_l, uncurry weak_equality_spec) \<in> sorted_poly_rel \<times>\<^sub>r sorted_poly_rel \<rightarrow>\<^sub>f \<langle>bool_rel\<rangle>nres_rel"
schur: shows "unitary_schur_decomposition A3 es = (B, P', Q')"
Weierstrass_m_test_ev: fixes A :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. norm (f n x) \<le> M n"   and "summable M" shows "uniform_limit A (\<lambda>n x. \<Sum>i<n. f i x) (\<lambda>x. \<Sum>i. f i x) sequentially"
vimage_eq_UN: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'b set" shows "f -` B = (\<Union>y\<in>B. f -` {y})"
infs_coinduct: fixes R :: "'b stream \<Rightarrow> bool"   and w :: "'b stream"   and A :: "'b set" assumes "R w"   and "\<And>w. R w \<Longrightarrow> \<exists>u v. w = u @- v \<and> set u \<inter> A \<noteq> {} \<and> R v" shows "infs A w"
finite_imp_card_positive: assumes "finite (carrier G)" shows "0 < card {\<one>}"
split_face_edges_f12_f21_sym: fixes f :: "face"   and g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and a :: "nat"   and b :: "nat" assumes "f \<in> \<F> g"   and "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs" shows "((a, b) \<in> \<E> f12 \<or> (a, b) \<in> \<E> f21) = ((a, b) \<in> \<E> f \<or> ((b, a) \<in> \<E> f12 \<or> (b, a) \<in> \<E> f21) \<and> ((a, b) \<in> \<E> f12 \<or> (a, b) \<in> \<E> f21))"
l13: shows "e\<^sup>2 + d\<^sup>2 \<in> radical_sqrt"
uminus_uminus_mat: fixes A :: "'b mat" shows "- (- A) = A"
basis_sub: shows "basis \<subseteq> sub.tangent_space p"
fps_of_poly_1': shows "fps_of_poly [:1::'b:] = 1"
simps_2: fixes f :: "'fsym"   and Tl :: "'fsym trm list" shows "Ik.wt (Fn f Tl) = (wtFsym f \<and> list_all Ik.wt Tl \<and> arOf f = map Ik.tpOf Tl)"
int_g: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" shows "GE.int \<xi> (gT T) = Ik.int \<xi> T"
bij_swap_comp: fixes p :: "'c \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "bij p" shows "a \<rightleftharpoons>\<^sub>F b \<circ> p = Fun.swap (inv p a) (inv p b) p"
inject_2: fixes x3 :: "qtname"   and y3 :: "qtname" shows "(ClassT x3 = ClassT y3) = (x3 = y3)"
d: shows "matrix_sum d (\<lambda>l. tensor_P (proj_psi_l (l + 1)) (proj_k (l + 1))) R \<in> carrier_mat d d"
split_half_ts_2: shows "drop (length ts div 2) ts = (sub, sep) # rs"
sel_9: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "getId (Recv x31 x32 x33 x34 x35 x36) = x31"
assms_2: shows "rk {A, C, \<beta>} = 2"
multiplicity_squarefree_part: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "multiplicity p (\<Prod>(squarefree_part n)) = (if p \<in> squarefree_part n then 1 else 0)"
read_only_reads_unowned_nth_update: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and sb' :: "'a memref list"   and \<O>' :: "nat set"   and p' :: "'a"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<O>') (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<subseteq> read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"   and "\<O>' \<union> all_acquired sb' \<subseteq> \<O> \<union> all_acquired sb" shows "read_only_reads_unowned (ts\<^sub>s\<^sub>b[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')])"
A1'_def: shows "A1' \<equiv> \<lambda>view (m0, m1). coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. S1_OT12 (S2, S3) () \<bind> (\<lambda>a. S1_OT12 (S4, S5) () \<bind> (\<lambda>b. Let (M, (S2, S3, S4, S5, m0, m1), a, b, view) D))))))"
idem_F: fixes P :: "'a process" shows "P \<sqsubseteq>\<^sub>F P"
event_stays_valid_if_no_occurrence: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> occurs_on ev'"   and "can_occur ev' c" shows "can_occur ev' c'"
extract_splitres_r_def: fixes r :: "('a, 'b) splitres" shows "extract_splitres_r r \<equiv> case Rep_splitres r of (l, a, r) \<Rightarrow> Abs_FingerTree r"
isShortestPath_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isShortestPath c u p v \<equiv> Graph.isPath c u p v \<and> (\<forall>p'. Graph.isPath c u p' v \<longrightarrow> length p \<le> length p')"
eval_VcompNml: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml t"   and "Nml u"   and "VSeq t u" shows "\<lbrace>t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u\<rbrace> = \<lbrace>t\<rbrace> \<cdot> \<lbrace>u\<rbrace>"
integrable_restrict_space: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integrable (restrict_space M \<Omega>) f = integrable M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x)"
inorder_delete: fixes t :: "'t"   and x :: "'a" assumes "inv t \<and> sorted (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
take_not_empty_conv: fixes n :: "nat"   and xs :: "'d list" shows "(xs \<down> n \<noteq> []) = (0 < n \<and> xs \<noteq> [])"
r10: fixes x :: "name"   and y :: "name"   and s :: "trm"   and u :: "trm"   and t :: "trm" assumes "x \<sharp> (y, s, u)"   and "y \<sharp> (s, t)" shows "(s to x in t) to y in u \<mapsto> s to x in t to y in u"
anormal_dnf_nnf: fixes as :: "'a list"   and \<phi> :: "'a fm"   and a :: "'a" assumes "as \<in> set (dnf (nnf \<phi>))"   and "qfree \<phi>"   and "normal \<phi>"   and "a \<in> set as" shows "anormal a"
\<phi>_local_bij: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "bij_betw (\<phi> (b, a)) (C.hom b a) (Hom.set (b, a))"
WT_resource_\<I>_uniform_UNIV: fixes A :: "'h set"   and res :: "('h, 'i) resource" shows "\<I>_uniform A UNIV \<turnstile>res res \<surd>"
Le_enat_imp_ile: fixes m :: "enat"   and n :: "enat" assumes "(m, n) \<in> Le_enat" shows "m \<le> n"
induct: fixes P :: "ternaryformula \<Rightarrow> bool"   and a0 :: "ternaryformula" assumes "\<And>v. P (TernaryValue v)"   and "\<And>t1 t2. \<lbrakk>P t1; P t2\<rbrakk> \<Longrightarrow> P (TernaryAnd t1 t2)"   and "\<And>t1 t2. \<lbrakk>P t1; P t2\<rbrakk> \<Longrightarrow> P (TernaryOr t1 t2)"   and "\<And>t. P t \<Longrightarrow> P (TernaryNot (TernaryNot t))"   and "\<And>v. P (TernaryNot (TernaryValue v))"   and "\<And>t1 t2. \<lbrakk>P (TernaryNot t1); P (TernaryNot t2)\<rbrakk> \<Longrightarrow> P (TernaryNot (TernaryAnd t1 t2))"   and "\<And>t1 t2. \<lbrakk>P (TernaryNot t1); P (TernaryNot t2)\<rbrakk> \<Longrightarrow> P (TernaryNot (TernaryOr t1 t2))" shows "P a0"
pref_profile_eqI: fixes agents :: "'c set"   and alts :: "'d set"   and R1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "pref_profile_wf agents alts R1"   and "pref_profile_wf agents alts R2"   and "\<And>x. x \<in> agents \<Longrightarrow> R1 x = R2 x" shows "R1 = R2"
nat_of_times_sat: fixes x :: "'b sat"   and y :: "'b sat" shows "nat_of (x * y) = min (nat_of x * nat_of y) LENGTH('b)"
integral_FTC_Icc: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {a..b})"   and "continuous_on {a..b} f" shows "LBINT x. indicat_real {a..b} x *\<^sub>R f x = F b - F a"
T3: shows "y = univ_for_pr (c_pair (c_snd (c_snd n)) x)"
mu_G_spec: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. borel))" shows "\<mu>G (emb I J X) = emeasure (P J) X"
l1_inv5_trans: shows "{l1_inv5} TS.trans l1 {> l1_inv5}"
mertens_third_theorem_strong: shows "\<forall>\<^sub>F x in at_top. \<bar>(\<Prod>p\<in>{p. prime p \<and> real p \<le> x}. 1 - 1 / real p) - third_mertens_const / ln x\<bar> \<le> 10 * third_mertens_const / (ln x)\<^sup>2"
simps_2: fixes f :: "'c"   and ss :: "('c, 'e) Term.term list"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "Fun f ss \<cdot> \<sigma> = Fun f (ss \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<sigma>)"
split_sign_changes_correct_nbh: fixes x\<^sub>0 :: "real" assumes "poly (hd ps) x\<^sub>0 \<noteq> 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. (\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x) = sign_changes ps x"
assms: fixes k :: "'s \<rightarrow> ('a \<times> 's)\<cdot>'f" shows "P (StateT\<cdot>k)"
leg1_composite: shows "\<mu>\<nu>.leg1 = \<mu>.leg1 \<cdot> prj\<^sub>1"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "'a poly" shows "smult a (smult b x) = smult (a * b) x"
pos_integrable_to_top: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> real"   and l :: "real" assumes "\<And>i. A i \<in> sets M"   and "incseq A"   and "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x"   and "\<And>i. set_integrable M (A i) f"   and "(\<lambda>i. set_lebesgue_integral M (A i) f) \<longlonglongrightarrow> l" shows "set_integrable M (\<Union> (range A)) f"
t1: shows "chosen s' \<noteq> NotAnInput \<longrightarrow> valueChosen s' (chosen s')"
descendants_bounds_snd: fixes k :: "'b"   and i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and v :: "nat"   and v' :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b" assumes "descendants k i branch xs"   and "(v, v') \<in> xs"   and "branch !. v = Some (ps, a)" shows "v' < length ps"
mat_of_cblinfun_inverse: fixes B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_of_mat (mat_of_cblinfun B) = B"
rel_prod_conv: fixes R1 :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
norm: shows "fls_base_factor_to_fps (f * g) = fls_base_factor_to_fps f * fls_base_factor_to_fps g"
elims: fixes x :: "'b extB"   and xa :: "'b extB"   and y :: "'b extB" assumes "x + xa = y" obtains "xa__ = extB.Bot" and "xa = extB.Bot" and "ya__ = extB.Val top"   | x :: "'b" where "xa__ = extB.Bot" and "xa = extB.Val xb__" and "ya__ = extB.Val xb__"   | x :: "'b" where "xa__ = extB.Val xc__" and "xa = extB.Bot" and "ya__ = extB.Val xc__"   | x :: "'b" and y :: "'b" where "xa__ = extB.Val x" and "xa = extB.Val y" and "ya__ = extB.Val (x + y)"
keys_default: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "keys (Mapping.default k v m) = insert k (keys m)"
rel_transp: fixes R1 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "transp R1"   and "transp R2" shows "transp (rel_sum R1 R2)"
dim_row_transfer_rule: fixes A :: "'e mat"   and A' :: "(('e, 'f) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_row A = CARD('g)"
sel_2: fixes x1 :: "'g"   and x2 :: "'f" shows "foo.field_2 (make_foo x1 x2) = x2"
entropy_density_def: fixes b :: "real"   and M :: "'b measure"   and N :: "'b measure" shows "entropy_density b M N = log b \<circ> enn2real \<circ> RN_deriv M N"
ts_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : ts'' _> ts'''"
wt_st: shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (App s t))) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t))"
hom: shows "canonical_proj_ext (\<lambda>i. I (Suc i)) n \<in> ring_hom R (RDirProd_list (map (\<lambda>i. R Quot I (Suc i)) [0..<Suc n]))"
cos_of_real_pi: shows "cos (of_real pi) = - (1::'b)"
Red_I_\<G>_L_q_def: fixes q :: "'q"   and N :: "('f \<times> 'l) set" shows "Red_I_\<G>_L_q q N = {\<iota> \<in> Inf_FL. \<G>_I_L_q q \<iota> \<noteq> None \<and> the (\<G>_I_L_q q \<iota>) \<subseteq> Red_I_q q (\<Union> (\<G>_F_L_q q ` N)) \<or> \<G>_I_L_q q \<iota> = None \<and> \<G>_F_L_q q (concl_of \<iota>) \<subseteq> \<Union> (\<G>_F_L_q q ` N) \<union> Red_F_q q (\<Union> (\<G>_F_L_q q ` N))}"
valueChosen_def: fixes s :: "state"   and v :: "InputsOrNi" shows "valueChosen s v = (\<exists>b\<in>\<Union> (range Ballot). maxBalInp s b v \<and> (\<exists>p. \<exists>D\<in>MajoritySet. \<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))))"
simps: fixes a :: "'a llist" shows "lfinite a = (a = LNil \<or> (\<exists>xs x. a = LCons x xs \<and> lfinite xs))"
reflp_mono: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp R"   and "\<And>x y. R x y \<longrightarrow> Q x y" shows "reflp Q"
f8: shows "\<forall>x0 x1. (\<exists>v2. (pre v2 \<and> (v2, x1) \<in> V) \<and> \<not> x0 v2 \<le> (if nofail (x0 v2) then M v2 else FAIL)) = ((pre (xx x0 x1) \<and> (xx x0 x1, x1) \<in> V) \<and> \<not> x0 (xx x0 x1) \<le> (if nofail (x0 (xx x0 x1)) then M (xx x0 x1) else FAIL))"
poly_order_exists: fixes p :: "'a list"   and d :: "nat"   and a :: "'a" assumes "length p = d"   and "poly p \<noteq> poly []" shows "\<exists>n. [- a, 1::'a] %^ n divides p \<and> \<not> [- a, 1::'a] %^ Suc n divides p"
ipresCons_imp_ipresSubstAbs: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "ipresCons h hA MOD"   and "igSubstCls MOD"   and "igConsIPresIGWls MOD"   and "igFreshCls MOD" shows "ipresSubstAbs h hA MOD"
poly_to_row_def: fixes ts :: "'b list"   and p :: "'b \<Rightarrow>\<^sub>0 'a" shows "poly_to_row ts p = vec_of_list (map (lookup p) ts)"
ennreal_tendsto_top_eq_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" shows "((\<lambda>z. ennreal (f z)) \<longlongrightarrow> \<top>) F = filterlim f at_top F"
NE_intT: fixes \<sigma> :: "'tp" shows "\<exists>a. eintT \<sigma> a"
assms: shows "s' = snd (fst (read_state_reg_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
rev_nth: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
rat_number_expand_2: shows "1 = Fract 1 1"
terminating_path_root_end_forward_terminating: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e" shows "backward_terminating_path_root e (x\<^sup>T)"
LNil_le: fixes s :: "'b llist" shows "LNil \<le> s"
ordsetmap_smap: fixes X :: "'w set"   and Y :: "'w set" assumes "X \<in> \<P>"   and "Y \<in> \<P>"   and "order.greater_eq X Y" shows "order.greater_eq (dual_order.poset_simplex_map \<P> Y) (dual_order.poset_simplex_map \<P> X)"
non_winning_moves_remains_non_winning_Even: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a"   and m :: "'a" assumes "even (length p)"   and "\<forall>\<sigma>. \<not> strategy_winning_by_Even \<sigma> p" shows "\<not> strategy_winning_by_Even \<sigma> (p @ [m])"
face_cycle_set_self: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "digraph_map G M" shows "a \<in> pre_digraph_map.face_cycle_set M a"
hyps: assumes "distincts xss"   and "xs \<in> set xss" shows "lists_succ (xs # remove1 xs xss) = lists_succ xss"
is_nextElem_circ: fixes xs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "distinct xs"   and "is_nextElem xs a b"   and "is_nextElem xs b a" shows "|xs| \<le> 2"
PSP_Stable: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "F \<in> Stable B" shows "F \<in> A \<inter> B \<longmapsto>w A' \<inter> B"
ocomplete_no_cast: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and R :: "nat set"   and m :: "'f"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and T :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" shows "((\<sigma>, s), R:*cast(m), \<sigma>', s') \<notin> ocnet_sos T"
ccpo_Sup_least: fixes A :: "'b set"   and z :: "'b" assumes "Complete_Partial_Order.chain ordb A"   and "\<And>x. x \<in> A \<Longrightarrow> ordb x z" shows "ordb (lubb A) z"
pmdl_comp_red_basis_subset: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pmdl (set (comp_red_basis xs)) \<subseteq> pmdl (set xs)"
distinct_15: shows "IndexOfOp \<noteq> EqualsIgnoreCaseOp"
eventually_nonzero_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" shows "eventually_nonzero F f = (\<forall>\<^sub>F x in F. f x \<noteq> (0::'d))"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "MIR.num" shows "maxcoeff (CN n c t) = max \<bar>c\<bar> (maxcoeff t)"
bexI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
flowsto_congI1: fixes A :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and B :: "real set"   and C :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and D :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set" assumes "flowsto A B C D" shows "c1_on_open_euclidean.flowsto g f' Y A B C D"
map_comp: fixes g1 :: "'d \<Rightarrow> 'a"   and g2 :: "'e \<Rightarrow> 'b"   and g3 :: "'f \<Rightarrow> 'c"   and f1 :: "'g \<Rightarrow> 'd"   and f2 :: "'h \<Rightarrow> 'e"   and f3 :: "'i \<Rightarrow> 'f"   and v :: "('g, 'h, 'i) generat" shows "map_generat g1 g2 g3 (map_generat f1 f2 f3 v) = map_generat (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
lt_lower_less: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "lower p v \<noteq> 0" shows "lt (lower p v) \<prec>\<^sub>t v"
maxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "folding.gallery (x # xs)" shows "folding.gallery xs"
wpt_dual_assertion_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.wpt one times inf top (times (dual x) y) = sup (mbt_algebra.neg_assert one times inf dual bot x) y"
const_in: fixes x :: "'b" assumes "x \<in> Field s" shows "const x = r.zero"
comm_neutral: fixes a :: "'a" shows "a \<squnion> \<bottom> = a"
is_path_memb: fixes v :: "'v"   and p :: "('v \<times> 'v) list"   and v' :: "'v" assumes "is_path v p v'" shows "v \<in> V \<and> v' \<in> V"
modular_2': fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y ; x \<cdot> z = (y \<cdot> z ; x\<^sup>\<smile>) ; x \<cdot> z"
l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs get_ancestors_di get_ancestors_di_locs Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs Shadow_DOM.remove"
ord_spmf_conv: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "ord_spmf R = rel_spmf R OO ord_spmf (=)"
linear_surjective_isomorphism: fixes f :: "'a \<Rightarrow> 'b" assumes "clinear f"   and "surj f"   and "cdim UNIV = cdim UNIV" shows "\<exists>f'. clinear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
D: shows "map_pmf con s = Sum_pmf (8 / 10) Da Db"
insertAt_def: fixes n :: "nat"   and x :: "'a"   and ls :: "'a list" shows "insertAt n x ls = take n ls @ x # drop n ls"
image_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and r :: "'d set" shows "f ` g ` r = (f \<circ> g) ` r"
zero_rel_def: shows "cq.zero_rel = cq.frac \<zero> \<one>"
hoare_patial_seq_assoc_sem: fixes A :: "complex mat"   and S1 :: "com"   and S2 :: "com"   and S3 :: "com"   and B :: "complex mat" shows "(\<Turnstile>\<^sub>p {A} S1;; S2;; S3 {B}) = (\<Turnstile>\<^sub>p {A} S1;; (S2;; S3) {B})"
tell_env_def: fixes w :: "'w"   and m :: "('c, 'm) envT" shows "local.tell_env w m = EnvT (\<lambda>r. tell w (run_env m r))"
list_update_id: fixes xs :: "'k list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
minimal_antichain_path_weightp_distinct: fixes l1 :: "'vtx"   and l2 :: "'vtx" shows "minimal_antichain {xs. path_weightp l1 l2 xs} = minimal_antichain {xs. path_weightp_distinct l1 l2 xs}"
partition_onD1: fixes A :: "'b set"   and P :: "'b set set" assumes "partition_on A P" shows "A = \<Union> P"
real_of_card: fixes A :: "'b set" shows "real (card A) = (\<Sum>x\<in>A. 1)"
fls_lr_inverse_power_ring1_2: fixes x :: "'a"   and f :: "'a fls"   and n :: "nat" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * x = (1::'a)" shows "fls_right_inverse (f ^ n) (x ^ n) = fls_right_inverse f x ^ n"