hcju/isabelleps huggingface dataset:
possible_steps_not_empty_iff: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and a :: "String.literal"   and b :: "value list" assumes "step e s r a b \<noteq> None" shows "\<exists>aa ba. (aa, ba) |\<in>| possible_steps e s r a b"
as0_16: assumes "S_M S M (D + negs (mset As)) \<noteq> {#}" shows "negs (mset As0) = S DA0"
assms: shows "i < card {a. a < m \<and> a \<in> I}"
ring_r_one: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d" assumes "Ring R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> 1\<^sub>r\<^bsub>R\<^esub> = x"
N_pos: fixes n :: "int" assumes "n \<noteq> 0" shows "0 < N n"
class_add_sees_methods_Obj: fixes P :: "jvm_prog"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> jvm_method) \<times> char list) option"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> Object sees_methods Mm"   and "C \<noteq> Object" shows "class_add P (C, cdec) \<turnstile> Object sees_methods Mm"
intros_3: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "primerec (recf.id m n) m"
suminf_bound: fixes f :: "nat \<Rightarrow> ereal"   and x :: "ereal" assumes "\<forall>N. sum f {..<N} \<le> x"   and "\<And>n. 0 \<le> f n" shows "suminf f \<le> x"
drop_bit_half: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One))) = divide (drop_bit n a) (numeral.numeral one plus (num.Bit0 num.One))"
sum_moebius_mu_aux1: shows "(\<lambda>x. sum_upto moebius_mu x / x - sum_upto (\<lambda>n. moebius_mu n * ln (real n)) x / (x * ln x)) \<in> O(\<lambda>x. 1 / ln x)"
uminus_in_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "((\<lambda>x. - f x) \<in> o[F](g)) = (f \<in> o[F](g))"
fmapU_eq_bindU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'d \<rightarrow> udom\<cdot>'d"   and returnU :: "udom \<rightarrow> udom\<cdot>'d"   and bindU :: "udom\<cdot>'d \<rightarrow> (udom \<rightarrow> udom\<cdot>'d) \<rightarrow> udom\<cdot>'d"   and f :: "udom \<rightarrow> udom"   and xs :: "udom\<cdot>'d" assumes "class.monad fmapU returnU bindU" shows "fmapU\<cdot>f\<cdot>xs = bindU\<cdot>xs\<cdot>(\<Lambda> x. returnU\<cdot>(f\<cdot>x))"
llist_all2_expand: fixes xs :: "'c llist"   and ys :: "'d llist"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "lnull xs = lnull ys"   and "\<lbrakk>\<not> lnull xs; \<not> lnull ys\<rbrakk> \<Longrightarrow> P (lhd xs) (lhd ys) \<and> llist_all2 P (ltl xs) (ltl ys)" shows "llist_all2 P xs ys"
err_step_defs_3: fixes n :: "nat" shows "error n = map (\<lambda>x. (x, Err)) [0..<n]"
assms_3: fixes x :: "'a" assumes "x \<in> S" shows "onorm (f' x) \<le> B"
beta_def: shows "\<beta> \<in> carrier Zp \<and> f \<bullet> \<beta> = \<zero>"
ereal_0_gt_inverse: fixes x :: "ereal" shows "(0 < inverse x) = (x \<noteq> \<infinity> \<and> 0 \<le> x)"
I_in_cb_def: fixes s :: "'b list"   and j :: "nat" shows "I_in_cb s j \<equiv> \<lambda>i. if j = 1 then i = 0 else strict_border (take (i - 1) s) (take (j - 1) s) \<and> \<ff> s j \<le> i + 1"
PRelQ: shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
restrict_continuous_map: fixes X :: "'c topology"   and S :: "'c set"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "topspace X \<subseteq> S" shows "continuous_map X X' (restrict f S) = continuous_map X X' f"
trss_stack_comp: 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 r :: "'a list" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "((s @ r, c), w, s' @ r, c') \<in> trcl (trss fg)"
nontrans2: fixes u :: "'a"   and r :: "'a" assumes "u \<parallel> r" shows "\<nexists>t. r \<parallel> t \<and> t \<parallel> u"
uminus_ms_aux_MSLNil_iff: fixes xs :: "('a \<times> real) msllist" shows "(uminus_ms_aux xs = MSLNil) = (xs = MSLNil)"
cr': shows "P,\<Phi> |- (None, h, f' # frs', sh) [ok]"
bounded_countable_web_axioms: shows "bounded_countable_web \<Gamma>"
take_bit_eq_0_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "(take_bit n a = zero) = dvd.dvd times (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) a"
measure_differentiable_image: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S" shows "Sigma_Algebra.measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
assms_3: shows "coincides_on Mkt Mkt2 (stocks Mkt)"
cos_angle_eqD: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'c"   and e :: "'c"   and f :: "'c" assumes "cos (angle a b c) = cos (angle d e f)" shows "angle a b c = angle d e f"
nths_nths: fixes xs :: "'g list"   and A :: "nat set"   and B :: "nat set" shows "nths (nths xs A) B = nths xs {i \<in> A. \<exists>j\<in>B. card {i' \<in> A. i' < i} = j}"
subseqs_refl: fixes xs :: "'a list" shows "xs \<in> set (subseqs xs)"
finite_group_axioms_def: fixes G :: "('c, 'd) monoid_scheme" shows "finite_group_axioms G \<equiv> finite (carrier G)"
ref_validity_def: fixes A :: "'b attree" shows "\<turnstile>\<^sub>V A \<equiv> \<exists>A'. A \<sqsubseteq>\<^sub>V A'"
pairs_uminus_closed: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter" assumes "triple phi" shows "triple.pairs_uminus phi (x, y) \<in> triple.pairs phi"
assms_3: shows "\<forall>x. S1 x \<le> S2 x"
simps_2: fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and s\<^sub>1 :: "'e net_state"   and s\<^sub>2 :: "'e net_state" shows "netgmap sr (SubnetS s\<^sub>1 s\<^sub>2) = (let (\<sigma>\<^sub>1, ss) = netgmap sr s\<^sub>1; (\<sigma>\<^sub>2, tt) = netgmap sr s\<^sub>2 in (\<sigma>\<^sub>1 ++ \<sigma>\<^sub>2, SubnetS ss tt))"
less_imp_production_subseteq_interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C < D" shows "production C \<subseteq> interp D"
add_clock_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3 :: "'b state \<times> (v, v) result" assumes "evaluate ck env s e r3" shows "\<forall>s' r'. r3 = (s', r') \<and> ck = False \<longrightarrow> (\<exists>c. evaluate True env (update_clock (\<lambda>_. c) s) e (update_clock (\<lambda>_. 0) s', r'))"
SPLIT_def: fixes v :: "'c"   and w :: "'c" shows "SPLIT v w \<equiv> v = w"
discr0_transT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.discr0 tval aval indis c"   and "PL.mustT tval aval c s"   and "PL.transT tval aval (c, s) s'" shows "(s, s') \<in> indis"
IsoInvId1: fixes f :: "'b" assumes "ciso f" shows "Cinv f ;; f = id (cod f)"
asymmetric: shows "\<forall>x\<in>T. \<forall>y\<in>T. x \<prec> y \<longrightarrow> \<not> y \<prec> x"
CQ1Concl_def: shows "CQ1Concl = (Geq (Differential (Var vid1)) (Differential (Const 0)) \<leftrightarrow> Geq ($' vid1) (Differential (Const 0)))"
is_none_code_2: fixes x :: "'a" shows "Option.is_none (Some x) = False"
Method_i_i_i_o_o_o_o_def: shows "Method_i_i_i_o_o_o_o \<equiv> \<lambda>x xa xb. pred.Pred (\<lambda>(xc, xd, xe, xf). x \<turnstile> xa sees xb: xc\<rightarrow>xd = xe in xf)"
insertE: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
maintained_preserved_by_isomorphism: fixes G :: "('c, 'b) labeled_graph"   and f :: "'d \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'd"   and r :: "('c, 'e) labeled_graph \<times> ('c, 'e) labeled_graph" assumes "\<And>x. x \<in> vertices G \<Longrightarrow> (f \<circ> g) x = x"   and "graph G"   and "maintained r (map_graph_fn G g)" shows "maintained r G"
simps: fixes a :: "'a list" shows "assoc_list_of_array (Array a) = zip [0..<length a] a"
CA_Says_6_lemma: fixes i :: "nat"   and C :: "agent"   and NC3 :: "nat"   and NonceCCA :: "nat"   and evs :: "event list" assumes "Crypt (priSK (CA i)) (Hash \<lbrace>Agent C, Nonce NC3, Agent (CA i), Nonce NonceCCA\<rbrace>) \<in> parts (knows Spy evs)"   and "evs \<in> set_cr"   and "CA i \<notin> bad" shows "\<exists>Y K. Says (CA i) C (Crypt K \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC3, Agent (CA i), Nonce NonceCCA\<rbrace>, Y\<rbrace>) \<in> set evs"
smallest_ccdi_sets: shows "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
A: shows "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
nxtActI: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<and> \<parallel>c\<parallel>\<^bsub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> \<and> \<not> (\<exists>k\<ge>n. sb.latestAct_cond c t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> k)"
is_ground_cls_imp_is_ground_lit: fixes L :: "'a literal"   and C :: "'a literal multiset" assumes "L \<in># C"   and "is_ground_cls C" shows "is_ground_lit L"
ideal_prod_assocTr: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and n :: "nat" assumes "ideal R A"   and "ideal R B"   and "ideal R C" shows "\<forall>f. f \<in> {j. j \<le> n} \<rightarrow> set_mult R (A \<diamondsuit>\<^sub>r B) C \<longrightarrow> \<Sigma>\<^sub>e R f n \<in> A \<diamondsuit>\<^sub>r (B \<diamondsuit>\<^sub>r C)"
surjective: fixes r :: "('a, 'b) processCalculus_scheme" shows "r = \<lparr>Reductions = Reductions r, \<dots> = processCalculus.more r\<rparr>"
eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(x = y) = (x \<le>f y \<and> y \<le>f x)"
pr_gr_value: fixes n :: "nat"   and x :: "nat" shows "c_assoc_value (pr_gr (loc_upb n x)) (c_pair n x) = univ_for_pr (c_pair n x)"
NTD: shows "NTCatDom \<eta> = CatDom F"
simps: fixes a :: "'b"   and r :: "('c \<times> 'b) set" shows "(a \<in> Range r) = (\<exists>a b. a = b \<and> (a, b) \<in> r)"
ext_inject: fixes states :: "hf"   and init :: "hf"   and final :: "hf"   and "next" :: "hf \<Rightarrow> 'b \<Rightarrow> hf \<Rightarrow> bool"   and more :: "'c"   and states' :: "hf"   and init' :: "hf"   and final' :: "hf"   and next' :: "hf \<Rightarrow> 'b \<Rightarrow> hf \<Rightarrow> bool"   and more' :: "'c" shows "(\<lparr>states = states, init = init, final = final, next = next, \<dots> = more\<rparr> = \<lparr>states = states', init = init', final = final', next = next', \<dots> = more'\<rparr>) = (states = states' \<and> init = init' \<and> final = final' \<and> next = next' \<and> more = more')"
R_ball: shows "proots (p - [:b:]) \<subseteq> ball z0 R"
group_def: fixes G :: "('c, 'd) monoid_scheme" shows "Group.group G \<equiv> Group.monoid G \<and> Group.group_axioms G"
eq: fixes b :: "'b" assumes "b \<in> B" shows "f b = g b"
apply_outputs_def: fixes p :: "'a aexp list"   and s :: "'a \<Rightarrow> value option" shows "apply_outputs p s = map (\<lambda>p. aval p s) p"
basis_indep: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep_system.basis carrier indep X" shows "indep X"
transitive: shows "\<forall>x y z. le x y \<and> le y z \<longrightarrow> le x z"
rep_eq: fixes x :: "'a \<Rightarrow> bool"   and xa :: "('a, 'b) fmap" shows "fmlookup (fmfilter x xa) = map_filter x (fmlookup xa)"
Apps_inj: fixes f :: "expr"   and ts :: "expr list"   and g :: "expr" shows "(f \<bullet> ts = g \<bullet> ts) = (f = g)"
hyp: shows "matrix_to_iarray (Gram_Schmidt_upt_k A k) = Gram_Schmidt_upt_k_iarrays (matrix_to_iarray A) k"
scale_right_distrib: fixes x :: "'b"   and y :: "'b"   and a :: "'a" assumes "x \<in> U\<^sub>M"   and "y \<in> U\<^sub>M" shows "s a (x + y) = s a x + s a y"
mono: fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "is_filter F"   and "\<forall>x. P x \<longrightarrow> Q x"   and "F P" shows "F Q"
rtrancl_on_induct: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> rtrancl_on F r"   and "a \<in> F \<Longrightarrow> P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> rtrancl_on F r; (y, z) \<in> r; y \<in> F; z \<in> F; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
emb_step_at: shows "emb_step_at p dir.Right t = s"
iso_inv_iso: fixes f :: "'a" assumes "local.iso f" shows "local.iso (local.inv f)"
hoareD: fixes I :: "'a set"   and R :: "('a \<times> 'a) set"   and J :: "'a set"   and s :: "'a"   and s' :: "'a" assumes "{I} R {> J}"   and "s \<in> I"   and "(s, s') \<in> R" shows "s' \<in> J"
\<tau>red: shows "s2 -\<tau>2\<rightarrow>* s2'"
seq: shows "\<forall>i<n. (f i, f (Suc i)) \<in> r"
measure_ge_1_iff: fixes M :: "'a pmf"   and A :: "'a set" shows "(1 \<le> measure_pmf.prob M A) = (measure_pmf.prob M A = 1)"
assms_3: fixes loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "P c0"   and "next_change_multiplicity' c0 c1 loc t n" shows "P c1"
assms: shows "(xs, X) \<in> \<langle>A\<rangle>list_wset_rel"
HInfinite_diff_HFinite_Infinitesimal_disj: fixes x :: "'a star" assumes "x \<notin> Infinitesimal" shows "x \<in> HInfinite \<or> x \<in> HFinite - Infinitesimal"
assms_3: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
IrrFinGroupRep_trivialGSubspace: shows "IrrFinGroupRepresentation G (\<cdot>) 0"
assms_2: shows "put \<sigma> u = put \<rho> v"
his_determ_ex: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n' :: "nat"   and nid' :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "(n', nid') \<in> Blockchain.his active cmp bc mining t n nid" shows "\<exists>!nid'. (n', nid') \<in> Blockchain.his active cmp bc mining t n nid"
gf: fixes x :: "'a" assumes "x \<in> affine hull S" shows "f x *\<^sub>R z + \<xi> = x"
init_q: fixes v :: "real"   and s0 :: "real" shows "movement.q v s0 0 = s0"
eqvtI: fixes X :: "'d set"   and x :: "'d"   and p :: "(name \<times> name) list" assumes "eqvt X"   and "x \<in> X" shows "p \<bullet> x \<in> X"
n_def: shows "n = (LEAST n. 0 < n \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
characteristic_weak_formula_def: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "indexed_weak_nominal_ts.characteristic_weak_formula \<tau> satisfies transition P \<equiv> Conj (Abs_bset {indexed_weak_nominal_ts.distinguishing_weak_formula \<tau> satisfies transition P Q |Q. \<not> indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q})"
p: shows "g \<turnstile> n-ns @ [m']\<rightarrow>m'"
herbrand: fixes Cs :: "fterm literal set set" assumes "\<forall>G. \<not> eval\<^sub>c\<^sub>s HFun G Cs"   and "finite Cs"   and "\<forall>C\<in>Cs. finite C" shows "\<exists>T. closed_tree T Cs"
assms_6: fixes i :: "nat" assumes "0 < i"   and "i < n" shows "(deriv ^^ i) f \<xi> = 0"
interp_ins_def: fixes ops :: "('a \<times> 'a option) list" shows "interp_ins ops \<equiv> foldl insert_spec [] ops"
pullback_algebra_def: fixes M :: "'b measure"   and f :: "'a \<Rightarrow> 'b"   and \<Omega>' :: "'a set" shows "sets.pullback_algebra M f \<Omega>' = sigma_sets \<Omega>' {f -` A \<inter> \<Omega>' |A. A \<in> sets M}"
irbd_surj: shows "top_pres (\<P> bd\<^sup>-\<^sub>\<R>)"
upto_Nil: fixes i :: "int"   and j :: "int" shows "([i..j] = []) = (j < i)"
cases: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" assumes "ord_resolve a1 a2 a3 a4 a5 a6" obtains CAs :: "'a literal multiset list" and n :: "nat" and Cs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and \<sigma> :: "'s" and D :: "'a literal multiset" where "a1 = CAs" and "a2 = D + negs (mset As)" and "a3 = AAs" and "a4 = As" and "a5 = \<sigma>" and "a6 = (\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma>" and "length CAs = n" and "length Cs = n" and "length AAs = n" and "length As = n" and "n \<noteq> 0" and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)" and "\<forall>i<n. AAs ! i \<noteq> {#}" and "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))" and "eligible \<sigma> As (D + negs (mset As))" and "\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)" and "\<forall>i<n. S (CAs ! i) = {#}"
split_sym_rel: fixes G :: "('b \<times> 'b) set" assumes "sym G"   and "irrefl G" obtains E :: "('b \<times> 'b) set" where "E \<inter> E\<inverse> = {}" and "G = E \<union> E\<inverse>"
simps_2: fixes s :: "'b"   and h :: "'b \<rightarrow> ('c, 'b) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>h\<cdot>s = (case h\<cdot>s of Done \<Rightarrow> LNil | Skip\<cdot>s' \<Rightarrow> unfold\<cdot>h\<cdot>s' | Yield\<cdot>x\<cdot>s' \<Rightarrow> LCons\<cdot>x\<cdot>(unfold\<cdot>h\<cdot>s'))"
pathstart_in_path_image: fixes g :: "real \<Rightarrow> 'b" shows "pathstart g \<in> path_image g"
g3up_def: shows "g3up \<equiv> {(xa, x). g3upp xa x}"
simps: fixes v :: "'c"   and uu :: "'c Tree"   and uv :: "'c Tree" shows "val (T v uu uv) = v"
ltake_is_lprefix: fixes n :: "enat"   and xs :: "'c llist" shows "ltake n xs \<sqsubseteq> xs"
mangoldt_non_primepow: fixes n :: "nat" assumes "\<not> primepow n" shows "mangoldt n = (0::'a)"
is_empty_def: fixes m :: "('c, 'd) mapping" shows "is_empty m = (keys m = {})"
finite_ran: fixes p :: "'c \<Rightarrow> 'd option" assumes "finite (dom p)" shows "finite (ran p)"
iTILL_iMODb_subset_conv: fixes n' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "0 < n'" shows "([\<dots>n'] \<subseteq> [ r, mod m, c ]) = (r = 0 \<and> m = Suc 0 \<and> n' \<le> r + m * c)"
I'_def: shows "I' \<equiv> \<lambda>(ps, n). if pbval b ps then wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n) else Q (ps, n)"
cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "\<Omega>[F](f) = \<Omega>[F](g)"
SFAssThrow\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>2 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
complex_poly_def: fixes re :: "int poly"   and im :: "int poly" shows "complex_poly re im = (let i = [:1, 0, 1:] in factors_of_int_poly (poly_add re (poly_mult im i)))"
l_def: shows "l = sum_list (concat (map (\<lambda>i. if i < 0 then [i] else []) is))"
simps_1: fixes uu :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'g \<Rightarrow> 'e"   and x :: "'g" shows "ap_sum uu (pure f) (pure x) = pure (f x)"
closure_p_eq_true: fixes P :: "'b \<Rightarrow> bool" shows "(\<forall>A. P A) = (P = true)"
distinct_7: fixes x2 :: "name"   and x3 :: "(Term_Class.term \<times> sterm) list" shows "Svar x2 \<noteq> Sabs x3"
cols_upt_k_insert: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "Suc k < ncols A" shows "cols_upt_k A (Suc k) = insert (column (mod_type_class.from_nat (Suc k)) A) (cols_upt_k A k)"
n'_eq: fixes N :: "'a stream measure"   and M :: "'a stream measure" assumes "R N M" shows "N = measure_pmf (p N M) \<bind> n' N M"
stateful_strand_step_subst_inI_2: fixes t :: "('e, 'f) Term.term"   and A :: "('e, 'f) stateful_strand_step list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" assumes "receive\<langle>t\<rangle> \<in> set A" shows "receive\<langle>t \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
input_stored_in_reg_aux_is_generalisation_aux: fixes t' :: "transition"   and t :: "transition"   and mr :: "nat"   and mi :: "nat"   and i :: "nat"   and r :: "nat" assumes "input_stored_in_reg_aux t' t mr mi = Some (i, r)" shows "is_generalisation_of t' t i r"
a: shows "nil \<sqinter> (c::'a::refinement_lattice) ; c\<^sup>\<omega> ; c\<^sup>\<omega> \<sqsubseteq> nil \<sqinter> c ; c\<^sup>\<omega> \<sqinter> c ; c\<^sup>\<omega> ; c\<^sup>\<omega>"
l6_11_existence: fixes R :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "R \<noteq> A"   and "B \<noteq> C" shows "\<exists>X. A Out X R \<and> Cong A X B C"
B_not_empty: fixes V :: "'a set"   and f :: "'a \<Rightarrow> real" shows "0 \<in> B V f"
4: shows "is_non_overlapping (Range (a - (X \<union> {i}) \<times> Range a)) \<and> is_non_overlapping (a `` (X \<union> {i}))"
simps: fixes a1 :: "'b list"   and a2 :: "'c sequent" shows "((a1, a2) \<in> Ax) = ((\<exists>i. a1 = [] \<and> a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)) \<or> a1 = [] \<and> a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>))"
vbrelation_vintersectionI: fixes s :: "V" shows "vbrelation (r \<inter>\<^sub>\<circ> s)"
familyUnionFiniteEverySetFinite: fixes XX :: "'c set set" assumes "finite (\<Union> XX)" shows "\<forall>X\<in>XX. finite X"
eb: fixes x :: "'a" assumes "x \<in> path_image p" shows "ball x e \<subseteq> S"
l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"
dom_hcomp: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.src \<nu> = Span.trg \<mu>" shows "Span.dom (\<nu> \<circ> \<mu>) = Span.dom \<nu> \<circ> Span.dom \<mu>"
con_rews_1: shows "ErrorT\<cdot>\<bottom> = \<bottom>"