hcju/isabelleps huggingface dataset:
ih: fixes ua :: "('s, 'v) tm"   and ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#size ua, size ta, size sa#} < {#size u, size t, size s#}"   and "ua >\<^sub>t ta"   and "ta >\<^sub>t sa" shows "ua >\<^sub>t sa"
loop_stepD: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "B\<^sub>1 \<noteq> {}"   and "u \<in> S"   and "real c < W B\<^sub>1 + w u"   and "W B\<^sub>2 + w u \<le> real c" shows "inv\<^sub>3 (P\<^sub>1 \<union> wrap B\<^sub>1) P\<^sub>2 {} (B\<^sub>2 \<union> {u}) (S - {u}) L"
proj_stoch_measurable_if_adapted: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure"   and n :: "nat" assumes "Filtration.filtration M F"   and "adapt_stoch_proc F f N" shows "proj_stoch_proc f n \<in> M \<rightarrow>\<^sub>M stream_space N"
length_butlast: fixes xs :: "'a list" shows "length (butlast xs) = length xs - 1"
rep_eq: fixes x :: "'c stone_phi_pair"   and xa :: "'c stone_phi_pair" shows "(x \<le> xa) = stone_phi.pairs_less_eq (Rep_stone_phi_pair x) (Rep_stone_phi_pair xa)"
inv_into_def: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'e" shows "inv_into A f = (\<lambda>x. SOME y. y \<in> A \<and> f y = x)"
simps_1: fixes ao :: "'a option"   and ys :: "'a list" shows "set_less_eq_aux_Compl ao [] ys = True"
nxtActive_no_active: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>!i. n \<le> i \<and> \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<not> (\<exists>i'\<ge>Suc \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. \<parallel>c\<parallel>\<^bsub>t i'\<^esub>)"
nact_exists: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<and> \<not> (\<exists>k\<ge>n. sb.latestAct_cond c t i k)"
totalE: fixes f :: "recf"   and n :: "nat"   and xs :: "nat list" assumes "Partial_Recursive.total f"   and "recfn n f"   and "length xs = n" shows "eval f xs \<down>"
induct: fixes P :: "'b prog"   and x1 :: "char list"   and x2 :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and Pa :: "char list \<Rightarrow> ((char list \<times> char list) \<times> staticb \<times> ty) list \<Rightarrow> bool" assumes "P \<turnstile> x1 has_fields x2"   and "\<And>C D fs ms FDTs FDTs'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D has_fields FDTs; Pa D FDTs; FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs\<rbrakk> \<Longrightarrow> Pa C FDTs'"   and "\<And>D fs ms FDTs. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs\<rbrakk> \<Longrightarrow> Pa Object FDTs" shows "Pa x1 x2"
splitAt_rotate_pair_conv: fixes xs :: "'a list"   and x :: "'a"   and n :: "nat" assumes "distinct xs"   and "x \<in> set xs" shows "snd (splitAt x (rotate n xs)) @ fst (splitAt x (rotate n xs)) = snd (splitAt x xs) @ fst (splitAt x xs)"
atMost_UNIV_triv: shows "{..UNIV} = UNIV"
solvable_imp_trivial_derived_seq: fixes H :: "'a set" assumes "solvable_seq G H" shows "\<exists>n. (derived G ^^ n) H = {\<one>}"
list_rel_induct: fixes l :: "'d list"   and l' :: "'e list"   and R :: "('d \<times> 'e) set"   and P :: "'d list \<Rightarrow> 'e list \<Rightarrow> bool" assumes "(l, l') \<in> \<langle>R\<rangle>list_rel"   and "P [] []"   and "\<And>x x' l l'. \<lbrakk>(x, x') \<in> R; (l, l') \<in> \<langle>R\<rangle>list_rel; P l l'\<rbrakk> \<Longrightarrow> P (x # l) (x' # l')" shows "P l l'"
that: fixes a :: "atom" assumes "a \<notin> X"   and "sort_of a = s" shows "thesis"
diff_gt_0_iff_gt: 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 a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (minus a b) = less b a"
widen_Boolean: fixes G :: "prog"   and T :: "ty" assumes "G\<turnstile>PrimT Boolean\<preceq>T" shows "T = PrimT Boolean"
assms: fixes ns :: "('node \<times> 'val) set" assumes "Mapping.lookup nodes_of_phis (snd next) = Some ns" shows "finite ns"
simps_2: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm"   and n :: "nat" shows "extend S f (Suc n) = (let Sn = extend S f n in if consistent ({f n} \<union> Sn) then witness (params ({f n} \<union> Sn)) (f n) \<union> {f n} \<union> Sn else Sn)"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "le (local.Min A) a"
emb: fixes J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>j. M))" shows "emeasure (lim_sequence x) (PF.emb UNIV J X) = emeasure (CI J) X"
sum_with_infinite: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "comm_monoid_add_ow S pls z"   and "infinite A" shows "sum_with pls z g A = z"
cnf_of_encode_initial_state_set_ii: fixes \<Pi> :: "'b strips_problem" assumes "is_valid_problem_strips \<Pi>" shows "cnf (\<Phi>\<^sub>I \<Pi>) = (\<Union>v\<in>set (\<Pi>\<^sub>\<V>). {{literal_formula_to_literal (encode_state_variable 0 (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>I) v))}})"
at_within_union: fixes x :: "'c"   and S :: "'c set"   and T :: "'c set" shows "at x within S \<union> T = sup (at x within S) (at x within T)"
bsingleton_def: fixes x :: "'d" shows "bsingleton x = binsert x bempty"
composite_simps_4: shows "cod (f \<star> g) = cod f \<star> cod g"
mem_ball_leI: fixes x :: "'a"   and y :: "'a"   and e :: "real"   and f :: "real" assumes "x \<in> ball y e"   and "e \<le> f" shows "x \<in> ball y f"
pre_set: fixes c :: "('a, 'b, 'c) ann" shows "pre c = \<Inter> (pre_set c)"
A_splay_max_def: fixes t :: "'b tree" shows "A_splay_max t = real (T_splay_max t) + \<Phi> (splay_max t) - \<Phi> t"
insertCI: fixes a :: "'e"   and B :: "'e set"   and b :: "'e" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
poly_mult_closed: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2" shows "polynomial K (poly_mult p1 p2)"
struct_subst_type_2: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta>1 :: "nat \<Rightarrow> Syntax.type"   and c :: "cmd"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and U :: "Syntax.type"   and T1 :: "Syntax.type"   and \<alpha> :: "nat"   and \<beta> :: "nat" assumes "\<Gamma> , \<Delta>1 \<turnstile>\<^sub>C c"   and "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : U \<Leftarrow> T1"   and "\<Delta>1 = \<Delta>\<langle>\<alpha>:T1\<rangle>" shows "\<Gamma> , \<Delta>\<langle>\<beta>:U\<rangle> \<turnstile>\<^sub>C c[\<alpha>=\<beta> liftM_ctxt E \<beta>]\<^sup>C"
int_distrib_4: fixes w :: "int"   and z1 :: "int"   and z2 :: "int" shows "w * (z1 - z2) = w * z1 - w * z2"
map_ident: fixes t :: "'a fset" shows "(\<lambda>x. x) |`| t = t"
Phi_bij: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "is_bij (\<Phi> w x y z)"
induct: fixes prog :: "cmd"   and x1 :: "w_node"   and x2 :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and x3 :: "w_node"   and P :: "w_node \<Rightarrow> (w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list \<Rightarrow> w_node \<Rightarrow> bool" assumes "While_CFG.path prog x1 x2 x3"   and "\<And>n. While_CFG.valid_node prog n \<Longrightarrow> P n [] n"   and "\<And>n'' as n' a n. \<lbrakk>While_CFG.path prog n'' as n'; P n'' as n'; valid_edge prog a; sourcenode a = n; targetnode a = n''\<rbrakk> \<Longrightarrow> P n (a # as) n'" shows "P x1 x2 x3"
cancel_div_mod_rules_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "plus (plus (times (divide a b) b) (modulo a b)) c = plus a c"
find_max_correct: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite S" shows "find_max P S \<le> SPEC (\<lambda>\<sigma>. case \<sigma> of None \<Rightarrow> \<forall>x\<in>S. \<not> P x | Some y \<Rightarrow> P y \<and> y \<in> S \<and> (\<forall>x\<in>S. P x \<longrightarrow> x \<le> y))"
ineq_2: shows "real_of_1 x \<le> real_of_rat r'"
has_laplace_cmul: fixes f :: "real \<Rightarrow> complex"   and F :: "complex"   and S :: "complex"   and r :: "real" assumes "(f has_laplace F) S" shows "((\<lambda>x. r *\<^sub>R f x) has_laplace r *\<^sub>R F) S"
fls_subdegree_geI: fixes f :: "'a fls"   and n :: "int" assumes "f \<noteq> 0"   and "\<And>k. k < n \<Longrightarrow> fls_nth f k = (0::'a)" shows "n \<le> fls_subdegree f"
is_ext_def: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" shows "\<eta>y.is_ext y' g f \<equiv> \<guillemotleft>f : F y \<rightarrow>\<^sub>C y'\<guillemotright> \<and> g = G f \<cdot>\<^sub>D \<eta> y"
finite_Pow_iff: fixes A :: "'c set" shows "finite (Pow A) = finite A"
apath_append_iff: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" shows "G.apath u (p @ q) v = (G.apath u p (G.awlast u p) \<and> G.apath (G.awlast u p) q v \<and> set (G.awalk_verts u p) \<inter> set (tl (G.awalk_verts (G.awlast u p) q)) = {})"
uminus_add_minus_vec: fixes l :: "'a vec"   and n :: "nat"   and r :: "'a vec" assumes "l \<in> carrier_vec n"   and "r \<in> carrier_vec n" shows "- (l + r) = - l - r"
min_cons_hyp_in_P1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<psi> \<in> \<R>\<^sup>2" shows "min_cons_hyp \<psi> \<in> \<P>"
induced_vector_decomp: fixes vs :: "'v list"   and hs :: "'g list"   and cs :: "'f list"   and v :: "'v" assumes "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs \<and> R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) vs"   and "Supgroup.is_rcoset_replist G ((0::'g) # hs)"   and "length cs = length vs"   and "v = cs \<bullet>\<sharp>\<cdot> vs" shows "induced_vector v = (cs @ replicate (length hs * length vs) (0::'f)) \<bullet>\<currency>\<currency> concat (negHorbit_list ((0::'g) # hs) induced_vector vs)"
qtable_project_fv: fixes n :: "nat"   and \<phi> :: "Formula.formula"   and R :: "'a list set"   and P :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" assumes "qtable (Suc n) (fv \<phi>) (mem_restr (lift_envs R)) P X" shows "qtable n (Formula.fvi (Suc 0) \<phi>) (mem_restr R) (\<lambda>v. \<exists>x. P ((if 0 \<in> fv \<phi> then Some x else None) # v)) (tl ` X)"
outsideOutside: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" shows "(P outside X) outside Y = P outside (X \<union> Y)"
restrict_singleton: fixes k :: "'d"   and f :: "'d \<times> 'd \<Rightarrow> 'c"   and l :: "'d"   and i :: "'d"   and j :: "'d" shows "([k] \<langle>f\<rangle> [l]) (i, j) = (if i = k \<and> j = l then f (i, j) else bot)"
6: shows "(\<lambda>x. F (g2 x, x) \<bullet> j) integrable_on cbox a b"
rat_to_real_def: fixes x :: "rat" shows "to_real x = real_of_rat x"
mono_body_fixp: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lub :: "'c set \<Rightarrow> 'c"   and F :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c" assumes "partial_function_definitions leq lub"   and "\<And>x. monotone (fun_ord leq) leq (\<lambda>f. F f x)" shows "ccpo.fixp (fun_lub lub) (fun_ord leq) F = F (ccpo.fixp (fun_lub lub) (fun_ord leq) F)"
hyps: fixes Ts :: "ty list"   and vs :: "val list"   and l :: "sfields"   and l' :: "sfields" assumes "length ps' = length Ts"   and "length ps' = length vs"   and "P \<turnstile> \<langle>blocks (ps', Ts, vs, e),(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>" shows "\<exists>l''. P \<turnstile> \<langle>e,(h, l(ps' [\<mapsto>] vs), sh)\<rangle> \<Rightarrow> \<langle>e',(h', l'', sh')\<rangle>"
dist_bounded: fixes f :: "'a \<Rightarrow>\<^sub>C 'b"   and x :: "'a"   and g :: "'a \<Rightarrow>\<^sub>C 'b" shows "dist (apply_bcontfun f x) (apply_bcontfun g x) \<le> dist f g"
sg_sg: fixes G :: "('e, 'f) Group_scheme"   and K :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> K "   and "G \<guillemotright> H "   and "H \<subseteq> K" shows "\<natural>\<^bsub>G\<^esub>K \<guillemotright> H "
natural_isomorphism_\<rr>: shows "natural_isomorphism Right_a.comp Right_a.comp Right_a.R Right_a.map Right_a.\<rr>"
fold_empty: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "Finite_Set.fold f z {} = z"
normalize_mult_unit_right: fixes b :: "'b poly"   and a :: "'b poly" assumes "is_unit b" shows "smult (inverse (lead_coeff (a * b))) (a * b) = smult (inverse (lead_coeff a)) a"
power_gt1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "less one (power.power one times a (Suc n))"
pr_comp3_2': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and k :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "pr_conv_3_to_1 f \<in> PrimRec1'"   and "pr_conv_2_to_1 g \<in> PrimRec1'"   and "pr_conv_2_to_1 h \<in> PrimRec1'"   and "pr_conv_2_to_1 k \<in> PrimRec1'" shows "pr_conv_2_to_1 (\<lambda>x y. f (g x y) (h x y) (k x y)) \<in> PrimRec1'"
bind_ask2: fixes m :: "'m"   and f :: "'a list \<Rightarrow> 'b \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
index_pref: fixes u :: "'b list"   and w :: "'b list" assumes "\<^bold>|u\<^bold>| \<le> \<^bold>|w\<^bold>|"   and "\<forall>i<\<^bold>|u\<^bold>|. u ! i = w ! i" shows "u \<le>p w"
cindex_path_def: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "cindex_path g z = cindex 0 1 (\<lambda>t. Im (g t - z) / Re (g t - z))"
prime_dvd_prod_mset_iff: fixes p :: "'a"   and A :: "'a multiset" assumes "prime p" shows "(p dvd \<Prod>\<^sub># A) = (\<exists>x. x \<in># A \<and> p dvd x)"
hyps: assumes "p \<noteq> 0" shows "proots_count p B = proots_count (p \<circ>\<^sub>p q) A"
add_four_block_mat: fixes A1 :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B1 :: "'b mat"   and nc2 :: "nat"   and C1 :: "'b mat"   and nr2 :: "nat"   and D1 :: "'b mat"   and A2 :: "'b mat"   and B2 :: "'b mat"   and C2 :: "'b mat"   and D2 :: "'b mat" assumes "A1 \<in> carrier_mat nr1 nc1"   and "B1 \<in> carrier_mat nr1 nc2"   and "C1 \<in> carrier_mat nr2 nc1"   and "D1 \<in> carrier_mat nr2 nc2"   and "A2 \<in> carrier_mat nr1 nc1"   and "B2 \<in> carrier_mat nr1 nc2"   and "C2 \<in> carrier_mat nr2 nc1"   and "D2 \<in> carrier_mat nr2 nc2" shows "four_block_mat A1 B1 C1 D1 + four_block_mat A2 B2 C2 D2 = four_block_mat (A1 + A2) (B1 + B2) (C1 + C2) (D1 + D2)"
inducts: fixes x :: "'a set \<Rightarrow> 'a set"   and P :: "('a set \<Rightarrow> 'a set) \<Rightarrow> bool" assumes "IK_nf x"   and "P id"   and "P I"   and "P K"   and "P (I \<circ> K)"   and "P (K \<circ> I)"   and "P (I \<circ> K \<circ> I)"   and "P (K \<circ> I \<circ> K)" shows "P x"
impl_of_RBT_HM_cases: fixes y :: "(uint32, ('a, 'b) assoc_list) RBT.rbt" assumes "y \<in> {hm. HashMap_Impl.invar hm}" obtains x :: "('a, 'b) hashmap" where "y = impl_of_RBT_HM x"
spr_simVal_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "('b, 'e, 'f) BEState list"   and envAction :: "('b, 'e, 'f) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> ('b, 'e, 'f) BEState"   and envVal :: "('b, 'e, 'f) BEState \<Rightarrow> 'c \<Rightarrow> bool"   and agents :: "'b odlist"   and envObsC :: "'e \<Rightarrow> 'h" assumes "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC" shows "FiniteDetBroadcastEnvironment.spr_simVal envVal \<equiv> envVal \<circ> sprLst"
row_space_is_preserved: fixes P :: "rat mat"   and m :: "nat"   and A :: "rat mat" assumes "invertible_mat P"   and "P \<in> carrier_mat m m"   and "A \<in> carrier_mat m n" shows "gs.row_space (P * A) = gs.row_space A"
divide_out_primepow: fixes n :: "'a" assumes "n \<noteq> (0::'a)"   and "\<not> is_unit n" obtains p :: "'a" and k :: "nat" and n' :: "'a" where "prime p" and "p dvd n" and "\<not> p dvd n'" and "0 < k" and "n = p ^ k * n'"
sg_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "sg plus = (plus = (+) \<and> is_sg TYPE('b))"
thr_a_conv: fixes ls :: "'j lock_actions"   and nts :: "('g, 'h, 'i) new_thread_action list"   and jswss :: "'g conditional_action list \<times> ('g, 'k) wait_set_action list \<times> 'g interrupt_action list \<times> 'l list" shows "\<lbrace>(ls, nts, jswss)\<rbrace>\<^bsub>t\<^esub> = nts"
measurable_limsup: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>n. A n \<in> sets M" shows "limsup A \<in> sets M"
CP_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "CP_wrt \<psi> \<equiv> {U. \<exists>s. learn_cp \<psi> U s}"
neg_le_minus_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq a (uminus (divide b c)) = less_eq (uminus b) (times a c)"
gen_iapath_sd_path: fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "V \<subseteq> pverts H"   and "H.gen_iapath V u p v" shows "pre_digraph.gen_iapath (with_proj (subdivide H e w)) V u (sd_path e w p) v"
take_prefix: fixes a :: "'a list"   and b :: "'a list" assumes "is_prefix a b" shows "take (length a) b = a"
exec_bind: fixes io\<^sub>1 :: "'b io"   and io\<^sub>2 :: "'b \<Rightarrow> 'a io"   and world :: "\<^url>" shows "exec (io\<^sub>1 \<bind> io\<^sub>2) world = exec (io\<^sub>2 (eval io\<^sub>1 world)) (exec io\<^sub>1 world)"
UNIV_I: fixes x :: "'b::type" shows "(x::'b::type) \<in> UNIV"
eq_iff_8: fixes term1 :: "Syntax.term"   and var :: "Syntax.var"   and term2 :: "Syntax.term"   and term1a :: "Syntax.term"   and vara :: "Syntax.var"   and term2a :: "Syntax.term" shows "(Syntax.Let term1 var term2 = Syntax.Let term1a vara term2a) = ([[atom var]]lst. term2 = [[atom vara]]lst. term2a \<and> term1 = term1a)"
hmac_trans_lkr_aux: fixes X :: "msg"   and K :: "msg"   and s :: "l2_state" assumes "hmac X K \<in> parts {M. \<exists>x A B. Chan x A B M \<in> chan s}"   and "K \<notin> synth (analz (extr (bad s) (ik s) (chan s)))"   and "s \<in> l2_inv3" shows "hmac X K \<in> parts (extr (bad s) (ik s) (chan s))"
continuous_constant_on_closure: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and a :: "'e"   and x :: "'d" assumes "continuous_on (closure S) f"   and "\<And>x. x \<in> S \<Longrightarrow> f x = a"   and "x \<in> closure S" shows "f x = a"
index_zero_vec_1: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "0\<^sub>v n $ i = (0::'b)"
simps_1: fixes M :: "nat" shows "output_size (Input M) = M"
pnPlus_PLUS: fixes xs1 :: "'a rexp list"   and xs2 :: "'a rexp list" assumes "xs1 \<noteq> []"   and "xs2 \<noteq> []"   and "\<forall>x\<in>set (xs1 @ xs2). \<not> is_Zero x \<and> \<not> is_Plus x"   and "sorted xs2"   and "distinct xs2" shows "pnPlus (PLUS xs1) (PLUS xs2) = flatten PLUS (set (xs1 @ xs2))"
wf_atom_mono: fixes tys :: "'b \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and tys' :: "'b \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "'b atom" assumes "tys \<subseteq>\<^sub>m tys'"   and "wf_atom tys a" shows "wf_atom tys' a"
sat_fv_cong: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and v :: "'d list"   and v' :: "'d list"   and \<sigma> :: "'c trace"   and i :: "nat" assumes "fo_spec nfv fv sat"   and "\<And>x. x \<in> fv \<Longrightarrow> v ! x = v' ! x" shows "sat \<sigma> v i = sat \<sigma> v' i"
groups_not_empty: shows "{G \<times> I |G. G \<in> \<G>} \<noteq> {}"
simps_3: fixes w :: "nat"   and v :: "nat"   and va :: "'a tree"   and vb :: "'a tree"   and t\<^sub>2 :: "'a tree"   and a :: "'a" shows "mergeSibling (Node w (Node v va vb) t\<^sub>2) a = Node w (mergeSibling (Node v va vb) a) (mergeSibling t\<^sub>2 a)"
scale_right_diff_distrib: fixes a :: "real"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
A_1: fixes x :: "'a" assumes "x \<in># A" shows "prime x"
hyps: shows "list_constr I (length qs) \<longrightarrow> (0 < poly (prod_list (retrieve_polys qs I)) x) = (z I sign = 1)"
must_have_one_false_contra: fixes G :: "'b gexp list"   and s :: "'b \<Rightarrow> value option" assumes "\<forall>g\<in>set G. gval g s \<noteq> false" shows "gval (fold gAnd G (Bc True)) s \<noteq> false"
comm_ring_hom_axioms: shows "comm_ring_hom (inv hom)"
cInf_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 A :: "'b set"   and x :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq A" shows "less_eq (Inf A) x = (\<forall>y. less x y \<longrightarrow> (\<exists>a\<in>A. less a y))"
proj_setD_2: fixes y :: "'c"   and x :: "'c"   and A :: "'c set" assumes "y \<in> proj_set x A" shows "dist x y = infdist x A"
set_incl_pred3: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(A \<le> B) = (\<forall>a1 a2 a3. A a1 a2 a3 \<longrightarrow> B a1 a2 a3)"
h2: assumes "\<exists>I \<omega>. term_sem I \<theta> \<omega> \<noteq> \<omega> x" shows "x \<in> BVG (x := \<theta>)"
assms_1: shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
Suc_1: assumes "mod_type_class.to_nat i < k"   and "\<not> is_zero_row i A"   and "k \<le> nrows A" shows "Hermite_of_upt_row_i A k ass res $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range (res (Hermite_of_upt_row_i A k ass res $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))))"
top_right_normalize: fixes a :: "'b" shows "gcd a (0::'b) = normalize a"
vector_eq_rdot: fixes x :: "'a"   and y :: "'a" shows "(\<forall>z. x \<bullet> z = y \<bullet> z) = (x = y)"