hcju/isabelleps huggingface dataset:
latestActEx: fixes n :: "nat"   and nid :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<exists>n'. latestAct_cond nid t n n' \<and> (\<forall>n''. latestAct_cond nid t n n'' \<longrightarrow> n'' \<le> n')"
bounded_by_bound_of: fixes P :: "'c \<Rightarrow> real" assumes "Expectations.bounded P" shows "bounded_by (bound_of P) P"
resl_top: fixes x :: "'a" shows "\<top> \<leftarrow> x = \<top>"
node_rel_sv: fixes Re :: "('a \<times> 'b) set"   and R :: "('c \<times> 'd) set" assumes "single_valued Re"   and "single_valued R" shows "single_valued (\<langle>Re, R\<rangle>node_rel)"
assms: shows "(X1 + X2, Y) \<in> multpw ns"
minus_unique: fixes y :: "'a vec"   and x :: "'a vec"   and y' :: "'a vec" assumes "y + x = 0\<^sub>v n"   and "x + y' = 0\<^sub>v n"   and "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "y' \<in> carrier_vec n" shows "y = y'"
bijective_closed_imp_open_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "closed_map X X' f"   and "f ` topspace X = topspace X'"   and "inj_on f (topspace X)" shows "open_map X X' f"
cases: fixes x :: "'b Trace" obtains s :: "'b" where "x = tInit s__"   | t :: "'b Trace" and s :: "'b" where "x = t \<leadsto> s"
map_sel_1: fixes a :: "'a literal"   and f :: "'a \<Rightarrow> 'b" assumes "is_pos a" shows "atm_of (map_literal f a) = f (atm_of a)"
finite_pdigraph_dg_prod_3: shows "finite_pdigraph \<alpha> (3\<^sub>\<nat>) (\<lambda>i. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | i = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>)"
g_def: shows "g = vec.construct (set_of_vector X) f"
RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m_is_RPHIL: fixes i :: "nat" assumes "0 < i" shows "RPHIL i = RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m i 0"
L: shows "Says m run 5 X (Card m) (Crypt (sesK (c * f)) (Passwd m)) \<in> set evsFR5 \<and> Says m run 5 (Card m) X (Crypt (sesK (c * f)) (Number 0)) \<in> set evsFR5"
ivl_disj_un_singleton_6: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{l..<u} \<union> {u} = {l..u}"
sel_1: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "shd (siterate f x) = x"
lconf_def: fixes P :: "'a prog"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and E :: "String.literal \<Rightarrow> ty option" shows "P,h \<turnstile> l (:\<le>) E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
poly_vars_def: fixes p :: "('c monom \<times> 'd) list" shows "poly_vars p = set (concat (map (monom_vars_list \<circ> fst) p))"
constant_rep_design_axioms: shows "constant_rep_design \<V> \<B> (\<Lambda> * (\<v> - 1) div (\<k> - 1))"
bounded_Int: fixes S :: "'c set"   and T :: "'c set" assumes "bounded S \<or> bounded T" shows "bounded (S \<inter> T)"
cd_3: shows "(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p' / u powr (p + 1)))) \<in> \<Theta>(\<lambda>x. d * real x powr p + c * real x powr p')"
inv_inv_next_plane: fixes p :: "nat" shows "invariant Invariants.inv next_plane\<^bsub>p\<^esub>"
other_preds_dominated: fixes m :: "'node" assumes "m \<in> set (old.predecessors g (defNode g v))"   and "v' \<notin> phiUses g m" shows "old.dominates g (defNode g v) m"
run_asset_iv_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "let (s', S, A, B, C, D) = run_asset_iv n s; SK = (Some S, {A, B}, {C, D}) in insert (Owner n, SesKey SK) ({Asset n} \<times> {SesKey SK, Token n (Auth_PriK n) B C SK}) \<subseteq> s'"
of_nat_mono: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and i :: "nat"   and j :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "i \<le> j" shows "less_eq (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j)"
stat_subprob: fixes C :: "'s set" assumes "essential_class C"   and "countable C"   and "Ball C pos_recurrent" shows "emeasure (stat C) C \<le> 1"
comp_fun_commute_axioms: shows "comp_fun_commute MaxR_f"
assms: fixes x :: "'a"   and y :: "'a" assumes "le x y"   and "le y x" shows "f x = f y"
locally_respects_next_state: shows "\<forall>s u execs. \<not> ifp (current s) u \<longrightarrow> vpeq u s (next_state s execs)"
execute_make: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and h :: "heap" shows "execute (Array.make n f) h = Some (Array.alloc (map f [0..<n]) h)"
AxWVTSym: fixes m :: "Body"   and k :: "Body"   and y :: "'a Point"   and x :: "'a Point" assumes "IOb m"   and "IOb k" shows "(y = wvt k m x) = (x = wvt m k y)"
in_gamma_update: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and s :: "char list \<Rightarrow> int"   and S :: "'a st"   and i :: "int"   and a :: "'a"   and x :: "char list" assumes "Gamma_semilattice num' plus' \<gamma>"   and "s \<in> \<gamma>_st \<gamma> S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>_st \<gamma> (update S x a)"
pcompose_1: fixes p :: "'b poly" shows "1 \<circ>\<^sub>p p = 1"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n"   and "m < j" shows "p A i j"
UP_car_memE_2: fixes g :: "nat \<Rightarrow> 'a"   and x :: "nat" assumes "g \<in> carrier (UP R)"   and "deg R g < x" shows "g x = \<zero>"
llist_all2_llist_of: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "llist_all2 P (llist_of xs) (llist_of ys) = list_all2 P xs ys"
r_null: fixes x :: "'a" shows "x * (0::'a) = (0::'a)"
sup_continuous_at_left: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "sup_continuous f" shows "continuous (at_left x) f"
prime_elem_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.prime_elem times one zero p = (p \<noteq> zero \<and> \<not> dvd.dvd times p one \<and> (\<forall>a b. dvd.dvd times p (times a b) \<longrightarrow> dvd.dvd times p a \<or> dvd.dvd times p b))"
cbnd_grow: fixes k :: "nat"   and d :: "nat" shows "k \<le> cbnd k d"
label_incrs_def: fixes as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and i :: "nat" shows "as \<oplus>s i \<equiv> map (\<lambda>(n, et, n'). (n \<oplus> i, et, n' \<oplus> i)) as"
distinct_no_loop1: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "1 < |vertices f|" shows "f \<bullet> v \<noteq> v"
matrix_blockmat: fixes ls :: "brick list" shows "mat (rat_poly.row_length (blockmat ls)) (length (blockmat ls)) (blockmat ls)"
final: shows "FL.passive_subset (Liminf_llist (lmap lclss_of_state Sts)) = {}"
Lcm_real_def: shows "Lcm = normalization_euclidean_semiring_class.Lcm"
Call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "'d"   and n :: "nat"   and t :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> =Suc n\<Rightarrow> t"
map_append: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list" shows "map f (xs @ ys) = map f xs @ map f ys"
1: fixes x :: "nat"   and y :: "nat" assumes "x < y" shows "f x < f y"
l3: shows "length (zip (map (unity_root k) (map int [0..<k])) ws) = k"
lemma2a: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> (eps L)\<^sup>*" shows "(True # p, True # q) \<in> (eps (or L R))\<^sup>*"
induct: fixes x1 :: "'vtx"   and x2 :: "'vtx"   and x3 :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and P :: "'vtx \<Rightarrow> 'vtx \<Rightarrow> ('vtx \<times> 'lbl \<times> 'vtx) list \<Rightarrow> bool" assumes "path x1 x2 x3"   and "\<And>l1 l2. l1 = l2 \<Longrightarrow> P l1 l2 []"   and "\<And>l1 l2 xs lbl l3. \<lbrakk>path l1 l2 xs; P l1 l2 xs; lbl \<in>\<^sub>A weights l2 l3\<rbrakk> \<Longrightarrow> P l1 l3 (xs @ [(l2, lbl, l3)])" shows "P x1 x2 x3"
Compl_set_less_eq_Compl: fixes A :: "'a set"   and B :: "'a set" shows "(- A \<sqsubseteq> - B) = (B \<sqsubseteq> A)"
add_match_match_not_cases: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match (MatchNot m) rs, Undecided\<rangle> \<Rightarrow> Undecided" shows "matches \<gamma> m p \<or> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"
deg_pm_mono: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" assumes "s adds t" shows "deg_pm s \<le> deg_pm t"
f1_def: shows "PROP f1 \<equiv> (\<And>k v kvs. P (Suc 0) ((k, v) # kvs))"
count_list_append: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" shows "count_list (xs @ ys) x = count_list xs x + count_list ys x"
RESTRICT_def: fixes \<phi> :: "('a, 'k) aformula" shows "RESTRICT \<phi> \<equiv> RESTRICT_VARS enum_class.enum (sorted_list_of_set \<circ> FV \<phi>) \<phi>"
ars1: fixes x :: "('b \<times> 'b) set" shows "rel_ad x O x = {}"
uminus_sum_list_alternating_order2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b"   and n :: "nat" assumes "class.group_add minus plus zero uminus"   and "plus s s = zero"   and "plus t t = zero" shows "uminus (monoid_add.sum_list plus zero (alternating_list n s t)) = monoid_add.sum_list plus zero (if gcd_nat.greater_eq n 2 then alternating_list n t s else alternating_list n s t)"
quotient_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "A // r = (\<Union>x\<in>A. {r `` {x}})"
adopt_node_locs_def: shows "Shadow_DOM.adopt_node_locs = adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
R17_d: shows "pmf (sds R17) d = 1 / 2"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "local.comp g f = C g f"
exponential_order_def: fixes M :: "real"   and c :: "real"   and f :: "real \<Rightarrow> 'a" shows "exponential_order M c f = (0 < M \<and> (\<forall>\<^sub>F t in at_top. norm (f t) \<le> M * exp (c * t)))"
univalent_bot_closed: shows "matrix_stone_relation_algebra.univalent mbot"
wordinterval_CIDR_split_prefixmatch: fixes r :: "'a wordinterval" shows "wordinterval_to_set r = \<Union> (prefix_to_wordset ` set (wordinterval_CIDR_split_prefixmatch r))"
span_empty: shows "span {} = {\<zero>\<^bsub>V\<^esub>}"
img_bound: shows "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t (S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S')"
fmrestrict_fset_dom: fixes m :: "'d \<rightharpoonup>\<^sub>f 'e" shows "fmrestrict_fset (fmdom m) m = m"
gen_def: shows "gen \<equiv> \<lambda>P I. {x. genp (\<lambda>x. x \<in> P) (\<lambda>x. x \<in> I) x}"
prime_dvd_power_iff: 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"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "0 < n" shows "dvd.dvd times p (power.power one times x n) = dvd.dvd times p x"
int: fixes S :: "'a set" assumes "S \<in> \<F>"   and "interior S = {}" shows "S = {}"
assms_1: shows "\<LM> A  \<RM> = \<Delta> + \<Delta>'"
ipd_unique: fixes y :: "'n"   and x :: "'n"   and y' :: "'n" assumes "y ipd\<rightarrow> x"   and "y' ipd\<rightarrow> x" shows "y = y'"
column_def: fixes j :: "'d"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "column j A = (\<chi>i. A $h i $h j)"
OV_inv1D: fixes x :: "'a obsv_state_scheme"   and Q :: "process set"   and v :: "val"   and r :: "nat"   and Q' :: "process set" assumes "x \<in> OV_inv1"   and "quorum_for Q v (votes x r)"   and "Q' \<in> Quorum" shows "quorum_for Q' v (obsv_state.obs x r)"
inj_on_\<alpha>': shows "inj_on (\<lambda>(k, v). (k, \<alpha> v)) (set lc)"
verit_negate_coefficient_3: fixes a :: "'b"   and b :: "'b" assumes "a = b" shows "- a = - b"
aff_dim_inner_basis_exists: fixes V :: "'a set" shows "\<exists>B\<subseteq>V. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
infTp: fixes \<sigma> :: "'tp"   and intT :: "'tp \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'fsym \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'psym \<Rightarrow> univ list \<Rightarrow> bool" assumes "infTp \<sigma>"   and "MModel intT intF intP" shows "infinite {a. intT \<sigma> a}"
sub_block_id: fixes block :: "('c, 'd) fm list \<times> 'd" shows "sub_block id block = block"
bound_of_sound: fixes P :: "'c \<Rightarrow> real" assumes "sound P" shows "0 \<le> bound_of P"
add_true_asm: fixes b :: "assn"   and p :: "'a Heap"   and a :: "'a \<Rightarrow> assn" assumes "<b * true> p <a>\<^sub>t" shows "<b> p <a>\<^sub>t"
simps_2: fixes f :: "'a"   and ss :: "('a, 'c) Term.term list"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "Fun f ss \<cdot> \<sigma> = Fun f (map (\<lambda>t. t \<cdot> \<sigma>) ss)"
IH: fixes n :: "nat"   and p :: "'a poly" assumes "k < n" shows "poly ((pderiv ^^ k) ([:- c, 1::'a:] ^ n * p)) c = (0::'a)"
simps_2: fixes x :: "'a"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and v :: "'b"   and xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList ((x, e, v) # xs) C l s = (l x = preT C e s \<and> preList xs C l s)"
propa_dom_invariant: fixes ss :: "nat list list"   and w :: "nat list"   and ss' :: "nat list list"   and w' :: "nat list" assumes "wf_dom ss w"   and "w \<noteq> []"   and "propa f (step (hd w) (ss ! hd w)) ss (tl w) = (ss', w')" shows "wf_dom ss' w'"
c1: assumes "ip \<notin> kD rt" shows "P (rt(ip \<mapsto> r))"
lemma_2_5_18: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b \<le> c * a r\<rightarrow> c * b"
mkPinj_def: fixes x :: "nat"   and P :: "pol" shows "mkPinj x P = (case P of Pc c \<Rightarrow> Pc c | Pinj y P \<Rightarrow> Pinj (x + y) P | PX p1 y p2 \<Rightarrow> Pinj x P)"
no_spoofing': shows "(common_matcher, in_doubt_allow),p\<lparr>p_iiface := iface_sel iface\<rparr>\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface)))"
coprime_mult_left_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (times a b) c = (algebraic_semidom.coprime times one a c \<and> algebraic_semidom.coprime times one b c)"
Fields_i_i_oE: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "pred.eval (Fields_i_i_o x xa) xb" obtains "x \<turnstile> xa has_fields xb"
choose_pivot_def: fixes uu :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and uv :: "'b \<Rightarrow> 'a"   and uw :: "'b list"   and lo :: "nat"   and hi :: "nat" shows "choose_pivot uu uv uw lo hi = SPEC (\<lambda>k. lo \<le> k \<and> k \<le> hi)"
makespan_correct_2: fixes m :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "LoadBalancing m" shows "\<exists>x\<in>{1..m}. T x = f_Max\<^sub>0 T m"
induct: fixes P :: "('a, 'b) aformula \<Rightarrow> bool"   and a0 :: "('a, 'b) aformula" assumes "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (FOr \<phi> \<psi>)"   and "\<And>v. P (FBool v)"   and "\<And>v. P (FBase v)"   and "\<And>v. P (FNot v)"   and "\<And>v va. P (FAnd v va)"   and "\<And>v va. P (FEx v va)"   and "\<And>v va. P (FAll v va)" shows "P a0"
init_fin_silent_moveD: fixes t :: "'t"   and s :: "status"   and x :: "'x"   and m :: "'m"   and s' :: "status"   and x' :: "'x"   and m' :: "'m" assumes "\<tau>trsys.silent_move (init_fin t) init_fin_\<tau>move ((s, x), m) ((s', x'), m')" shows "silent_move t (x, m) (x', m') \<and> s = s' \<and> s' = Running"
fls_X_intpow_base_factor: fixes i :: "int" shows "fls_base_factor (fls_X_intpow i) = 1"
vid_onD: fixes a :: "V"   and A :: "V" assumes "\<langle>a, a\<rangle> \<in>\<^sub>\<circ> vid_on A" shows "a \<in>\<^sub>\<circ> A"
IH: shows "fsafe \<phi> \<longrightarrow> (\<forall>i. fsafe (\<sigma> i)) \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (PFsubst \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (PFadjoint I \<sigma>) \<phi>))"
n_eq_1_iff: fixes s :: "'s" assumes "s \<in> S" shows "(n s = 1) = (\<forall>cfg\<in>cfg_on s. v cfg = 1)"
diamond_Un: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "diamond R"   and "diamond S"   and "commute R S" shows "diamond (sup R S)"
allDefs_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and n :: "'node" shows "uninst_code.ssa.allDefs p g n \<equiv> defs g n \<union> uninst_code.ssa.phiDefs p g n"
tllist_all2_tnthD2: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist"   and n :: "nat" assumes "tllist_all2 P Q xs ys"   and "enat n < tlength ys" shows "P (tnth xs n) (tnth ys n)"
dual_sup: 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" shows "dual (sup x y) = inf (dual x) (dual y)"
finalStateNonApplicable: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "isFinalState state F0 decisionVars = (\<not> applicableDecide state decisionVars \<and> \<not> applicableUnitPropagate state F0 \<and> \<not> applicableBacktrack state F0)"
val_poss_n_val_poss: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(0 < v x) = (0 < n_val K v x)"
cring_simprules_25: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs H" shows "rev_H' a \<noteq> a"
assms_2: shows "A\<^sub>F \<sharp>* \<Phi>"
hyps_2: shows "hereditarily_fs (S_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
c_graph_lm_2: fixes y :: "nat"   and f :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "y = f x" shows "c_pair x y \<in> c_graph f"
ce_set_by_minsky: fixes A :: "nat set" assumes "A \<in> ce_sets" obtains M :: "(nat \<times> (nat, nat) Op) set" where "finite M" and "deterministic M" and "0 \<notin> fst ` M" and "\<And>x vs. \<lbrakk>vs 0 = x; vs 1 = 0; x \<in> A\<rbrakk> \<Longrightarrow> \<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>*" and "\<And>x vs vs'. \<lbrakk>vs 0 = x; vs 1 = 0; ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>*\<rbrakk> \<Longrightarrow> x \<in> A"
lunit_coherence1: fixes f :: "'a" assumes "C.ide f" shows "\<l>\<^sub>D[F f] \<cdot>\<^sub>D D.inv (unit (trg\<^sub>C f) \<star>\<^sub>D F f) = lF f"
cobounded1: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<subseteq># a \<union># b"
block: fixes tf :: "tf"   and tn :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and es :: "b_e list" assumes "tf = tn _> tm"   and "\<C>\<lparr>label := [tm] @ label \<C>\<rparr> \<turnstile> es : tn _> tm" shows "\<C> \<turnstile> [Block tf es] : tn _> tm"
cases: fixes a :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state" assumes "Sinca.final a" obtains F :: "'fenv_inca" and f :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef" and pc :: "nat" and H :: "'henv" and \<Sigma> :: "'dyn list" where "a = State F H [Frame f pc \<Sigma>]" and "Finca_get F f = Some fd" and "pc = length (body fd)"
Collect_member: fixes dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "Collect (DList_Set.member dxs) = set (list_of_dlist dxs)"
sorted_iff_nth_mono: fixes xs :: "'a word list" shows "signed.sorted xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le>s xs ! j)"
top_le: fixes a :: "'a" assumes "(1::'a) \<le> a" shows "a = (1::'a)"
fds_fds_nth: fixes f :: "'b fds" shows "fds (($) f) = f"
solve_depressed_cubic_Cardano_real: fixes e :: "real"   and f :: "real"   and y :: "real" assumes "solve_depressed_cubic_Cardano_real e f = Some y" shows "{y. y ^ 3 + e * y + f = 0} = {y}"
labeled_sat: shows "{\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> NL} \<subseteq> Red_I NL"
ide_u: shows "ide t\<^sub>0u\<^sub>1.p\<^sub>0"
older_seniors_card_le: fixes x :: "nat"   and n :: "nat"   and y :: "nat" shows "(card (older_seniors x n) < card (older_seniors y n)) = (older_seniors x n \<subset> older_seniors y n)"
reduced_GB_D3: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_reduced_GB G" shows "punit.is_monic_set G"
csl: shows "cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>\<^esup> k @ xs @ [\<pi> l]"
scalar_matrix_mult_row_def: fixes c :: "'c"   and A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e" shows "scalar_matrix_mult_row c A i = (\<chi>j. c * A $ i $ j)"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "GE.wtE \<xi>" shows "GE.satPB \<xi> gPB"
sat_And_Not: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (MFOTL.And_Not \<phi> \<psi>) = (MFOTL.sat \<sigma> v i \<phi> \<and> \<not> MFOTL.sat \<sigma> v i \<psi>)"
obj_fields_marked_mark_field_done: fixes s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state" assumes "obj_at_field_on_heap (\<lambda>r. marked r s) (gc_tmp_ref s) (gc_field s) s"   and "obj_fields_marked s" shows "obj_fields_marked (s(gc := s gc\<lparr>field_set := gc_field_set s - {gc_field s}\<rparr>))"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> \<GG> = vcard \<GG>"
not_choice_simps_10: fixes pn :: "'c" shows "not_choice (call(pn))"
suminf_multr: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "(\<Sum>n. f n * c) = (\<Sum>n. f n) * c"