hcju/isabelleps huggingface dataset:
indep_vars_iff_distr_eq_PiM: fixes I :: "'c set"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> 'd"   and M' :: "'c \<Rightarrow> 'd measure" assumes "I \<noteq> {}"   and "\<And>i. random_variable (M' i) (X i)" shows "indep_vars M' X I = (distr M (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x) = Pi\<^sub>M I (\<lambda>i. distr M (M' i) (X i)))"
assms: shows "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>n. A)"
ltrm_inv_simp: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "monom (UP R) (monom (UP R) (f (deg R f)) (deg R f) (deg R (monom (UP R) (f (deg R f)) (deg R f)))) (deg R (monom (UP R) (f (deg R f)) (deg R f))) = monom (UP R) (f (deg R f)) (deg R f)"
case_eq_if: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
lc: shows "lead_coeff p \<noteq> (0::'a)"
floor_le_zero: fixes x :: "'b" shows "(\<lfloor>x\<rfloor> \<le> 0) = (x < (1::'b))"
primes_psi_def: shows "\<psi> = sum_upto mangoldt"
subst_indep: fixes \<Gamma>' :: "(name, 'a) fmap"   and t :: "'a"   and \<Gamma> :: "(name, 'a) fmap" assumes "closed_env \<Gamma>'" shows "subst t (\<Gamma> ++\<^sub>f \<Gamma>') = subst (subst t \<Gamma>') \<Gamma>"
Domain_empty_iff: fixes r :: "('a \<times> 'b) set" shows "(Domain r = {}) = (r = {})"
the_coext_unique: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f"   and "\<epsilon>x.is_coext x' f g" shows "g = \<epsilon>x.the_coext x' f"
1: shows "f \<in> space M \<rightarrow> space M1 \<times> space M2"
step_Suc: fixes i :: "nat"   and t :: "('a, 'b, 'c) event list"   and c :: "('a, 'b, 'c) configuration"   and c' :: "('a, 'b, 'c) configuration" assumes "i < length t"   and "trace c t c'" shows "s c t i \<turnstile> t ! i \<mapsto> s c t (Suc i)"
restrict0_const_diff_fun_space: fixes c :: "real" shows "restrict0 carrier (\<lambda>x. c) \<in> diff_fun_space"
euclidean_closure_of: fixes S :: "'b set" shows "euclidean closure_of S = closure S"
asymp_equiv_at_top_imp_at_left: fixes a :: "real" assumes "(\<lambda>x. f (a - inverse x)) \<sim>[at_top] (\<lambda>x. g (a - inverse x))" shows "f \<sim>[at_left a] g"
echelon_form_upt_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "echelon_form_upt_row A k = ((\<forall>i. mod_type_class.to_nat i < k \<and> is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. mod_type_class.to_nat j < k \<and> \<not> is_zero_row j A)) \<and> (\<forall>i j. i < j \<and> mod_type_class.to_nat j < k \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ j $ n \<noteq> (0::'a))))"
strict_coboundedI1: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset" assumes "a \<subset># c" shows "a \<inter># b \<subset># c"
nn_integral_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = set_nn_integral M \<Omega> f"
stable_no_blocking_onI2: fixes ds :: "'b set"   and X :: "'a set" assumes "\<And>h X''. Pds'.blocking_on ds X h X'' \<Longrightarrow> False" shows "Pds'.stable_no_blocking_on ds X"
ren_proj: fixes x :: "'sz"   and y :: "'sz"   and a :: "(real, 'sz) vec"   and z :: "'sz" shows "RSadj x y a $ z = a $ swap x y z"
supr_under: fixes a :: "nat + 'a Bonk_Schramm_extension_unfolded set" assumes "a \<in> Field wo_aux" shows "a = supr wo_aux (under wo_aux a)"
fst_Gauss_Jordan_PA: fixes A :: "(('b, 'c) vec, 'd) vec" shows "fst (Gauss_Jordan_PA A) ** A = snd (Gauss_Jordan_PA A)"
point_replication_number_def: fixes B :: "'b set multiset"   and x :: "'b" shows "B rep x \<equiv> int (size (filter_mset ((\<in>) x) B))"
split_mult_pos_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero" shows "less_eq zero (times a b)"
LeftDerivation_append: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and c :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.LeftDerivation \<NN> \<TT> \<RR> a (D @ E) c = (\<exists>b. CFG.LeftDerivation \<NN> \<TT> \<RR> a D b \<and> CFG.LeftDerivation \<NN> \<TT> \<RR> b E c)"
assms: fixes x :: "'n" assumes "x \<in> cbox a b" shows "f x = (0::'a)"
A_Issues_B: fixes A :: "agent"   and B :: "agent"   and servTicket :: "msg"   and servK :: "nat"   and T3 :: "nat"   and evs :: "event list" assumes "Says A B \<lbrace>servTicket, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "A Issues B with Crypt servK \<lbrace>Agent A, Number T3\<rbrace> on evs"
n_def: shows "n \<equiv> \<Prod>A1 * s1\<^sup>2"
eq: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>xa. \<lbrakk>xa = x; xa \<notin> S\<rbrakk> \<Longrightarrow> f xa = g xa"   and "\<And>xa. \<lbrakk>xa \<in> S; xa \<noteq> x\<rbrakk> \<Longrightarrow> f xa = g xa"   and "\<And>xa. \<lbrakk>xa \<notin> S; xa \<noteq> x\<rbrakk> \<Longrightarrow> f xa = g xa" shows "f = g"
integral_mult_right: fixes c :: "'a"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a) \<Longrightarrow> integrable M f" shows "LINT x|M. c * f x = c * integral\<^sup>L M f"
assms_6: shows "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle> \<rightarrow>* \<langle>throw a,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
assms_1: fixes u :: "'t" assumes "u \<in> keys p" shows "v \<preceq>\<^sub>t u"
simps_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(allow x1 = allow y1) = (x1 = y1)"
that: fixes n' :: "nat" assumes "steps0 (1, tp) (A |+| B) n' = (Suc (length A div 2), tp')" shows "thesis"
nonzcoeffs: fixes a :: "real"   and b :: "real"   and c :: "real"   and r :: "real" assumes "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0" shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0"
psubst_eq_rawpsubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)"   and "\<And>i j. \<lbrakk>i < j; j < length txs\<rbrakk> \<Longrightarrow> snd (txs ! j) \<notin> FvarsT (fst (txs ! i))" shows "psubst \<phi> txs = rawpsubst \<phi> txs"
cInf_greatest: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set"   and z :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq z x" shows "less_eq z (Inf X)"
case4: fixes y :: "game" assumes "zin y (right_options x)" shows "\<not> ge_game (plus_game x (neg_game y), zero_game)"
cases: fixes a :: "trm" assumes "val a" obtains x :: "name" and e :: "trm" where "a = Lam [x].e"
separate_point_closed: fixes s :: "'c set"   and a :: "'c" assumes "closed s"   and "a \<notin> s" shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"
infinity_evalUni: fixes At :: "atomUni"   and x :: "real" shows "(\<exists>x. \<forall>y<x. aEvalUni At y) = evalUni (substNegInfinityUni At) x"
qr: shows "[a, b] = q \<otimes>\<^bsub>K [X]\<^esub> r"
typing_closed_in: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : T" shows "T closed_in \<Gamma>"
factors_equal: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a = b"   and "c = d" shows "a + c = b + d"
nhop_update_changed_kno_val: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and dsn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt ip (dsn, kno, val, hops, nhip)" shows "the (nhop (update rt ip (dsn, kno, val, hops, nhip)) ip) = nhip"
tmadd_nb: fixes n :: "nat"   and t :: "tm"   and s :: "tm" assumes "tmbound n t"   and "tmbound n s" shows "tmbound n (tmadd t s)"
enat_set_eq_UNIV: shows "enat_set = UNIV"
assms_2: fixes g :: "'a" shows "Mapping.keys (uses g) \<subseteq> set (\<alpha>n g)"
consec_add: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" assumes "left r = left s \<and> right r = right t \<and> right s = left t" shows "\<parallel>r\<parallel> = \<parallel>s\<parallel> + \<parallel>t\<parallel>"
finite_constant: fixes C :: "'b set" assumes "finite C" shows "finite (constant_rules C)"
prime_card_def: shows "class.prime_card TYPE('b) \<equiv> prime CARD('b)"
f_shrink_last_assoc: fixes xs :: "'b list"   and a :: "nat"   and b :: "nat" shows "xs \<div>\<^sub>l a \<div>\<^sub>l b = xs \<div>\<^sub>l (a * b)"
dim_sumlist: fixes xs :: "int vec list" assumes "\<forall>x\<in>set xs. dim_vec x = n" shows "dim_vec (M.sumlist xs) = n"
intF: fixes T :: "'a set" assumes "T \<in> \<C>" shows "U \<inter> F T = T"
add_sup_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semilattice_sup_ab_group_add plus zero minus uminus less_eq less sup" shows "plus a (sup b c) = sup (plus a b) (plus a c)"
index_ordered_le: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "index i \<le> index j"
assms: shows "root (traverse_tree path (pure rat_of \<diamondop> stern_brocot_recurse)) = root (traverse_tree path' (pure rat_of \<diamondop> stern_brocot_recurse))"
times_poly_def: fixes p :: "'a poly"   and q :: "'a poly" shows "p * q = fold_coeffs (\<lambda>a p. Polynomial.smult a q + pCons (0::'a) p) p 0"
less_then_max_plus3: fixes n :: "nat"   and x :: "nat" shows "n < max (n + 3) x"
dsj_commute: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" assumes "H \<turnstile> B OR A" shows "H \<turnstile> A OR B"
extensional_set: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and b :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.ide S a"   and "partial_magma.ide S b"   and "set_category_data.set S img a = set_category_data.set S img b" shows "a = b"
subterms_subst_subset: fixes \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and t :: "('a, 'c) Term.term" shows "subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma> \<subseteq> subterms (t \<cdot> \<sigma>)"
not_frequently: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<not> frequently P F) = (\<forall>\<^sub>F x in F. \<not> P x)"
lex_pp_lin: fixes s :: "('c, 'd) pp"   and t :: "('c, 'd) pp" shows "lex_pp s t \<or> lex_pp t s"
IH: fixes s :: "'variable \<Rightarrow> bool option" assumes "are_all_operators_applicable s \<pi>\<^sub>1"   and "are_all_operators_non_interfering \<pi>\<^sub>1" shows "execute_serial_plan s (\<pi>\<^sub>1 @ \<pi>\<^sub>2) = execute_serial_plan (execute_serial_plan s \<pi>\<^sub>1) \<pi>\<^sub>2"
SUP_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 bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
simps: fixes t :: "'b"   and a1 :: "'c"   and a2 :: "'a" shows "proj_thd (t, a1, a2) = a2"
lesseq_vecI: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n"   and "\<And>i. i < n \<Longrightarrow> v $ i \<le> w $ i" shows "v \<le> w"
P4: shows "Bet A D' B'' \<and> Cong D' B'' D B"
polypoly'_def: fixes bs :: "'a list"   and p :: "poly" shows "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
simps_1: fixes uu :: "trilean" shows "uu \<and>? invalid = invalid"
ant_0_in_Ainteg: shows "(0, 0) \<in> Ainteg"
iMin_equality: fixes k :: "'a"   and I :: "'a set" assumes "k \<in> I"   and "\<And>x. x \<in> I \<Longrightarrow> k \<le> x" shows "iMin I = k"
assms: shows "R \<subseteq> 1\<^sub>\<sigma>"
is_extensional: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c"   and \<tau> :: "'a \<Rightarrow> 'c"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "\<not> partial_magma.arr A f" shows "\<tau> f = partial_magma.null B"
final_state_correct1: fixes ts' :: "'a list"   and ms :: "nat \<Rightarrow> 'a option"   and r :: "'b" assumes "invar (ts', ms, r)"   and "final (ts', ms, r)" shows "r = fold_mset f start (mset ts)"
lb_ln_1: fixes y :: "float"   and prec :: "nat"   and x :: "float" assumes "Some y = lb_ln prec x" shows "real_of_float y \<le> ln (real_of_float x)"
IHes: fixes l :: "char list \<Rightarrow> val option" assumes "P \<turnstile> h \<surd>"   and "P \<turnstile> E \<surd>"   and "\<D>s es \<lfloor>dom l\<rfloor>"   and "\<not> finals es" shows "\<exists>es' s'. P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"
abs_eq: fixes x :: "('b, 'c) RBT_Impl.rbt" assumes "eq_onp is_rbt x x" shows "RBT.lookup (RBT x) = rbt_lookup x"
assms: shows "\<forall>var\<in>vars (poly c). v1 var = v2 var"
diag_open: shows "\<forall>l. branch (SOME b. branch b (deeptree l) \<and> \<not> falsifies\<^sub>c\<^sub>s b Cs) (deeptree l) \<and> \<not> falsifies\<^sub>c\<^sub>s (SOME b. branch b (deeptree l) \<and> \<not> falsifies\<^sub>c\<^sub>s b Cs) Cs"
helper_I_closed: fixes a :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<in> carrier R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "a \<otimes> x \<in> I" shows "a \<otimes> (x \<otimes> y) \<in> I"
P_def: fixes p :: "nat"   and a :: "int"   and x :: "int"   and y :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "P p a x y = ([x * y = a] (mod int p) \<and> y \<in> S1 p)"
proper_sub_incidence_system_def: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" shows "proper_sub_incidence_system \<U> \<A> \<V> \<B> \<equiv> proper_sub_set_system \<U> \<A> \<V> \<B> \<and> incidence_system \<U> \<A>"
maximal_independent_subset: fixes V :: "'d set" obtains B :: "'d set" where "B \<subseteq> V" and "independent B" and "V \<subseteq> span B"
intros: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set"   and c :: "'c"   and s :: "('b \<times> 'c) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
psi_def: fixes s :: "nat \<Rightarrow> nat option"   and i :: "nat"   and x :: "nat" assumes "cons_lim s" shows "cons_lim.psi s i x \<equiv> Some (last (cons_lim.prefixes s i x))"
r_find_ksb: shows "eval r_find_k [b] \<down>= pdec1 (the (eval (Mn 1 f) [b]))"
oalist_inv_raw_map2_val_pair: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and g :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'd) list"   and h :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "tc.map2_val_compat g"   and "tc.map2_val_compat h" shows "tc.oalist_inv_raw (map2_val_pair_tc f g h xs ys)"
nat_mult_less_cancel_disj: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m < k * n) = (0 < k \<and> m < n)"
OF_match_linear_match_allsameaction: fixes gr :: "'b set"   and oms :: "'b set list"   and \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c"   and pri :: "16 word"   and act :: "'d" assumes "gr \<in> set oms"   and "\<gamma> gr p = True" shows "OF_match_linear \<gamma> (map (\<lambda>x. split3 OFEntry (pri, x, act)) oms) p = Action act"
coeff_sym_mpoly_monom_of_set_eq_0': fixes X :: "nat set"   and Y :: "nat set"   and k :: "nat" assumes "finite X"   and "\<not> Y \<subseteq> X"   and "finite Y" shows "MPoly_Type.coeff (sym_mpoly X k) (monom_of_set Y) = (0::'b)"
is_continuous: fixes U :: "'b set" assumes "is_open' U" shows "is_open (f \<^sup>\<inverse> S U)"
directly_subsumes_def: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and s2 :: "nat"   and t1 :: "transition"   and t2 :: "transition" shows "directly_subsumes e1 e2 s1 s2 t1 t2 \<equiv> \<forall>c1 c2 t. obtains s1 c1 e1 0 <> t \<and> obtains s2 c2 e2 0 <> t \<longrightarrow> subsumes t1 c2 t2"
Liminf_mono: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Liminf F f \<le> Liminf F g"
measurable_measure_subprob_algebra: fixes a :: "'c set"   and A :: "'c measure" assumes "a \<in> sets A" shows "(\<lambda>M. Sigma_Algebra.measure M a) \<in> borel_measurable (subprob_algebra A)"
ang_vec_sym: fixes z1 :: "complex"   and z2 :: "complex" assumes "\<angle> z1 z2 \<noteq> pi" shows "\<angle> z1 z2 = - \<angle> z2 z1"
solution_usolves_on_X: shows "(solution usolves_ode f from t_max) (cball t_max et) X"
I_K_I_K: shows "I \<circ> K \<circ> I \<circ> K = I \<circ> K"
T_A'_eq: shows "T_A' (length qs) = int (T init qs acts)"
right_idem: fixes a :: "'a word"   and b :: "'a word" shows "signed.max (signed.max a b) b = signed.max a b"
dvd_eq_mod_eq_0: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times a b = (modulo b a = zero)"
distrib_inf_le: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup" shows "less_eq (sup (inf x y) (inf x z)) (inf x (sup y z))"
assms: fixes x :: "V" assumes "\<And>y. y \<in> elts x \<Longrightarrow> P y" shows "P x"
globEANow: fixes n :: "nat"   and i :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "n \<le> i"   and "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<box>\<^sub>b\<gamma>)" shows "eval c t t' i \<gamma>"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a + (a + b) = a + b"
cut_less_singleton: fixes a :: "'a"   and t :: "'a" shows "{a} \<down>< t = (if a < t then {a} else {})"
alphaBoundOutputChain': fixes xvec :: "name list"   and yvec :: "name list"   and B :: "('a, 'b, 'c) boundOutput" assumes "length xvec = length yvec"   and "yvec \<sharp>* B"   and "yvec \<sharp>* xvec"   and "distinct yvec" shows "\<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*yvec\<rparr>[xvec yvec] \<bullet>\<^sub>v B"
L_B_def: fixes x :: "'a"   and y :: "'a" shows "L_B x y = seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y]"
pIH_2: fixes q :: "'b poly" shows "map_poly h (p + q) = map_poly h p + map_poly h q"
column_add_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and n :: "'e"   and m :: "'e"   and q :: "'d" shows "column_add A n m q = (\<chi>i j. if j = n then A $h i $h n + A $h i $h m * q else A $h i $h j)"
ennreal_less_top: fixes x :: "real" shows "ennreal x < top"
exhaust_disc: fixes process :: "'a process" obtains "isAction process"   | "isChoice process"
mono_plus': fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'a"   and b2 :: "'a" assumes "a1 \<sqsubseteq> b1"   and "a2 \<sqsubseteq> b2" shows "plus' a1 a2 \<sqsubseteq> plus' b1 b2"
open_submanifold: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set" assumes "submanifold charts k S" shows "open S"
f1: shows "4 \<le> rk {A, B, C, D}"
eq_iff_4: fixes name :: "name"   and fmla :: "fmla"   and namea :: "name"   and fmlaa :: "fmla" shows "(exi name fmla = exi namea fmlaa) = ([[atom name]]lst. fmla = [[atom namea]]lst. fmlaa)"
dim_vec_of_list: fixes as :: "'a list" shows "dim_vec (vec_of_list as) = length as"
lookup_present_eq_key': fixes k :: "'b"   and al :: "('b \<times> 'a) list"   and v :: "'a" assumes "lookup (\<lambda>x. x = k) al = Some v" shows "(k, v) \<in> set al"
fresh_name_fresh: fixes S :: "char list set" assumes "finite S" shows "fresh_name S \<notin> S"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>* v"   and "v \<rightarrow>\<^sup>* w" shows "u \<rightarrow>\<^sup>* w"
normalize_lcm_left: fixes a :: "'a"   and b :: "'a" shows "lcm (normalize a) b = lcm a b"
l'_def: shows "l' = proj2_line_through (Rep_hyp2 a') (Rep_hyp2 b')"
image_poly_subst_ideal_subset: fixes g :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and F :: "(('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "poly_subst g ` ideal F \<subseteq> ideal (poly_subst g ` F)"
nGt_le_conv: fixes x :: "'a"   and y :: "'a" shows "(comparator_of x y \<noteq> Gt) = tc.le x y"
assoc_simps_4: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "local.dom \<a>[f, g, h] = (local.dom f \<star> local.dom g) \<star> local.dom h"
HVZK: shows "local.\<Sigma>_protocols_base.HVZK"
reachable_awalkI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "H.awalk u p v" shows "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
oalist_inv_raw_Nil: fixes ko :: "'a key_order" shows "ko.oalist_inv_raw ko []"
eq_dual_iff: fixes x :: "'a"   and y :: "'a" shows "(dual x = dual y) = (x = y)"
pathfinish_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "pathfinish (reversepath g) = pathstart g"
map_pmf_cong: fixes p :: "'e pmf"   and q :: "'e pmf"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "map_pmf f p = map_pmf g q"
nonzero_inverse_inverse_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse (inverse a) = a"