hcju/isabelleps huggingface dataset:
in_keys_map_indetsE: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'d \<Rightarrow> 'b"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys (map_indets f p)" obtains s :: "'d \<Rightarrow>\<^sub>0 nat" where "s \<in> keys p" and "t = (\<Sum>x\<in>keys s. monomial (lookup s x) (f x))"
\<i>_simps_5: fixes a :: "'a" assumes "B.obj a" shows "cod (UP \<i>\<^sub>B[a] \<cdot> cmp\<^sub>U\<^sub>P (a, a)) = UP a"
Subset_Eats_sf: fixes t :: "tm"   and u :: "tm"   and i :: "name" assumes "\<And>j. Sigma_fm (Var j IN t)"   and "\<And>k. Sigma_fm (Var k EQ u)" shows "Sigma_fm (Var i SUBS Eats t u)"
pred_prod_beta: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and xy :: "'c \<times> 'd" shows "pred_prod P Q xy = (P (fst xy) \<and> Q (snd xy))"
one_fresh_preserv: fixes t :: "lam"   and s :: "lam"   and a :: "name" assumes " t \<longrightarrow>\<^sub>1 s"   and "a \<sharp> t" shows "a \<sharp> s"
select_convs_2: fixes ifaces :: "idecl list"   and classes :: "cdecl list"   and more :: "'a" shows "classes \<lparr>ifaces = ifaces, classes = classes, \<dots> = more\<rparr> = classes"
carrier_subgroup_generated: fixes G :: "('c, 'd) monoid_scheme"   and S :: "'c set" shows "carrier (subgroup_generated G S) = generate G (carrier G \<inter> S)"
br_run_sim: fixes \<alpha> :: "('c, 'ls, 'gs) pid_global_config \<Rightarrow> ('c, 'ls, 'gs) global_config"   and I :: "('c, 'ls, 'gs) pid_global_config \<Rightarrow> bool"   and r :: "nat \<Rightarrow> ('c, 'ls, 'gs) pid_global_config" assumes "br pidgc_\<alpha> (\<lambda>_. True) = br \<alpha> I"   and "pid.is_run r" shows "sa.is_run (\<alpha> \<circ> r)"
abs_of_nat: 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (semiring_1.of_nat one plus zero n) = semiring_1.of_nat one plus zero n"
cospan_\<mu>\<nu>: shows "C.cospan \<mu>\<nu>.leg0 \<pi>.leg1"
state_set_thm: fixes s1 :: "'a list"   and s2 :: "'a list" shows "(s1 \<in> state_set s2) = (prefix s1 s2 \<and> s1 \<noteq> [])"
U_def: shows "U \<equiv> map (\<lambda>i. SOME u. (T ! i, u) \<in> timpl_closure' c \<and> (S ! i, u) \<in> timpl_closure' c) [0..<length T]"
equivp_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "equivp R = (\<forall>x y. R x y = (R x = R y))"
find_first_none: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "find_first x xs = None"
col: shows "addcol a l k A = A * addrow_mat n a k l"
insert_prepend: fixes i :: "nat"   and j :: "nat"   and a :: "'c"   and l :: "'c list"   and b :: "'c" assumes "i = Suc j" shows "list_insert (a # l) i b = a # list_insert l j b"
trivial_phis_def: fixes g :: "'g" shows "trivial_phis g = Set.filter (\<lambda>(n, v). trivial_code v (the (Mapping.lookup (phis g) (n, v)))) (Mapping.keys (phis g))"
left_commute: fixes b :: "'b upred"   and a :: "'b upred"   and c :: "'b upred" shows "(b \<and> a \<and> c) = (a \<and> b \<and> c)"
card2_sunflower: fixes S :: "'b set set" assumes "finite S"   and "card S \<le> 2" shows "sunflower S"
get_ancestors_same_root_node: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and ptr'' :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors"   and "ptr'' \<in> set ancestors" shows "h \<turnstile> Shadow_DOM.get_root_node ptr' \<rightarrow>\<^sub>r root_ptr = h \<turnstile> Shadow_DOM.get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr"
int_fw_upd: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "dbm_int m n"   and "k \<le> n"   and "i \<le> n"   and "j \<le> n"   and "i' \<le> n"   and "j' \<le> n"   and "fw_upd m k i j i' j' \<noteq> \<infinity>" shows "get_const (fw_upd m k i j i' j') \<in> \<int>"
diff_diff_ennreal: fixes a :: "ennreal"   and b :: "ennreal" assumes "a \<le> b"   and "b \<noteq> \<infinity>" shows "b - (b - a) = a"
assms_3: fixes n :: "nat" shows "f \<triangleright> n = g \<triangleright> n"
left_idem: fixes a :: "'a"   and b :: "'a" shows "sup a (sup a b) = sup a b"
carrier_possible: fixes t :: "'b ttree"   and x :: "'b" assumes "possible t x" shows "x \<in> carrier t"
A: shows "0 < offs_num (length ns) xs index key mi ma j"
linear_cmul: fixes f :: "'a \<Rightarrow> 'b"   and r :: "real"   and b :: "'a" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
elims_3: fixes x :: "const"   and xa :: "const" assumes "\<not> x \<le> xa" obtains x :: "const" and y :: "const" where "xa__ = x" and "xa = y" and "\<not> (y = Any \<or> x = y)"
g2_scale_j_contin: shows "continuous_on (cbox a b) (\<lambda>x. (0::'a, g2 x))"
rel_4: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" assumes "yun_rel F c f" shows "F \<noteq> 0"
nfa_imp_regular: fixes M :: "'b nfa"   and L :: "'b list set" assumes "nfa M"   and "nfa.language M = L" shows "regular L"
LR: shows "(l, l') \<in> \<langle>R\<rangle>list_rel"
assms_2: shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"
WHILET_def: shows "WHILET \<equiv> WHILEIT (\<lambda>_. True)"
sin_arcsin: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "sin (arcsin y) = y"
segment_split: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and z :: "'b" assumes "y \<in> segment f x z" shows "segment f x z = segment f x y \<union> {y} \<union> segment f y z"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('e, 'c) resource"   and b :: "('e, 'd) resource" shows "rel_resource' R a b = rel_fun (=) (rel_spmf (rel_prod R (rel_resource' R))) (run_resource a) (run_resource b)"
one_mult_mat_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "1\<^sub>m n *\<^sub>v v = v"
h_def: shows "h \<equiv> snd (normalize_quot (c, b))"
finite_image_set: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "finite {x. P x}" shows "finite {f x |x. P x}"
INum_int_1: fixes i :: "int" shows "INum (i)\<^sub>N = of_int i"
degree_less_sum': fixes p :: "real mpoly"   and var :: "nat"   and n :: "nat"   and q :: "real mpoly"   and m :: "nat" assumes "MPoly_Type.degree p var = n"   and "MPoly_Type.degree q var = m"   and "n < m" shows "MPoly_Type.degree (p + q) var = m"
connected_continuous_image: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "connected s" shows "connected (f ` s)"
ipd_1: shows "\<pi> m = ipd (\<pi> k)"
K_cfg_valid_cfgD_1: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and cfg' :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' = cont cfg (state cfg')"
left_hom_is_subcategory: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "subcategory (\<cdot>) (left \<mu>)"
snoc_outer_2: shows "ts @ [t] = ts1 @ ts2"
has_derivative_at': fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "(f has_derivative f') (at x) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow> norm (f x' - f x - f' (x' - x)) / norm (x' - x) < e))"
comp_null_1: fixes f :: "'a" shows "H.null \<star> f = H.null"
le_fin_ind': fixes a :: "'item"   and w :: "'item list"   and v :: "'item list" assumes "[a] \<preceq>\<^sub>F w"   and "v \<preceq>\<^sub>F w"   and "a \<notin> set v" shows "Ind {a} (set v)"
one_neq_zero: shows "(1::'b) \<noteq> (0::'b)"
\<alpha>\<beta>_trans: fixes s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s'' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" assumes "s \<approx> s'"   and "s' \<approx> s''" shows "s \<approx> s''"
Suc: shows "{l \<in> lists A. length l = n} \<lesssim> A"
new_shadow_root_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'm" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
rel_spmf_bind_reflI: fixes p :: "'h spmf"   and P :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i spmf"   and g :: "'h \<Rightarrow> 'j spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf P (f x) (g x)" shows "rel_spmf P (p \<bind> f) (p \<bind> g)"
ran_map_inv: fixes f :: "'c \<Rightarrow> 'd option" assumes "inj_on f (dom f)" shows "ran (map_inv f) = dom f"
reward_until_unfold: fixes X :: "'s set"   and s :: "'s"   and \<omega> :: "'s stream" shows "reward_until X s \<omega> = (if s \<in> X then 0 else \<rho> s + \<iota> s (shd \<omega>) + reward_until X (shd \<omega>) (stl \<omega>))"
s\<^sub>1s\<^sub>2': shows "s\<^sub>1 \<leadsto> s\<^sub>2'"
proves_descend_abs_rule_iff: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and t :: "Core.term"   and x :: "variable"   and \<tau>' :: "typ"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "is_closed s"   and "is_closed t"   and "(x, \<tau>') \<notin> FV \<Gamma>"   and "typ_ok \<Theta> \<tau>'"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "(\<Theta>,\<Gamma> \<turnstile> mk_eq s t) = (\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' s) (Abs_fv x \<tau>' t))"
m_closed: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d" assumes "subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
t_App_inversion: fixes \<Gamma> :: "(name \<times> ty) list"   and t1 :: "lam"   and t2 :: "lam"   and T :: "ty" assumes "\<Gamma> \<turnstile> App t1 t2 : T" shows "\<exists>T'. \<Gamma> \<turnstile> t1 : T' \<rightarrow> T \<and> \<Gamma> \<turnstile> t2 : T'"
not_\<omega>_lt_of_nat: fixes n :: "nat" shows "\<not> \<omega> < of_nat n"
assms_1: shows "x \<^sup>@ j = y \<^sup>@ k"
multi_to_arith_prog: fixes start :: "int"   and steps :: "nat \<Rightarrow> nat"   and c :: "nat \<Rightarrow> nat" shows "multi_arith_prog 1 start steps c = arith_prog start (steps 0) (c 0)"
parallel_reduction_unitE: fixes A :: "'a trm" assumes "Unit >> A" shows "A = Unit"
qtable_cong: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and B :: "nat set"   and Q' :: "'a option list \<Rightarrow> bool" assumes "qtable n A P Q X"   and "A = B"   and "\<And>v. P v \<Longrightarrow> Q v = Q' v" shows "qtable n B P Q' X"
is_OK_returns_heap_I: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and h' :: "'l" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h \<turnstile> ok f"
ivp_solsD_1: fixes X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "X \<in> Sols f U S t\<^sub>0 s" shows " D X = (\<lambda>t. f t (X t)) on U s"
h0: shows "\<forall>sgn. sgn \<in> set signs \<and> sgn \<notin> consistent_sign_vec qs ` set (characterize_root_list_p p) \<and> 0 < rat_of_nat (card {xa. poly p xa = 0 \<and> consistent_sign_vec qs xa = sgn}) \<longrightarrow> z_R I sgn = 0"
st_HFinite: fixes x :: "real star" assumes "x \<in> HFinite" shows "st x \<in> HFinite"
simps_442: fixes x10 :: "'b ltlc" shows "atoms_ltlc (G\<^sub>c x10) = atoms_ltlc x10"
AXIOMSc_def: fixes B :: "ty" shows "AXIOMSc B \<equiv> {<a>:Ax y b |a y b. True}"
isgcd_def: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and a :: "'c"   and b :: "'c" shows "x gcdof\<^bsub>G\<^esub> a b = (x divides\<^bsub>G\<^esub> a \<and> x divides\<^bsub>G\<^esub> b \<and> (\<forall>y\<in>carrier G. y divides\<^bsub>G\<^esub> a \<and> y divides\<^bsub>G\<^esub> b \<longrightarrow> y divides\<^bsub>G\<^esub> x))"
eqpi: fixes i :: "'b" assumes "i \<in> L" shows "fi i * (log b (1 / (1 / b powr real (cw_len i))) + log b (fi i)) = fi i * log b (fi i / (1 / b powr real (cw_len i)))"
pmf_expectation_bind: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c pmf"   and p :: "'b pmf"   and h :: "'c \<Rightarrow> 'd" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))"   and "set_pmf p \<subseteq> A" shows "measure_pmf.expectation (p \<bind> f) h = (\<Sum>a\<in>A. pmf p a *\<^sub>R measure_pmf.expectation (f a) h)"
A1: shows "A1 = mat (dim_row A) (dim_col A) (($$) (four_block_mat A B C D))"
emb_eq: shows "prod_emb UNIV M J (Pi\<^sub>E J F') = prod_emb UNIV M {..n} (Pi\<^sub>E {..n} F)"
diff_fun_on_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c" shows "diff_fun_on A f = (\<exists>W. A \<subseteq> W \<and> W \<subseteq> carrier \<and> open W \<and> (\<exists>f'. diff_fun k (charts_submanifold W) f' \<and> (\<forall>x\<in>A. f x = f' x)))"
bisim1BinOpThrow2: fixes P :: "'b prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and e1 :: "'addr expr1"   and bop :: "bop"   and v1 :: "'addr val" assumes "P,e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "P,e1 \<guillemotleft>bop\<guillemotright> e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk @ [v1], loc, length (compE2 e1) + pc, \<lfloor>a\<rfloor>)"
uncurry_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b \<times> 'c" shows "uncurry f a \<equiv> case a of (x, y) \<Rightarrow> f x y"
prod_closed: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "\<And>x. x \<in> X \<Longrightarrow> f x \<in> A" shows "prod f X \<in> A"
class_of_zero_rng_of_frac: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set"   and s :: "'c" assumes "eq_obj_rng_of_frac R S"   and "s \<in> S" shows "(\<zero>\<^bsub>R\<^esub> |\<^bsub>\<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>\<^esub> s) = \<zero>\<^bsub>eq_obj_rng_of_frac.rec_rng_of_frac R S\<^esub>"
r_list_encode_def: fixes n :: "nat" shows "r_list_encode n \<equiv> Cn (Suc n) S [Cn (Suc n) r_prod_encode [r_constn n n, r_tuple_encode n]]"
nrename_fresh_interesting2: fixes c :: "coname"   and M :: "trm"   and x :: "name"   and y :: "name" assumes "c \<sharp> M[x\<turnstile>n>y]" shows "c \<sharp> M"
right_inverse_linear: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and g :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "linear (*s) (*s) f"   and "f \<circ> g = id" shows "linear (*s) (*s) g"
eq_act: shows "vpeq (Step.partition tid) s1 s2"
times_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<cdot> B = {u \<cdot> v |u v. u \<in> A \<and> v \<in> B}"
equality: shows "k * (p ^ n choose k) = p ^ n * (p ^ n - 1 choose (k - 1))"
Gromov_extension_isometry_composition: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "isometry f"   and "isometry g" shows "Gromov_extension (g \<circ> f) = Gromov_extension g \<circ> Gromov_extension f"
a: shows " t1 \<longrightarrow>\<^sub>1\<^sup>* t2"
divisor_count_upper_bound_aux: fixes n :: "nat" shows "divisor_count n \<le> 2 * card {d. d dvd n \<and> real d \<le> sqrt (real n)}"
phis'_aux_finite: fixes phis :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and g :: "'g"   and v :: "'var"   and ns :: "'node set" assumes "finite (Mapping.keys phis)" shows "finite (Mapping.keys (phis'_aux g v ns phis))"
stake_strict_2: fixes i :: "Integer" shows "stake\<cdot>i\<cdot>\<bottom> = \<bottom>"
extended_complem: fixes S :: "'b formula set"   and F :: "'b formula" assumes "fin_sat S" shows "(F \<in> extended S) \<noteq> (\<^bold>\<not> F \<in> extended S)"
simps_4: fixes x :: "octo"   and y :: "octo" shows "Im3 (x + y) = Im3 x + Im3 y"
assms: shows "local.att (\<pi> i) \<noteq> None"
process_rule_4_futures: fixes ys :: "'e list"   and Y :: "'e set"   and P :: "'e process"   and xs :: "'e list"   and x :: "'e" assumes "(ys, Y) \<in> futures P xs" shows "(ys @ [x], {}) \<in> futures P xs \<or> (ys, insert x Y) \<in> futures P xs"
uniformly_continuous_imp_Cauchy_continuous: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and \<sigma> :: "nat \<Rightarrow> 'a" assumes "uniformly_continuous_on S f"   and "Cauchy \<sigma>"   and "\<And>n. \<sigma> n \<in> S" shows "Cauchy (f \<circ> \<sigma>)"
intro: fixes G :: "('a, 'b) Graph_scheme"   and v0 :: "'a"   and v1 :: "'a" assumes "Digraph G"   and "v0_v1_Digraph_axioms G v0 v1" shows "v0_v1_Digraph G v0 v1"
leq_refl: fixes x :: "'b llist" shows "lprefix x x"
dns1'': fixes x :: "('a \<times> 'a) set" shows "rel_ad (rel_ad x) ; x = x"
dfs_subset_reachable_visit_nodes: fixes ys :: "'b"   and xs :: "'a list" assumes "invariant ys"   and "list_all isNode xs" shows "nodeAbs ` set_of (dfs ys xs) \<subseteq> nodeAbs ` (reachable (set xs) \<union> set_of ys)"
invariant: fixes a :: "'b"   and b :: "'b" assumes "PB a"   and "B a b" shows "PB b"
sin_times_pi_eq_0: fixes x :: "real" shows "(sin (x * pi) = 0) = (x \<in> \<int>)"
VLambda_eqI: fixes X :: "V"   and Y :: "V"   and f :: "V \<Rightarrow> V"   and g :: "V \<Rightarrow> V" assumes "X = Y"   and "\<And>x. x \<in>\<^sub>\<circ> X \<Longrightarrow> f x = g x" shows "VLambda X f = VLambda Y g"
w_2: shows "norm (w - k1) < norm (k1 - k2) / 2"
l4_ports_negate_one_not_has_disc_negated_1: fixes ports :: "ipt_l4_ports" shows "\<not> has_disc_negated is_Src_Ports False (l4_ports_negate_one Src_Ports ports)"
fund_circuit_def: fixes x :: "'a"   and B :: "'a set" shows "fund_circuit x B \<equiv> THE C. circuit C \<and> C \<subseteq> insert x B"
sep: shows "separating \<Gamma> (TER (current_of_bipartite f))"
index_add_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>1 :: "'a vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 + v\<^sub>2) $ i = v\<^sub>1 $ i + v\<^sub>2 $ i"
min_galleryI_betw: fixes X :: "'b set set"   and x :: "'b set"   and y :: "'b set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "x \<noteq> y"   and "SimplicialComplex.maxsimpchain X (x # xs @ [y])"   and "\<And>ys. SimplicialComplex.maxsimpchain X (x # ys @ [y]) \<Longrightarrow> order.greater_eq (length ys) (length xs)" shows "SimplicialComplex.min_maxsimpchain X (x # xs @ [y])"
sign_S5_thm_3: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<diamond>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>& \<psi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<diamond>((\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<exists>\<alpha>. \<psi> \<alpha>)) in v]"
while_fixp_induct: fixes P :: "(bool \<times> nat \<Rightarrow> (bool \<times> nat) spmf) \<Rightarrow> bool" assumes "spmf.admissible P"   and "P (\<lambda>while. return_pmf None)"   and "\<And>while'. P while' \<Longrightarrow> P (\<lambda>s. if fst s then (case s of (b, x) \<Rightarrow> map_spmf (\<lambda>b'. (\<not> b', x + (if b' then 0 else 1))) (bernoulli p)) \<bind> while' else return_spmf s)" shows "P local.while"
pow_ord_eq_ord_iff: fixes a :: "'a"   and k :: "nat" assumes "finite (carrier (mult_of R))"   and "a \<in> carrier (mult_of R)" shows "(G.ord (a [^] k) = G.ord a) = coprime k (G.ord a)"
sub_inc_sysI: fixes \<U> :: "'a set"   and \<A> :: "'a set multiset" assumes "incidence_system \<U> \<A>"   and "\<U> \<subseteq> \<V>"   and "\<A> \<subseteq># \<B>" shows "sub_incidence_system \<U> \<A> \<V> \<B>"
of_int_abs: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sgn :: "'k \<Rightarrow> 'k"   and abs :: "'k \<Rightarrow> 'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and x :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "ring_1.of_int one plus zero minus \<bar>x\<bar> = abs (ring_1.of_int one plus zero minus x)"
cases: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" obtains "interaction_bound consider gpv \<le> n"
stalk_axioms: shows "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x"
gcd_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and a' :: "'a"   and b' :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "gcd a b \<noteq> zero"   and "a = times a' (gcd a b)"   and "b = times b' (gcd a b)" shows "algebraic_semidom.coprime times one a' b'"
cset_seq_ran: fixes A :: "'c cset" shows "ran (cset_seq A) = rcset A"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
diff_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "less_eq a b" shows "plus (minus b a) a = b"
b_aux_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "b_aux a b c = b \<sqinter> e_aux a b c \<squnion> d_aux a b c"
pth_b_2: fixes c :: "real"   and x :: "'c"   and z :: "'c"   and d :: "real"   and y :: "'c" shows "c *\<^sub>R x + z + d *\<^sub>R y \<equiv> c *\<^sub>R x + (z + d *\<^sub>R y)"
wrapC_def: shows "wrapC \<equiv> \<Lambda> g e. g\<cdot>e\<cdot>Just\<cdot>Nothing"
pw_EASSERT_2: fixes \<Phi> :: "bool"   and u :: "unit" shows "inres (EASSERT \<Phi>) (Inr u)"
distinct_1: fixes x1 :: "int"   and x2 :: "char list" shows "A x1 \<noteq> B x2"
bernoulli_pmf_0: shows "bernoulli_pmf 0 = return_pmf False"
N_gso: fixes i :: "nat" assumes "i < length fs" shows "\<parallel>gso i\<parallel>\<^sup>2 \<le> N"