hcju/isabelleps huggingface dataset:
nneg_bounded_by_nneg: fixes b :: "real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "0 \<le> b"
long_divides_def: fixes p :: "'a list list"   and q :: "'a list list"   and t :: "'a list list \<times> 'a list list" shows "UP.long_divides p q t = (t \<in> carrier (poly_ring (K [X])) \<times> carrier (poly_ring (K [X])) \<and> p = q \<otimes>\<^bsub>poly_ring (K [X])\<^esub> fst t \<oplus>\<^bsub>poly_ring (K [X])\<^esub> snd t \<and> (snd t = [] \<or> degree (snd t) < degree q))"
sorted_list_psub: fixes w :: "'a list" shows "Sorted_Less.sorted w \<longrightarrow> w \<noteq> [] \<longrightarrow> (sorted_list_of_set (set (tl w)), w) \<in> (\<lambda>(x, y). (sorted_list_of_set x, sorted_list_of_set y)) ` {(A, B). A \<subset> B \<and> finite B}"
field_poly_irreducible_dvd_mult: fixes p :: "'c poly"   and q :: "'c poly"   and r :: "'c poly" assumes "irreducible p" shows "(p dvd q * r) = (p dvd q \<or> p dvd r)"
defined_offending': fixes m :: "'b SecurityInvariant_configured"   and G :: "'b graph" assumes "configured_SecurityInvariant m"   and "wf_graph G"   and "\<not> c_sinvar m G" shows "c_offending_flows m G \<noteq> {}"
simps_6: fixes C' :: "char list"   and V :: "'a"   and e :: "'a exp" shows "not_init C' (V:=e) = not_init C' e"
sample_vars_insert2: fixes A :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and x :: "'v"   and m :: "'m" assumes "finite A" shows "sample_vars X (insert x A) m = sample_vars X A (bind (memo (sample_var X) x) (\<lambda>_. m))"
select_convs_1: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'a" shows "break_status \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = break_status"
fps_XD_add: fixes a :: "'a fps"   and b :: "'a fps" shows "fps_XD (a + b) = fps_XD a + fps_XD b"
well_order_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"
inf_def: fixes x :: "'b"   and y :: "'b" shows "inf x y = (if x \<le> y then x else y)"
induct: fixes P :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> bool"   and a0 :: "('b \<times> 'c) tree"   and a1 :: "'b" assumes "\<And>x. P \<langle>\<rangle> x"   and "\<And>l a uu_ r x. \<lbrakk>cmp x a = LT \<Longrightarrow> P l x; cmp x a = GT \<Longrightarrow> P r x\<rbrakk> \<Longrightarrow> P \<langle>l, (a, uu_), r\<rangle> x" shows "P a0 a1"
nat_add_left_cancel_le: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "order.greater_eq (k + n) (k + m) = order.greater_eq n m"
remdups_adj_obtain_adjacency: fixes i :: "nat"   and xs :: "'c list" assumes "i + 1 < length (remdups_adj xs)"   and "0 < length xs" obtains j :: "nat" where "j + 1 < length xs" and "remdups_adj xs ! i = xs ! j" and "remdups_adj xs ! (i + 1) = xs ! (j + 1)"
induct: fixes P :: "'b hp list \<Rightarrow> bool"   and a0 :: "'b hp list" assumes "\<And>x hsl hsr. \<lbrakk>P hsl; P hsr\<rbrakk> \<Longrightarrow> P (Hp x hsl # hsr)"   and "P []" shows "P a0"
a_de_morgan_var_3: fixes x :: "'a"   and y :: "'a" shows "ad (d x + d y) = ad x \<cdot> ad y"
map_spmf_const: fixes x :: "'a"   and p :: "'b spmf" shows "map_spmf (\<lambda>_. x) p = scale_spmf (weight_spmf p) (return_spmf x)"
assms: shows "\<exists>c>0. \<exists>n. \<forall>m>n. f m \<le> c * g m"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom \<t>[f] = local.map (local.dom f)"
measurable_restrict_space3: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and A :: "'d set"   and B :: "'e set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> A \<rightarrow> B" shows "f \<in> restrict_space M A \<rightarrow>\<^sub>M restrict_space N B"
index_of_pr_is_real: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1" shows "nat_to_pr (index_of_pr f) = f"
components_are_iso: fixes a :: "'b" assumes "D.ide a" shows "D.iso (trg\<^sub>D a)"
fromArr_def: fixes Arr :: "'a set"   and x :: "'a" shows "quiver.fromArr Arr x \<equiv> if x \<in> Arr then MkArr False True [x] else partial_magma.null (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g))"
Inter_greatest: fixes A :: "'d set set"   and C :: "'d set" assumes "\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X" shows "C \<subseteq> \<Inter> A"
assms_1: shows "injective (w \<sqinter> - q \<squnion> (w \<sqinter> q)\<^sup>T)"
ty_expr_is_type: fixes E :: "env"   and e :: "expr"   and T :: "ty" assumes "E\<turnstile>e\<Colon>-T"   and "wf_prog (prg E)" shows "is_type (prg E) T"
wf_qorder_axioms: shows "class.wf_qorder (\<le>) (<)"
subgroup_generated_minimal: fixes H :: "'a set"   and S :: "'a set" assumes "subgroup H G"   and "S \<subseteq> H" shows "carrier (subgroup_generated G S) \<subseteq> H"
T_vrestr_0_1: fixes A :: "'a set" assumes "A \<in> events" shows "(T ^^ 0) --` A = A"
rep_eq: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'd"   and xa :: "('e, 'c) uexpr"   and xb :: "('f, 'c) uexpr"   and xc :: "('g, 'c) uexpr"   and xd :: "('h, 'c) uexpr" shows "\<lbrakk>qtop x xa xb xc xd\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b) (\<lbrakk>xc\<rbrakk>\<^sub>e b) (\<lbrakk>xd\<rbrakk>\<^sub>e b))"
max_Sup_enat2: fixes Y :: "enat set"   and x :: "enat" assumes "Y \<noteq> {}" shows "max x (\<Squnion> Y) = \<Squnion> (max x ` Y)"
wf_def: fixes r :: "('k \<times> 'k) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
\<pi>_bounds_aux: fixes n :: "nat" assumes "1 \<le> n" shows "ln (fact (2 * n)) - 2 * ln (fact n) = prime_sum_upto (\<lambda>p. real_of_int (S n p) * ln (real p)) (real (2 * n))"
continuous_on_eq_continuous_within: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>x\<in>s. continuous (at x within s) f)"
minus_mod_eq_div_mult: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (modulo a b) = times (divide a b) b"
uncountable_disjoint_family_then_exists_zero_measure: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"   and "uncountable I"   and "disjoint_family_on A I" shows "\<exists>i\<in>I. Sigma_Algebra.measure M (A i) = 0"
flowD_KIR: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "flow \<Delta> f"   and "x \<noteq> source \<Delta>"   and "x \<noteq> sink \<Delta>" shows "KIR f x"
sum_mset_distrib_right: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and f :: "'l \<Rightarrow> 'k"   and M :: "'l multiset"   and c :: "'k" assumes "class.semiring_0 plus zero times" shows "times (comm_monoid_add.sum_mset plus zero (image_mset f M)) c = comm_monoid_add.sum_mset plus zero {#times (f x) c. x \<in># M#}"
rel_option_None1: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e option" shows "rel_option P None x = (x = None)"
insert_nth_induct: fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b"   and a2 :: "'b list" assumes "\<And>x xs. P 0 x xs"   and "\<And>n x y ys. P n x ys \<Longrightarrow> P (Suc n) x (y # ys)"   and "\<And>n x. P (Suc n) x []" shows "P a0 a1 a2"
arr_cod: fixes f :: "'a2" assumes "C2.arr f" shows "C2.arr (C2.cod f)"
lemma_2: fixes PROB :: "(('d, bool) fmap \<times> ('d, bool) fmap) set"   and s :: "('d, bool) fmap"   and as :: "(('d, bool) fmap \<times> ('d, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "2 ^ card (fmdom' s) - 1 < length as" shows "\<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []"
ate: shows "approx_tse prec x s c 1 f vs = Some ivl"
wadjust_le_def: shows "wadjust_le \<equiv> inv_image lex_square wadjust_measure"
group_isomorphisms_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b" shows "group_isomorphisms G H f g \<equiv> f \<in> hom G H \<and> g \<in> hom H G \<and> (\<forall>x\<in>carrier G. g (f x) = x) \<and> (\<forall>y\<in>carrier H. f (g y) = y)"
map_simp_ide: fixes a :: "'a" assumes "S.ide a" shows "\<ll>.map a = \<l>[a]"
interchange_rows_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and a :: "'d"   and b :: "'d" shows "interchange_rows A a b = (\<chi>i j. if i = a then A $ b $ j else if i = b then A $ a $ j else A $ i $ j)"
diff_fun_def: fixes k :: "enat"   and charts :: "('d, 'e) chart set"   and f :: "'d \<Rightarrow> 'f" shows "diff_fun k charts f \<equiv> diff k charts charts_eucl f"
assms_3: assumes "i \<notin>\<^sub>\<circ> I" shows "P"
degree_sum_le: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b poly"   and n :: "nat" assumes "finite S"   and "\<And>p. p \<in> S \<Longrightarrow> degree (f p) \<le> n" shows "degree (sum f S) \<le> n"
wfactors_factors: fixes as :: "'a list"   and a :: "'a" assumes "wfactors G as a"   and "set as \<subseteq> carrier G" shows "\<exists>a'. factors G as a' \<and> a' \<sim> a"
invar_reachable: fixes I :: "nat \<times> nat \<times> (nat \<Rightarrow> thread) \<Rightarrow> bool"   and s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "B.is_invar I"   and "B.reachable s" shows "I s"
NotBox: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<not> \<box>F) = (\<diamond>\<not> F)"
fact_dvd_poly_higher_pderiv_aux': fixes m :: "nat"   and n :: "nat"   and p :: "int poly"   and x :: "int" assumes "m \<le> n" shows "fact m dvd poly ((pderiv ^^ n) p) x"
mat_row_last_carrier: fixes A :: "'b mat"   and n :: "nat" shows "mat_row_last A n \<in> carrier_mat n (dim_col A)"
simps_10: fixes x5 :: "String.literal" shows "Class x5 \<noteq> Void"
eqvt_fresh_cong2: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "atom"   and x :: "'a"   and y :: "'b" assumes "\<And>p x y. p \<bullet> f x y = f (p \<bullet> x) (p \<bullet> y)"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> f x y"
countably_subadditive_outer_measure: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "increasing M f" shows "countably_subadditive (Pow \<Omega>) (outer_measure M f)"
group_axioms_def: fixes G :: "('d, 'e) monoid_scheme" shows "Group.group_axioms G \<equiv> carrier G \<subseteq> Units G"
en_def: shows "\<langle>c #\<^bsub>enat k\<^esub>t\<rangle> = enat en"
spans_image: fixes V :: "'b set"   and B :: "'b set" assumes "V \<subseteq> span B" shows "f ` V \<subseteq> span (f ` B)"
case_2: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat" shows "(case Read\<^sub>s\<^sub>b x21 x22 x23 x24 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f2 x21 x22 x23 x24"
spaceN_iff: fixes x :: "'b"   and N :: "'b quasinorm" shows "(x \<in> space\<^sub>N N) = (eNorm N x < \<infinity>)"
r_leap: fixes i :: "nat"   and f :: "recf"   and x :: "nat"   and t :: "nat" assumes "i = encode f"   and "recfn (e_length x) f" shows "eval r_leap [t, i, x] \<down>= encode_config (iterate t step ([(f, Partial_Recursive.list_decode x, [])], None))"
Mapping_lookup_filter_None: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'b"   and m :: "('b, 'a) mapping" assumes "\<And>v. \<not> P k v" shows "Mapping.lookup (Mapping.filter P m) k = None"
summable_iff_suminf_neq_top: fixes f :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> f n"   and "\<not> summable f" shows "(\<Sum>i. ennreal (f i)) = top"
DSourcesA31_L1: shows "DSources level1 sA31 = {sA22}"
has_integral_f: fixes n :: "nat" assumes "a \<le> real n" shows "(f n has_integral F (real n) - F a) {a..}"
multiplicity_normalize_left: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one (normalize p) x = factorial_semiring.multiplicity times one p x"
simps_14: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x1 :: "'d" shows "tmap f1 f2 (TNil x1) = TNil (f2 x1)"
represents_nth_root: fixes y :: "'a"   and x :: "'a"   and p :: "int poly" assumes "y ^ n = x"   and "p represents x" shows "local.poly_nth_root p represents y"
is_ring: fixes R :: "('a, 'b) ring_scheme" assumes "field R" shows "ring R"
int_of_uint32_def: shows "int_of_uint32 \<equiv> map_fun Rep_uint32 id uint"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Network c s t"   and "Preflow c s t f" shows "NPreflow c s t f"
less_\<omega>1_imp_countable: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>1" shows "countable (elts \<alpha>)"
sorted_descending: fixes xs :: "'c list" shows "sorted_descending (rev xs) = sorted xs"
to_bl_of_bin: fixes bin :: "int" shows "to_bl (word_of_int bin) = bin_to_bl LENGTH('b) bin"
normalize_quot_0_1: fixes x :: "'a" shows "normalize_quot (0::'a, x) = (0::'a, 1::'a)"
l_set_child_nodes_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<equiv> l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
simps_1: fixes Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Length :: "'b \<Rightarrow> nat"   and satisfies0 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'b"   and b :: "bool" shows "Formula_Operations.satisfies_gen Extend Length satisfies0 r \<AA> (FBool b) = b"
agree_state_succ_idempot: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" assumes "a \<in> PROB"   and "s \<in> valid_states PROB"   and "agree (snd a) s" shows "state_succ s a = s"
cauchy_ine\<^sub>N_def: fixes N :: "'a quasinorm"   and u :: "nat \<Rightarrow> 'a" shows "cauchy_ine\<^sub>N N u = (\<forall>e>0. \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. eNorm N (u n - u m) < e)"
of_int_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 1 = one"
red_dfs_inv_def: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and reds :: "'a set"   and onstack :: "'a set" shows "red_dfs_inv E U reds onstack \<equiv> E `` U \<subseteq> U \<and> finite U \<and> reds \<subseteq> U \<and> E `` reds \<subseteq> reds \<and> E `` reds \<inter> onstack = {}"
transitionI_1: fixes P :: "pi"   and P''' :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'''"   and "P''' \<longmapsto> \<alpha> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<alpha> \<prec> P'"
sos: shows "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"
permutes_lt: fixes p :: "'a \<Rightarrow> 'a"   and n :: "'a"   and i :: "'a" assumes "p permutes {..<n}"   and "i < n" shows "p i < n"
is_run_def: fixes r :: "nat \<Rightarrow> 'v" shows "is_run r \<equiv> r 0 \<in> V0 \<and> ipath E r"
simps_3: fixes v :: "nat"   and va :: "'a"   and vb :: "'a list" shows "chop (Suc v) (va # vb) = take (Suc v) (va # vb) # chop (Suc v) (drop (Suc v) (va # vb))"
write_cpu_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and cr :: "CPU_register"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu w cr s1)) \<and> t2 = snd (fst (write_cpu w cr s2))" shows "low_equal t1 t2"
add_match_simp: fixes m :: "'c match_expr" shows "add_match m = map (\<lambda>r. Rule (MatchAnd m (get_match r)) (get_action r))"
IH: fixes q :: "'c" assumes "set xs \<subseteq> inputs M"   and "completely_specified M"   and "q \<in> nodes M" shows "LS\<^sub>i\<^sub>n M q {xs} \<noteq> {}"
distrib: fixes g :: "'l \<Rightarrow> 'k"   and h :: "'l \<Rightarrow> 'k"   and A :: "'l set" shows "(\<Sum>x\<in>A. g x + h x) = sum g A + sum h A"
fst_snd_Gauss_Jordan_column_k_iarrays_PA_eq: fixes P :: "'b iarray iarray"   and i :: "nat"   and A :: "'b iarray iarray"   and k :: "nat" shows "fst (snd (Gauss_Jordan_column_k_iarrays_PA (P, i, A) k)) = fst (Gauss_Jordan_column_k_iarrays (i, A) k)"
card_of_UNIV2: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "r \<le>o |UNIV|"
Rel_def: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "Rel r \<equiv> r"
large: shows "max \<bar>resultant f (pderiv f)\<bar> (max \<bar>lead_coeff f\<bar> \<bar>lead_coeff (pderiv f)\<bar>) < int CARD('a)"
map_id: fixes t :: "('d, 'e, 'f) cntxt" shows "\<R>\<^sub>C id id t = t"
bis_coind2: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set"   and theta :: "('b \<times> 'b) set" assumes "Bisim.mono Retr"   and "theta \<subseteq> Retr (theta \<union> bis Retr)"   and "theta\<inverse> \<subseteq> Retr (theta\<inverse> \<union> bis Retr)" shows "theta \<subseteq> bis Retr"
length_prefix: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "length (prefix f n) = Suc n"
inf_bot_right: fixes x :: "'f multiset" shows "x \<inter># {#} = {#}"
has_products_preserved_by_bijection: fixes I :: "'b set"   and \<phi> :: "'b \<Rightarrow> 'd"   and I' :: "'d set" assumes "has_products I"   and "bij_betw \<phi> I I'"   and "I' \<noteq> UNIV" shows "has_products I'"
filter_iii: fixes F :: "'a set" assumes "F \<in> filters" shows "(1::'a) \<in> F"
intros_2: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and conf' :: "'b"   and T :: "'b list"   and final :: "'b"   and conf :: "'b" assumes "traces.trace step conf' T final"   and "step conf conf'" shows "traces.trace step conf (conf' # T) final"
top_lifted_boolean_algebra_def: shows "top \<equiv> Abs_lifted_boolean_algebra (\<lambda>f. top)"
eq_tpoly_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and q :: "('a pvar, hmultiset) tpoly"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.eq_tpoly ground_heads_var \<delta> arity_sym wt_sym q p = (\<forall>A. tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A \<longrightarrow> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of q) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p))"
Bag_s_mul_ext: fixes xs :: "('b, nat) alist"   and ys :: "('b, nat) alist"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" shows "((Bag xs, Bag ys) \<in> s_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = fst (mul_ext f (ass_list_to_single_list (alist.impl_of xs)) (ass_list_to_single_list (alist.impl_of ys)))"
assms: shows "k < length (trace_parallel_plan_sas_plus I \<pi>) - 1"
assms: shows "homology_group 0 X \<cong> homology_group 0 Y"
Ps_eq: fixes cf1 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and cf2 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and s :: "'state" assumes "AE bT in T cf1. qsend (cf1 ## bT) < \<infinity>"   and "AE bT in T cf2. qsend (cf2 ## bT) < \<infinity>"   and "proper (fst cf1)"   and "proper (fst cf2)"   and "fst cf1 \<approx>01 fst cf2"   and "snd cf1 \<approx> snd cf2" shows "T.prob cf1 {bT \<in> space (T cf1). \<exists>n. qsend (cf1 ## bT) = enat n \<and> snd ((cf1 ## bT) !! n) \<approx> s} = T.prob cf2 {bT \<in> space (T cf2). \<exists>n. qsend (cf2 ## bT) = enat n \<and> snd ((cf2 ## bT) !! n) \<approx> s}"
induct: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat \<Rightarrow> nat" assumes "\<And>d. P 0 d"   and "\<And>n d. P n (d \<circ> Suc) \<Longrightarrow> P (Suc n) d" shows "P a0 a1"