hcju/isabelleps huggingface dataset:
less_then_max_plus2: fixes n :: "nat"   and x :: "nat" shows "n + 2 < max (n + 3) x"
StaticDownCastReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs @ [C] @ Cs'),s'\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs @ [C]),s'\<rangle>"
ran_def: fixes m :: "'e \<Rightarrow> 'd option" shows "ran m = {b. \<exists>a. m a = Some b}"
add_mult_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A + B) *\<^sub>v v = A *\<^sub>v v + B *\<^sub>v v"
is_processT7: fixes P :: "'c process" shows "\<forall>s t. s \<in> D P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> D P"
dg_Obj_if_Cod_vrange: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>)" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
intersects_sctns_spec_def: fixes X :: "'c set"   and sctns :: "'c sctn set" shows "intersects_sctns_spec X sctns = SPEC (\<lambda>b. b \<or> X \<inter> \<Union> (plane_of ` sctns) = {})"
open_segment_eq_empty': fixes a :: "'b"   and b :: "'b" shows "({} = {a<--<b}) = (a = b)"
complete_lattice_ow_axioms_def: fixes U :: "'a set"   and Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and top :: "'a" shows "complete_lattice_ow_axioms U Inf Sup le bot top \<equiv> ((\<forall>A x. A \<subseteq> U \<longrightarrow> x \<in> A \<longrightarrow> le (Inf A) x) \<and> (\<forall>A z. A \<subseteq> U \<longrightarrow> z \<in> U \<longrightarrow> (\<forall>x. x \<in> A \<longrightarrow> le z x) \<longrightarrow> le z (Inf A)) \<and> (\<forall>x A. x \<in> A \<longrightarrow> A \<subseteq> U \<longrightarrow> le x (Sup A))) \<and> (\<forall>A z. A \<subseteq> U \<longrightarrow> z \<in> U \<longrightarrow> (\<forall>x. x \<in> A \<longrightarrow> le x z) \<longrightarrow> le (Sup A) z) \<and> Inf {} = top \<and> Sup {} = bot"
Rep_range: shows "range Rep = {a. sort_of a \<in> range sort_fun}"
listset_RModule_Rsmult_closed: fixes Ms :: "'m set list"   and r :: "'r"   and ms :: "'m list" assumes "\<forall>M\<in>set Ms. RModule R (\<cdot>) M"   and "r \<in> R"   and "ms \<in> listset Ms" shows "map ((\<cdot>) r) ms \<in> listset Ms"
is_unit_gcd: fixes a :: "'a"   and b :: "'a" shows "is_unit (gcd a b) = coprime a b"
strict_implies_order: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<prec> a" shows "b \<preceq> a"
3: shows "to_nat (from_nat c) = to_nat (from_nat c)"
almost_full_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "almost_full_on P A = (\<forall>f\<in>SEQ A. good P f)"
simps: fixes a1 :: "'c list"   and a2 :: "'d list" shows "same_length a1 a2 = (a1 = [] \<and> a2 = [] \<or> (\<exists>xs ys x y. a1 = x # xs \<and> a2 = y # ys \<and> same_length xs ys))"
keys_restrict_indets_subset: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "keys (restrict_indets p) \<subseteq> restrict_indets_pp ` keys p"
valueChosen_equal_case: fixes s :: "state"   and b :: "nat"   and v :: "InputsOrNi"   and D :: "Disk set"   and p :: "Proc"   and ba :: "nat"   and w :: "InputsOrNi"   and Da :: "Disk set"   and pa :: "Proc" assumes "maxBalInp s b v"   and "D \<in> MajoritySet"   and "\<forall>d\<in>D. b \<le> bal (disk s d p)"   and "maxBalInp s ba w"   and "Da \<in> MajoritySet"   and "\<forall>d\<in>Da. ba \<le> bal (disk s d pa)"   and "b \<le> ba" shows "v = w"
GF_advice_a1: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and X :: "'b ltln set" assumes "\<F> \<phi> w \<subseteq> X"   and "w \<Turnstile>\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>"
G'_no_deadends: fixes v :: "'a" assumes "v \<in> V\<^bsub>G'\<^esub>" shows "\<not> G'.deadend v"
proper_designI: assumes "int (size \<B>\<^sup>C) \<noteq> 0" shows "proper_design \<V> \<B>\<^sup>C"
order_iff_strict: fixes a :: "'g"   and b :: "'g" shows "(a \<le> b) = (a < b \<or> a = b)"
scalars_zero_if_basis: fixes X :: "(('a, 'b) vec, 'b) vec"   and f :: "'b \<Rightarrow> 'a" assumes "is_basis (set_of_vector X)"   and "(\<Sum>i\<in>UNIV. f i *s X $ i) = 0" shows "\<forall>i\<in>UNIV. f i = (0::'a)"
term_to_nterm_alpha_equiv0: fixes \<Gamma>1 :: "name list"   and \<Gamma>2 :: "name list"   and t1 :: "Term_Class.term"   and s1 :: "name"   and s2 :: "name" assumes "length \<Gamma>1 = length \<Gamma>2"   and "distinct \<Gamma>1"   and "distinct \<Gamma>2"   and "wellformed' (length \<Gamma>1) t1"   and "fBall (frees t1 |\<union>| fset_of_list \<Gamma>1) (\<lambda>y. y \<le> s1)"   and "fdisjnt (fset_of_list \<Gamma>1) (frees t1)"   and "fBall (frees t1 |\<union>| fset_of_list \<Gamma>2) (\<lambda>y. y \<le> s2)"   and "fdisjnt (fset_of_list \<Gamma>2) (frees t1)" shows "alpha_equiv (fmap_of_list (zip \<Gamma>1 \<Gamma>2)) (fst (run_state (term_to_nterm \<Gamma>1 t1) s1)) (fst (run_state (term_to_nterm \<Gamma>2 t1) s2))"
drop_bv_prepend_help4: fixes x :: "nat"   and y :: "nat"   and b :: "'a" assumes "x \<le> y" shows "drop x (bv_prepend y b []) = bv_prepend (y - x) b []"
elims: fixes x :: "Core.term"   and y :: "(variable \<times> typ) set" assumes "fv x = y" obtains uu_ :: "String.literal" and uv_ :: "typ" where "x = Ct uu_ uv_" and "y = {}"   | v :: "variable" and T :: "typ" where "x = Fv v T" and "y = {(v, T)}"   | uw_ :: "nat" where "x = Bv uw_" and "y = {}"   | ux_ :: "typ" and body :: "Core.term" where "x = Abs ux_ body" and "y = fv body"   | t :: "Core.term" and u :: "Core.term" where "x = t $ u" and "y = fv t \<union> fv u"
mem_simps_9: fixes A :: "'c"   and C :: "'c set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
size_cartesian_product: fixes A :: "'b multiset"   and B :: "'c multiset" shows "size (A \<times># B) = size A * size B"
lm041: fixes X :: "'c set" shows "({X} \<in> partitionsUniverse) = (X \<noteq> {})"
algebraic_imp_algebraic_int: fixes p :: "int poly"   and x :: "'b" assumes "ipoly p x = (0::'b)"   and "p \<noteq> 0" shows "algebraic_int (of_int (lead_coeff p) * x)"
range_subset_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a \<le> b" shows "({a..b} \<subseteq> {c..d}) = (c \<le> a \<and> b \<le> d)"
nat_interval_union: fixes m :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "m \<le> n" shows "{i. i \<le> n \<and> P i} = {i. i \<le> m \<and> P i} \<union> {i. m < i \<and> i \<le> n \<and> P i}"
islimptI: fixes x :: "'c"   and S :: "'c set" assumes "\<And>T. \<lbrakk>x \<in> T; open T\<rbrakk> \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x" shows "x islimpt S"
intro: fixes M :: "'b measure" assumes "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)" shows "sigma_finite_measure M"
assms_3: shows "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root \<in> set ancestors"
xp: shows "fst (exec_step P h stk loc C M pc ics frs sh) = \<lfloor>xcp\<rfloor>"
Gcd_set_eq_fold: fixes xs :: "'a list" shows "normalization_euclidean_semiring_class.Gcd (set xs) = fold normalization_euclidean_semiring_class.gcd xs (0::'a)"
extremum_unique: fixes a :: "'b list" shows "(a \<le>p \<epsilon>) = (a = \<epsilon>)"
induct: fixes x1 :: "'a"   and x2 :: "'a"   and succs :: "'a \<Rightarrow> 'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x1 [succs]\<rightarrow>* x2"   and "\<And>g. P g g"   and "\<And>g g' g''. \<lbrakk>g' \<in> set (succs g); g' [succs]\<rightarrow>* g''; P g' g''\<rbrakk> \<Longrightarrow> P g g''" shows "P x1 x2"
M_supset: shows "timpl_closure t T \<subseteq> M"
aezdeltafun_decomp: fixes b :: "'a"   and a :: "'b" shows "b \<delta>\<delta> a = b \<delta>\<delta> (0::'b) * ((1::'a) \<delta>\<delta> a)"
sorted_descending_tail: fixes xs :: "'c list"   and x :: "'c" shows "sorted_descending (xs @ [x]) = (sorted_descending xs \<and> (\<forall>y\<in>set xs. x \<le> y))"
hs_of_list_def: fixes l :: "'c list" shows "hs_of_list l = heapify (of_list_tree l)"
fold_graph_determ: fixes z :: "'a"   and B :: "'a set"   and x :: "'a"   and y :: "'a" assumes "fold_graph f z B x"   and "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "y = x"
eucl_of_list_prod: fixes xs :: "real list" shows "eucl_of_list xs = (eucl_of_list (take DIM('b) xs), eucl_of_list (drop DIM('b) xs))"
summable_rabs: fixes f :: "nat \<Rightarrow> real" assumes "summable (\<lambda>n. \<bar>f n\<bar>)" shows "\<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
Suc: shows "deriv (monom P a n) b = [n] \<cdot> a \<otimes> b [^] (n - 1)"
is12_def_3: shows "is2 \<lhd> input_sizes m2"
partition_on_the_part_eq: fixes A :: "'c set"   and P :: "'c set set"   and x :: "'c"   and X :: "'c set" assumes "partition_on A P"   and "x \<in> X"   and "X \<in> P" shows "(THE X. x \<in> X \<and> X \<in> P) = X"
accp_wfPD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "wfP r" shows "Wellfounded.accp r x"
bisim_can_sync_preserve1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and t :: "'b"   and x1 :: "'a"   and m1 :: "'c"   and x2 :: "'g"   and m2 :: "'h"   and LT :: "('d + 'b + 'b) set" assumes "FWbisimulation final1 r1 final2 r2 bisim"   and "bisim t (x1, m1) (x2, m2)"   and "multithreaded_base.can_sync r1 t x1 m1 LT" shows "multithreaded_base.can_sync r2 t x2 m2 LT"
Ico_subset_Ioo_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linorder less_eq less" shows "(ord.atLeastLessThan less_eq less a b \<subseteq> ord.greaterThanLessThan less c b) = (less_eq b a \<or> less c a)"
continuous_on_sum: fixes I :: "'b set"   and S :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_on S (f i)" shows "continuous_on S (\<lambda>x. \<Sum>i\<in>I. f i x)"
simps_1: fixes f :: "'f" shows "mark_guards f Skip = Skip"
select_convs_1: fixes carrier :: "'d set"   and mult :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and more :: "'e" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
axioms_1: fixes f :: "'c \<Rightarrow> 'd"   and H :: "'c set"   and G :: "('c, 'e) monoid_scheme" assumes "weak_group_morphism f H G" shows "H \<lhd> G"
create_insert_p_s_ordered: fixes N :: "('c, 'd) message set"   and s :: "('c, 'd option) insert_message list"   and a :: "'c extended \<Rightarrow> 'c sort_key extended"   and m :: "('c, 'd) insert_message"   and n :: "nat"   and \<sigma> :: "'d"   and new_id :: "'c" assumes "is_associated_string N s"   and "a_conditions (insert_messages N) a"   and "Inr (Insert m) = create_insert s n \<sigma> new_id" shows "a (P m) < a (S m)"
IH: shows "(\<Union>x\<in>set (subst_list (subst v t) E). fv (fst x) \<union> fv (snd x)) \<subseteq> fv t \<union> (\<Union>x\<in>set E. fv (fst x) \<union> fv (snd x))"
mono_lub: fixes Y :: "('c \<Rightarrow> 'a) set"   and le_b :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "Complete_Partial_Order.chain (fun_ord (\<le>)) Y"   and "\<And>f. f \<in> Y \<Longrightarrow> monotone le_b (\<le>) f" shows "monotone le_b (\<le>) (fun_lub Sup Y)"
hyps: shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
le_of_comp_of_ords_gen: fixes lt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. lt x y \<Longrightarrow> le x y" shows "le_of_comp (comp_of_ords le lt) = le"
monoid_list_axioms: shows "monoid_list (+) (0::'a)"
permutes_image: fixes p :: "'d \<Rightarrow> 'd"   and S :: "'d set" assumes "p permutes S" shows "p ` S = S"
reachable_def: fixes c :: "com" shows "reachable c = {c'. \<exists>s t. (c, s) \<rightarrow>* (c', t)}"
le_imp_Ej_subset: fixes u :: "nat list"   and x :: "nat list"   and j :: "nat" assumes "u \<le>\<^sub>v x" shows "Ej j u \<subseteq> Ej j x"
fls_X_intpow_nonzero: fixes i :: "int" shows "fls_X_intpow i \<noteq> 0"
lfp_ordinal_induct_set: fixes f :: "'c set \<Rightarrow> 'c set"   and P :: "'c set \<Rightarrow> bool" assumes "mono f"   and "\<And>S. P S \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (\<Union> M)" shows "P (lfp f)"
path2_prefix_ex: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and m' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "m' \<in> set ns" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>m'" and "prefix ns' ns" and "m' \<notin> set (butlast ns')"
singleton_add_means_empty: fixes A :: "'a"   and \<Gamma> :: "'a multiset"   and B :: "'a" assumes "\<LM> A  \<RM> = \<Gamma> \<oplus> B" shows "\<Gamma> = \<Empt>"
nat_k_induct: fixes k :: "nat"   and P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<forall>i<k. P i"   and "\<forall>n\<^sub>0. (\<forall>i<k. P (n\<^sub>0 + i)) \<longrightarrow> P (n\<^sub>0 + k)" shows "P n"
as_1: shows "\<forall>i\<in>Basis. 0 < x \<bullet> i"
FiniteBroadcastEnvironment_axioms: shows "FiniteBroadcastEnvironment jkbp"
sort_key_msort_key_conv: fixes key :: "'b \<Rightarrow> 'c" shows "sort_key key = msort_key key"
succ_le_TC_iff: fixes j :: "V"   and i :: "V" assumes "Ord j" shows "(ZFC_in_HOL.succ i \<sqsubseteq> j) = (i \<sqsubset> j)"
fill_count_none: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and ns :: "nat list"   and ub :: "nat" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "ns \<noteq> []"   and "offs_pred ns ub xs index key mi ma"   and "length xs \<le> ub" shows "count (mset (fill xs ns index key ub mi ma)) None = ub - length xs"
Collect_extract_image: fixes g :: "'e \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "{g (f x y) |x y. P x y} = g ` {f x y |x y. P x y}"
cones_map_ide: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "diagram J C D"   and "\<chi> \<in> {\<chi>. Limit.cone J C D a \<chi>}" shows "(\<lambda>\<chi>\<in>{\<chi>. Limit.cone J C D (partial_magma.cod C a) \<chi>}. \<lambda>j. if partial_magma.arr J j then C (\<chi> j) a else partial_magma.null C) \<chi> = \<chi>"
simps: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" shows "Optimal_BST.min_wpl w i j = (if j < i then 0 else (MIN k\<in>{i..j}. Optimal_BST.min_wpl w i (k - 1) + Optimal_BST.min_wpl w (k + 1) j) + w i j)"
vwb_lens_wb: fixes x :: "'c \<Longrightarrow> 'd" assumes "vwb_lens x" shows "wb_lens x"
hom_dvd_hom: fixes x :: "'a poly poly"   and y :: "'a poly poly" shows "(poly_y_x x dvd poly_y_x y) = (x dvd y)"
eats_cong_ax_def: shows "eats_cong_ax = Var X1 EQ Var X2 AND Var X3 EQ Var X4 IMP Eats (Var X1) (Var X3) EQ Eats (Var X2) (Var X4)"
image_triv: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Reflect Bet Cong A A A B"
that: fixes A :: "'a multiset"   and C :: "'a zmultiset"   and B :: "'a multiset" assumes "M = zmset_of A + C"   and "N = zmset_of B + C" shows "thesis"
bounded_diff: fixes S :: "'b set"   and T :: "'b set" assumes "bounded S" shows "bounded (S - T)"
integral: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>"
simps_1: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and allowed :: "'a word set"   and denied1 :: "'a word set" shows "no_spoofing_algorithm iface ipassmt [] allowed denied1 = (allowed - denied1 \<subseteq> ipcidr_union_set (set (the (ipassmt iface))))"
simps_2: fixes x :: "'b"   and first :: "indices"   and "next" :: "indices list"   and u :: "'b trm"   and S :: "'b trm" shows "replace_subterm (Var x) (first # next) u S = False"
subtensor_def: fixes A :: "'b tensor"   and i :: "nat" shows "subtensor A i = tensor_from_vec (tl (dims A)) (fixed_length_sublist (vec A) (prod_list (tl (dims A))) i)"
prefix_fin_prefix_fininf_trans: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a stream" assumes "u \<le> v"   and "v \<le>\<^sub>F\<^sub>I w" shows "u \<le>\<^sub>F\<^sub>I w"
list_all_append: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'e list" shows "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
below_rel_def: fixes A :: "('c \<times> 'd) set" shows "\<langle>A\<rangle>below_rel = \<langle>A\<rangle>sctn_rel O br below_halfspace top"
assms_3: fixes A :: "'a set" assumes "A \<in> sets M" shows "emeasure M (T -` A \<inter> space M) = emeasure M A"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "dnf (\<phi> and\<^sub>n \<psi>) = dnf \<phi> \<otimes> dnf \<psi>"
funion_least: fixes A :: "'a fset"   and C :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| C"   and "B |\<subseteq>| C" shows "A |\<union>| B |\<subseteq>| C"
count_mset_set_leq: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "count (mset_set A) x \<le> 1"
cobounded2: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "linordered_monoid.min a b \<preceq> b"
stut_noteq: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "STUTINV F"   and "STUTINV G" shows "STUTINV F \<noteq> G"
mono_generate_field: fixes I :: "'a set"   and J :: "'a set" assumes "I \<subseteq> J"   and "J \<subseteq> carrier R" shows "generate_field R I \<subseteq> generate_field R J"
a_mod_le_b_s: shows "a mod m \<le> b"
gen_sqrt_upper_scaled: fixes x :: "real"   and u :: "real"   and n :: "nat" assumes "0 \<le> x"   and "0 < u" shows "sqrt x \<le> sqrtu (x * u\<^sup>2) n / u"
sees_methods_rec: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option"   and Mm' :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D sees_methods Mm"   and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)" shows "P \<turnstile> C sees_methods Mm'"
induct: fixes x :: "real"   and P :: "real \<Rightarrow> bool" assumes "x \<in> radical_sqrt"   and "\<And>x. x \<in> \<rat> \<Longrightarrow> P x"   and "\<And>x. \<lbrakk>x \<in> radical_sqrt; P x\<rbrakk> \<Longrightarrow> P (- x)"   and "\<And>x. \<lbrakk>x \<in> radical_sqrt; P x; x \<noteq> 0\<rbrakk> \<Longrightarrow> P (1 / x)"   and "\<And>x y. \<lbrakk>x \<in> radical_sqrt; P x; y \<in> radical_sqrt; P y\<rbrakk> \<Longrightarrow> P (x + y)"   and "\<And>x y. \<lbrakk>x \<in> radical_sqrt; P x; y \<in> radical_sqrt; P y\<rbrakk> \<Longrightarrow> P (x * y)"   and "\<And>x. \<lbrakk>x \<in> radical_sqrt; P x; 0 \<le> x\<rbrakk> \<Longrightarrow> P (sqrt x)" shows "P x"
simps_3: fixes n :: "'a itrm"   and n' :: "'a itrm"   and x :: "'a itrm" shows "norm_nn n (n' \<diamondop> x) = norm_nn (norm_pn \<B> n) n' \<diamondop> x"
topology_bases_eq_aux: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and Q :: "'c set \<Rightarrow> bool" assumes "(arbitrary union_of P) S"   and "\<And>U x. \<lbrakk>P U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>V. Q V \<and> x \<in> V \<and> V \<subseteq> U" shows "(arbitrary union_of Q) S"
INF_mono: 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 B :: "'c set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. less_eq (f n) (g m)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
ab_group_add_ow_axioms: shows "ab_group_add_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w (-\<^sub>o\<^sub>w) uminus"
degree_map_poly_hom: fixes p :: "int poly" shows "degree (of_int_poly p) = degree p"
Nat2nat_def: shows "Nat2nat \<equiv> inv nat2Nat"
protCl_prot: fixes \<sigma> :: "'tp" assumes "protCl \<sigma>" shows "prot \<sigma>"
InitState_States: fixes A :: "('a, 'b, 'c) seqauto" shows "InitState A \<in> States A"
lexord_def: fixes r :: "('g \<times> 'g) set" shows "lexord r = {(x, y). \<exists>a v. y = x @ a # v \<or> (\<exists>u a b v w. (a, b) \<in> r \<and> x = u @ a # v \<and> y = u @ b # w)}"
simps_2: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and k' :: "'a"   and v' :: "'c"   and xs :: "('a \<times> 'c) list" shows "update_by_fun_pair k f ((k', v') # xs) = (case comp k k' of Eq \<Rightarrow> let v = f v' in if v = (0::'c) then xs else (k, v) # xs | Lt \<Rightarrow> let v = f (0::'c) in if v = (0::'c) then (k', v') # xs else (k, v) # (k', v') # xs | Gt \<Rightarrow> (k', v') # update_by_fun_pair k f xs)"
simps_2: fixes A :: "'b mat"   and d :: "nat"   and j :: "nat"   and pv :: "'b poly vec"   and n :: "nat" shows "mk_poly2_row A d j pv (Suc n) = mk_poly2_row A d j pv n |\<^sub>v n \<mapsto> pv $ n + monom (A $$ (n, j)) d"
take_bit_of_int: fixes n :: "nat"   and k :: "int" shows "take_bit n (of_int k) = of_int (take_bit n k)"
gFreshInp_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme"   and ys :: "'c"   and y :: "'f"   and inp' :: "'j \<Rightarrow> ('a, 'b, 'c, 'f, 'e) qTerm set option"   and inp :: "'j \<Rightarrow> 'g option" shows "gFreshInp MOD ys y inp' inp \<equiv> liftAll2 (gFresh MOD ys y) inp' inp"
to_nat_le: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and y :: "'d"   and k :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less y (mod_type.from_nat Abs k)" shows "mod_type.to_nat Rep y < k"