hcju/isabelleps huggingface dataset:
of_int_uint: fixes w :: "'b word" shows "of_int (uint w) = unsigned w"
language_state_prefix: fixes w1 :: "('d \<times> 'e) list"   and w2 :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f" assumes "w1 @ w2 \<in> LS M q" shows "w1 \<in> LS M q"
exists_def: fixes f :: "'a \<Rightarrow> ??.Quickcheck_Narrowing.property" shows "??.Quickcheck_Narrowing.exists f = (case narrowing 100 of ??.Quickcheck_Narrowing.narrowing_cons.Narrowing_cons ty cs \<Rightarrow> Existential ty (\<lambda>t. f (??.Quickcheck_Narrowing.conv cs t)) (partial_term_of TYPE('a)))"
Pair_inject: fixes a :: "'a"   and b :: "'c"   and a' :: "'a"   and b' :: "'c" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
tendsto_abs_ereal: fixes u :: "'a \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'a filter" assumes "(u \<longlongrightarrow> l) F" shows "((\<lambda>n. \<bar>u n\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
funpow_dist1_le_self: fixes m :: "nat"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "(f ^^ m) x = x"   and "0 < m"   and "y \<in> orbit f x" shows "funpow_dist1 f x y \<le> m"
Roots_def: shows "Roots \<equiv> \<lambda>p. {\<alpha>. ipoly p \<alpha> = 0}"
sublens'_prop3: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and \<rho> :: "'e"   and v :: "'d" assumes "vwb_lens X"   and "vwb_lens Y"   and "X \<subseteq>\<^sub>L' Y" shows "put\<^bsub>Y\<^esub> \<sigma> (get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<rho> (get\<^bsub>Y\<^esub> \<sigma>)) v)) = put\<^bsub>X\<^esub> \<sigma> v"
alphaAbs_ex_equal_or_qAFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_ex_equal_or_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
nn_integral_dominated_convergence_norm: fixes u :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure"   and u' :: "'c \<Rightarrow> 'd"   and w :: "'c \<Rightarrow> real" assumes "\<And>i. u i \<in> borel_measurable M"   and "u' \<in> borel_measurable M"   and "w \<in> borel_measurable M"   and "\<And>j. AE x in M. norm (u j x) \<le> w x"   and "\<integral>\<^sup>+ x. ennreal (w x) \<partial>M < \<infinity>"   and "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x" shows "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (u' x - u i x)) \<partial>M) \<longlonglongrightarrow> 0"
ipset_from_cidr_not_empty: fixes base :: "'a word"   and len :: "nat" shows "ipset_from_cidr base len \<noteq> {}"
thmD11: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and X :: "symbol \<times> character list"   and T :: "(symbol \<times> character list) set"   and q :: "(symbol \<times> character list) list"   and N :: "symbol"   and \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and r :: "nat"   and \<gamma> :: "symbol list"   and k' :: "nat"   and x :: "item"   and I :: "item set" assumes "p \<in> \<PP>"   and "charslength p = k"   and "X \<in> T"   and "T \<subseteq> \<X> k"   and "q = p @ [X]"   and "(N, \<alpha> @ \<beta>) \<in> \<RR>"   and "r \<le> length q"   and "leftderives [\<SS>] (terminals (take r q) @ [N] @ \<gamma>)"   and "leftderives \<alpha> (terminals (drop r q))"   and "k' = k + length (chars_of_token X)"   and "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r q)) k'"   and "I = items_le k' (\<pi> k' {} (Scan T k (Gen (Prefixes p))))" shows "x \<in> I"
call_return_edges: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a" shows "\<exists>Q' f'. kind a' = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"
intros: fixes xss :: "'d list list"   and yss :: "'d list list" assumes "list_all2 (\<lambda>xs ys. xs = ys \<or> xs = ys @ ys) xss yss" shows "pcancel1 (concat xss) (concat yss)"
vinsert_def: fixes x :: "V"   and y :: "V" shows "vinsert x y \<equiv> set (insert x (elts y))"
lsum_simps_1: fixes f :: "'a \<Rightarrow> nat" shows "lsum f [] = 0"
simplicial_chain_def: fixes p :: "nat"   and S :: "('d \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real) \<Rightarrow>\<^sub>0 'e" shows "simplicial_chain p S c \<equiv> Poly_Mapping.keys c \<subseteq> Collect (simplicial_simplex p S)"
fun_left_comm: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y \<^bold>* (x \<^bold>* z) = x \<^bold>* (y \<^bold>* z)"
SCConca: fixes p :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "singleCombinators p"   and "singleCombinators [a]" shows "singleCombinators (a # p)"
strict_wary_subst_def: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" shows "strict_wary_subst \<rho> = (\<forall>x. wary (\<rho> x) \<and> arity (\<rho> x) \<in> {arity_var x, \<infinity>} \<and> local.ground_heads (head (\<rho> x)) \<subseteq> ground_heads_var x)"
intro: fixes \<G> :: "'b cyclic_group" assumes "cyclic_group \<G>" shows "list_cdh \<G>"
lmap_lappend_distrib: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist"   and ys :: "'c llist" shows "lmap f (lappend xs ys) = lappend (lmap f xs) (lmap f ys)"
lcm_mult_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "lcm (a * c) (b * c) = normalize (lcm b a * c)"
closed_Collect_imp: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}"   and "topological_space.closed open {x. Q x}" shows "topological_space.closed open {x. P x \<longrightarrow> Q x}"
empty_iff: fixes c :: "'c" shows "(c \<in> {}) = False"
ht_hash_update: fixes ls :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b" assumes "ht_hash ls" shows "ht_hash (abs_update k v ls)"
LIMSEQ_imp_Suc: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l" shows "f \<longlonglongrightarrow> l"
size_height_log: fixes kdt :: "'b kdt" shows "log 2 (real (KD_Tree.size_kdt kdt)) \<le> real (height kdt)"
tree_of_embed_commute: fixes e :: "'c \<Rightarrow> 'a"   and p :: "('c \<times> 'c rose_tree list \<times> 'c rose_tree list) list"   and t :: "'c rose_tree" shows "tree_of_zipper\<^sub>m (embed_path e p, embed_source_tree e t) = embed_source_tree e (tree_of_zipper (p, t))"
infer_from_def: fixes CC :: "'b literal multiset set"   and \<gamma> :: "'b Inference_System.inference" shows "old_infer_from CC \<gamma> = (set_mset (old_side_prems_of \<gamma> + {#old_main_prem_of \<gamma>#}) \<subseteq> CC)"
assms_1: shows "(f \<longlongrightarrow> a) F"
NotDSource: fixes S :: "CSet"   and i :: "AbstrLevelsID" assumes "\<forall>x\<in>OUT S. \<forall>Z\<in>AbstrLevel i. x \<notin> IN Z" shows "\<forall>C\<in>AbstrLevel i. S \<notin> DSources i C"
simps_1: fixes S :: "Runs.var set"   and x :: "Runs.var" shows "in_progress (Some S) x = (x \<in> S)"
sgn_inf_sym: fixes p :: "real poly" shows "sgn_pos_inf (p \<circ>\<^sub>p [:0, - 1:]) = sgn_neg_inf p"
two_prod_lists_cases_snoc_Cons: fixes as :: "('a \<times> 'b) list"   and bs :: "('c \<times> 'd) list" obtains xs :: "('a \<times> 'b) list" where "as = xs__" and "bs = []"   | ys :: "('c \<times> 'd) list" where "as = []" and "bs = ys__"   | xs :: "('a \<times> 'b) list" and aa :: "'a" and ba :: "'b" and ab :: "'c" and bb :: "'d" and ys :: "('c \<times> 'd) list" where "as = xs @ [(aa, ba)] \<and> bs = (ab, bb) # ys"
inext_mono2_infin: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I"   and "infinite I" shows "n < inext n I"
mignotte_helper_coeff: fixes h :: "complex poly"   and i :: "nat" shows "cmod (poly.coeff h i) \<le> real (degree h - 1 choose i) * mahler_measure_poly h + real (min i 1 * (degree h - 1 choose (i - 1))) * cmod (lead_coeff h)"
K: fixes x :: "comb"   and y :: "comb" shows "K \<bullet> x \<bullet> y \<rightarrow>\<^sup>1 x"
mono_neutral_cong_right: fixes T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'c)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g T = sum h S"
mat_diag_mult_left: fixes A :: "'a mat"   and n :: "nat"   and nr :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<in> carrier_mat n nr" shows "mat_diag n f * A = Matrix.mat n nr (\<lambda>(i, j). f i * A $$ (i, j))"
remove_max_multiset: fixes l :: "'b"   and m :: "'a"   and l' :: "'b" assumes "\<not> is_empty l"   and "inv l"   and "(m, l') = remove_max l" shows "add_mset m (multiset l') = multiset l"
simps_1: fixes x :: "Preliminaries.var" shows "GE.tpOf (Var x) = GE.tpOfV x"
lexmax_le_multiset: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list" assumes "trans r" shows "r|\<sigma>| \<subseteq># mset \<sigma>"
UNpart01_cases: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "II \<in> UNpart01 cl dl P" obtains "II = BrnFT cl dl"   | n :: "nat" and I :: "nat set" where "n \<in> theNFTBoth cl dl" and "I \<in> P n" and "II = local.shift cl n I" and "II \<in> UNpart1 cl dl P"
discI: fixes option :: "'h option"   and x2 :: "'h" assumes "option = Some x2" shows "option \<noteq> None"
unique_expression6: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and t :: "nat \<Rightarrow> 'b" assumes "free_generator R M H"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "inj_on f {j. j \<le> n}"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "g \<in> {j. j \<le> m} \<rightarrow> H"   and "inj_on g {j. j \<le> m}"   and "t \<in> {j. j \<le> m} \<rightarrow> carrier R"   and "l_comb R M n s f = l_comb R M m t g"   and "\<forall>j\<in>{j. j \<le> n}. s j \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<forall>k\<in>{j. j \<le> m}. t k \<noteq> \<zero>\<^bsub>R\<^esub>" shows "f ` {j. j \<le> n} = g ` {j. j \<le> m}"
prems: fixes i :: "'a"   and f' :: "('a + 'b, 'b) trm" assumes "SFunctions \<sigma> i = Some f'" shows "dfree f'"
abs_eq: fixes xa :: "nat \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'a" assumes "Cauchy xa \<and> Cauchy xa \<and> (\<lambda>n. dist (xa n) (xa n)) \<longlonglongrightarrow> 0"   and "Cauchy x \<and> Cauchy x \<and> (\<lambda>n. dist (x n) (x n)) \<longlonglongrightarrow> 0" shows "dist (abs_metric_completion xa) (abs_metric_completion x) = lim (\<lambda>n. dist (xa n) (x n))"
pred_map: fixes Q :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e list" shows "list_all Q (map f x) = list_all (Q \<circ> f) x"
simps_3: fixes x41 :: "'a"   and x42 :: "('a, 'b) MWLfCom"   and x43 :: "('a, 'b) MWLfCom"   and y41 :: "'a"   and y42 :: "('a, 'b) MWLfCom"   and y43 :: "('a, 'b) MWLfCom" shows "(if x41 then x42 else x43 fi = if y41 then y42 else y43 fi) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
representation_def: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and basis :: "'e set"   and v :: "'e" assumes "module scale" shows "module.representation scale basis v = (if \<not> module.dependent scale basis \<and> v \<in> module.span scale basis then SOME f. (\<forall>v. f v \<noteq> (0::'d) \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> (0::'d)} \<and> (\<Sum>v | f v \<noteq> (0::'d). scale (f v) v) = v else (\<lambda>b. 0::'d))"
sum_eq: shows "(\<Sum>j = 0..<n. 1 / real (f (j' + j)) * real_of_float x ^ j) = (\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (f (j' + j))) * real_of_float (- x) ^ j)"
objectsCounterModel: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "objects (counterModel f) = {z. \<exists>i. z = obj i}"
supp_set_elem_finite: fixes S :: "'a set"   and m :: "'a"   and y :: "atom" assumes "finite S"   and "m \<in> S"   and "y \<in> supp m" shows "y \<in> supp S"
len_compose_hchop: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=v1\<parallel>v2 \<and> len v1 ts c = ext v1 \<and> len v2 ts c = ext v2 \<longrightarrow> len v ts c = ext v"
prl: fixes g :: "'f"   and f :: "'f" assumes "prl g" shows "snd (prc f g) = True"
A: shows "c_fst (c_fst key) mod 7 = 3"
eval_ubx_to_inl: fixes opubx :: "opubx"   and \<Sigma> :: "(dynamic, nat, bool) unboxed list"   and z :: "(dynamic, nat, bool) unboxed" assumes "eval_ubx opubx \<Sigma> = Some z" shows "inl (deinl (deubx opubx)) (map unboxed_dynamic.norm_unboxed \<Sigma>) = Some (deubx opubx)"
star_cases: fixes x :: "'b star" obtains X :: "nat \<Rightarrow> 'b" where "x = star_n X"
sub_nominals: fixes f :: "'d \<Rightarrow> 'b"   and p :: "('c, 'd) fm" shows "nominals (sub f p) = f ` nominals p"
abs_eq: shows "0 = Abs_fls (\<lambda>_. 0::'a)"
BoxConst: fixes P :: "bool" shows "\<turnstile> (\<box>#P) = #P"
has_bochner_integral_FTC_Icc_nonneg: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> real" assumes "f \<in> borel_measurable borel"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (F has_real_derivative f x) (at x)"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x"   and "a \<le> b" shows "has_bochner_integral lborel (\<lambda>x. f x * indicat_real {a..b} x) (F b - F a)"
simps_3: fixes x :: "nat"   and e :: "exp" shows "FV (ELam x e) = FV e - {x}"
affine_hull_3_imp_collinear: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<in> affine hull {a, b}" shows "collinear {a, b, c}"
l1_inv2D: fixes Rb :: "fid_t"   and B :: "agent"   and A :: "agent"   and x :: "l1_state"   and gnx :: "msg" assumes "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "in_progressS (progress x Rb) {xny, xgnx, xgny, xsk}"   and "guessed_frame Rb xgnx = Some gnx"   and "can_signal x A B"   and "x \<in> l1_inv2" shows "0 < signalsInit x (Running A B (Exp gnx (NonceF (Rb $ ny))))"
WHILE_annotVI_def: fixes V :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> int"   and I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "WHILE_annotVI V I \<equiv> While"
elims_2: fixes x :: "real_max_bot"   and xa :: "real_max_bot" assumes "x \<le> xa" obtains uu_ :: "real_max_bot" where "xa__ = MInfty" and "xa = uu_"   | x :: "real" and y :: "real" where "xa__ = R x" and "xa = R y" and "x \<le> y"
prefix_le_fin_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a list" assumes "traces ind"   and "w\<^sub>1 \<le> w\<^sub>2"   and "traces.le_fin ind w\<^sub>2 w\<^sub>3" shows "traces.le_fin ind w\<^sub>1 w\<^sub>3"
Cv1_inter_Nv2_subsetof_Upsilon1: shows "C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>1\<^esub>"
extend_basis_def: fixes B :: "'b set" shows "vs1.extend_basis B = (SOME B'. B \<subseteq> B' \<and> vs1.independent B' \<and> vs1.span B' = UNIV)"
front_tickFree_implies_tickFree: fixes t :: "'a event list"   and a :: "'a event" assumes "front_tickFree (t @ [a])" shows "tickFree t"
simps_1: fixes xs :: "'a list" shows "proper_interval_set_aux xs [] = False"
sum_list_nth_eqI: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys"   and "\<And>x y. (x, y) \<in> set (zip xs ys) \<Longrightarrow> x = y" shows "sum_list xs = sum_list ys"
vdiff_vinsert_right_in: fixes A :: "V"   and a :: "V"   and B :: "V" shows "A -\<^sub>\<circ> vinsert a B = A -\<^sub>\<circ> B -\<^sub>\<circ> set {a}"
left_idem: fixes a :: "'g"   and b :: "'g" shows "min a (min a b) = min a b"
simple_integral_cong_AE: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "simple_function M f"   and "simple_function M g"   and "AE x in M. f x = g x" shows "integral\<^sup>S M f = integral\<^sup>S M g"
eq_\<I>_converter_mono: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>2 :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and conv' :: "('i, 'j, 'k, 'l) converter"   and \<I>1' :: "('i, 'j) \<I>"   and \<I>2' :: "('k, 'l) \<I>" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1',\<I>2' \<turnstile>\<^sub>C conv \<sim> conv'"
SD_preorderI: fixes p :: "'alt pmf"   and q :: "'alt pmf" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "\<And>x. x \<in> alts \<Longrightarrow> measure_pmf.prob q (preferred_alts (Pareto R) x) \<le> measure_pmf.prob p (preferred_alts (Pareto R) x)" shows "SD (Pareto R) q p"
prob_conj_imp2: fixes Pr :: "'key \<times> 'message list \<Rightarrow> bool"   and Q :: "'key \<times> 'message list \<Rightarrow> bool" assumes "prob ({x. Pr x} \<inter> msgs) = 0" shows "prob ({x. Pr x \<and> Q x} \<inter> msgs) = 0"
sinvar_mono_imp_is_offending_flows_mono: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "'b graph"   and FF :: "('b \<times> 'b) set"   and nP :: "'b \<Rightarrow> 'c"   and F :: "('b \<times> 'b) set" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph G"   and "SecurityInvariant_withOffendingFlows.is_offending_flows sinvar FF G nP" shows "SecurityInvariant_withOffendingFlows.is_offending_flows sinvar (FF \<union> F) G nP"
WbisT_Sym: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>wT d" shows "d \<approx>wT c"
gSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstCls MOD \<equiv> gSubstGVar1 MOD \<and> gSubstGVar2 MOD \<and> gSubstGAbs MOD \<and> gSubstGOp MOD"
right_idem: fixes a :: "'b"   and b :: "'b" shows "(a AND b) AND b = a AND b"
indep_sets_mono: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set"   and J :: "'c set"   and G :: "'c \<Rightarrow> 'a set set" assumes "indep_sets F I"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> G i \<subseteq> F i" shows "indep_sets G J"
one_plus_x_powr_approx_ex: fixes x :: "real"   and p :: "real" assumes "\<bar>x\<bar> \<le> 1 / 2" obtains t :: "real" where "\<bar>t\<bar> < 1 / 2" and "(1 + x) powr p = 1 + p * x + p * (p - 1) * (1 + t) powr (p - 2) / 2 * x\<^sup>2"
Id_onI: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "(a, a) \<in> Id_on A"
smult_closed: fixes r :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "m \<in> V" shows "r \<cdot> m \<in> V"
Lim_in_closed_set: fixes S :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and F :: "'d filter"   and l :: "'c" assumes "closed S"   and "\<forall>\<^sub>F x in F. f x \<in> S"   and "F \<noteq> bot"   and "(f \<longlongrightarrow> l) F" shows "l \<in> S"
LNil_eq_lmap: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "(LNil = lmap f xs) = (xs = LNil)"
Diff_infinite_finite: fixes T :: "'c set"   and S :: "'c set" assumes "finite T"   and "infinite S" shows "infinite (S - T)"
bit_minus_exp_iff: fixes m :: "nat"   and n :: "nat" shows "bit (- ((2::'a) ^ m)) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> m \<le> n)"
NewFail: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list"   and l :: "char list \<Rightarrow> val option" assumes "new_Addr h = None" shows "P \<turnstile> \<langle>new C,(h, l)\<rangle> \<Rightarrow> \<langle>THROW OutOfMemory,(h, l)\<rangle>"
minarc_min: fixes y :: "'a"   and x :: "'a" shows "arc y \<and> y \<sqinter> x \<noteq> bot \<longrightarrow> sum_class.sum (minarc x \<sqinter> x) \<le> sum_class.sum (y \<sqinter> x)"
int_nand_same: fixes x :: "int" shows "x AND NOT x = 0"
val_Zp_ultrametric0: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>"   and "y \<in> carrier Zp"   and "y \<noteq> \<zero>"   and "x \<oplus> y \<noteq> \<zero>" shows "min (val_Zp x) (val_Zp y) \<le> val_Zp (x \<oplus> y)"
orientable_2_3: fixes x :: "'a" shows "(\<exists>y. x \<le> y \<squnion> id y \<and> y \<sqinter> id y \<le> x \<sqinter> top) = (\<exists>y. x \<le> y \<squnion> id y \<and> y \<sqinter> id y = x \<sqinter> top)"
SD_inefficientE: fixes p :: "'alt pmf" assumes "\<not> SD_efficient R p" obtains q :: "'alt pmf" and i :: "'agent" where "q \<in> lowered.lotteries" and "\<And>i. i \<in> agents' \<Longrightarrow> SD (R i) p q" and "i \<in> agents'" and "p \<prec>[SD (R i)] q"
dghm_ObjMap_vrange: shows "\<R>\<^sub>\<circ> (dghm_inc \<BB> \<CC>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
r_mod2_def: shows "r_mod2 \<equiv> Cn 1 r_sub [recf.Id 1 0, Cn 1 r_mul [r_const 2, r_div2]]"
in_orbitE: fixes b :: "'a"   and f :: "'a perm"   and a :: "'a" assumes "b \<in> orbit f a" obtains n :: "nat" where "b = (f ^ n) \<langle>$\<rangle> a" and "n < order f a"
red_unionI1: fixes F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and G :: "('a \<Rightarrow>\<^sub>0 'd) set" assumes "punit.red F p q" shows "punit.red (F \<union> G) p q"
InitBlockRedsFinal: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and V :: "char list"   and v :: "val"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V \<mapsto> v), sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>"   and "final e'" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l'(V := l V), sh'),b'\<rangle>"
of_nat_power: fixes p :: "nat"   and x :: "nat" assumes "p < 2 ^ x"   and "x < LENGTH('b)" shows "word_of_nat p < 2 ^ x"
finite_state: fixes \<T> :: "'a stateful_policy" assumes "wf_stateful_policy \<T>" shows "finite (flows_state \<T>)"
term_is_le_rel_minus_minus: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'a"   and t :: "'a"   and b :: "'a"   and u :: "'t"   and v :: "'t" assumes "ord_term_lin.is_le_rel rel"   and "a adds t"   and "b adds t" shows "rel ((t - a) \<oplus> u) ((t - b) \<oplus> v) = rel (b \<oplus> u) (a \<oplus> v)"
is_bezout_ext_euclid_ext2: shows "is_bezout_ext euclid_ext2"
in_out_arcs_reflexive: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a"   and e :: "'b" assumes "sym_digraph G"   and "v \<in> verts G"   and "e \<in> in_arcs G v" shows "\<exists>e'. e' \<in> out_arcs G v \<and> head G e' = tail G e"
qconv_id: fixes x :: "'a['b, 'c]" shows "qconv id\<^sub>C x = x"
Left_Coset_Class_unit: fixes g :: "'a" assumes "g \<in> G" shows "g \<cdot>| L = contained.Class g"
simps_2: fixes f :: "'c \<rightarrow> 'b \<rightarrow> 'b"   and d :: "'b"   and x :: "'c"   and xs :: "['c]" shows "foldr\<cdot>f\<cdot>d\<cdot>(x : xs) = f\<cdot>x\<cdot>(foldr\<cdot>f\<cdot>d\<cdot>xs)"
sc_mks: shows "strongly_connected (with_proj (mk_symmetric G))"
basis_subset_carrier: fixes X :: "(nat \<times> int \<times> nat) set" assumes "s.SpanningForest X" shows "X \<subseteq> E"
std_apart_apart'': fixes x :: "char list"   and t :: "fterm"   and y :: "char list" assumes "x \<in> vars\<^sub>t (t \<cdot>\<^sub>t (\<lambda>x. \<epsilon> (y @ x)))" shows "\<exists>x'. x = y @ x'"
exceptionalVertex_def: fixes g :: "graph"   and v :: "nat" shows "exceptionalVertex g v \<equiv> except g v \<noteq> 0"
Nil: fixes uu :: "nat list" shows "omega_sum_aux 0 uu = 0"
pset_pnTimes: fixes r :: "'a rexp"   and s :: "'a rexp" shows "pset (pnTimes r s) = pset (Times r s)"
star_plus_without_loops: fixes x :: "'a" shows "top \<sqinter> - top = x \<sqinter> top \<sqinter> - top"