hcju/isabelleps huggingface dataset:
that: fixes \<B> :: "'a set set" assumes "countable \<B>"   and "{} \<notin> \<B>"   and "\<And>C. C \<in> \<B> \<Longrightarrow> openin (top_of_set S) C"   and "\<And>T. openin (top_of_set S) T \<Longrightarrow> \<exists>\<U>\<subseteq>\<B>. T = \<Union> \<U>" shows "thesis"
Vagree_trans: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and \<nu>'' :: "variable \<Rightarrow> real"   and W :: "variable set" assumes "Vagree \<nu> \<nu>' V"   and "Vagree \<nu>' \<nu>'' W" shows "Vagree \<nu> \<nu>'' (V \<inter> W)"
MatchOr_MatchNot: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m1 :: "'a match_expr"   and m2 :: "'a match_expr"   and a :: "action"   and p :: "'b" shows "matches \<gamma> (MatchNot (MatchOr m1 m2)) a p = (matches \<gamma> (MatchNot m1) a p \<and> matches \<gamma> (MatchNot m2) a p)"
multiplicity_Lcm_atLeast1AtMost_Suc: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "0 < n" shows "multiplicity p (Lcm {1..Suc n}) = (if \<exists>k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})"
that: fixes ids :: "nat list" assumes "distinct ids"   and "set ids \<subseteq> {..<length ls}"   and "ss = map ((!) ls) ids" shows "thesis"
suffix_Cons': fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "suffix [x] (y # ys)" shows "suffix [x] ys \<or> y = x \<and> ys = []"
full_elims_2: fixes n :: "nat"   and l :: "'b tree23"   and p :: "'b"   and r :: "'b tree23" assumes "full n \<langle>l, p, r\<rangle>" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na r"
step1_converse: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "step1 r\<inverse>\<inverse> = (step1 r)\<inverse>\<inverse>"
adj_cancel1: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "f \<circ> g \<le> id"
schroeder_2_var: fixes y :: "'a" shows "is_conjugation (\<lambda>x. x ; y) (\<lambda>x. x ; y\<^sup>\<smile>)"
4: shows "irreflexive (t\<^sup>+ * e \<squnion> e)"
val_zero: fixes a :: "('a, 'b) IEEE.float" assumes "is_zero a" shows "valof a = 0"
interpret_floatariths_fold_const_fa: fixes ds :: "floatarith list" shows "interpret_floatariths (map fold_const_fa ds) = interpret_floatariths ds"
euclidean_size_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(euclidean_size b = 0) = (b = zero)"
2: shows "prv (imp (cnj (PP \<langle>\<phi>\<rangle>) (PP \<langle>neg \<phi>\<rangle>)) (cnj (PP \<langle>\<phi>\<rangle>) (PP (encF N \<langle>\<phi>\<rangle>))))"
prime_aux: fixes p :: "nat" assumes "prime p" shows "(if 1 < 1 then 0::'a else (1::'a)) = (1::'a)"
eq: fixes w :: "'a" shows "f w = eval_poly g (q w)"
strand_subst_hom_1: fixes S :: "('c, 'd) strand_step list"   and S' :: "('c, 'd) strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "S @ S' \<cdot>\<^sub>s\<^sub>t \<theta> = (S \<cdot>\<^sub>s\<^sub>t \<theta>) @ (S' \<cdot>\<^sub>s\<^sub>t \<theta>)"
lehmers_weak_theorem: fixes p :: "nat"   and a :: "nat" assumes "2 \<le> p"   and "\<And>x. \<lbrakk>0 < x; x < p - 1\<rbrakk> \<Longrightarrow> [a ^ x \<noteq> 1] (mod p)"   and "[a ^ (p - 1) = 1] (mod p)" shows "prime p"
eventually_nhds_in_open: fixes "open" :: "'e set \<Rightarrow> bool"   and s :: "'e set"   and x :: "'e" assumes "class.topological_space open"   and "open s"   and "x \<in> s" shows "\<forall>\<^sub>F y in topological_space.nhds open x. y \<in> s"
projection_idem: fixes M :: "'a set"   and x :: "'a" assumes "closed_csubspace M" shows "projection M (projection M x) = projection M x"
inputRenameSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* K" shows "\<Psi> \<rhd> P \<longmapsto> K\<lparr>N\<rparr> \<prec> P'"
sigma_finite: fixes s :: "'s" obtains A :: "nat \<Rightarrow> 's stream set" where "range A \<subseteq> T.events s" and "\<Union> (range A) = space (T s)" and "\<And>i. emeasure (T s) (A i) \<noteq> \<infinity>"
Gromov_converging_at_boundary_def: fixes u :: "nat \<Rightarrow> 'b" shows "Gromov_converging_at_boundary u = (\<forall>a M. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n))"
CSHORun_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and HOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set"   and SHOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set"   and coords :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" shows "CSHORun A rho HOs SHOs coords \<equiv> CHOinitConfig A (rho 0) (coords 0) \<and> (\<forall>r. CSHOnextConfig A r (rho r) (HOs r) (SHOs r) (coords (Suc r)) (rho (Suc r)))"
quantify_nfa_def: fixes i :: "nat" shows "quantify_nfa i = (\<lambda>(bdds, as). (map (quantify_bdd i) bdds, as))"
returns_toI: fixes t :: "real"   and x :: "'d"   and P :: "'d set" assumes "0 < t"   and "t \<in> existence_ivl0 x"   and "flow0 x t \<in> P"   and "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P"   and "closed P" shows "returns_to P x"
simps_5: fixes v :: "val" shows "INIT_class (Val v) = None"
distinct_1: fixes x2 :: "'b" shows "Err \<noteq> OK x2"
least_Upper_cong_r: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and A' :: "'a set"   and x :: "'a" assumes "weak_partial_order L"   and "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=}\<^bsub>L\<^esub> A'" shows "is_lub L x A = is_lub L x A'"
HPow_mono_strict_iff: fixes x :: "hf"   and y :: "hf" shows "(HPow x < HPow y) = (x < y)"
ncs_1: shows "fst ` set ncs = fst ` set cs"
wt_method_def2: fixes C' :: "char list"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "wt_method P C' b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s = (is \<noteq> [] \<and> length \<tau>s = length is \<and> OK ` set \<tau>s \<subseteq> states P mxs mxl \<and> wt_start P C' b Ts mxl\<^sub>0 \<tau>s \<and> wt_app_eff (sup_state_opt P) app eff \<tau>s)"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<^bold>\<sqinter> b) \<^bold>\<sqinter> c = a \<^bold>\<sqinter> b \<^bold>\<sqinter> c"
emb_step_subst: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm"   and \<rho> :: "'b \<Rightarrow> ('a, 'b) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "subst \<rho> t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b subst \<rho> s"
exec_gpv_parametric': fixes S :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and CALL :: "'j \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'n \<Rightarrow> bool"   and A :: "'o \<Rightarrow> 'p \<Rightarrow> bool" shows "((S ===> CALL ===> rel_spmf (rel_prod R S)) ===> rel_gpv'' A CALL R ===> S ===> rel_spmf (rel_prod A S)) exec_gpv exec_gpv"
k_score_def: fixes k :: "nat"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and strat :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat" shows "k_score k e strat = (let states = Inference.S e; pairs_to_score = ffilter (\<lambda>(x, y). x < y) (states |\<times>| states); paths = (\<lambda>(s1, s2). (s1, s2, paths_of_length k e s1, paths_of_length k e s2)) |`| pairs_to_score in Let ((\<lambda>(s1, s2, p1, p2). \<lparr>Score = score_from_list p1 p2 e strat, S1 = s1, S2 = s2\<rparr>) |`| paths) (ffilter (\<lambda>x. 0 < Score x)))"
comp_id: fixes f :: "'d \<Rightarrow> 'e" shows "f \<circ> id = f"
matrix_to_iarray_snd_snd_Gauss_Jordan_column_k_det_P: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and n :: "'b" assumes "i \<le> nrows A"   and "k < ncols A" shows "matrix_to_iarray (snd (snd (Gauss_Jordan_column_k_det_P (n, i, A) k))) = snd (snd (Gauss_Jordan_column_k_det_P_iarrays (n, i, matrix_to_iarray A) k))"
divide_eq_eq_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(divide b a = one) = (a \<noteq> zero \<and> a = b)"
lfinite_ldropn: fixes n :: "nat"   and xs :: "'c llist" shows "lfinite (ldropn n xs) = lfinite xs"
insert_Collect: fixes a :: "'k"   and P :: "'k \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
set_ConsD: fixes y :: "'k"   and x :: "'k"   and xs :: "'k list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
Pd_range': fixes x :: "'a"   and d :: "'b" assumes "x \<in> Field (Pd d)" shows "Xd x = d"
gen_g_impl_rel_ext_internal_def: fixes Rm :: "('j \<times> 'l) set"   and Rv :: "('g \<times> 'k set) set"   and Re :: "('h \<times> ('k \<times> 'k) set) set"   and Rv0 :: "('i \<times> 'k set) set" shows "gen_g_impl_rel_ext Rm Rv Re Rv0 \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
lists_succ_in_conv: fixes xss :: "'b list list"   and x :: "'b" shows "(lists_succ xss x \<in> \<Union> (sset xss)) = (x \<in> \<Union> (sset xss))"
binrelchain_append_reduce1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "binrelchain P (xs @ ys)" shows "binrelchain P xs"
semiring_ow: shows "class.semiring = semiring_ow UNIV"
disjSD_2IDa: fixes x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'd) Combinators"   and a :: "'b set set"   and b :: "'b set set" assumes "disjSD_2 x y"   and "\<not> member DenyAll x"   and "\<not> member DenyAll y"   and "a = first_srcNet x"   and "b = first_destNet x" shows "disjSD_2 (DenyAllFromTo a b \<oplus> DenyAllFromTo b a \<oplus> x) y"
Suc_3: shows "(r - 1) ^ Suc k * fact (Suc k) < card F"
csim: shows "s\<inverse> O m \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>"
rel_frontier_affine_hull: fixes S :: "'c set" shows "rel_frontier S \<subseteq> affine hull S"
lve_edge: fixes valid_edge :: "'f \<Rightarrow> bool"   and a :: "'f"   and src :: "'f \<Rightarrow> 'g"   and E :: "'g"   and trg :: "'f \<Rightarrow> 'g"   and X :: "'g"   and e :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node"   and knd :: "'f \<Rightarrow> 'h edge_kind" assumes "valid_edge a"   and "src a \<noteq> E \<or> trg a \<noteq> X"   and "e = (Node (src a), knd a, Node (trg a))" shows "lift_valid_edge valid_edge src trg knd E X e"
ostep_invariantE: fixes A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "('e \<times> 'f) \<times> 'g \<times> 'e \<times> 'f \<Rightarrow> bool"   and \<sigma> :: "'e"   and s :: "'f"   and a :: "'g"   and \<sigma>' :: "'e"   and s' :: "'f" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" obtains "P ((\<sigma>, s), a, \<sigma>', s')"
IH: assumes "wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t S \<subseteq> V" shows "wf'\<^sub>s\<^sub>s\<^sub>t V S"
a_m_pos: fixes m :: "int" assumes "0 < m" shows "- \<infinity> * ant m = - \<infinity>"
sum_UNIV_type_def: fixes f :: "'d \<Rightarrow> 'c"   and uu :: "'d itself" shows "sum_UNIV_type f uu = sum f UNIV"
rel_fun_neg_distr_cond_def: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and uu :: "('j \<times> 'k \<times> 'l) itself" shows "rel_fun_neg_distr_cond A A' uu = (\<forall>B B'. A OO A' ===> B OO B' \<le> (A ===> B) OO (A' ===> B'))"
path_entry_unconc: fixes la :: "nat list"   and lb :: "nat list"   and w :: "nat" assumes "path_entry (g_E G) (la @ lb) w" obtains v :: "nat" where "path_entry (g_E G) lb v"
fmdom'_fmrestrict_set: fixes X :: "'a set"   and f :: "('a, 'b) fmap" shows "fmdom' (fmrestrict_set X f) = X \<inter> fmdom' f"
sturm_neq_Nil: fixes p :: "real poly" shows "sturm p \<noteq> []"
comp_simp_2: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" assumes "J.seq g f"   and "g = J.One" shows "g \<cdot>\<^sub>J f = f"
dependent_zero: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "Modules.module scale"   and "(0::'b) \<in> A" shows "module.dependent scale A"
LIMSEQ_I: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r" shows "X \<longlonglongrightarrow> L"
has_integral_vec1_I: fixes f :: "(real, 1) vec \<Rightarrow> 'c"   and y :: "'c"   and S :: "(real, 1) vec set" assumes "(f has_integral y) S" shows "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
add_neg_numeral_special_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus (numeral.numeral one plus m)) one = neg_numeral.sub minus plus one num.One m"
list_of_oalist_insert: fixes x :: "'a \<times> 'b"   and xs :: "'x" shows "list_of_oalist (local.insert x xs) = update_by_raw x (list_of_oalist xs)"
power_numeral_even: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and w :: "num" assumes "class.monoid_mult one times" shows "power.power one times z (numeral (num.Bit0 w)) = (let w = power.power one times z (numeral w) in times w w)"
rep_eq: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "'e cset" shows "rcset (cimage x xa) = x ` rcset xa"
ctr_1: fixes A :: "'b llist set" assumes "\<forall>x\<in>A. lnull x" shows "lSup A = <>"
cblinfun_compose_def: shows "(o\<^sub>C\<^sub>L) \<equiv> map_fun (*\<^sub>V) (map_fun (*\<^sub>V) CBlinfun) (\<circ>)"
simps_4: fixes C :: "ty"   and D :: "ty"   and X :: "ctrm set" shows "NEGn (C OR D) X = AXIOMSn (C OR D) \<union> BINDINGn (C OR D) X \<union> ORLEFT (C OR D) (\<parallel>(C)\<parallel>) (\<parallel>(D)\<parallel>)"
Sum_Ico_nat: fixes m :: "nat"   and n :: "nat" shows "\<Sum> {m..<n} = (n * (n - 1) - m * (m - 1)) div 2"
cmp_interpretation: shows "identity_arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (f \<star> g) Tfg.\<rho>\<sigma>.tab (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) (tab\<^sub>1 f \<star> Tfg.p\<^sub>1) (f \<star> g) Tfg.tab (tab\<^sub>0 (f \<star> g)) (tab\<^sub>1 (f \<star> g)) (f \<star> g)"
transversal: fixes x :: "'a" assumes "x \<in> S"   and "s x = (0::'b)" shows "blinfun_apply (Ds x) (f x) \<noteq> (0::'b)"
QSS: fixes q :: "'Q" assumes "q \<in> ta_rstates complementTA" shows "q \<in> Q"
t_covering_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" shows "t_covering_design point_set block_collection u_block_size grouping min_index \<equiv> block_design point_set block_collection u_block_size \<and> t_covering_design_axioms point_set block_collection u_block_size grouping min_index"
tendsto_norm_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
oInv_equality: fixes x :: "ordinal"   and a :: "ordinal" assumes "F x \<le> a"   and "a < F (oSuc x)" shows "oInv F a = x"
ex_rcoset_replist: fixes H :: "'g set" assumes "Subgroup H" shows "\<exists>gs. is_rcoset_replist H gs"
nth_prime_Suc: fixes n :: "nat" shows "nth_prime (Suc n) = smallest_prime_beyond (Suc (nth_prime n))"
proj_eq: shows "proj_onto (column b G) {column i G |i. i < b} = (\<Sum>x\<leftarrow>map (\<lambda>n. column (mod_type_class.from_nat n) G) [0..<mod_type_class.to_nat b]. (column b G \<bullet> x / (x \<bullet> x)) *\<^sub>R x)"
dimM_1: shows "0 < card {i. i < 2 ^ n \<and> f i}"
igConsIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igConsIPresIGWls MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWls MOD \<and> igOpIPresIGWls MOD"
foldr_correct: fixes f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) FingerTreeStruc"   and \<sigma> :: "'a" shows "FingerTreeStruc.foldr f t \<sigma> = foldr f (FingerTreeStruc.toList t) \<sigma>"
CauchyI: fixes X :: "nat \<Rightarrow> 'a" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e" shows "Cauchy X"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i |i. Inl i \<in> SIGO (OProd ODE1 ODE2)}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
dagger_trans_eq: fixes x :: "('a \<times> 'a) set" shows "x\<^sup>* O x\<^sup>* = x\<^sup>*"
simps_4: fixes b :: "bexp"   and c1 :: "'b acom"   and c2 :: "'b acom"   and S :: "'b"   and b' :: "bexp"   and c1' :: "'b acom"   and c2' :: "'b acom"   and S' :: "'b" shows "(IF b THEN c1 ELSE c2 {S} \<sqsubseteq> IF b' THEN c1' ELSE c2' {S'}) = (b = b' \<and> c1 \<sqsubseteq> c1' \<and> c2 \<sqsubseteq> c2' \<and> S \<sqsubseteq> S')"
conn1_4: shows "gs.lin_indpt (set (map of_int_hom.vec_hom fs))"
cblinfun_of_mat_def: shows "cblinfun_of_mat \<equiv> map_fun id CBlinfun (\<lambda>M v. if M \<in> carrier_mat (length canonical_basis) (length canonical_basis) then basis_enum_of_vec (M *\<^sub>v vec_of_basis_enum v) else (0::'c))"
K_rep2: fixes pat :: "[:'r:]"   and us :: "[:'r:]"   and vs :: "[:'r:]" assumes "pat \<noteq> \<bottom>"   and "us :@ vs = pat" shows "K (rep2\<cdot>pat\<cdot>(us, vs))"
rev_nth_mapi_block: fixes ps :: "'b list"   and v' :: "nat"   and p :: "'b"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> ('c, 'd) fm"   and a :: "'d" assumes "ps !. v' = Some p" shows "f v' p on (mapi f ps, a)"
residues_nat_def: fixes n :: "nat" shows "residues_nat n \<equiv> 1 < n"
hyps_2: shows "fst ((\<alpha> \<circ> f) e) \<inter> mon_pl (map (\<alpha> \<circ> f) w1) = {}"
bind_advantage: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> ('grp \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "okamoto_\<Sigma>_commit.abstract_com.bind_advantage \<eta> \<A> \<le> okamoto_\<Sigma>_commit.rel_advantage \<eta> (okamoto_\<Sigma>_commit.adversary \<eta> \<A>)"
ln_primefact: fixes n :: "nat" assumes "n \<noteq> 0" shows "ln (real n) = (\<Sum>d = 1..n. if primepow d \<and> d dvd n then ln (real (aprimedivisor d)) else 0)"
ex_dense_closure_interE: fixes T :: "'b set set"   and S :: "'b set" assumes "topology T"   and "S \<subseteq> carrier\<^bsub>T\<^esub>" obtains D :: "'b set" and C :: "'b set" where "D \<subseteq> carrier\<^bsub>T\<^esub>" and "C \<subseteq> carrier\<^bsub>T\<^esub>" and "D dense\<^bsub>T\<^esub>" and "C closed\<^bsub>T\<^esub>" and "S = D \<inter> C"
theorem24xxiii: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<sqinter> - (x * z) = x * (y \<sqinter> - z) \<sqinter> - (x * z)"
assms_2: fixes \<alpha> :: "'act" assumes "\<alpha>\<^sub>S = S_action.Act \<alpha>"   and "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>" shows "R"
mono_onD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and r :: "'a"   and s :: "'a" assumes "mono_on f A"   and "r \<in> A"   and "s \<in> A"   and "r \<le> s" shows "f r \<le> f s"
qsend_eq_0: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cf :: "('d, 'c, 'a) cmd \<times> 'b"   and \<omega> :: "(('d, 'c, 'a) cmd \<times> 'b) stream" assumes "PL_Indis cval indis"   and "PL_Indis.discrCf aval tval cval indis cf" shows "sfirst (holds (PL_Indis.discrCf aval tval cval indis)) (cf ## \<omega>) = 0"
irreflpI: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>a. \<not> R a a" shows "irreflp R"
Impl: fixes A :: "triple set"   and P :: "state \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Q :: "state \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Ms :: "(cname \<times> mname) set"   and Z :: "state" assumes "\<forall>Z. A \<union> (\<Union>Z. (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm)) ` Ms) |\<turnstile> (\<lambda>Cm. (P Z Cm, body Cm, Q Z Cm)) ` Ms" shows "A |\<turnstile> (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm)) ` Ms"
meld_correct: fixes q :: "('a, 'b) SkewBinomialHeap"   and q' :: "('a, 'b) SkewBinomialHeap" shows "SkewBinomialHeap_loc.to_mset (SkewBinomialHeap_loc.meld q q') = SkewBinomialHeap_loc.to_mset q + SkewBinomialHeap_loc.to_mset q'"
succ_it_is_iterator: fixes g :: "'G"   and v :: "'V" assumes "invar g" shows "set_iterator (succ_it g v) (succ (\<alpha> g) v)"
irrefl: fixes a :: "'b multiset" shows "\<not> a \<subset># a"
intros_3: fixes i :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a formula"   and a :: "'b"   and y :: "'a formula"   and b :: "'b" assumes "i \<notin> S"   and "hom_graph f S x a"   and "hom_graph f S y b" shows "hom_graph f (insert i S) (ifte (var i) x y) (ifte (f i) a b)"
Smith_1xn_two_matrices: fixes A :: "'a mat"   and n :: "nat"   and P :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and Smith_1xn :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat" assumes "A \<in> carrier_mat 1 n"   and "(P, S, Q) = Smith_1xn A"   and "is_SNF A (Smith_1xn A)" shows "\<exists>Smith_1xn'. is_SNF A (1\<^sub>m 1, Smith_1xn' A)"
hyps: fixes xs :: "'b list" assumes "map f xs = map g ys" shows "length xs = length ys"
smult_as_map_poly: fixes a :: "'a" shows "smult a = map_poly ((*) a)"
tauLaw1StatImpLeft: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q"   and "(\<Psi>, \<tau>.P, Q) \<in> Rel" shows "\<Psi> \<rhd> \<tau>.P \<lessapprox>\<^sub>w<Rel> Q"
sel_2: fixes x1 :: "('c \<times> 'b \<times> 'b) set"   and x2 :: "'b set" shows "vertices (LG x1 x2) = x2"
imply_ImpE: fixes ps :: "('c, 'd) fm list"   and p :: "('c, 'd) fm"   and q :: "('c, 'd) fm" shows "\<turnstile> ps \<^bold>\<leadsto> p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> q"
distinct_5: fixes x3 :: "'b" shows "dFAILi \<noteq> dRETURN x3"
Complete_Lattice_axioms: fixes c :: "com" shows "Complete_Lattice {C. strip C = c} (Inf_acom c)"
that: fixes n :: "nat"   and r :: "real"   and g :: "complex \<Rightarrow> complex" assumes "0 < n"   and "0 < r"   and "ball \<xi> r \<subseteq> S"   and "g holomorphic_on ball \<xi> r"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w = (w - \<xi>) ^ n * g w"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0" shows "thesis"
tendsto_of_real_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter" shows "((\<lambda>x. of_real (f x)) \<longlongrightarrow> of_real c) F = (f \<longlongrightarrow> c) F"