hcju/isabelleps huggingface dataset:
interior_simplex_nonempty: fixes S :: "'c set" assumes "independent S"   and "finite S"   and "card S = DIM('c)" obtains a :: "'c" where "a \<in> interior (convex hull insert (0::'c) S)"
simps: fixes a :: "'a set \<Rightarrow> 'a set" shows "CK_nf a = (a = id \<or> a = C \<or> a = K \<or> a = C \<circ> K \<or> a = K \<circ> C \<or> a = C \<circ> K \<circ> C \<or> a = K \<circ> C \<circ> K \<or> a = C \<circ> K \<circ> C \<circ> K \<or> a = K \<circ> C \<circ> K \<circ> C \<or> a = C \<circ> K \<circ> C \<circ> K \<circ> C \<or> a = K \<circ> C \<circ> K \<circ> C \<circ> K \<or> a = C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K \<or> a = K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C \<or> a = C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C)"
assms_3: shows "net_ips s1 = net_tree_ips n1"
set_sel_1: fixes a :: "'c list" assumes "a \<noteq> []" shows "lead_coeff a \<in> set a"
less_irrefl: fixes x :: "'k multiset" shows "\<not> x \<subset># x"
mod_bound_instance: fixes j :: "nat"   and i :: "nat"   and k :: "nat" assumes "j < i" obtains j' :: "nat" where "k < j'" and "j' mod i = j"
eventually_times_inverse_1: fixes f :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> (0::'b)" shows "\<forall>\<^sub>F x in F. inverse (f x) * f x = (1::'b)"
case_cong: fixes memref :: "'b memref"   and memref' :: "'b memref"   and 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> 'c"   and g1 :: "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> 'c"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and g2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> instr list \<Rightarrow> 'c"   and g3 :: "'b \<Rightarrow> 'b \<Rightarrow> instr list \<Rightarrow> 'c"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'c"   and g4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'c" assumes "memref = memref'"   and "\<And>x11 x12 x13 x14 x15 x16 x17 x18. memref' = Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<Longrightarrow> f1 x11 x12 x13 x14 x15 x16 x17 x18 = g1 x11 x12 x13 x14 x15 x16 x17 x18"   and "\<And>x21 x22 x23 x24. memref' = Read\<^sub>s\<^sub>b x21 x22 x23 x24 \<Longrightarrow> f2 x21 x22 x23 x24 = g2 x21 x22 x23 x24"   and "\<And>x31 x32 x33. memref' = Prog\<^sub>s\<^sub>b x31 x32 x33 \<Longrightarrow> f3 x31 x32 x33 = g3 x31 x32 x33"   and "\<And>x41 x42 x43 x44. memref' = Ghost\<^sub>s\<^sub>b x41 x42 x43 x44 \<Longrightarrow> f4 x41 x42 x43 x44 = g4 x41 x42 x43 x44" shows "(case memref of Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<Rightarrow> f1 x11 x12 x13 x14 x15 x16 x17 x18 | Read\<^sub>s\<^sub>b x21 x22 x23 x24 \<Rightarrow> f2 x21 x22 x23 x24 | Prog\<^sub>s\<^sub>b x31 x32 x33 \<Rightarrow> f3 x31 x32 x33 | Ghost\<^sub>s\<^sub>b x41 x42 x43 x44 \<Rightarrow> f4 x41 x42 x43 x44) = (case memref' of Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<Rightarrow> g1 x11 x12 x13 x14 x15 x16 x17 x18 | Read\<^sub>s\<^sub>b x21 x22 x23 x24 \<Rightarrow> g2 x21 x22 x23 x24 | Prog\<^sub>s\<^sub>b x31 x32 x33 \<Rightarrow> g3 x31 x32 x33 | Ghost\<^sub>s\<^sub>b x41 x42 x43 x44 \<Rightarrow> g4 x41 x42 x43 x44)"
path_split_2: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>* n'" shows "valid_edge a"
A: shows "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')"
rb_aux_inv_D2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and u :: "'t" assumes "rb_aux_inv (bs, ss, ps)"   and "u \<in> set ss" shows "is_syz_sig dgrad u"
rawpsubstT_compose_freshVar2: fixes r :: "'trm"   and ts :: "'trm list"   and xs :: "'var list"   and us :: "'var list"   and vs :: "'var list" assumes "r \<in> atrm"   and "set ts \<subseteq> atrm"   and "set xs \<subseteq> var"   and "distinct xs"   and "set us \<subseteq> var"   and "distinct us"   and "set us \<inter> FvarsT r = {}"   and "set us \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set us \<inter> set xs = {}"   and "set vs \<subseteq> var"   and "distinct vs"   and "set vs \<inter> FvarsT r = {}"   and "set vs \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set vs \<inter> set xs = {}"   and "length us = length xs"   and "length vs = length xs"   and "length ts = length xs" shows "rawpsubstT (rawpsubstT r (zip (map Var us) xs)) (zip ts us) = rawpsubstT (rawpsubstT r (zip (map Var vs) xs)) (zip ts vs)"
ex_conjI: fixes P :: "'c \<Rightarrow> bool"   and c :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "P c"   and "Q c" shows "\<exists>c. P c \<and> Q c"
A_1: fixes x :: "'b"   and i :: "nat" shows "x \<in> A x i"
simps_1: fixes e :: "'a"   and f :: "udom \<rightarrow> udom\<cdot>'a error" shows "bindU\<cdot>(Err\<cdot>e)\<cdot>f = Err\<cdot>e"
fps_XD'_sum: fixes a :: "'a fps"   and f :: "'b \<Rightarrow> 'a fps"   and A :: "'b set" shows "fps_XD' a (sum f A) = (\<Sum>x\<in>A. fps_XD' a (f x))"
Fun: fixes x2a :: "('b, 'c) Term.term" assumes "x2a \<in> set ts" shows "vars_term_ms (x2a \<cdot> \<sigma>) = (\<Sum>x\<in>#vars_term_ms x2a. vars_term_ms (\<sigma> x))"
card_Un_le: fixes A :: "'c set"   and B :: "'c set" shows "card (A \<union> B) \<le> card A + card B"
Max_lit_eq_pos_or_neg_Max_atm: fixes C :: "'a literal multiset" assumes "C \<noteq> {#}" shows "Max_mset C = Pos (Max (atms_of C)) \<or> Max_mset C = Neg (Max (atms_of C))"
cinner_zero_right: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x zero = 0"
left_connected: fixes \<nu> :: "'b"   and \<nu>' :: "'b"   and \<mu> :: "'b" assumes "D.seq \<nu> \<nu>'" shows "(\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null) = (\<nu>' \<star>\<^sub>D \<mu> \<noteq> D.null)"
open_winding_number_levelsets: fixes \<gamma> :: "real \<Rightarrow> complex"   and k :: "complex" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>" shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
ncop_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "A \<noteq> B \<and> A \<noteq> C \<and> A \<noteq> D \<and> B \<noteq> C \<and> B \<noteq> D \<and> C \<noteq> D"
simps_5: fixes bbs :: "bool list"   and bs :: "int list"   and a :: "Cooper.num" shows "Ifm bbs bs (Le a) = (Inum bs a \<le> 0)"
N_below_pp: fixes x :: "'a" shows "- (- x \<sqinter> top) \<sqinter> top \<le> - - x"
low_proc_0_def: fixes p :: "prog" shows "low_proc_0 p \<equiv> \<forall>\<sigma>\<in>states p. \<forall>u. the (\<sigma> ''lowIn'') = 0 \<longrightarrow> (\<exists>\<sigma>'. body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<and> the (\<sigma>' ''lowOut'') = u)"
elims_3: fixes x :: "'b formula" assumes "\<not> is_cnf x" obtains F :: "'b formula" and G :: "'b formula" where "x = F \<^bold>\<and> G" and "\<not> (is_cnf F \<and> is_cnf G)"   | v :: "'b" where "x = Atom v__" and "\<not> is_disj (Atom v__)"   | "x = \<bottom>" and "\<not> is_disj \<bottom>"   | v :: "'b formula" where "x = \<^bold>\<not> va__" and "\<not> is_disj (\<^bold>\<not> va__)"   | v :: "'b formula" and va :: "'b formula" where "x = vb__ \<^bold>\<or> vaa__" and "\<not> is_disj (vb__ \<^bold>\<or> vaa__)"   | v :: "'b formula" and va :: "'b formula" where "x = v \<^bold>\<rightarrow> va" and "\<not> is_disj (v \<^bold>\<rightarrow> va)"
quotI: fixes a :: "'d" shows "{x. a \<sim> x} \<in> quot"
orb_def: fixes x :: "'a"   and a :: "'a" shows "(x \<in> Orb a) = (\<exists>s. s \<circ> a = x)"
decr_grading_term_idI: fixes d :: "'a \<Rightarrow> nat"   and v :: "'t"   and n :: "nat" assumes "hom_grading d"   and "d (pp_of_term v) \<le> n" shows "decr_grading_term d n v = v"
unitaryI2: fixes P :: "'b \<Rightarrow> real" assumes "nneg P"   and "bounded_by 1 P" shows "unitary P"
min_range_coeffs_le: fixes l :: "real"   and x :: "real"   and u :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and d :: "real" assumes "l \<le> x"   and "x \<le> u"   and "\<And>y. y \<in> {l..u} \<Longrightarrow> (f has_real_derivative f' y) (at y)"   and "\<And>y. y \<in> {l..u} \<Longrightarrow> f' y \<le> a"   and "(f l - f u + a * (u - l)) / 2 + \<bar>(f l + f u - a * (l + u)) / 2 - b\<bar> \<le> d" shows "\<bar>f x - (a * x + b)\<bar> \<le> d"
induct: fixes P :: "'a formula \<Rightarrow> bool"   and a0 :: "'a formula" assumes "\<And>k. P (Atom k)"   and "\<And>k. P (\<^bold>\<not> (Atom k))"   and "P \<bottom>"   and "P (\<^bold>\<not> \<bottom>)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>\<And>x xa xb. P F; \<And>x xa xb. P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>va. P (\<^bold>\<not> (\<^bold>\<not> va))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<and> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<or> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<rightarrow> vb))"   and "\<And>v va. P (v \<^bold>\<rightarrow> va)" shows "P a0"
poly_uminus_code: fixes p :: "'a poly" shows "poly_uminus p = poly_uminus_inner (coeffs p)"
meet_join_distrib1: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> (y \<squnion>\<^bsub>L\<^esub> z) = x \<sqinter>\<^bsub>L\<^esub> y \<squnion>\<^bsub>L\<^esub> x \<sqinter>\<^bsub>L\<^esub> z"
eval\<^sub>Cn: shows "(cms\<^sub>C ! n, mem\<^sub>C) \<leadsto>\<^sub>C (cmn', mem\<^sub>C')"
adds_pp_trans: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s adds t"   and "t adds\<^sub>p v" shows "s adds\<^sub>p v"
smult_pCons: fixes a :: "'c"   and b :: "'c"   and p :: "'c poly" shows "smult a (pCons b p) = pCons (a * b) (smult a p)"
compl_inf: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>-\<^bsub>L\<^esub> (x \<sqinter>\<^bsub>L\<^esub> y) = \<^bold>-\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> \<^bold>-\<^bsub>L\<^esub> y"
tag_name_eq2_h2: fixes ptr' :: "(_) element_ptr" shows "|h2 \<turnstile> get_tag_name ptr'|\<^sub>r = |h3 \<turnstile> get_tag_name ptr'|\<^sub>r"
q: shows "p = (\<Sum>x\<in>keys (mapping_of p). MPoly_Type.monom x (MPoly_Type.coeff p x))"
mins_subset: fixes i :: "int"   and j :: "int" shows "mins i j \<subseteq> {i + 1..j}"
trans_equiv_greater: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b = Equiv"   and "compare cmp b c = Greater" shows "compare cmp a c = Greater"
init: shows "FL.active_subset (lclss_of_state (lhd Sts)) = {}"
dom_in_domains: fixes f :: "(bool, bool list) concrete_category.arr" assumes "domains f \<noteq> {}" shows "local.dom f \<in> domains f"
sel_6: fixes x1 :: "((('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) list"   and x2 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x3 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x4 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x5 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x6 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_send (Transaction x1 x2 x3 x4 x5 x6) = x6"
RR_1: shows "[v \<oplus> real_of_int t]\<^sub>\<R> = region X I' r'"
integ_exis_horiz: fixes \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "\<exists>(k', \<gamma>'')\<in>two_chain_horizontal_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>'' = \<gamma>" shows "line_integral_exists F {j} \<gamma>"
independent_imp_trivial_combine: fixes Us :: "'a list"   and Ks :: "'a list" assumes "local.independent K Us"   and "set Ks \<subseteq> K"   and "combine Ks Us = \<zero>" shows "set (take (length Us) Ks) \<subseteq> {\<zero>}"
has_integral_integral_lebesgue_real: fixes f :: "'c \<Rightarrow> real" assumes "integrable lebesgue f" shows "(f has_integral integral\<^sup>L lebesgue f) UNIV"
fmempty_lookup: fixes x :: "'b" shows "fmlookup fmempty x = None"
sum_of_powers_nat: fixes x :: "nat"   and n :: "nat" shows "(x - 1) * sum ((^) x) {0..n} = x ^ Suc n - 1"
similar_nice_def: fixes x :: "Value"   and y :: "CValue" shows "(x \<triangleleft>\<triangleright> y) = (x = \<bottom> \<and> y = \<bottom> \<or> (\<exists>b. x = B\<cdot>(Discr b) \<and> y = CB\<cdot>(Discr b)) \<or> (\<exists>f g. x = Fn\<cdot>f \<and> y = CFn\<cdot>g \<and> (\<forall>a b. a \<triangleleft>\<triangleright> b\<cdot>C\<^sup>\<infinity> \<longrightarrow> f\<cdot>a \<triangleleft>\<triangleright> g\<cdot>b\<cdot>C\<^sup>\<infinity>)))"
hom_inf: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula"   and y :: "'b formula" shows "hom f (x \<sqinter> y) = hom f x \<sqinter> hom f y"
fermat_two_squares: fixes p :: "nat" assumes "prime p"   and "[p \<noteq> 3] (mod 4)" shows "is_sum2sq_nat p"
length_filter_conv_card: fixes p :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "|filter p xs| = card {i. i < |xs| \<and> p (xs ! i)}"
simps: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'c fset"   and b :: "'b" shows "ffold_ord f s b = (if s = {||} then b else let h = fMin s; t = s |-| {|h|} in ffold_ord f t (f h b))"
fold_simps_1: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b" shows "fold f [] s = s"
split_face_distinct2: fixes f12 :: "face"   and f21 :: "face"   and oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" assumes "(f12, f21) = split_face oldF ram1 ram2 newVertexList"   and "pre_split_face oldF ram1 ram2 newVertexList" shows "distinct (vertices f21)"
soundness: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and M :: "('d, 'c) model"   and g :: "'b \<Rightarrow> 'd"   and w :: "'d" assumes "A, n \<turnstile> branch" shows "\<exists>block\<in>set branch. \<exists>p. p on block \<and> \<not> M, g, w \<Turnstile> p"
gcd_idem_int: fixes x :: "int" shows "gcd x x = \<bar>x\<bar>"
infinite: fixes A :: "'k set" assumes "infinite A" shows "mset_set A = {#}"
eqv_sound: fixes \<phi> :: "WS1S_Nameful.ws1s"   and \<psi> :: "WS1S_Nameful.ws1s"   and I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset" assumes "eqv \<phi> \<psi>" shows "satisfies I1 I2 \<phi> = satisfies I1 I2 \<psi>"
Fract_cases: fixes q :: "'c fract" obtains a :: "'c" and b :: "'c" where "q = Fraction_Field.Fract a b" and "b \<noteq> (0::'c)"
deriv_bound_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and n :: "nat" shows "deriv_bound r a n = (\<nexists>b. (a, b) \<in> r ^^ Suc n)"
cycle_free_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat" shows "linordered_monoid.cycle_free m n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (\<forall>j\<le>n. linordered_monoid.len m i j (rem_cycles i j xs) \<preceq> linordered_monoid.len m i j xs) \<and> Le (0::'c) \<preceq> linordered_monoid.len m i i xs"
t1_2: shows "t1 \<in> existence_ivl0 x"
is_derivation_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" shows "src.is_derivation X p = (linear_on src.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X \<and> (\<forall>f g. f \<in> src.diff_fun_space \<longrightarrow> g \<in> src.diff_fun_space \<longrightarrow> X (f * g) = f p * X g + g p * X f))"
psi_odd_def: fixes n :: "nat" shows "psi_odd n = sum mangoldt_odd {1..n}"
cod_simp: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.cod f = cod f"
map_PP_sorted_list_of_set: fixes T :: "('x \<Rightarrow>\<^sub>0 nat) set" shows "map PP (pp_pm.ordered_powerprod_lin.sorted_list_of_set T) = ordered_powerprod_lin.sorted_list_of_set (PP ` T)"
infsetsum_nonneg: fixes A :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "0 \<le> infsetsum f A"
flush_all_until_volatile_write_Read_commute: fixes i :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and v :: "nat"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ls"   and "ls ! i = (p, Read volatile a t # is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ls[i := (p, is, \<theta>(t \<mapsto> v), sb @ [Read\<^sub>s\<^sub>b volatile a t v], \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ls m"
collapse: fixes labeled_graph :: "('e, 'f) labeled_graph" shows "LG (edges labeled_graph) (vertices labeled_graph) = labeled_graph"
dist_fs_cvec_zero: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" assumes "z \<noteq> vec_zero"   and "w \<noteq> vec_zero" shows "(dist_fs_cvec z w = 0) = ((cmod \<langle>z,w\<rangle>)\<^sup>2 = \<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2)"
mset_append: fixes xs :: "'a list"   and ys :: "'a list" shows "mset (xs @ ys) = mset xs + mset ys"
Tn_quasi_measure_preserving: fixes n :: "nat" shows "T ^^ n \<in> quasi_measure_preserving M M"
emb_prj_emb: fixes x :: "'c" assumes "DEFL('c) \<sqsubseteq> DEFL('d)" shows "EMB('d)\<cdot>(PRJ('d)\<cdot>(EMB('c)\<cdot>x)) = EMB('c)\<cdot>x"
content_ball_gt_0_iff: fixes z :: "'b"   and r :: "real" shows "(0 < content (ball z r)) = (0 < r)"
smult_r_distr: fixes a :: "'a"   and x :: "'c"   and y :: "'c" assumes "a \<in> carrier R"   and "x \<in> carrier M1"   and "y \<in> carrier M1" shows "a \<odot>\<^bsub>M1\<^esub> (x \<oplus>\<^bsub>M1\<^esub> y) = a \<odot>\<^bsub>M1\<^esub> x \<oplus>\<^bsub>M1\<^esub> a \<odot>\<^bsub>M1\<^esub> y"
rep_eq: fixes x :: "'a covector"   and xa :: "'a covector" shows "Rep_covector (x \<squnion> xa) = Rep_covector x \<squnion> Rep_covector xa"
1: shows "z \<squnion> z\<^sup>T = y \<and> asymmetric z \<and> injective z"
partition_\<omega>\<omega>_aux: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>" shows "Partitions.partn_lst (lenlex less_than) WW [\<omega> \<up> \<omega>, \<alpha>] 2"
sup_absorb2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x \<squnion> y = y"
Abs_perm_inverse: fixes y :: "'b \<Rightarrow> 'b" assumes "y \<in> perms" shows "Rep_perm (Abs_perm y) = y"
zero_less_iff_neq_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
simps_1: fixes uu :: "'a tree23" shows "len (T uu) = 1"
vsv_vcard_vdomain: fixes r :: "V" assumes "vsv r" shows "vcard (\<D>\<^sub>\<circ> r) = vcard r"
prems_3: shows "(t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
abelian_groupE_2: fixes R :: "('a, 'b) ring_scheme" assumes "abelian_group R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
assms: fixes i :: "'a" assumes "i \<in> I" shows "x i \<in> space\<^sub>N N"
EX: fixes wa :: "('p, 'ba) label list"   and wb :: "('p, 'ba) label list"   and uh :: "'n"   and ch :: "'n list multiset" assumes "w = wa @ LCall p # wb"   and "(([sp], c), wa, [uh], ch) \<in> trcl (trss fg)"   and "(([uh], ch), LCall p, [entry fg p, u'], ch) \<in> trss fg"   and "(uh, Call p, u') \<in> edges fg"   and "(([entry fg p], ch), wb, [return fg p], c') \<in> trcl (trss fg)" shows "P"
keys_pp_to_list_def: fixes t :: "('a, 'b) pp" shows "keys_pp_to_list t = sorted_list_of_set (keys_pp t)"
l11_31_1: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "B Out A C"   and "D \<noteq> E"   and "F \<noteq> E" shows "A B C LeA D E F"
thetaSDone_Sretr: shows "thetaSDone \<subseteq> Sretr thetaSDone"
valid_sops_expr_mono: fixes t :: "nat"   and e :: "expr"   and t' :: "nat" assumes "valid_sops_expr t e"   and "t \<le> t'" shows "valid_sops_expr t' e"
continuous_on_inverse: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "\<forall>x\<in>s. f x \<noteq> (0::'d)" shows "continuous_on s (\<lambda>x. inverse (f x))"
le_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
Rep_hyp2_inverse: fixes x :: "hyp2" shows "Abs_hyp2 (Rep_hyp2 x) = x"
assms_3: shows "B.isomorphic (F f) (F f')"
len_b: shows "Suc (Suc n) < e_length (local.prefixes j (Suc (Suc n)))"
ordered_semiring_record_simps_1: fixes carrier :: "'a set"   and more :: "'b" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
order_tendstoI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and y :: "'b"   and f :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "class.order_topology less_eq less open"   and "\<And>a. less a y \<Longrightarrow> \<forall>\<^sub>F x in F. less a (f x)"   and "\<And>a. less y a \<Longrightarrow> \<forall>\<^sub>F x in F. less (f x) a" shows "filterlim f (topological_space.nhds open y) F"
in_prime_factors_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and x :: "'c" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "(p \<in># factorial_semiring.prime_factorization zero times one normalize x) = (x \<noteq> zero \<and> dvd.dvd times p x \<and> normalization_semidom.prime zero times one normalize p)"
axis_nth: fixes i :: "'b"   and x :: "'a" shows "axis i x $ i = x"
preserves_seq: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VV.seq g f" shows "C.VV.seq (C.VV.map g) (C.VV.map f)"
Dup_block: fixes p :: "('b, 'c) fm"   and i :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and ps' :: "('b, 'c) fm list" assumes "Dup p i ((ps, a) # branch) xs" shows "Dup p i ((ps' @ ps, a) # branch) xs"