hcju/isabelleps huggingface dataset:
open_prod_def: fixes S :: "('a \<times> 'b) set" shows "open S = (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
simps_2: fixes shared :: "bool"   and r0 :: "nat"   and r1 :: "nat"   and m :: "(nat \<times> nat) convnet"   and w :: "nat \<Rightarrow> real" shows "insert_weights shared (Conv (r0, r1) m) w = Conv (extract_matrix w r0 r1) (insert_weights shared m (\<lambda>i. w (i + r0 * r1)))"
norm_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C" shows "free_monoidal_category.norm C t = monoidal_language.Inv C (monoidal_language.red C (monoidal_language.Cod C t)) \<^bold>\<cdot> monoidal_language.Diagonalize C t \<^bold>\<cdot> monoidal_language.red C (monoidal_language.Dom C t)"
0: shows "LBINT x=ereal a..ereal b. g' x *\<^sub>R f (g x) = integral {a..b} (\<lambda>x. g' x *\<^sub>R f (g x))"
rep_eq: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "L_transform_formula x = L_transform_Tree\<^sub>\<alpha> (FL_Formula.formula.Rep_formula x)"
chain_boundary_cmul: fixes p :: "nat"   and k :: "int"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p (frag_cmul k c) = frag_cmul k (chain_boundary p c)"
select_convs_4: fixes c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_glob \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = c_glob"
simps_4: fixes x2 :: "ref_ty"   and x1 :: "prim_ty" shows "RefT x2 \<noteq> PrimT x1"
lock_thread_ok_has_lockE: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option"   and l :: "'d"   and t :: "'e" assumes "lock_thread_ok ls ts"   and "has_lock (ls $ l) t" obtains x :: "'f" and ln' :: "'d \<Rightarrow>f nat" where "ts t = \<lfloor>(x, ln')\<rfloor>"
esssup_eq_AE: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "esssup M f = Inf {z. AE x in M. f x \<le> z}"
eq_refl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
swapEnv_substEnv: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good Y"   and "goodEnv rho" shows "rho &[Y / y]_ys &[z1 \<and> z2]_zs = rho &[z1 \<and> z2]_zs &[Y #[z1 \<and> z2]_zs / y @ys[z1 \<and> z2]_zs]_ys"
wreqweq: shows "{y \<in> \<omega>_limit_set x. f y \<noteq> (0::'a)} \<union> {y \<in> \<omega>_limit_set x. f y = (0::'a)} = \<omega>_limit_set x"
quasi_ordering_axioms: shows "quasi_ordering UNIV (\<le>) (<)"
3: assumes "i < n - degree u" shows "degree (u * monom (1::'a) (n - Suc (degree u + i))) = n - Suc i"
expectation_add_pair_pmf: fixes p :: "'a pmf"   and q :: "'b pmf"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite (set_pmf p)"   and "finite (set_pmf q)" shows "measure_pmf.expectation (pair_pmf p q) (\<lambda>(x, y). f x + g y) = measure_pmf.expectation p f + measure_pmf.expectation q g"
parSym: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<sim>\<^sub>l Q \<parallel> P"
comp_arr_inv': fixes f :: "'a" assumes "L.iso f" shows "L.comp f (L.inv f) = L.cod f"
is_scc_closed: fixes E :: "('c \<times> 'c) set"   and U :: "'c set"   and x :: "'c"   and y :: "'c" assumes "is_scc E U"   and "x \<in> U"   and "(x, y) \<in> E\<^sup>*"   and "(y, x) \<in> E\<^sup>*" shows "y \<in> U"
pr_conv_2_to_1_lm: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec2" shows "pr_conv_2_to_1 f \<in> PrimRec1"
OclAnd_true1: fixes X :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(true and X) = X"
assoc'_naturality: fixes f0 :: "'d"   and f1 :: "'d"   and f2 :: "'d" assumes "D.arr f0"   and "D.arr f1"   and "D.arr f2" shows "((f0 \<otimes>\<^sub>D f1) \<otimes>\<^sub>D f2) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[D.dom f0, D.dom f1, D.dom f2] = \<a>\<^sub>D\<^sup>-\<^sup>1[D.cod f0, D.cod f1, D.cod f2] \<cdot>\<^sub>D (f0 \<otimes>\<^sub>D f1 \<otimes>\<^sub>D f2)"
in_netmap: fixes i :: "nat"   and ns :: "'c net_state" assumes "i \<in> net_ips ns" shows "netmap ns i \<noteq> None"
list_all2_trans: fixes P1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and P2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and P3 :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and as :: "'g list"   and bs :: "'h list"   and cs :: "'i list" assumes "\<And>a b c. \<lbrakk>P1 a b; P2 b c\<rbrakk> \<Longrightarrow> P3 a c"   and "list_all2 P1 as bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 as cs"
group_axioms_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b" shows "group_axioms f z inverse \<equiv> (\<forall>a. f z a = a) \<and> (\<forall>a. f (inverse a) a = z)"
C: fixes y :: "xml" assumes "y \<in> set B" shows "monotone sum_bot.le_fun sum_bot_ord (C y)"
bernpoly_def: fixes n :: "nat" shows "bernpoly n = (\<lambda>x. \<Sum>k\<le>n. real (n choose k) * bernoulli k * x ^ (n - k))"
transpose_iff: fixes A :: "(('a, 'c) vec, 'b) vec"   and B :: "(('a, 'c) vec, 'b) vec" shows "(Finite_Cartesian_Product.transpose A = Finite_Cartesian_Product.transpose B) = (A = B)"
C0: shows "C \<ge>\<^sub>m 0\<^sub>m n n"
single_add: fixes k :: "'a"   and a :: "'b"   and b :: "'b" shows "monomial (a + b) k = monomial a k + monomial b k"
bigtheta_powr: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and p :: "real" assumes "f \<in> \<Theta>[F](g)" shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> \<Theta>[F](\<lambda>x. \<bar>g x\<bar> powr p)"
indep_not_basis: fixes X :: "'a set" assumes "indep X"   and "\<not> basis X" shows "\<exists>x\<in>carrier - X. indep (insert x X)"
nat_distrib_2: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a * (b + c) = a * b + a * c"
induct: fixes x :: "('s, 'v) tm"   and P :: "('s, 'v) tm \<Rightarrow> bool" assumes "wary x"   and "\<And>\<zeta>. P (Hd \<zeta>)"   and "\<And>s t. \<lbrakk>wary s; P s; wary t; P t; enat (num_args s) < arity_hd (head s)\<rbrakk> \<Longrightarrow> P (App s t)" shows "P x"
hom_0_iff: fixes x :: "'b poly" shows "(pCons (0::'b) x = 0) = (x = 0)"
sinvar_mono_imp_sinvar_mono: fixes N :: "'v set"   and E :: "('v \<times> 'v) set"   and E' :: "('v \<times> 'v) set"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "sinvar \<lparr>nodes = N, edges = E\<rparr> nP" shows "sinvar \<lparr>nodes = N, edges = E'\<rparr> nP"
prefixes_Suc_last: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and t :: "nat" assumes "r1_bc s" shows "e_nth (r1_bc.prefixes s j (Suc t)) (e_length (r1_bc.prefixes s j (Suc t)) - 1) = 1"
wf_prog_wwf_prog: fixes 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" assumes "wf_C_prog P" shows "wwf_prog P"
mgu_img_subterm_subst: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and u :: "('b, 'a) Term.term" assumes "mgu s t = Some \<delta>"   and "u \<in> subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<delta>) - range Var" shows "u \<in> subterms s \<union> subterms t - range Var \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
iso_lunit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.iso (\<F>C.lunit a)"
assoc: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
real_ex_int_real01: fixes P :: "real \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>i u. 0 \<le> u \<and> u < 1 \<and> P (real_of_int i + u))"
one_le_inverse_iff: 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 x :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq one (inverse x) = (less zero x \<and> less_eq x one)"
unitr_ex: fixes x :: "'c" shows "\<exists>e\<in>E. D x e \<and> x \<oplus> e = x"
trms\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) strand_step list" shows "trms\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>t\<^sub>p ` set S)"
union_code: fixes xs :: "'a list"   and ys :: "'a list" shows "mset xs + mset ys = mset (xs @ ys)"
simps: fixes t :: "'t"   and a1 :: "(status \<times> 'x1) \<times> 'm1"   and a2 :: "(status \<times> 'x2) \<times> 'm2" shows "t \<turnstile> a1 \<approx>i a2 = ((\<exists>x1 m1 x2 m2. a1 = ((PreStart, x1), m1) \<and> a2 = ((PreStart, x2), m2) \<and> t \<turnstile> (x1, m1) \<approx> (x2, m2)) \<or> (\<exists>x1 m1 x2 m2. a1 = ((Running, x1), m1) \<and> a2 = ((Running, x2), m2) \<and> t \<turnstile> (x1, m1) \<approx> (x2, m2)) \<or> (\<exists>x1 m1 x2 m2. a1 = ((Finished, x1), m1) \<and> a2 = ((Finished, x2), m2) \<and> t \<turnstile> (x1, m1) \<approx> (x2, m2) \<and> final1 x1 \<and> final2 x2))"
append_eq_Cons: fixes x :: "'d"   and xs :: "'d list" shows "[x] @ xs = x # xs"
card_columns_le_ncols: fixes A :: "(('a, 'c) vec, 'b) vec" shows "card (columns A) \<le> ncols A"
simps_5: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "w \<Turnstile> \<phi> and \<psi> = (w \<Turnstile> \<phi> \<and> w \<Turnstile> \<psi>)"
list_of_dlist_empty: shows "list_of_dlist Dlist.empty = []"
card_farey_fractions: fixes N :: "nat" shows "card (farey_fractions N) = sum totient {0<..N}"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "y \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> y \<in> carrier (G\<lparr>carrier := S\<rparr>)"
real_of_ereal_inverse: fixes a :: "ereal" shows "real_of_ereal (inverse a) = 1 / real_of_ereal a"
lossless_iff_pmf_None: fixes p :: "'h spmf" shows "lossless_spmf p = (pmf p None = 0)"
sid_le_nc_var: fixes x :: "'a" shows "1\<^sub>\<sigma> \<sqinter> x \<le> 1\<^sub>\<sigma> \<sqinter> x \<parallel> nc"
position: shows "position_of (prio_emb_step prio t s) (prio_emb_pos prio (prio_emb_step prio t s) s)"
transfer_rule_numeral: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R (0::'a) (0::'b)"   and "R (1::'a) (1::'b)"   and "rel_fun R (rel_fun R R) (+) (+)" shows "rel_fun (=) R numeral numeral"
cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^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_def: fixes ptr :: "(_) object_ptr" shows "cast ptr = (case ptr of object_ptr.Ext (Inl x) \<Rightarrow> Map.empty x | object_ptr.Ext (Inr (Inl x)) \<Rightarrow> Map.empty x | object_ptr.Ext (Inr (Inr (Inl shadow_root_ptr))) \<Rightarrow> Some shadow_root_ptr | object_ptr.Ext (Inr (Inr (Inr x))) \<Rightarrow> Map.empty x)"
prems_2: shows "combine Ks (u # Us) = \<zero>"
sel_2: fixes x1 :: "'l"   and x2 :: "'k" shows "snd (x1, x2) = x2"
gval_fold_gAnd_append_singleton: fixes a :: "'a gexp list"   and G :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gAnd (a @ [G]) (Bc True)) s = gval (fold gAnd a (Bc True)) s \<and>? gval G s"
mplus_mzero_right: fixes m :: "'a\<cdot>'b" shows "mplus\<cdot>m\<cdot>mzero = m"
snd_eqvt: fixes p :: "perm"   and x :: "'b \<times> 'a" shows "p \<bullet> snd x = snd (p \<bullet> x)"
c_2: shows "\<forall>i. \<psi> i = \<chi> (the (c i))"
rewrite_iter_slow_sound: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n rewrite_iter_slow \<phi> = w \<Turnstile>\<^sub>n \<phi>"
distinct_tl: fixes xs :: "'c list" assumes "distinct xs" shows "distinct (tl xs)"
cMismatch: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and b :: "name"   and c :: "name"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "\<And>C. F C P a x P'"   and "b \<noteq> c" shows "F C ([b\<noteq>c]P) a x P'"
all_s_s_mul_ext: fixes A :: "'a multiset"   and B :: "'a multiset"   and s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "A \<noteq> {#}"   and "\<forall>b. b \<in># B \<longrightarrow> (\<exists>a. a \<in># A \<and> (a, b) \<in> s)" shows "(A, B) \<in> s_mul_ext ns s"
Problem_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" shows "M.Problem wtFsym wtPsym arOf resOf parOf \<Phi> \<equiv> Sig.Signature TYPE('c) wtFsym wtPsym \<and> M.Problem_axioms wtFsym wtPsym arOf resOf parOf \<Phi>"
ord_eq_less_trans: fixes a :: "'k"   and b :: "'k"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and c :: "'k" assumes "a = b"   and "less b c" shows "less a c"
diff_diff_add_mset: fixes M :: "'k multiset"   and N :: "'k multiset"   and P :: "'k multiset" shows "M - N - P = M - (N + P)"
map_upd_eqD1: fixes m :: "'c \<Rightarrow> 'd option"   and a :: "'c"   and x :: "'d"   and n :: "'c \<Rightarrow> 'd option"   and y :: "'d" assumes "m(a \<mapsto> x) = n(a \<mapsto> y)" shows "x = y"
K_consistency: shows "consistency {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}"
vector_mult_closed: fixes top :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and y :: "'c"   and x :: "'c" assumes "class.bounded_pre_left_semiring top one times sup less_eq less bot"   and "times y top = y" shows "times (times x y) top = times x y"
assms: shows "rel_spmf (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) p q"
inv_one: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.monoid G" shows "inv\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>G\<^esub>"
prems: shows "Subobjs P C (rev (C' # Cs'))"
\<psi>_invertible: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<exists>!g. \<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright> \<and> \<psi> x g = f"
scale_left_commute: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and b :: "'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" shows "a \<odot> (b \<odot> x) = b \<odot> (a \<odot> x)"
cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "o[F](f) = o[F](g)"
viewRn_empty: fixes t :: "('c, 'd) FingerTreeStruc" shows "(t \<noteq> Empty) = (viewRn t \<noteq> None)"
gauss_sum_int_def: fixes k :: "int" shows "gauss_sum_int k = (\<Sum>m = 1..n. \<chi> m * unity_root n (int m * k))"
assms_3: shows "(v, a') \<in> dom (\<phi>\<^sub>S \<Psi> s)"
sets_Sup_eq: fixes M :: "'b measure set"   and X :: "'b set" assumes "\<And>m. m \<in> M \<Longrightarrow> space m = X"   and "M \<noteq> {}" shows "sets (Sup M) = sigma_sets X (\<Union> (sets ` M))"
is_syz_sigI: fixes s :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t"   and d :: "'a \<Rightarrow> nat" assumes "s \<noteq> 0"   and "lt s = u"   and "s \<in> dgrad_sig_set' (length fs) d"   and "rep_list s = 0" shows "is_syz_sig d u"
digit_to_char_simps_4: shows "digit_to_char 3 = CHR ''3''"
zmod_imp_ZMod: fixes a :: "int"   and m :: "int"   and b :: "int" assumes "a mod m = b mod m" shows "ZMod m a = ZMod m b"
assms: shows "mi.restrict_top_impl p var val bdd = Some (r, bdd')"
arev_arev: fixes a :: "'b" shows "rev_I (rev_I a) = a"
Abs_freelist_inverse: fixes as :: "'b list" shows "freeword (Abs_freelist as) = map pairtrue as"
blocking_on_def: fixes ds :: "'d set"   and X :: "'x set"   and h :: "'h"   and X' :: "'x set" shows "blocking_on ds X h X' = (X' \<noteq> Ch h X \<and> X' = Ch h (X \<union> X') \<and> X' \<subseteq> CD_on ds (X \<union> X'))"
Rep_cfun_inject: fixes x :: "'b \<rightarrow> 'c"   and y :: "'b \<rightarrow> 'c" shows "(Rep_cfun x = Rep_cfun y) = (x = y)"
intros_1: fixes T :: "ty"   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 v :: "val" assumes "\<forall>C. T \<noteq> Class C" shows "P \<turnstile> T casts v to v "
conjugate_conjugate_sprod: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (conjugate v \<bullet> w) = inner_prod w v"
absolutely_integrable_imp_integrable: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S"   and "S \<in> sets lebesgue" shows "integrable (lebesgue_on S) f"
Meet_greatest: fixes A :: "'b set"   and x :: "'b" assumes "\<And>a. a \<in> A \<Longrightarrow> x \<sqsubseteq> a" shows "x \<sqsubseteq> \<Sqinter>A"
sums_mult2: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b" assumes "f sums a" shows "(\<lambda>n. f n * c) sums (a * c)"
Gromov_completion_boundary_limit: fixes x :: "'a Gromov_completion"   and u :: "'b \<Rightarrow> 'a Gromov_completion"   and F :: "'b filter" assumes "x \<in> Gromov_boundary" shows "(u \<longlongrightarrow> x) F = ((\<lambda>n. extended_Gromov_product_at basepoint (u n) x) \<longlongrightarrow> \<infinity>) F"
assms: shows "\<forall>n''>n'. \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>"
rcsTr7: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot> \<rho> a \<in> H" shows "H \<bullet> a \<subseteq> H \<bullet> b"
Var\<^sub>0_def: fixes n :: "'b" shows "Var\<^sub>0 n \<equiv> monomial (1::'c) (monomial 1 n)"
right_totalE: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and y :: "'f" assumes "right_total A" obtains x :: "'e" where "A x y"
poincare_distance_formula'_def: fixes u :: "complex"   and v :: "complex" shows "poincare_distance_formula' u v = 1 + 2 * ((cmod (u - v))\<^sup>2 / ((1 - (cmod u)\<^sup>2) * (1 - (cmod v)\<^sup>2)))"
simps_11: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and l2 :: "value" shows "AExp_Lexorder.less_aexp_aux (Plus e1 e2) (L l2) = False"
arr_Par_ArrVal_vdomain: shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrDom\<rparr>"
continuous_on_e2ennreal: fixes A :: "ereal set" shows "continuous_on A e2ennreal"
abs_eq: fixes xa :: "'b fset"   and x :: "'c fset" shows "xa |\<times>| x = Abs_fset (fset xa \<times> fset x)"
eqvt_bound: fixes p :: "perm"   and x :: "'e" shows "p \<bullet> unpermute p x \<equiv> x"
in_homE: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" obtains "D.arr f" and "D.dom f = a" and "D.cod f = b"
integral_nonneg: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>C f = n \<and> f \<cdot>\<^sub>C n = n"
to_total_type_wellformed: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and x :: "'a" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.types_wellformed bexp_vars \<C>_vars \<Gamma>" shows "sifum_types.type_wellformed bexp_vars \<C>_vars (sifum_types.to_total dma_type \<Gamma> x)"
\<tau>exec_move_into_\<tau>exec_moves: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and es :: "'addr expr1 list" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_moves ci P t (e # es) h s s'"
small_imp_big: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)" shows "f \<in> \<Omega>[F](g)"
path_inf_priorities_def: fixes P :: "'a llist" shows "path_inf_priorities P \<equiv> {k. \<forall>n. k \<in> lset (ldropn n (lmap \<omega> P))}"
HFail_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2b s"   and "HFail s s' p" shows "Inv2b s'"
the_\<theta>_props_1: shows "\<guillemotleft>w.the_\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright>"
upt_Suc: fixes i :: "nat"   and j :: "nat" shows "[i..<Suc j] = (if i \<le> j then [i..<j] @ [j] else [])"
canonical_basis_cblinfun_def: shows "canonical_basis = [1]"
inverse_arrowsE: fixes f :: "'a"   and g :: "'a" assumes "A.inverse_arrows f g" obtains "A.ide (g \<cdot>\<^sub>A f)" and "A.ide (f \<cdot>\<^sub>A g)"
snapshot_step_cs_preservation_q: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and q :: "'a"   and cid :: "nat"   and p :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "\<not> regular_event ev"   and "occurs_on ev = q"   and "channel cid = Some (p, q)"   and "Marker \<notin> set (msgs c cid)"   and "\<not> ps c q \<noteq> None" shows "map Msg (fst (cs c cid)) @ takeWhile ((\<noteq>) Marker) (msgs c cid) = map Msg (fst (cs c' cid)) @ takeWhile ((\<noteq>) Marker) (msgs c' cid) \<and> snd (cs c' cid) = Recording"
integer_of_uint16_def: shows "integer_of_uint16 = integer_of_int \<circ> int_of_uint16"
P2: fixes i :: "nat"   and xs :: "nat list"   and ys :: "nat list" assumes "i \<le> k"   and "js = xs @ ys" shows "\<not> (i \<in> set xs \<and> i \<in> set ys)"
strategyproofness: shows "pmf (sds R33) b + pmf (sds R33) a \<le> pmf (sds R5) b + pmf (sds R5) a"
au: shows "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
plossless_INF_le_expectation_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and f :: "'c \<Rightarrow> ennreal"   and fail :: "ennreal" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<Sqinter> (f ` results_gpv \<I> gpv) \<le> expectation_gpv fail \<I> f gpv"
that: fixes z :: "'b" assumes "y = Some z"   and "R x z" shows "thesis"
upper_uminus: fixes A :: "'a interval" shows "upper (- A) = - lower A"
Rep_formula_inverse: fixes x :: "'a formula" shows "Abs_formula (Rep_formula x) = x"
implInsec_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> sym.valid \<union> range LtK \<union> Tags"   and "sym.implInsec A B M \<in> synth (analz H)" shows "sym.implInsec A B M \<in> H \<or> M \<in> synth (analz H)"