hcju/isabelleps huggingface dataset:
real_transfer_operator_intg_fpos_1: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. f (T x) * g x)"   and "\<And>x. 0 \<le> f x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "integrable M (\<lambda>x. f x * real_transfer_operator g x)"
dims_tensor_from_lookup: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" shows "dims (tensor_from_lookup ds e) = ds"
invertible_commutes_with_inverse: fixes a :: "'b" assumes "A.additive.invertible a"   and "a \<in> A" shows "f (A.additive.inverse a) = B.additive.inverse (f a)"
drop_Cons': fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
bot_eq: shows "\<bottom> = - (\<iota> \<squnion> - \<iota>)"
comp_def: fixes f :: "'m \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'm" shows "f \<circ> g = (\<lambda>x. f (g x))"
assms: shows "run_one_step d i (s, vs, ves, $Br x8) = (s', vs', res)"
vcomp_assoc: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<rho> :: "'c \<Rightarrow> 'd"   and H :: "'c \<Rightarrow> 'd"   and \<sigma> :: "'c \<Rightarrow> 'd"   and K :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<rho>"   and "natural_transformation A B G H \<sigma>"   and "natural_transformation A B H K \<tau>" shows "vertical_composite.map A B (vertical_composite.map A B \<rho> \<sigma>) \<tau> = vertical_composite.map A B \<rho> (vertical_composite.map A B \<sigma> \<tau>)"
maybe_arith_int_None: fixes f :: "int \<Rightarrow> int \<Rightarrow> int"   and a1 :: "value option"   and a2 :: "value option" shows "(maybe_arith_int f a1 a2 = None) = (\<nexists>n n'. a1 = Some (Num n) \<and> a2 = Some (Num n'))"
accepts_star: fixes A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "accepts (RegExp2NA.star A) w = (\<exists>us. (\<forall>u\<in>set us. accepts A u) \<and> w = concat us)"
solve_depressed_cubic_real: fixes y :: "real"   and e :: "real"   and f :: "real" shows "(y \<in> set (solve_depressed_cubic_real e f)) = (y ^ 3 + e * y + f = 0)"
rep_in_hom: shows "\<guillemotleft>\<rho>'.T0.trnr\<^sub>\<epsilon> r' \<rho>' : g \<star> f\<^sup>* \<Rightarrow> r'\<guillemotright>"
comp_arr_inv': fixes f :: "'c local.term set" assumes "\<F>C.iso f" shows "f \<cdot> \<F>C.inv f = \<F>C.cod f"
inducts: fixes P :: "'g list \<Rightarrow> bool"   and list :: "'g list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
rcosets_finite: fixes R :: "'a set"   and H :: "'a set" assumes "R \<in> rcosets H"   and "H \<subseteq> carrier G"   and "finite H" shows "finite R"
less_eq_prod_def: fixes leq_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and leq_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "less_eq_prod leq_a less_a leq_b = (\<lambda>(x1, x2) (y1, y2). less_a x1 y1 \<or> leq_a x1 y1 \<and> leq_b x2 y2)"
pre_ext: shows "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (mk_eq' \<tau>' (Abs_fv x \<tau> s $ Bv 0) (Abs_fv x \<tau> t $ Bv 0))"
j_eq: shows "mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0..<nrows_iarray (matrix_to_iarray A)] ! i) (matrix_to_iarray A))"
subst_idemI: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "subst_domain \<sigma> \<inter> range_vars \<sigma> = {}" shows "subst_idem \<sigma>"
set_of_interval_union: fixes A :: "'b interval"   and B :: "'b interval" shows "set_of A \<union> set_of B \<subseteq> set_of (sup A B)"
rel_gpv_conv_rel_gpv'': fixes A :: "'f \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_gpv A C = rel_gpv'' A C (=)"
meet_cont': fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'c"   and y :: "'c" assumes "class.cont_binary_meet below"   and "chain Y" shows "(\<Squnion>i. Y i) \<sqinter> y = (\<Squnion>i. Y i \<sqinter> y)"
uppert_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "local.uppert A i j \<equiv> j < i \<longrightarrow> A $$ (i, j) = (0::'a)"
andalso_thms_3: fixes y :: "tr" shows "(\<bottom> andalso y) = \<bottom>"
i: shows "ideal_generated {a1, b1} = ideal_generated {1::'a}"
butlast_bin_rest: fixes bl :: "bool list" shows "butlast bl = bin_to_bl (length bl - Suc 0) (bl_to_bin bl div 2)"
unshared_share_in: fixes a :: "nat"   and S :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list" assumes "a \<in> dom S"   and "a \<notin> all_unshared sb" shows "a \<in> dom (share sb S)"
\<b>_one: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "finite X"   and "valid_decomp X ps"   and "standard_decomp k ps" shows "\<b> ps (Suc 0) = (if ps\<^sub>+ = [] then 0 else Suc (Max (poly_deg ` fst ` set (ps\<^sub>+))))"
sint_1_cases: fixes a :: "'b word" obtains "LENGTH('b) = 1" and "a = 0" and "sint a = 0"   | "LENGTH('b) = 1" and "a = 1" and "sint 1 = - 1"   | "1 < LENGTH('b)" and "sint 1 = 1"
integral_nonneg_eq_0_iff_AE: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 \<le> f x" shows "(integral\<^sup>L M f = 0) = (AE x in M. f x = 0)"
nontriv_solution_altdef: fixes z :: "'a \<times> 'a" shows "nontriv_solution z = (solution z \<and> z \<noteq> (1::'a, 0::'a))"
word_numeral_alt: fixes b :: "num" shows "numeral b = word_of_int (numeral b)"
dd: shows "direct_decomp (ideal (insert f0 F) \<inter> P[X]) [ideal {f0} \<inter> P[X], T]"
ANNOT_def: fixes x :: "'c"   and a :: "annot" shows "Autoref_Tagging.ANNOT x a \<equiv> x"
ide_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "ide f = (f \<noteq> Null \<and> Dom f \<in> {False, True} \<and> Cod f = Dom f \<and> Map f = [])"
in_oreturn: fixes x :: "'c"   and s :: "'d"   and v :: "'c" shows "(oreturn x s = Some v) = (v = x)"
rep_equivalent: fixes H :: "'a set"   and x :: "'b"   and g :: "'a" assumes "H \<in> carrier (G LMod stabiliser x)"   and "g \<in> H" shows "H = g <# stabiliser x"
stream_all2_refl: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a stream" shows "stream_all2 P x x = pred_stream (\<lambda>x. P x x) x"
rep_eq: fixes x :: "\<I>" shows "right x = snd (Rep_\<I> x)"
measurable_distr2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and L :: "'b measure"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'b \<Rightarrow> 'c measure" assumes "(\<lambda>(x, y). f x y) \<in> L \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N"   and "g \<in> L \<rightarrow>\<^sub>M subprob_algebra M" shows "(\<lambda>x. distr (g x) N (f x)) \<in> L \<rightarrow>\<^sub>M subprob_algebra N"
MGF_Impl: fixes M :: "cname \<times> mname"   and Z :: "state" shows "{} |\<turnstile> {MGT (Impl M) Z}"
invalidate_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" shows "invalidate rt dests \<equiv> \<lambda>ip. case (rt ip, dests ip) of (None, x) \<Rightarrow> None | (Some (a, ab, ad, af, ah, ba), None) \<Rightarrow> Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) \<Rightarrow> Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)"
compl_top_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus top = bot"
f_1not0: shows "f 1 \<noteq> f 0"
trace_rev_induct: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and P :: "('p, 's, 'm) configuration \<Rightarrow> ('p, 's, 'm) event list \<Rightarrow> ('p, 's, 'm) configuration \<Rightarrow> bool" assumes "trace c t c'"   and "\<And>c. P c [] c"   and "\<And>c t c' ev c''. \<lbrakk>trace c t c'; P c t c'; c' \<turnstile> ev \<mapsto> c''\<rbrakk> \<Longrightarrow> P c (t @ [ev]) c''" shows "P c t c'"
fls_zero_eqI: fixes f :: "'b fls" assumes "\<And>n. f $$ n = (0::'b)" shows "f = 0"
set_primes_upto_sieve: fixes n :: "nat" shows "set (primes_upto n) = numbers_of_marks 2 (sieve 1 (replicate (n - 1) True))"
keys_eq_empty_iff: fixes c :: "'b \<Rightarrow>\<^sub>0 'c" shows "(keys c = {}) = (c = 0)"
lprefix_ltlI: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys" shows "ltl xs \<sqsubseteq> ltl ys"
inject_1: fixes x1 :: "int"   and y1 :: "int" shows "(A x1 = A y1) = (x1 = y1)"
hisPrev_prop_2: fixes n' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid" shows "\<exists>nid'. (hisPred t n nid n', nid') \<in> his t n nid"
left_commute: fixes b :: "'a upred"   and a :: "'a upred"   and c :: "'a upred" shows "(b \<or> a \<or> c) = (a \<or> b \<or> c)"
geodesic_subsegment_def: fixes G :: "'b set"   and x :: "'b"   and s :: "real"   and t :: "real" shows "geodesic_subsegment G x s t = G \<inter> {z. s \<le> dist x z \<and> dist x z \<le> t}"
butlast_rev: fixes xs :: "'a list" shows "butlast (rev xs) = rev (tl xs)"
intros_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> =n\<Rightarrow> Normal (f s)"
rep_eq: fixes x :: "'c dense"   and xa :: "'c dense" shows "Rep_dense (x \<sqinter> xa) = Rep_dense x \<sqinter> Rep_dense xa"
1: shows "Cring_Poly.compose R (monom (UP R) a n) g = a \<odot>\<^bsub>UP R\<^esub> g [^]\<^bsub>UP R\<^esub> n"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "TE.wtE \<xi>" shows "Ik.satPB \<xi> \<Phi>"
set_cols_in: shows "set (cols A) \<subseteq> carrier_vec n"
normalize_quot_swap: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> (0::'b)"   and "b \<noteq> (0::'b)" shows "normalize_quot (b, a) = (snd (normalize_quot (a, b)) div unit_factor (fst (normalize_quot (a, b))), fst (normalize_quot (a, b)) div unit_factor (fst (normalize_quot (a, b))))"
simps_1: fixes a :: "'d"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "PsubstFO ($\<alpha> a) \<sigma> = $\<alpha> a"
fsigma_imp_gdelta: fixes S :: "'b set" assumes "fsigma S" shows "gdelta (- S)"
permutes_conv_has_dom: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" shows "(f permutes S) = (bij f \<and> has_dom f S)"
sorted_wrt_irrefl_distinct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "irreflp r" shows "sorted_wrt r xs \<longrightarrow> distinct xs"
ceiling_minus: fixes x :: "'a" shows "\<lceil>- x\<rceil> = - \<lfloor>x\<rfloor>"
parent_child_rel_node_ptr: fixes parent :: "(_) object_ptr"   and child :: "(_) object_ptr"   and h :: "(_) heap" assumes "(parent, child) \<in> Shadow_DOM.parent_child_rel h" shows "is_node_ptr_kind child"
msetext_dersh_singleton: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" shows "msetext_dersh gt [y] [x] = gt y x"
1: shows "prv (imp \<phi> (imp (all x (neg \<phi>)) fls))"
is_weak_composition_with_homs: shows "weak_composition_with_homs (\<cdot>) (\<star>) src trg"
Cprops_2: shows "support_set p \<union> C = A"
isomorphicI: fixes f :: "'a" assumes "B.iso f" shows "B.dom f \<cong>\<^sub>B B.cod f"
field_nilp_zero: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier K"   and "x^\<^bsup>K n\<^esup> = \<zero>" shows "x = \<zero>"
binsert_sorted: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and x :: "'b" shows "sortedTree h t \<longrightarrow> sortedTree h (binsert h x t)"
assms_6: fixes s :: "('i, 'a) state" shows "P s = P' (\<lambda>x y. y < x) \<B>\<^sub>i\<^sub>l \<B>\<^sub>i\<^sub>u \<B>\<^sub>l \<B>\<^sub>u \<B>\<^sub>i\<^sub>l_update \<I>\<^sub>l \<I>\<^sub>u Geq Leq s"
simple_function_compose2: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "simple_function M f"   and "simple_function M g" shows "simple_function M (\<lambda>x. h (f x) (g x))"
coeffs_map_poly_hom: fixes p :: "real poly" shows "coeffs (map_poly of_real p) = map of_real (coeffs p)"
simps_5: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'m" shows "RDocument.more \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = more"
antisym_conv2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "(\<not> y < x) = (x = y)"
f_def: fixes n :: "nat"   and i :: "nat" shows "f n i = e i * (1 - 1 / real (Suc (Suc n)))"
assms_7: shows "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)"
7: shows "(\<lambda>x. F (g1 x, x) \<bullet> j) integrable_on cbox a b"
is_Pure_map_generat: fixes f :: "'j \<Rightarrow> 'g"   and g :: "'k \<Rightarrow> 'h"   and h :: "'l \<Rightarrow> 'i"   and x :: "('j, 'k, 'l) generat" shows "is_Pure (map_generat f g h x) = is_Pure x"
psiCasesFreshChain_1: fixes xvec :: "name list"   and Cs :: "('f \<times> ('d, 'e, 'f) psi) list" shows "xvec \<sharp>* psiCases Cs = xvec \<sharp>* Cs"
arctan_upper_32_def: shows "arctan_lower_33 \<equiv> \<lambda>x. 7 * (33 * x ^ 4 + 170 * x\<^sup>2 + 165) * x / (5 * (5 * x ^ 6 + 105 * x ^ 4 + 315 * x\<^sup>2 + 231))"
inverse_mult_eq_1': fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "f * inverse f = 1"
mset_sort: fixes cmp :: "'b comparator"   and xs :: "'b list" shows "mset (Sorting_Algorithms.sort cmp xs) = mset xs"
subseq_Cons2_neq: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" assumes "subseq (x # xs) (y # ys)"   and "x \<noteq> y" shows "subseq (x # xs) ys"
eventually_conj_iff: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "(\<forall>\<^sub>F x in F. P x \<and> Q x) = (eventually P F \<and> eventually Q F)"
None_add2: fixes x :: "'a option" shows "None ++ x = None"
drop_cancel_suffix: fixes a :: "'a list"   and c :: "'a list"   and n :: "nat"   and b :: "'a list" assumes "a @ c = drop n (b @ c)" shows "a = drop n b"
goinitial_impl_def: shows "goinitial_impl \<equiv> (None, Map.empty)"
simps_1: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "meldUniq [] bq = bq"
copysign_1_0_1: shows "copysign_double 1 0 = 1"
a1: fixes x :: "'a" assumes "x \<in> S" shows "a x = 1"
clock_simAbs_inj_on: shows "inj_on clock_simAbs (rel_ext (\<lambda>x. clock_simAbs x \<in> (\<Union>a. Clock.sim_equiv_class a ` Clock.jkbpC)))"
even_and_iff_int: fixes k :: "int"   and l :: "int" shows "even (k AND l) = (even k \<or> even l)"
ctr_2: fixes n :: "enat"   and xs :: "'c llist" assumes "n \<noteq> 0"   and "\<not> lnull xs" shows "ltake n xs = LCons (lhd xs) (ltake (epred n) (ltl xs))"
prv_eql_trans: fixes t1 :: "'trm"   and t2 :: "'trm"   and t3 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "t3 \<in> trm"   and "prv (eql t1 t2)"   and "prv (eql t2 t3)" shows "prv (eql t1 t3)"
iwlsFSbSwTR_iterSTR_ipresVar: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "ipresVar (iterSTR MOD) MOD"
unique_moebius_three_points: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and M1 :: "moebius"   and w1 :: "complex_homo"   and w2 :: "complex_homo"   and w3 :: "complex_homo"   and M2 :: "moebius" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3"   and "moebius_pt M1 z1 = w1"   and "moebius_pt M1 z2 = w2"   and "moebius_pt M1 z3 = w3"   and "moebius_pt M2 z1 = w1"   and "moebius_pt M2 z2 = w2"   and "moebius_pt M2 z3 = w3" shows "M1 = M2"
simps: fixes sr :: "nat"   and i :: "nat" shows "Entry sr i = lo sr (Pi (Suc i))"
ca_tendsto_zero: shows "(\<lambda>n. real_of_int (c (Suc n)) / real_of_int (a n)) \<longlonglongrightarrow> 0"
cod_in_codomains: fixes f :: "'a" assumes "H.codomains f \<noteq> {}" shows "H.cod f \<in> H.codomains f"
b'x: shows "Derives1 b' (fst d - 1 + length (snd r)) (snd d) x"
DOM_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.DOM (\<F>C.mkarr t) = Dom t"
sgn_real_alg_rat: fixes x :: "real_alg" shows "real_of_rat (sgn_real_alg_rat x) = sgn (real_of x)"
assms_3: shows "card {z \<in> A. z < x} = card {z \<in> A. z < y}"
unitor_coincidence_1: fixes a :: "'a" assumes "obj a" shows "\<l>[a] = \<i>[a]"
scale_sum_left: fixes f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and x :: "('b, 'c) vec" shows "sum f A *s x = (\<Sum>a\<in>A. f a *s x)"
eq_refl: fixes x :: "'b"   and y :: "'b"   and ko :: "'b key_order" assumes "x = y" shows "ko.le ko x y"
refine_list_2: fixes R :: "('a \<times> 'b) set" shows "((#), (#)) \<in> R \<rightarrow> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
assms: shows "FlexRayController (nReturn k) recv (nC k) (nStore k) (nSend k) (nGet k)"
distinct_43: fixes x4 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type" shows "Collection x4 \<noteq> Set x5"
orthogonal_matrix_orthonormal_columns: fixes A :: "((real, 'a) vec, 'a) vec" shows "orthogonal_matrix A = ((\<forall>i. norm (column i A) = 1) \<and> (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (column i A) (column j A)))"
id_eq_prop_prop_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1" shows "[F \<^bold>= G \<^bold>\<rightarrow> G \<^bold>= F in v]"
h8c: shows "(- Bv * sqrt Cv)\<^sup>2 = Bv\<^sup>2 * Cv"
len_decomp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and xs :: "nat list"   and ys :: "nat list"   and y :: "nat"   and zs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and x :: "nat"   and z :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "xs = ys @ y # zs" shows "linordered_ab_monoid_add.len plus m x z xs = plus (linordered_ab_monoid_add.len plus m x y ys) (linordered_ab_monoid_add.len plus m y z zs)"
3: shows "fmlookup (action_needed_asses a (f ++ g)) v = fmlookup f v"
next'_def: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration" shows "next' c0 c1 = (next_propagate c0 c1 \<or> next_change_multiplicity c0 c1 \<or> c1 = c0)"
iTrigger_False_left: fixes I :: "nat set"   and Q :: "nat \<Rightarrow> bool" shows "(False. t' \<T> t I. Q t) = (\<box> t I. Q t)"
borel_measurable_indicator': fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a \<Rightarrow> 'b set" assumes "{x \<in> space M. f x \<in> A x} \<in> sets M" shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
Thin: fixes Gxy :: "'a set"   and x :: "'a"   and y :: "'a"   and Gxz :: "'a set"   and z :: "'a"   and Gyz :: "'a set"   and w :: "'a" assumes "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gxz x z"   and "geodesic_segment_between Gyz y z"   and "w \<in> Gxy" shows "infdist w (Gxz \<union> Gyz) \<le> M"
h: shows "hd sl1 \<in> \<delta> (last sl)"
rotate_B_def: shows "rotate_B = (\<lambda>v. case v of A \<Rightarrow> D | B \<Rightarrow> B | C \<Rightarrow> A | D \<Rightarrow> C)"
lower_max_interval: fixes x :: "'a interval"   and y :: "'a interval" shows "lower (max_interval x y) = max (lower x) (lower y)"
induct: fixes P :: "modality \<Rightarrow> safety \<Rightarrow> 'a regex \<Rightarrow> bool"   and a0 :: "modality"   and a1 :: "safety"   and a2 :: "'a regex" assumes "\<And>m uu_ n. P m uu_ (Skip n)"   and "\<And>m g \<phi>. P m g (Test \<phi>)"   and "\<And>m g r s. \<lbrakk>P m g r; P m g s\<rbrakk> \<Longrightarrow> P m g (Plus r s)"   and "\<And>g r s. \<lbrakk>P Futu g s; P Futu Lax r\<rbrakk> \<Longrightarrow> P Futu g (Times r s)"   and "\<And>g r s. \<lbrakk>P Past g r; P Past Lax s\<rbrakk> \<Longrightarrow> P Past g (Times r s)"   and "\<And>m g r. P m g r \<Longrightarrow> P m g (Star r)" shows "P a0 a1 a2"
ovl_shorten: fixes f :: "face"   and v :: "nat"   and va :: "nat option"   and vol :: "nat option list" assumes "distinct (verticesFrom f v)"   and "filter (\<lambda>v. v \<in> set (removeNones (va # vol))) (verticesFrom f v) = removeNones (va # vol)" shows "filter (\<lambda>v. v \<in> set (removeNones vol)) (verticesFrom f v) = removeNones vol"
atomic_step_def: fixes s :: "state_t"   and ipt :: "int_point_t" shows "atomic_step s ipt \<equiv> case ipt of SK_IPC dir stage partner page \<Rightarrow> atomic_step_ipc (current s) dir stage partner page s | SK_EV_WAIT EV_FINISH EV_CONSUME_ALL \<Rightarrow> atomic_step_ev_wait_all (current s) s | SK_EV_WAIT EV_FINISH EV_CONSUME_ONE \<Rightarrow> atomic_step_ev_wait_one (current s) s | SK_EV_WAIT _ consume \<Rightarrow> s | SK_EV_SIGNAL EV_SIGNAL_PREP partner \<Rightarrow> s | SK_EV_SIGNAL EV_SIGNAL_FINISH partner \<Rightarrow> atomic_step_ev_signal (current s) partner s | NONE \<Rightarrow> s"
log_convex_G: shows "convex_on {0<..} (ln \<circ> G)"
check_eqv_soundness: fixes \<AA> :: "interp"   and idx :: "idx"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and \<psi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" assumes "#\<^sub>V \<AA> = idx"   and "WS1S_Alt.check_eqv idx \<phi> \<psi>" shows "WS1S_Alt.sat \<AA> \<phi> = WS1S_Alt.sat \<AA> \<psi>"
ipurge_tr_set: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list" shows "set (ipurge_tr I D u xs) \<subseteq> set xs"
mirlfr: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (mirlfr p) \<and> Ifm bs (mirlfr p) = Ifm bs (E p)"
semiring_1_ow_def: fixes U :: "'b set"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "semiring_1_ow U one times plus zero \<equiv> zero_neq_one_ow U one zero \<and> semiring_0_ow U plus zero times \<and> monoid_mult_ow U one times"
mcdj': shows "m cd\<^bsup>\<pi>\<^esup>\<rightarrow> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> \<pi> m)"
f\<C>_A: shows "f\<C>_A \<equiv> {F \<in> \<C>_A. face_distance z F = chamber_distance C F}"
trg_dom: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.trg (Span.dom \<mu>) = Span.trg \<mu>"
d: shows "set lI = {x \<in> I. x \<notin> s}"
strong_convergence: fixes i :: "example_peers \<times> nat"   and j :: "example_peers \<times> nat" assumes "example.is_valid_state_id i"   and "example.is_valid_state_id j"   and "set (example.received_messages i) = set (example.received_messages j)" shows "example.state i = example.state j"
symm_preorder_is_equivalence: fixes Rel :: "('a \<times> 'a) set" assumes "preorder Rel"   and "sym Rel" shows "equivalence Rel"
fin_dim_li_fin: fixes A :: "'a vec set" assumes "fin_dim"   and "lin_indpt A"   and "A \<subseteq> carrier_vec n" shows "finite A"
diff_def: shows "diff = (\<lambda>w. f w - g w)"
simps_2: fixes n1 :: "('c, 'd) Node" shows "is_measured_ftree (Single n1) = is_measured_node n1"
rational_poly_cond_iff: fixes p :: "int poly"   and x :: "'a" assumes "poly_cond p"   and "ipoly p x = (0::'a)"   and "1 < degree p" shows "(x \<in> \<rat>) = (degree p = 1)"
type_aexpr_Load: fixes v :: "Eg1.var"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option" assumes "v \<notin> dom \<Gamma>" shows "type_aexpr \<Gamma> (Load v) (dma_type v)"