hcju/isabelleps huggingface dataset:
oseqpTEs_4: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}send(s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = send (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "q = p"
False: shows "B $$ (i, i) \<noteq> (0::'a)"
Radj_repd2: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and r :: "real" shows "Radj x y (repd \<nu> x r) = repd (Radj x y \<nu>) y r"
order_tendstoD_1: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "(f \<longlongrightarrow> y) F"   and "a < y" shows "\<forall>\<^sub>F x in F. a < f x"
nxt_mono: fixes t :: "'b ttree"   and t' :: "'b ttree"   and x :: "'b" assumes "paths t \<subseteq> paths t'" shows "paths (nxt t x) \<subseteq> paths (nxt t' x)"
assms_2: shows "set_mset K \<subseteq> r \<down>m J"
elims_3: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "\<not> inv_begin1 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "(l, r) \<noteq> ([], Oc \<up> n)"
discrim_Crypt: fixes K :: "nat"   and X :: "msg" shows "discrim (Crypt K X) = discrim X + 2"
cases: fixes a1 :: "'a list"   and a2 :: "'a \<times> 'a" assumes "list_pairs a1 a2" obtains xs :: "'a list" and p :: "'a \<times> 'a" and x :: "'a" where "a1 = x__ # xs__" and "a2 = p" and "list_pairs xs__ p"   | y :: "'a" and xs :: "'a list" and x :: "'a" where "a1 = x # xs" and "a2 = (x, y)" and "y \<in> set xs"
inf_fun_def: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" shows "inf f g = (\<lambda>x. inf (f x) (g x))"
right_stoch_mat_mult_stoch_vec: fixes a :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('a, 'b) Finite_Cartesian_Product.vec" assumes "right_stoch_mat a"   and "stoch_vec v" shows "stoch_vec (a *v v)"
eval_wlp_Skip: fixes P :: "'a \<Rightarrow> real" shows "wlp Skip P = P"
ell2_of_vec_def: shows "ell2_of_vec = basis_enum_of_vec"
N_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and n :: "nat"   and bT :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" shows "N cf n bT \<equiv> \<not> discrCf ((cf ## bT) !! n)"
fSup_distr: fixes f :: "'e \<Rightarrow> 'c" shows "Sup_pres (\<lambda>x. x \<circ> f)"
nn: fixes b :: "'b" assumes "b \<in> B - A" shows "(0::'a) \<le> f b"
order_antisym: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
inter_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (rel_fun (rel_set A) (rel_set A)) (\<inter>) (\<inter>)"
sup_union_right2: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<in># N" shows "N \<union># add_mset x M = add_mset x ((N - {#x#}) \<union># M)"
reflT: shows "rel_reflects_binary_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
Refl: fixes x :: "'a language"   and y :: "'a language"   and R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "x = y" shows "Plus_cong R x y"
pars_mhead: fixes m :: "methd" shows "pars (mhead m) = pars m"
weave_def: fixes A :: "nat set"   and xs :: "'a list"   and ys :: "'a list" shows "weave A xs ys = map (\<lambda>i. if i \<in> A then xs ! card {a \<in> A. a < i} else ys ! card {a \<in> - A. a < i}) [0..<length xs + length ys]"
count_notin: fixes x :: "'c"   and xs :: "'c list" assumes "x \<notin> set xs" shows "count_list xs x = 0"
assms_2: shows "s \<in> ast_problem.valid_states prob"
substC_o: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm"   and c :: "('c, 'd) lit list" shows "substC (subst \<pi>1 \<circ> \<pi>2) c = substC \<pi>1 (substC \<pi>2 c)"
the_def: fixes option :: "'e option" shows "the option = (case option of Some x2 \<Rightarrow> x2)"
prim_vc_3: fixes minarc :: "'b \<Rightarrow> 'b"   and sum :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and t :: "'b"   and v :: "'b"   and g :: "'b"   and r :: "'b" assumes "class.m_kleene_algebra minarc sum plus inf less_eq less sup bot top uminus one times conv star"   and "m_kleene_algebra.prim_spanning_invariant inf less_eq top uminus one times conv star t v g r"   and "inf (times v (uminus (conv v))) g = bot" shows "m_kleene_algebra.spanning_tree inf less_eq uminus one times conv star t g r"
element_ptr_kinds_eq_h2: shows "element_ptr_kinds h3 = element_ptr_kinds h2"
uminus: shows "fs ! i - c \<cdot>\<^sub>v fs ! j = fs ! i + - c \<cdot>\<^sub>v fs ! j"
vertex_quotient_webD: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "vertex (quotient_web \<Gamma> f) x" shows "vertex \<Gamma> x \<and> x \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"
openin_subtopology_Int_subset: fixes u :: "'c set"   and S :: "'c set"   and v :: "'c set" assumes "openin (top_of_set u) (u \<inter> S)"   and "v \<subseteq> u" shows "openin (top_of_set v) (v \<inter> S)"
inv_unique: fixes y :: "'a set"   and x :: "'a set"   and y' :: "'a set" assumes "y \<otimes>\<^bsub>ideals_set R\<^esub> x = \<one>\<^bsub>ideals_set R\<^esub>"   and "x \<otimes>\<^bsub>ideals_set R\<^esub> y' = \<one>\<^bsub>ideals_set R\<^esub>"   and "x \<in> carrier (ideals_set R)"   and "y \<in> carrier (ideals_set R)"   and "y' \<in> carrier (ideals_set R)" shows "y = y'"
nsqn_sqn_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat" shows "nsqn rt dip = (if flag rt dip = Some val \<or> sqn rt dip = 0 then sqn rt dip else sqn rt dip - 1)"
downset_def: shows "prod.downset = prod.downset_set \<circ> \<eta>"
scaleR_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and a :: "real" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "less_eq x y"   and "0 \<le> a" shows "less_eq (scaleR a x) (scaleR a y)"
typeof_opubx_ar: fixes opubx :: "opubx" shows "length (fst (typeof_opubx opubx)) = ar (deinl (deubx opubx))"
upd_invs_Some: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and tas :: "('f, 'i, 'j) new_thread_action list"   and I :: "'f \<Rightarrow> 'k option"   and t :: "'f"   and i :: "'k"   and x :: "'g \<times> 'h \<Rightarrow>f nat"   and Q :: "'k \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "thread_oks ts tas"   and "I t = \<lfloor>i\<rfloor>"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_invs I Q tas t = \<lfloor>i\<rfloor>"
assms_1: shows "SN ((s \<union> q2)\<^sup>* O r O (s \<union> q2)\<^sup>* \<union> (s \<union> q2)\<^sup>* O q1 O (s \<union> q2)\<^sup>*)"
phiDefNodes_auxI: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and un :: "'node list"   and v :: "'var" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "unvisitedPath un ns"   and "\<forall>n\<in>set ns. v \<notin> defs g n"   and "length (predecessors g n) \<noteq> 1" shows "n \<in> phiDefNodes_aux g v un m"
pdc_def: fixes a :: "'a"   and b :: "'a" shows "(a \<leadsto> b) = (a \<rightarrow>\<^sub>\<S> b \<or> a \<rightarrow>\<^sub>\<E> b)"
numer_cf7_pos: fixes x :: "real" assumes "- (9943 / 10 ^ 3) \<le> x" shows "0 < numer_cf7 x"
count_mset_set': fixes A :: "'a set"   and x :: "'a" shows "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
inv_letter_def: fixes l :: "accent \<times> 'a" shows "inv_letter l \<equiv> case l of (a, x) \<Rightarrow> (case a of Acute \<Rightarrow> Grave | Grave \<Rightarrow> Acute | Macron \<Rightarrow> Macron, x)"
cmp_simps_2: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "src\<^sub>D (\<Phi> (f, g)) = src\<^sub>D (F g)"
degree_linear_power_neg: fixes a :: "'b"   and n :: "nat" shows "degree ([:a, - (1::'b):] ^ n) = n"
k_def: shows "k = (case z of (t, u) \<Rightarrow> t\<^sup>2 - int (u\<^sup>2 * D))"
rec_max2_def: fixes f :: "recf" shows "rec_max2 f = PR Z (CN rec_ifz [CN f [CN S [recf.Id 4 0], recf.Id 4 2, recf.Id 4 3], CN S [recf.Id 4 0], recf.Id 4 1])"
cases: fixes x :: "term" obtains name :: "name" where "x = Const name"   | v :: "name" where "x = Free v__"   | v :: "term" where "x = \<Lambda> va__"   | v :: "nat" where "x = Bound vb__"   | v :: "term" and va :: "term" where "x = v $ va"
isomorphic_symmetric: fixes f :: "'b"   and g :: "'b" assumes "D.isomorphic f g" shows "D.isomorphic g f"
powr_to_1: fixes z :: "complex" shows "z powr 1 = z"
accepting_pair\<^sub>R_simp: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and F :: "('d \<times> 'e \<times> 'd) set"   and I :: "('d \<times> 'e \<times> 'd) set"   and w :: "nat \<Rightarrow> 'e" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 (F, I) w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> F = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {}"
flows_fix_un_filternew_simp: shows "flows_fix \<T> \<union> filternew_flows_state \<T> = flows_fix \<T>"
sdrop_wait_least: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and n :: "nat" assumes "ev \<phi> xs"   and "\<phi> (sdrop n xs)" shows "wait \<phi> xs \<le> n"
simps: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and zs' :: "'f"   and z :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "qAbs zs' z X $[[x \<and> y]]_zs = qAbs zs' (z @zs'[x \<and> y]_zs) (X #[[x \<and> y]]_zs)"
bridge_on_Nom: fixes i :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and v :: "nat"   and xs :: "(nat \<times> nat) set"   and k :: "'c"   and j :: "'c" assumes "Nom i on (ps, a)" shows "Nom i on (mapi (bridge k j xs v) ps, a)"
Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inject: fixes x :: "\<langle>\<langle>'d set\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<langle>\<langle>'d set\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "x \<in> {X. X = \<bottom> \<or> X = null \<or> (\<forall>x\<in>\<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> \<bottom>)}"   and "y \<in> {X. X = \<bottom> \<or> X = null \<or> (\<forall>x\<in>\<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> \<bottom>)}" shows "(Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x = Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = (x = y)"
square_part_le_sqrt: fixes n :: "nat" shows "square_part n \<le> nat \<lfloor>sqrt (real n)\<rfloor>"
assms: shows "w \<le>p [a] \<^sup>@ m"
simps: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'b" shows "transition_system.path execute enabled a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> enabled a p \<and> transition_system.path execute enabled r (execute a p)))"
div_mult_le: fixes a :: "nat"   and b :: "nat" shows "a div b * b \<le> a"
Inl_Inr_False: fixes x :: "'a"   and y :: "'b" shows "(Inl x = Inr y) = False"
select_convs_5: fixes igWls :: "'g \<Rightarrow> 'c \<Rightarrow> bool"   and igWlsAbs :: "'f \<times> 'g \<Rightarrow> 'e \<Rightarrow> bool"   and igVar :: "'f \<Rightarrow> 'h \<Rightarrow> 'c"   and igAbs :: "'f \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> 'e"   and igOp :: "'a \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> ('d \<Rightarrow> 'e option) \<Rightarrow> 'c"   and igFresh :: "'f \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> bool"   and igFreshAbs :: "'f \<Rightarrow> 'h \<Rightarrow> 'e \<Rightarrow> bool"   and igSwap :: "'f \<Rightarrow> 'h \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> 'c"   and igSwapAbs :: "'f \<Rightarrow> 'h \<Rightarrow> 'h \<Rightarrow> 'e \<Rightarrow> 'e"   and igSubst :: "'f \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubstAbs :: "'f \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'i" shows "igOp \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igOp"
assms_1: shows "pd_kleene_allegory_class.acyclic t"
simE2_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel> S"   and "eqvt Rel"   and "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>\<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
exhaust: fixes y :: "pexpr1" obtains x1 :: "int" where "y = PCnst x1"   | x2 :: "nat" where "y = PVar x2"   | x31 :: "pexpr" and x32 :: "pexpr" where "y = PAdd x31 x32"   | x41 :: "pexpr" and x42 :: "pexpr" where "y = PSub x41 x42"   | x5 :: "pexpr" where "y = PNeg x5"
lang\<^sub>M\<^sub>2\<^sub>L_rexp_of': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang\<^sub>M\<^sub>2\<^sub>L n \<phi> = lang n (rexp_of' n \<phi>) - {[]}"
fill_offs_enum_stable: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and n :: "nat"   and k :: "'d" assumes "index_less index key"   and "index_same index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "0 < n" shows "filter (\<lambda>x. key x = k) (map the (fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma)) = filter (\<lambda>x. k = key x) xs"
qp_pre: shows "is_quantum_predicate (tensor_P pre (proj_k 0))"
set_rw: shows "{column i (Gram_Schmidt_upt_k A (Suc k)) |i. i < mod_type_class.from_nat (Suc k)} = {column i (Gram_Schmidt_upt_k A k) |i. i < mod_type_class.from_nat (Suc k)}"
funcset_carrier: fixes f :: "'c \<Rightarrow> 'd"   and X :: "('c, 'e) partial_object_scheme"   and Y :: "('d, 'f) partial_object_scheme"   and x :: "'c" assumes "f \<in> carrier X \<rightarrow> carrier Y"   and "x \<in> carrier X" shows "f x \<in> carrier Y"
interpolation_poly_int_def: fixes xs_ys :: "(int \<times> int) list"   and alg :: "interpolation_algorithm" assumes "distinct (map fst xs_ys)" shows "interpolation_poly_int alg xs_ys = (let rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ys; rp = interpolation_poly alg rxs_ys in if \<forall>x\<in>set (coeffs rp). is_int_rat x then Some (map_poly int_of_rat rp) else None)"
dist_half_times2: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "dist ((1 / 2) *\<^sub>R (a + b)) x * 2 = dist (a + b) (2 *\<^sub>R x)"
rel_map_2: fixes Sa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and g :: "'c \<Rightarrow> 'b"   and y :: "'c list" shows "list_all2 Sa x (map g y) = list_all2 (\<lambda>x y. Sa x (g y)) x y"
prime_factors_characterization_nat: fixes S :: "nat set"   and f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "n = (\<Prod>p\<in>S. p ^ f p)" shows "prime_factors n = S"
atLeast_iff: fixes i :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "(i \<in> ord.atLeast less_eq k) = less_eq k i"
map_comp_map: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "map f \<circ> map g = map (f \<circ> g)"
mem_Affine_project_coord_aformD: fixes x :: "'a"   and sctn :: "'a sctn"   and X :: "'a \<times> 'a pdevs" assumes "x \<in> Affine (project_coord_aform sctn X)"   and "\<bar>normal sctn\<bar> \<in> Basis" shows "x \<in> plane_of sctn"
poly_exp_constant_bound: fixes b :: "'a"   and c :: "'a"   and deg :: "nat" assumes "(0::'a) < b"   and "b < (1::'a)" shows "\<exists>p. \<forall>x. c * b ^ x * of_nat x ^ deg \<le> p"
nonprincipal: shows "\<chi> \<noteq> \<chi>\<^sub>0"
simps_1: fixes f\<^sub>1 :: "'b val \<Rightarrow> 'b val"   and f\<^sub>2 :: "'b val \<Rightarrow> 'b val"   and v\<^sub>1 :: "'b val"   and v\<^sub>2 :: "'b val" shows "F_Times f\<^sub>1 f\<^sub>2 (Seq v\<^sub>1 v\<^sub>2) = Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)"
iT_Mult_0: fixes I :: "nat set" assumes "I \<noteq> {}" shows "I \<otimes> 0 = [\<dots>0]"
assms_1: shows "\<forall>rs n. c \<noteq> Oc \<up> Suc rs @ Bk \<up> n"
simps_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "flow_network \<Delta> g" shows "flow_network.h_plus \<Delta> g 0 (x, y) = (if x = source \<Delta> then g (x, y) else 0)"
sym_lens_psym: fixes x :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "sym_lens x" shows "psym_lens x"
mult_im_sheaf_def: shows "mult_im_sheaf \<equiv> \<lambda>V. \<cdot>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
FEx_upward': fixes g :: "form"   and m :: "nat" assumes "infinite (deriv s)"   and "init s" shows "\<forall>n. contains f n (0, FEx g) \<longrightarrow> (\<exists>n'. contains f n' (m, FEx g))"
simps: fixes a :: "'b set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = aa \<triangleright> A \<and> finite A))"
num_of_nat_double: fixes n :: "nat" assumes "0 < n" shows "num_of_nat (n + n) = num.Bit0 (num_of_nat n)"
intros_12: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and es :: "char list exp list"   and Ts' :: "ty list" assumes "P,E,h \<turnstile> e :' Class C"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"   and "P,E,h \<turnstile> es [:'] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts" shows "P,E,h \<turnstile> e\<bullet>M(es) :' T"
exists_solve_consistent_rref: fixes A :: "(('d, 'e) vec, 'f) vec"   and j :: "'e" assumes "reduced_row_echelon_form A" shows "(\<exists>i. A $ i $ j = (1::'d) \<and> j = (LEAST n. A $ i $ n \<noteq> (0::'d))) = IArray.exists (\<lambda>i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'d) \<and> mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..<nrows_iarray (matrix_to_iarray A)])"
notifyWatches_def: fixes literal :: "Literal"   and state :: "State" shows "notifyWatches literal state \<equiv> notifyWatches_loop literal (getWatchList state literal) [] state"
o_assoc: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and h :: "'d \<Rightarrow> 'g" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
SDG: shows "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"
leg0_composite: shows "\<mu>\<nu>.leg0 = \<nu>.leg0 \<cdot> \<p>\<^sub>0[\<mu>.leg0, \<nu>.leg1]"
sams_lea456_suma2__lea: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a"   and G' :: "'a"   and H' :: "'a"   and I' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong D E F D' E' F'"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D' E' F'"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D' E' F' G' H' I'" shows "Tarski_neutral_dimensionless.LeA Bet Cong G H I G' H' I'"
assms: shows "segment G.face_cycle_succ a' a \<noteq> {}"
group_by_nonempty: fixes M :: "'a set"   and f :: "'a \<Rightarrow> 'b \<times> 'c" assumes "M \<noteq> {}" shows "group_by f M \<noteq> {}"
abc_list_crsp_steps: fixes lm :: "nat list"   and m :: "nat"   and aprog :: "abc_inst list"   and stp :: "nat"   and a :: "nat"   and lm' :: "nat list" assumes "abc_steps_l (0, lm @ 0 \<up> m) aprog stp = (a, lm')"   and "aprog \<noteq> []" shows "\<exists>lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and> abc_list_crsp lm' lma"
bigo_abs3: fixes f :: "'c \<Rightarrow> 'd" shows "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
A1: shows "c_fst (c_fst key) mod 7 = 5"
upper_triangularD: fixes A :: "'b mat"   and j :: "nat"   and i :: "nat" assumes "upper_triangular A"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'b)"
derS: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
sorted_less_set_eq: fixes xs :: "'b list" assumes "Sorted_Less.sorted xs" shows "xs = sorted_list_of_set (set xs)"
proper_def: fixes P :: "heap \<times> nat set \<Rightarrow> bool" shows "proper P \<equiv> \<forall>h h' as. (P (h, as) \<longrightarrow> in_range (h, as)) \<and> (P (h, as) \<and> relH as h h' \<and> in_range (h', as) \<longrightarrow> P (h', as))"
simps: fixes aa :: "'c"   and f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" shows "(aa \<in> segment f a b) = (aa = f a \<and> f a \<noteq> b \<or> (\<exists>x. aa = f x \<and> x \<in> segment f a b \<and> f x \<noteq> b))"
SecurityInvariant_complies_formal_def_def: fixes impl :: "'a SecurityInvariant"   and spec :: "'a SecurityInvariant_configured" shows "SecurityInvariant_complies_formal_def impl spec \<equiv> (\<forall>G. wf_list_graph G \<longrightarrow> implc_sinvar impl G = c_sinvar spec (list_graph_to_graph G)) \<and> (\<forall>G. wf_list_graph G \<longrightarrow> set ` set (implc_offending_flows impl G) = c_offending_flows spec (list_graph_to_graph G)) \<and> implc_isIFS impl = c_isIFS spec"
l11_51: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A \<noteq> C"   and "B \<noteq> C"   and "Cong A B A' B'"   and "Cong A C A' C'"   and "Cong B C B' C'" shows "Tarski_neutral_dimensionless.CongA Bet Cong B A C B' A' C' \<and> Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C' \<and> Tarski_neutral_dimensionless.CongA Bet Cong B C A B' C' A'"
next_dom_events_def: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list" shows "next_dom_events P D u xs \<equiv> {x. u = D x \<and> x \<in> next_events P xs}"
assms_3: shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"
vfsequence_vcons_ex: assumes "T \<noteq> []\<^sub>\<circ>" obtains xs' :: "V" and x :: "V" where "T = xs' #\<^sub>\<circ> x" and "vfsequence xs'"