Multilingual Mathematical Autoformalization
Paper
• 2311.03755 • Published
Statement: stringlengths 7 24.3k |
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lemma wprepare_goto_start_pos_Bk_nonempty_fst[simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []" |
lemma awalk_decomp:
assumes "awalk u p v"
assumes "w \<in> set (awalk_verts u p)"
shows "\<exists>q r. p = q @ r \<and> awalk u q w \<and> awalk w r v" |
lemma curry_uncurry:
assumes "functor A1 A2_B.comp F" and "functor A1 A2_B.comp G"
and "natural_transformation A1 A2_B.comp F G \<tau>"
shows "curry (uncurry F) (uncurry G) (uncurry \<tau>) = \<tau>" |
lemma connected_impl_maximally_connected:
assumes "connected_graph H"
assumes subgraph: "subgraph H G"
shows "maximally_connected H G" |
lemma torder_sync_order:
"torder_on (sactions P E) (sync_order P E)" |
lemma mset_ran_mem[simp, intro]: "finite r \<Longrightarrow> i\<in>r \<Longrightarrow> a i \<in># mset_ran a r" |
lemma numsubst0_numbound0:
assumes nb: "numbound0 t"
shows "numbound0 (numsubst0 t a)" |
lemma set_sel'_symm: "sel_symm f \<Longrightarrow> set_sel' f X Y = set_sel' f Y X" |
lemma deduct_proj_priv_term_prefix_ex:
assumes A: "ik\<^sub>s\<^sub>t (proj_unl l A) \<cdot>\<^sub>s\<^sub>e\<^sub>t I \<turnstile> t"
and t: "\<not>{} \<turnstile>\<^sub>c t"
shows "\<exists>B k s. (k = \<star> \<or> k = ln l) \<and> prefix (B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)]) A \<and>
declassified\<^sub>l\<^sub>s\<^sub>t ((B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)])) I = declassified\<^sub>l\<^sub>s\<^sub>t A I \<and>
ik\<^sub>s\<^sub>t (proj_unl l (B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)])) = ik\<^sub>s\<^sub>t (proj_unl l A)" |
lemma wprepare_loop_goon_Bk_nonempty[simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []" |
lemma rev_bl_order_simps:
"rev_bl_order F [] [] = F"
"rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> F)) xs ys" |
lemma consistent_signs_atw:
assumes "\<And>p. p \<in> set fs \<Longrightarrow> poly p x \<noteq> 0"
shows "consistent_sign_vec_copr fs x = signs_at fs x" |
lemma wf_rulesetD: assumes "wf_ruleset \<gamma> p (r # rs)" shows "wf_ruleset \<gamma> p [r]" and "wf_ruleset \<gamma> p rs" |
lemma arr_runit [simp]:
assumes "ide a"
shows "arr \<r>[a]" |
lemma sublist_appendI [simp, intro]: "sublist xs (ps @ xs @ ss)" |
lemma orth_at2_tsp__ts:
assumes "P \<noteq> Q" and
"P OrthAt A B C P X" and
"Q OrthAt A B C Q Y" and
"A B C TSP X Y"
shows "P Q TS X Y" |
theorem normal_form:
assumes "recfn n f"
obtains i where "\<forall>x. e_length x = n \<longrightarrow> eval r_normal_form [i, x] = eval f (list_decode x)" |
lemma trms\<^sub>s\<^sub>s\<^sub>t_proj_subset[simp]:
"trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n A) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (unlabel A)" (is ?A)
"trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m (proj n A)) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n A)" (is ?B)
"trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m (proj n A)) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m A)" (is ?C) |
lemma ctrm_trunc:
assumes "p \<in> carrier P"
assumes "degree p >0"
shows "zcf(trunc p) = zcf p" |
lemma fps_deriv_maclauren_0:
"(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) $ 0 = of_nat (fact k) * f $ k" |
lemma tLength_g0_conv:
"(tLength t > 0) \<longleftrightarrow> (\<exists>s t'. t = t' \<leadsto> s \<and> tLength t = Suc (tLength t'))" |
lemma sum_emeasure':
assumes [simp]: "finite A"
assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> B x \<in> sets M"
assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> emeasure M (B x \<inter> B y) = 0"
shows "(\<Sum>x\<in>A. emeasure M (B x)) = emeasure M (\<Union>x\<in>A. B x)" |
lemma ratfps_inverse_code [code abstract]:
"quot_of_ratfps (inverse x) =
(let (a,b) = quot_of_ratfps x
in if coeff a 0 = 0 then (0, 1)
else let u = unit_factor a in (b div u, a div u))" |
lemma nextVertex_in_edges: "v \<in> \<V> f \<Longrightarrow> (v, f \<bullet> v) \<in> edges f" |
lemma r_unit_add_rng_of_frac:
assumes "(r, s) \<in> carrier rel"
shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> \<zero>\<^bsub>rec_rng_of_frac\<^esub> = (r |\<^bsub>rel\<^esub> s)" |
lemma
assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})"
assumes deriv':
"\<And>z t x. z \<in> U \<Longrightarrow> x \<ge> a \<Longrightarrow> t \<in> {a..x} \<Longrightarrow>
((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)"
assumes cont: "continuous_on (U \<times> {of_int a..}) (\<lambda>(z, t). f' z t)"
assumes int: "\<And>b c z e. a \<le> b \<Longrightarrow> z \<in> U \<Longrightarrow>
(\<lambda>t. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}"
assumes int': "\<And>a' b y. y \<in> U \<Longrightarrow> a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow>
(\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
assumes conv: "convergent (\<lambda>y. G (real y))"
assumes bound: "eventually (\<lambda>x. \<forall>y\<in>U. norm (f y x) \<le> g x) at_top"
assumes "open U"
shows analytic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) analytic_on U"
and holomorphic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) holomorphic_on U" |
lemma iUntil_disj_distrib: "
(P t1. t1 \<U> t2 I. (Q1 t2 \<or> Q2 t2)) = ((P t1. t1 \<U> t2 I. Q1 t2) \<or> (P t1. t1 \<U> t2 I. Q2 t2))" |
lemma i_drop_the_conv: "
f \<Up> k = (THE g. (\<exists>xs. length xs = k \<and> xs \<frown> g = f))" |
lemma is_process3: "is_process P \<Longrightarrow> \<forall> s t. (s @ t,{}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P" |
lemma inj_map_prim: assumes "inj_on f A" and "u \<in> lists A" and
"primitive u"
shows "primitive (map f u)" |
lemma connecting_paths_sym_length: "i \<in> connecting_paths u v \<Longrightarrow> \<exists>j\<in>connecting_paths v u. (walk_length j) = (walk_length i)" |
lemma join_distr:
"x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
lemma inner_node_Entry_edge:
assumes "inner_node n"
obtains a where "valid_edge a" and "inner_node (targetnode a)"
and "sourcenode a = (_Entry_)" |
lemma asIMOD_igAbs2:
fixes SEM :: "('index,'bindex,'varSort,'sort,'opSym,'sTerm)semDom"
assumes *: "sWlsDisj SEM" and **: "igWls (asIMOD SEM) s X"
shows "igAbs (asIMOD SEM) xs x X =
(\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX
then X (val (x := sX)_xs)
else sDummy SEM s)
else undefined)" |
lemma fpxs_of_fls_compose_power [simp]:
"fpxs_of_fls (fls_compose_power f d) = fpxs_compose_power (fpxs_of_fls f) (of_nat d)" |
lemma both_mono2:
"paths t \<subseteq> paths t' \<Longrightarrow> paths (t'' \<otimes>\<otimes> t) \<subseteq> paths (t'' \<otimes>\<otimes> t')" |
lemma F_rec: "F n = F (n-1) + 1 + (\<Sum>i=m..<n. F (n-i-1))" if \<open>n>m\<close> "m > 0" |
lemma callee_of_rest_simps [simp]:
"callee_of_rest rest s (Inl iadv_rest) = map_spmf (apfst Inl) (rfunc_adv rest s iadv_rest)"
"callee_of_rest rest s (Inr iusr_rest) = map_spmf (apfst Inr) (rfunc_usr rest s iusr_rest)" |
lemma conditionalize:
assumes "finite A"
shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else \<^bold>1)" |
lemma bind_single:
"P \<bind> single = P" |
lemma [code]:
\<open>mask (Suc n) = push_bit n (1 :: 'a) OR mask n\<close>
\<open>mask 0 = (0 :: 'a)\<close> |
lemma last_ladder_\<gamma>:
assumes is_ladder: "is_ladder D L"
assumes ladder_last_n: "ladder_last_n L = length D"
shows "ladder_\<gamma> a D L (length L - Suc 0) = Derive a D" |
lemma USUP_empty [simp]: "(\<Squnion> i \<in> {} \<bullet> P(i)) = true" |
lemma rbt_eq_iff:
"t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2" |
lemma dsteps_to_steps: "a \<in> st.dstep ^^ n \<Longrightarrow> a \<in> st.step ^^ n" |
lemma PO_m1a_leak_refines_m1x_leak:
"{R1x1a}
(m1x_leak Rs), (m1a_leak Rs)
{> R1x1a}" |
lemma vec_contains_row_elements_mat:
assumes "i < dim_row M"
assumes "a \<in>$ row M i"
shows "a \<in> elements_mat M" |
lemma seq_filter_UNIV [simp]: "xs \<restriction>\<^sub>l UNIV = xs" |
lemma simplePath_empty_conv[simp]: "isSimplePath s [] t \<longleftrightarrow> s=t" |
lemma Some_in_opt [iff]:
"(Some x \<in> opt A) = (x\<in>A)" |
lemma interior_of_eq_empty:
"X interior_of S = {} \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<subseteq> S \<longrightarrow> T = {})" |
lemma deg_not_zero_imp_not_unit:
fixes f:: "'a::{idom_divide,semidom_divide_unit_factor} poly"
assumes deg_f: "degree f > 0"
shows "\<not> is_unit f" |
lemma wf_PLUS[simp]:
"wf n (PLUS xs) = (\<forall>r \<in> set xs. wf n r)" |
lemma inj_axis:
assumes "c \<noteq> 0"
shows "inj (\<lambda>k. axis k c :: ('a :: {zero}) ^ 'n)" |
lemma assumes ms: "measure_space M" and un: "(\<Union>i\<in>R. B i) = UNIV" and
fin: "finite (R::nat set)" and dis: "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> (B j1) \<inter> (B j2) = {}"
and meas: "\<forall>j\<in>R. B j \<in> measurable_sets M" and Ameas: "A \<in> measurable_sets M"
shows measure_split: "measure M A = (\<Sum>j\<in>R. measure M (A \<inter> B j))" |
lemma itypeI: "(c::'t) ::\<^sub>i I" |
lemma analz_insert_Crypt:
"Key (invKey K) \<notin> analz H
\<Longrightarrow> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)" |
lemma complement_p:
"x \<sqinter> y = bot \<Longrightarrow> x \<squnion> y = top \<Longrightarrow> -x = y" |
lemma trms\<^sub>s\<^sub>s\<^sub>t_memI[intro?]:
"send\<langle>ts\<rangle> \<in> set S \<Longrightarrow> t \<in> set ts \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"receive\<langle>ts\<rangle> \<in> set S \<Longrightarrow> t \<in> set ts \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"\<langle>ac: t \<doteq> s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"\<langle>ac: t \<doteq> s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"insert\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"insert\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"delete\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"delete\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
"\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" |
lemma it_context_true:
"-p * (-p \<rhd> x) = -p * x" |
lemma maddux2c: "(a \<lhd> x) \<sqinter> y \<le> a \<lhd> (x \<sqinter> (y \<rhd> a))" |
lemma valid_insert_both_member_options_pres: "invar_vebt t n \<Longrightarrow> x<2^n \<Longrightarrow> y < 2^n \<Longrightarrow> both_member_options t x
\<Longrightarrow> both_member_options (vebt_insert t y) x" |
lemma higher_deriv_diff:
fixes z::complex
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "z \<in> S"
shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z" |
theorem soundness:
"ir_hoare P c c' Q \<Longrightarrow> ir_valid P c c' Q" |
lemma no_microstep_broadcast: "\<not> (({l}broadcast(s\<^sub>m\<^sub>s\<^sub>g).p) \<leadsto>\<^bsub>\<Gamma>\<^esub> q)" |
lemma get_root_node_si_same_no_parent:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r cast child"
shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None" |
lemma hd_map: "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)" |
lemma asprod_pos_mono:"0 < w \<Longrightarrow> ((w *\<^sub>a x) \<le> (w *\<^sub>a y)) = (x \<le> y)" |
lemma lift_opts_exists:
"\<forall>x\<in>set ty_ty'_list. (\<lambda>(ty, ty'). is_sty_one P ty ty' = Some True) x \<Longrightarrow> \<exists>bools. lift_opts (map (\<lambda>(ty, ty'). is_sty_one P ty ty') ty_ty'_list) = Some bools" |
lemma before_vs_until:
"(before p q) = ((\<box>\<guillemotleft>p\<guillemotright>) U \<guillemotleft>q\<guillemotright>)" |
lemma (in SecurityInvariant_withOffendingFlows)
fixes "default_node_properties" :: "'a" ("\<bottom>")
shows "\<not> sinvar G nP \<Longrightarrow> sinvar_all_edges_normal_form P \<Longrightarrow>
(\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> (P (nP e1) (nP e2)) \<longrightarrow> \<not> (P \<bottom> (nP e2))) \<Longrightarrow>
(\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> (P (nP e1) (nP e2)) \<longrightarrow> \<not> (P (nP e1) \<bottom>)) \<Longrightarrow>
(\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> P \<bottom> \<bottom>)
\<Longrightarrow> \<not> sinvar G (nP(i := \<bottom>))" |
lemma "matches (\<beta>, \<alpha>) (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot (Match A))) a p = matches (\<beta>, \<alpha>) (MatchNot (Match A)) a p" |
lemma fields_declC:
"\<lbrakk>table_of (fields G C) efn = Some f; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow>
(\<exists>d. class G (declclassf efn) = Some d \<and>
table_of (cfields d) (fname efn)=Some f) \<and>
G\<turnstile>C \<preceq>\<^sub>C (declclassf efn) \<and> table_of (fields G (declclassf efn)) efn = Some f" |
lemma ET_target_source:
"\<lbrakk> TS \<subseteq> ET ST; t \<in> TS; target t \<in> States A; A \<in> SAs (HA ST) \<rbrakk> \<Longrightarrow> source t \<in> States A" |
lemma mcont_contD:
"\<lbrakk> mcont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y \<noteq> {} \<rbrakk>
\<Longrightarrow> f (luba Y) = lubb (f ` Y)" |
lemma d_delta_lnexp_cf3_nonneg: "numer_cf3 x > 0 \<Longrightarrow> numer_cf3 (-x) > 0 \<Longrightarrow> diff_delta_lnexp_cf3 x \<ge> 0" |
lemma "share_correct.sharing_correct input" |
lemma NormalizeD_sound:
assumes valid: "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P (normalize c) Q,A"
shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A" |
lemma FD1: "Der_1 \<D> \<Longrightarrow> Fr_1(\<F>\<^sub>D \<D>)" |
lemma vector_in_orthogonal_basis:
fixes a :: "'a::euclidean_space"
assumes "a \<noteq> 0"
obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S"
"span S = UNIV" "card S = DIM('a)" |
lemma wlp_is_weakest_liberal_precondition:
assumes "well_com S" and "is_quantum_predicate P"
shows "is_weakest_liberal_precondition (wlp S P) S P" |
lemma usemantics_UNIV: \<open>usemantics UNIV e f g p \<longleftrightarrow> semantics e f g p\<close> |
lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (sset s) P" |
lemma SN_on_imp_acc:
assumes "SN_on {(y, z). (z, y) \<in> r} {x}" shows "x \<in> Wellfounded.acc r" |
lemma subset_single [rewrite]: "{a} \<subseteq> B \<longleftrightarrow> a \<in> B" |
lemma inv_loop6_loop_Bk_Bk_drop[elim]: "\<lbrakk>0 < x; inv_loop6_loop x (b, Bk # list); b \<noteq> []; hd b = Bk\<rbrakk>
\<Longrightarrow> inv_loop6_loop x (tl b, Bk # Bk # list)" |
lemma solves_odeD:
assumes "(y solves_ode f) T X"
shows solves_ode_vderivD: "(y has_vderiv_on (\<lambda>t. f t (y t))) T"
and solves_ode_domainD: "\<And>t. t \<in> T \<Longrightarrow> y t \<in> X" |
lemma SAT_deduction: "SATAxiom x ==> x : deductions CutFreePC" |
lemma order_refl_type [iff]:
"\<tau> \<le> \<tau>"
for \<tau> :: "'a type" |
lemma connected_empty [simp]: "connected {}" |
lemma secureEQUIV:
"secure p G c H = (\<forall> q . secure1 p G c H \<and> secure2 q G c H)" |
lemma CE_symmetric: "CE r s \<Longrightarrow> CE s r" |
lemma const_transfer: "rel_fun (\<lambda>x y. x = c) (=) f (\<lambda>_. f c)" |
lemma COND_extr_prefix_path:
"\<lbrakk>hfs_valid ainfo uinfo l nxt; nxt = None\<rbrakk> \<Longrightarrow> prefix (extr_from_hd l) (AHIS l)" |
lemma segm_morph: "snd (Rep_segment x::('a::lip_order \<times> 'a::lip_order)) = fst (Rep_segment y)
\<Longrightarrow> segm x \<union> segm y = segm (x \<cdot> y)" |
lemma lcp_pref_monotone: assumes "w \<le>p r" and "w' \<le>p s" shows "w \<and>\<^sub>p w' \<le>p (r \<and>\<^sub>p s)" |
lemma list_max_is_max : "q \<in> set xs \<Longrightarrow> q \<le> list_max xs" |
lemma count_sum_mset_if_1_0: \<open>count M a = (\<Sum>x\<in>#M. if x = a then 1 else 0)\<close> |
lemma drop_upd_irrelevant: "m < n \<Longrightarrow> drop n (l[m:=x]) = drop n l" |
lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0" |
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About
This dataset is an amalgamation of every entry of the DQ Round Trip Problem Selection spreadsheet (https://docs.google.com/spreadsheets/d/1dEWWzjuEXwf9s4II0CixH4sqopc1flIMFx19UjHiyNU/edit?usp=sharing&resourcekey=0-_G7oxmbh7szV5jx-HxhepQ) with formal and informal columns combined and concatenated with the formal and informal statements from the Isabelle train and val sets from the Multilingual Mathematical Autoformalization dataset, to which a text column has been added in identical formatting.
Data Fields
Statement: either an informal statement or a formal statement in Isabelle
language: - en
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