Statement:
stringlengths 7
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lemma abs_ge_iff: "((x::real) \<le> abs y) = (x \<le> y \<or> x \<le> -y)" |
lemma honest_verifier_ZK:
shows "Schnorr_\<Sigma>.HVZK" |
lemma splitFace_edges_incr:
"pre_splitFace g ram1 ram2 f vs \<Longrightarrow>
(f\<^sub>1, f\<^sub>2, g') = splitFace g ram1 ram2 f vs \<Longrightarrow>
edges g \<subseteq> edges g'" |
lemma f_Exec_Stream_expand_aggregate_map_nth: "
\<lbrakk> 0 < k; n < length xs \<rbrakk> \<Longrightarrow>
f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot>\<^sub>f k) c)) k ag ! n =
ag (map f (f_Exec_Comp_Stream trans_fun (xs ! n # \<NoMsg>\<^bsup>k - Suc 0\<^esup>)
(f_Exec_Comp trans_fun (xs \<down> n \<odot>\<^sub>f k) c)))" |
lemma set_sort_on [simp]:
shows "set (sort_key_on r f xs) = set xs" |
lemma hpair_neq_Ord': assumes k: "Ord k" shows "k \<noteq> \<langle>x,y\<rangle>" |
lemma thermostat:
assumes "a > 0" and "0 < Tmin" and "Tmax < L"
shows "(\<lambda>s. Tmin \<le> s$1 \<and> s$1 \<le> Tmax \<and> s$4 = 0) \<le>
|LOOP
\<comment> \<open>control\<close>
((2 ::= (\<lambda>s. 0));(3 ::= (\<lambda>s. s$1));
(IF (\<lambda>s. s$4 = 0 \<and> s$3 \<le> Tmin + 1) THEN (4 ::= (\<lambda>s.1)) ELSE
(IF (\<lambda>s. s$4 = 1 \<and> s$3 \<ge> Tmax - 1) THEN (4 ::= (\<lambda>s.0)) ELSE skip));
\<comment> \<open>dynamics\<close>
(IF (\<lambda>s. s$4 = 0) THEN (x\<acute>= f a 0 & (\<lambda>s. s$2 \<le> - (ln (Tmin/s$3))/a))
ELSE (x\<acute>= f a L & (\<lambda>s. s$2 \<le> - (ln ((L-Tmax)/(L-s$3)))/a))) )
INV (\<lambda>s. Tmin \<le>s$1 \<and> s$1 \<le> Tmax \<and> (s$4 = 0 \<or> s$4 = 1))]
(\<lambda>s. Tmin \<le> s$1 \<and> s$1 \<le> Tmax)" |
lemma distinct_zipI1:
assumes "distinct xs"
shows "distinct (zip xs ys)" |
lemma invar_get_min_rest:
assumes "get_min_rest ts = (t',ts')"
assumes "ts\<noteq>[]"
assumes "invar ts"
shows "bheap t'" and "invar ts'" |
lemma Sup_eq_maximum_componentwise:
fixes s::"'a set"
assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> x \<bullet> b \<le> X \<bullet> b"
assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
shows "Sup s = X" |
lemma plus_inverse_ge_2:
fixes x :: real
assumes "x > 0"
shows "x + inverse x \<ge> 2" |
lemma next_map_generator [simp]:
"list.next (map_generator g) = apfst f \<circ> list.next g" |
lemma PO_m1_refines_a0ii_ri [iff]:
"refines
(R_a0iim1_ri \<inter> a0i_inv1_iagree \<times> (m1_inv1r_cache \<inter> m1_inv0_fin))
med_a0iim1_ri a0i m1" |
lemma below_Cinf[simp]: "r \<sqsubseteq> C\<^sup>\<infinity>" |
lemma exec_eval_mono [rule_format]:
"(s -c -n\<rightarrow> t \<longrightarrow> (\<forall>m. n \<le> m \<longrightarrow> s -c -m\<rightarrow> t)) \<and>
(s -e\<succ>v-n\<rightarrow> t \<longrightarrow> (\<forall>m. n \<le> m \<longrightarrow> s -e\<succ>v-m\<rightarrow> t))" |
lemma region_set_id:
fixes X k
defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}"
assumes "R \<in> \<R>" "v \<in> R" "finite X" "0 \<le> c" "c \<le> k x" "x \<in> X"
shows "[v(x := c)]\<^sub>\<R> = region_set R x c" "[v(x := c)]\<^sub>\<R> \<in> \<R>" "v(x := c) \<in> [v(x := c)]\<^sub>\<R>" |
lemma sum_list_listset_dirsum :
"add_independentS As \<Longrightarrow> as \<in> listset As \<Longrightarrow> sum_list as \<in> (\<Oplus>A\<leftarrow>As. A)" |
lemma l1_implements_a0i [iff]: "implements med01ia a0i l1" |
lemma per_exp: "u \<le>p r\<^sup>\<omega> \<Longrightarrow> u \<le>p r\<^sup>@k \<cdot> u" |
lemma converse_product_overlap:
"O x y \<Longrightarrow> O z (x \<otimes> y) \<Longrightarrow> O z y" |
lemma truncate_up_shift_nat: "truncate_up p (x * 2 powr real k) = truncate_up p x * 2 powr k" |
lemma eOp_nchotomy:
"(\<exists> inp binp. einp = OKI inp \<and> igWlsInp MOD delta inp \<and>
ebinp = OKI binp \<and> igWlsBinp MOD delta binp)
\<or>
(eOp MOD delta einp ebinp = ERR)" |
lemma res_to_fun_monom:
assumes "a \<in> carrier Zp"
assumes "b \<in> carrier Zp"
assumes "c \<in> carrier Zp"
assumes "a k = b k"
shows "(monom Zp_x c n \<bullet> a) k = (monom Zp_x c n \<bullet> b) k" |
lemma inj_Pair_const2: "inj (\<lambda>k. (k, C))" |
lemma "foldr\<cdot>trand\<cdot>TT = the_and" |
lemma verts_reachable_connected:
"verts G \<noteq> {} \<Longrightarrow> (\<forall>x\<in>verts G. \<forall>y\<in>verts G. x \<rightarrow>\<^sup>* y) \<Longrightarrow> connected G" |
lemma nth_subst:
"n < length A \<Longrightarrow> ($ S A)!n = $S (A!n)" |
lemma dgrad_set_le_Un: "dgrad_set_le d (S \<union> T) U \<longleftrightarrow> (dgrad_set_le d S U \<and> dgrad_set_le d T U)" |
lemma seqp_assignTE [elim]:
"\<lbrakk>((\<xi>, {l}\<lbrakk>u\<rbrakk> p), a, (\<xi>', q)) \<in> seqp_sos \<Gamma>; \<lbrakk>a = \<tau>; \<xi>' = u \<xi>; q = p\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
lemma auto_split_lemma:
"\<And>q ps res. auto_split A (delta A ps q) res ps xs =
maxsplit (\<lambda>ys. fin A (delta A ys q)) res ps xs" |
lemma constraint_model_Val_is_Value_term:
assumes "welltyped_constraint_model I A"
and "t \<cdot> I = Fun (Val n) []"
shows "t = Fun (Val n) [] \<or> (\<exists>m. t = Var (TAtom Value, m))" |
lemma deg_le_sectD:
assumes "t \<in> deg_le_sect X d"
shows "t \<in> .[X]" and "deg_pm t \<le> d" |
lemma symm_trans: assumes "symp R" shows "symp R^++" |
lemma eval_mod_exp_aux [simp]:
"mod_exp_aux m y x 0 = y"
"mod_exp_aux m y x (Suc 0) = (x * y) mod m"
"mod_exp_aux m y x (numeral (num.Bit0 n)) =
mod_exp_aux m y (x\<^sup>2 mod m) (numeral n)"
"mod_exp_aux m y x (numeral (num.Bit1 n)) =
mod_exp_aux m ((x * y) mod m) (x\<^sup>2 mod m) (numeral n)" |
lemma none_MT_rulessep[rule_format]: "none_MT_rules C p \<longrightarrow> none_MT_rules C (separate p)" |
lemma sq_mtx_minus_diag_diag[simp]: "sq_mtx_diag f - sq_mtx_diag g = (\<d>\<i>\<a>\<g> i. f i - g i)" |
lemma (in category) id_arrow [intro]:
assumes "A \<in> Ob"
shows "Id A \<in> Ar" |
lemma cs_sym:
assumes "x \<down>\<^sup>* y"
shows "y \<down>\<^sup>* x" |
lemma F_UU: "F \<bottom> = {(s,X). front_tickFree s}" |
lemma ESem_subst:
shows "\<lbrakk> e \<rbrakk>\<^bsub>\<sigma>(x := \<sigma> y)\<^esub> = \<lbrakk> e[x::= y] \<rbrakk>\<^bsub>\<sigma>\<^esub>" |
lemma [simp]:
fixes r s::real
shows is_interval_io: "is_interval {..<r}"
and is_interval_oi: "is_interval {r<..}"
and is_interval_oo: "is_interval {r<..<s}"
and is_interval_oc: "is_interval {r<..s}"
and is_interval_co: "is_interval {r..<s}" |
lemma prior_in_space:
"prior \<in> qbs_space (monadP_qbs (\<real>\<^sub>Q \<Rightarrow>\<^sub>Q \<real>\<^sub>Q))" |
lemma common_decision:
assumes run: "SHORun Ate_M rho HOs SHOs"
and comm: "SHOcommPerRd Ate_M (HOs r) (SHOs r)"
and nvp: "decide (rho r p) \<noteq> Some v"
and vp: "decide (rho (Suc r) p) = Some v"
and nwq: "decide (rho r q) \<noteq> Some w"
and wq: "decide (rho (Suc r) q) = Some w"
shows "w = v" |
lemma pop_array_rule' [hoare_triple]:
"n > 0 \<Longrightarrow> n \<le> length xs \<Longrightarrow>
<dyn_array_raw (xs, n) p>
pop_array p
<\<lambda>(x, r). dyn_array_raw (xs, n - 1) r * \<up>(x = xs ! (n - 1))>" |
lemma atm_imp_pos_or_neg_lit: "A \<in> atms_of C \<Longrightarrow> Pos A \<in># C \<or> Neg A \<in># C" |
lemma cf_const_if_HomDom_is_cat_1:
assumes "\<KK> : cat_1 \<aa> \<ff> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"
shows "\<KK> = cf_const (cat_1 \<aa> \<ff>) \<CC> (\<KK>\<lparr>ObjMap\<rparr>\<lparr>\<aa>\<rparr>)" |
lemma presCons_fromIMor[simp]:
assumes "ipresCons h hA (fromMOD MOD)"
shows "presCons (fromIMor h) (fromIMorAbs hA) MOD" |
lemma singular_relboundary_ss:
"singular_relboundary p X S x \<Longrightarrow> Poly_Mapping.keys x \<subseteq> singular_simplex_set p X" |
lemma SFD_validTrans:
assumes "validTrans trn"
and "UID' \<in>\<in> friendIDs (tgtOf trn) UID"
shows "\<not> SFD UID UID' trn \<and> \<not> SFD UID' UID trn" |
lemma simple_cg_closure_phase_1_helper'_validity_fst :
assumes "observable M1" and "observable M2"
and "\<And> u v . u |\<in>| x \<Longrightarrow> v |\<in>| x \<Longrightarrow> u \<in> L M1 \<Longrightarrow> u \<in> L M2 \<Longrightarrow> converge M1 u v \<and> converge M2 u v"
and "\<And> u v . u |\<in>| x1 \<Longrightarrow> v |\<in>| x1 \<Longrightarrow> u \<in> L M1 \<Longrightarrow> u \<in> L M2 \<Longrightarrow> converge M1 u v \<and> converge M2 u v"
and "\<And> x2 u v . x2 \<in> list.set xs \<Longrightarrow> u |\<in>| x2 \<Longrightarrow> v |\<in>| x2 \<Longrightarrow> u \<in> L M1 \<Longrightarrow> u \<in> L M2 \<Longrightarrow> converge M1 u v \<and> converge M2 u v"
and "u |\<in>| fst (snd (simple_cg_closure_phase_1_helper' x x1 xs))"
and "v |\<in>| fst (snd (simple_cg_closure_phase_1_helper' x x1 xs))"
and "u \<in> L M1" and "u \<in> L M2"
shows "converge M1 u v \<and> converge M2 u v" |
lemma cont_pathsCard[THEN cont_compose, cont2cont, simp]:
"cont pathsCard" |
lemma sat_push_edge_action_bound':
assumes "((f,l),p,(f',l')) \<in> trcl pr_algo_lts'"
shows "length (filter ((=) (SAT_PUSH' e)) p) \<le> 2*card V" |
lemma free_after_alloc:
assumes "alloc h c s = Success (h', cap)"
shows "\<exists>! ret. free h' cap = Success ret" |
lemma lnth_subset_Sup_llist: "enat i < llength Xs \<Longrightarrow> lnth Xs i \<subseteq> Sup_llist Xs" |
lemma UP_smult_zero [simp]:
"p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>" |
lemma (in group) generate_is_subgroup:
assumes "H \<subseteq> carrier G" shows "subgroup (generate G H) G" |
lemma vars_mset_psubset_uniflessI [intro]:
"vars_mset M \<subset> vars_mset N \<Longrightarrow> (M, N) \<in> unifless" |
lemma star_n_L_split_equal:
"n(x * y) \<le> n(y) \<Longrightarrow> x\<^sup>\<star> * n(y) * L = x\<^sup>\<star> * bot \<squnion> n(y) * L" |
lemma sublist_el': \<open>i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> x \<in> set (sublist xs i j) \<equiv> (\<exists> k. i\<le>k\<and>k\<le>j \<and> xs!k=x)\<close> |
lemma WeakCojunctive_Healthy_Refinement:
assumes "WeakConjunctive(HC)" and "P is HC"
shows "HC(P) \<sqsubseteq> P" |
lemma log_floor_sound: assumes "b > 1" "x > 0" "log_floor b x = y"
shows "b^y \<le> x" "x < b^(Suc y)" |
lemma enabled_ex_taken:
assumes \<open>epath steps\<close> \<open>Saturated steps\<close> \<open>enabled r (fst (shd steps))\<close>
shows \<open>\<exists>k. takenAtStep r (shd (sdrop k steps))\<close> |
lemma id_simps:
"(vid1 = vid2) = False" "(vid2 = vid3) = False" "(vid1 = vid3) = False"
"(fid1 = fid2) = False" "(fid2 = fid3) = False" "(fid1 = fid3) = False"
"(pid1 = pid2) = False" "(pid2 = pid3) = False" "(pid1 = pid3) = False"
"(pid1 = pid4) = False" "(pid2 = pid4) = False" "(pid3 = pid4) = False"
"(vid2 = vid1) = False" "(vid3 = vid2) = False" "(vid3 = vid1) = False"
"(fid2 = fid1) = False" "(fid3 = fid2) = False" "(fid3 = fid1) = False"
"(pid2 = pid1) = False" "(pid3 = pid2) = False" "(pid3 = pid1) = False"
"(pid4 = pid1) = False" "(pid4 = pid2) = False" "(pid4 = pid3) = False" |
lemma fixed_point_congruence:
assumes "order G = p ^ a"
assumes "prime p"
assumes finM:"finite M"
shows "card M mod p = card fixed_points mod p" |
lemma lowlink_path_complex:
assumes "(u,v) \<in> (tree_edges s)\<^sup>+"
and "u \<in> dom (finished s) \<or> (stack s \<noteq> [] \<and> u = hd (stack s))"
and "(v,w) \<in> cross_edges s \<union> back_edges s"
shows "\<exists>p. lowlink_path s u p w" |
lemma add\<^sub>O':
"\<langle>addO n m\<rangle>\<^sub>O = add\<^sub>O \<langle>n\<rangle>\<^sub>O \<langle>m\<rangle>\<^sub>O" |
lemma real_lim_then_eventually_real:
assumes "(u \<longlongrightarrow> ereal l) F"
shows "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" |
lemma eventually_nat_real:
assumes "eventually P (at_top :: real filter)"
shows "eventually (\<lambda>x. P (real x)) (at_top :: nat filter)" |
lemma ground_iff_no_vars: "ground (M::('a,'b) terms) \<longleftrightarrow> (\<forall>v. Var v \<notin> (\<Union>m \<in> M. subterms m))" |
lemma prefix_fin_prefix_fininf_trans[trans, intro]: "u \<le> v \<Longrightarrow> v \<le>\<^sub>F\<^sub>I w \<Longrightarrow> u \<le>\<^sub>F\<^sub>I w" |
lemma lang_lang_trad: "lang = to_language (lang_trad G)" |
lemma norm_respects_equiv:
assumes "equiv t u"
shows "\<^bold>\<parallel>t\<^bold>\<parallel> = \<^bold>\<parallel>u\<^bold>\<parallel>" |
lemma after_exists: "|~ (\<exists> x. P x)` = (\<exists> x. (P x)`)" |
theorem ltl_FG_logical_characterization:
"w \<Turnstile> F G \<phi> \<longleftrightarrow> (\<exists>\<G> \<subseteq> \<^bold>G (F G \<phi>). G \<phi> \<in> \<G> \<and> closed \<G> w)"
(is "?lhs \<longleftrightarrow> ?rhs") |
lemma sources_cod [simp]:
assumes "arr \<mu>"
shows "sources (cod \<mu>) = sources \<mu>" |
lemma (in qmpt) qmpt_factorI:
assumes "proj \<in> quasi_measure_preserving M M2"
"AE x in M. proj (T x) = T2 (proj x)"
"qmpt M2 T2"
shows "qmpt_factor proj M2 T2" |
lemma osubst_preserves_semBV:
shows "semBV I (OsubstFO ODE \<sigma>) = semBV (adjointFO I \<sigma> \<nu>) ODE" |
lemma has_integral_divide:
fixes c :: "_ :: real_normed_div_algebra"
shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x / c) has_integral (y / c)) S" |
lemma count_space_PiM_finite:
fixes B :: "'a \<Rightarrow> 'b set"
assumes "finite A" "\<And>i. countable (B i)"
shows "PiM A (\<lambda>i. count_space (B i)) = count_space (PiE A B)" |
lemma Cons_listrel1E2[elim!]:
assumes "(xs, y # ys) \<in> listrel1 r"
and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
shows R |
lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)" |
lemma stake_pos_minWait:
assumes rs: "fair rs" and m: "minWait rs s < pos rs r" and r: "r \<in> R" and s: "s \<in> S"
shows "pos (stl (trim rs s)) r = pos rs r - Suc (minWait rs s)" |
lemma mod_eq_imp_eq:
"\<lbrakk>b \<le> x; x < b + N; b \<le> y; y < b + N; x mod N = y mod N \<rbrakk> \<Longrightarrow> x=y" |
lemma ufa_init_correct: "ufa_\<alpha> [0..<n] = {(x,x) | x. x<n}" |
lemma isLimOrd_aboveS:
assumes l: isLimOrd and i: "i \<in> Field r"
shows "aboveS i \<noteq> {}" |
lemma option2set_None: "option2set None = {}" |
lemma Diff1_foldSetD:
"\<lbrakk>(A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D\<rbrakk> \<Longrightarrow>
(A, f x y) \<in> foldSetD D f e" |
lemma Qp_times_basic_semialg_left:
assumes "a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
shows "cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a) = basic_semialg_set (n+m) k (shift_vars n m a)" |
lemma ecut_apply: "y \<^bold>\<in> eclose x \<Longrightarrow> ecut f x y = f y" |
lemma run_reachable[simp]: "reachable (s i)" |
lemma hd_im_comm_eq:
assumes "g w = h w" and "w \<noteq> \<epsilon>" and comm: "g\<^sup>\<C> (hd w) \<cdot> h\<^sup>\<C>(hd w) = h\<^sup>\<C> (hd w) \<cdot> g\<^sup>\<C>(hd w)"
shows "g\<^sup>\<C> (hd w) = h\<^sup>\<C> (hd w)" |
lemma ball_simps [simp, no_atp]:
"\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
"\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
"\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
"\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
"\<And>P. fBall {||} P = True"
"\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
"\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
"\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)" |
lemma matches_iff:
"matches t p \<longleftrightarrow> (\<exists>\<sigma>. p \<cdot> \<sigma> = t)" |
lemma integrable_continuous:
fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
assumes "continuous_on (cbox a b) f"
shows "f integrable_on cbox a b" |
lemma wpo_ns_refl:
shows "s \<succeq> s" |
lemma find_index_append: "find_index P (xs @ ys) =
(if \<exists>x\<in>set xs. P x then find_index P xs else size xs + find_index P ys)" |
lemma has_integral_localized_vector_derivative:
"((\<lambda>x. f (g x) * vector_derivative p (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
((\<lambda>x. f (g x) * vector_derivative p (at x)) has_integral i) {a..b}" |
lemma cf_const_is_tiny_functor:
assumes "tiny_category \<alpha> \<CC>" and "tiny_category \<alpha> \<DD>" and "a \<in>\<^sub>\<circ> \<DD>\<lparr>Obj\<rparr>"
shows "cf_const \<CC> \<DD> a : \<CC> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<DD>" |
lemma alphaAbs_qAbs_imp_alphaAbs_all_distinct_qAFresh:
assumes "qGood X" and "qAbs xs x X $= qAbs xs' x' X'"
shows "alphaAbs_all_distinct_qAFresh xs x X xs' x' X'" |
lemma primitive_iff_separable_lemma:
assumes prod: "(\<forall>n. \<chi> n = \<Phi> n * \<chi>\<^sub>1 n) \<and> primitive_dchar d \<Phi>"
assumes \<open>d > 1\<close> \<open>0 < k\<close> \<open>d dvd k\<close> \<open>k > 1\<close>
shows "(\<Sum>m | m \<in> {1..k} \<and> coprime m k. \<Phi>(m) * unity_root d m) =
(totient k div totient d) * (\<Sum>m | m \<in> {1..d} \<and> coprime m d. \<Phi>(m) * unity_root d m)" |
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