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lemma align_sub: "aligned l (Node ts t) u \<Longrightarrow> (sub,sep) \<in> set ts \<Longrightarrow> \<exists>l' \<in> set (separators ts) \<union> {l}. aligned l' sub sep" |
lemma weak_unit_cancel_left:
assumes "weak_unit a" and "ide f" and "ide g"
and "a \<star> f \<cong> a \<star> g"
shows "f \<cong> g" |
lemma atLeastAtMost_split:
"{i..j} = {i..k} \<union> {k+1..j}" if "i \<le> k" "k \<le> j" for i j k :: int |
lemma upt_n_in_pow_nodes: "{0..<n} \<in> Pow (set nodes)" |
lemma InitConf_CompFun_Ancestor:
"\<lbrakk> S \<in> HAStates A; SA \<in> the (CompFun A S); T \<in> InitConf A; T \<in> States SA \<rbrakk>
\<Longrightarrow> S \<in> InitConf A" |
lemma bi_unique_rel_12_23 [simp, transfer_rule]: "bi_unique rel_12_23" |
lemma extensional'I [intro]:
assumes "\<And>x. x \<notin> A \<Longrightarrow> f x = unity"
shows "f \<in> extensional' A" |
lemma approx_from_below_dense_linorder:
fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}"
assumes "x > y"
shows "\<exists>u. (\<forall>n. u n < x) \<and> (u \<longlonglongrightarrow> x)" |
lemma ex_norm_eq_1: "\<exists>x. norm (x::'a::{real_normed_vector, perfect_space}) = 1" |
lemma epsclo_Reverse_nfa [simp]: "nfa.epsclo (Reverse_nfa M) Q = Q \<inter> dfa.states M" |
lemma PiE_dflt_empty_iff [simp]: "PiE_dflt A dflt B = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})" |
lemma ereal_closed_mono_set:
fixes S :: "ereal set"
shows "closed S \<and> mono_set S \<longleftrightarrow> S = {} \<or> S = {Inf S ..}" |
lemma (in infinite_coin_toss_space) spick_eq_pseudo_proj_True:
shows "spick w n True = pseudo_proj_True n w" |
lemma cltn2_abs_mult:
assumes "k \<noteq> 0" and "invertible A"
shows "cltn2_abs (k *\<^sub>R A) = cltn2_abs A" |
lemma lenlex_irreflexive: "(\<And>x. (x,x) \<notin> r) \<Longrightarrow> (xs,xs) \<notin> lenlex r" |
lemma finite_list':
assumes "finite A"
obtains xs where "A = set xs" "distinct xs" "length xs = card A" |
lemma lexLessBackjump:
assumes "p = prefixToLevel level a" and "level >= 0" and "level < currentLevel a"
shows "(p @ [(x, False)], a) \<in> lexLess" |
lemma is_const_sum_mod_2:
assumes "is_const"
shows "(f 0 + f 1) mod 2 = 0" |
lemma op_amtx_lin_get_aref: "(uncurry Array.nth, uncurry (RETURN oo PR_CONST op_amtx_lin_get)) \<in> [\<lambda>(_,i). i<N*M]\<^sub>a (is_amtx N M)\<^sup>k *\<^sub>a nat_assn\<^sup>k \<rightarrow> id_assn" |
lemma HOL_list_rotate1_hnr[sepref_fr_rules]: "(return \<circ> op_list_rotate1, RETURN \<circ> op_list_rotate1) \<in> (list_assn A)\<^sup>d \<rightarrow>\<^sub>a list_assn A" |
lemma distinctNets: "FWLink \<noteq> any \<and> FWLink \<noteq> i4_32 \<and> FWLink \<noteq> i10_32 \<and>
FWLink \<noteq> eth_intern \<and> FWLink \<noteq> eth_private \<and> any \<noteq> FWLink \<and> any \<noteq>
i4_32 \<and> any \<noteq> i10_32 \<and> any \<noteq> eth_intern \<and> any \<noteq> eth_private \<and> i4_32 \<noteq>
FWLink \<and> i4_32 \<noteq> any \<and> i4_32 \<noteq> i10_32 \<and> i4_32 \<noteq> eth_intern \<and> i4_32 \<noteq>
eth_private \<and> i10_32 \<noteq> FWLink \<and> i10_32 \<noteq> any \<and> i10_32 \<noteq> i4_32 \<and> i10_32
\<noteq> eth_intern \<and> i10_32 \<noteq> eth_private \<and> eth_intern \<noteq> FWLink \<and> eth_intern
\<noteq> any \<and> eth_intern \<noteq> i4_32 \<and> eth_intern \<noteq> i10_32 \<and> eth_intern \<noteq>
eth_private \<and> eth_private \<noteq> FWLink \<and> eth_private \<noteq> any \<and> eth_private \<noteq>
i4_32 \<and> eth_private \<noteq> i10_32 \<and> eth_private \<noteq> eth_intern " |
lemma eqButPID_cong[simp, intro]:
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>confIDs := uu1\<rparr>) (s1 \<lparr>confIDs := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>conf := uu1\<rparr>) (s1 \<lparr>conf := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>userIDs := uu1\<rparr>) (s1 \<lparr>userIDs := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>pass := uu1\<rparr>) (s1 \<lparr>pass := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>user := uu1\<rparr>) (s1 \<lparr>user := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>roles := uu1\<rparr>) (s1 \<lparr>roles := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>paperIDs := uu1\<rparr>) (s1 \<lparr>paperIDs := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> eeqButPID uu1 uu2 \<Longrightarrow> eqButPID (s \<lparr>paper := uu1\<rparr>) (s1 \<lparr>paper := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>pref := uu1\<rparr>) (s1 \<lparr>pref := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>voronkov := uu1\<rparr>) (s1 \<lparr>voronkov := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>news := uu1\<rparr>) (s1 \<lparr>news := uu2\<rparr>)"
"\<And> uu1 uu2. eqButPID s s1 \<Longrightarrow> uu1 = uu2 \<Longrightarrow> eqButPID (s \<lparr>phase := uu1\<rparr>) (s1 \<lparr>phase := uu2\<rparr>)" |
lemma wqo_on_sum_UNIV [intro]:
"wqo_on P UNIV \<Longrightarrow> wqo_on Q UNIV \<Longrightarrow> wqo_on (sum_le P Q) UNIV" |
lemma greater_than_two_values:
assumes "a \<noteq> b" "Value a \<le> z" "Value b \<le> z"
shows "z = Top" |
lemma set_of_paths_finite :
assumes "well_formed M"
and "q1 \<in> nodes M"
shows "finite { p . path M p q1 \<and> target p q1 = q2 \<and> length p \<le> k }" |
lemma not_bit_1_Suc [simp]:
\<open>\<not> bit 1 (Suc n)\<close> |
lemma SN_preserved[intro]:
assumes a: "SN t" "t \<mapsto> t'"
shows "SN t'" |
lemma arg_0_iff:
shows "z \<noteq> 0 \<and> Arg z = 0 \<longleftrightarrow> is_real z \<and> Re z > 0" |
lemma bg_type[autoref_itype]:
"bg_F ::\<^sub>i i_bg Ie Iv \<rightarrow>\<^sub>i \<langle>Iv\<rangle>\<^sub>ii_set"
"gb_graph_rec_ext ::\<^sub>i \<langle>\<langle>Iv\<rangle>\<^sub>ii_set\<rangle>\<^sub>ii_set \<rightarrow>\<^sub>i Ie \<rightarrow>\<^sub>i \<langle>Ie,Iv\<rangle>\<^sub>ii_bg_eext" |
lemma ereal_liminf_lim_mult:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
shows "liminf (\<lambda>n. u n * v n) = a * liminf v" |
lemma inv_ide [simp]:
assumes "ide a"
shows "inv a = a" |
lemma fieldT_size [simp]:
"(a, T) \<in> set fs \<Longrightarrow> size T < Suc (size_list (size_prod (\<lambda>x. 0) size) fs)" |
lemma siso_Par[simp]:
assumes "properL cl" and "sisoL cl"
shows "siso (Par cl)" |
lemma is_unit_Gcd_fin_iff [simp]:
"is_unit (Gcd\<^sub>f\<^sub>i\<^sub>n A) \<longleftrightarrow> Gcd\<^sub>f\<^sub>i\<^sub>n A = 1" |
lemma StaticUpCast':
"\<lbrakk> P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow>' \<langle>ref (a,Cs),s\<^sub>1\<rangle>; P \<turnstile> Path last Cs to C via Cs'; Ds = Cs@\<^sub>pCs' \<rbrakk>
\<Longrightarrow> P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<^sub>0\<rangle> \<Rightarrow>' \<langle>ref (a,Ds),s\<^sub>1\<rangle>" |
lemma compE\<^sub>2_not_Nil': "\<not>sub_RI e \<Longrightarrow> compE\<^sub>2 e \<noteq> []" |
lemma "DNI \<^bold>\<not> \<and> DNE \<^bold>\<not> \<longrightarrow> ECQ \<^bold>\<not>" |
lemma [simp]:
"(label0 = (\<lambda> x . if atom x then some else none)) = (atom0 = atom)" |
lemma FreeUltrafilterNat_HNatInfinite:
"\<forall>u. eventually (\<lambda>n. u < X n) \<U> \<Longrightarrow> star_n X \<in> HNatInfinite" |
lemma map_\<Gamma>_comp: "map_\<Gamma> g (map_\<Gamma> f s) = map_\<Gamma> (g \<circ> f) s" |
lemma Chn_triangle_eq:
shows "Chn (\<l>[f] \<bullet> (\<epsilon> \<star> f) \<bullet> \<a>\<^sup>-\<^sup>1[f, g, f] \<bullet> (f \<star> \<eta>) \<bullet> \<r>\<^sup>-\<^sup>1[f]) = gf.prj\<^sub>0 \<cdot> \<eta>.chine \<cdot> f.leg0"
and "Chn (\<r>[g] \<bullet> (g \<star> \<epsilon>) \<bullet> \<a>[g, f, g] \<bullet> (\<eta> \<star> g) \<bullet> \<l>\<^sup>-\<^sup>1[g]) = gf.prj\<^sub>1 \<cdot> \<eta>.chine \<cdot> g.leg1" |
lemma res_interior: "residuated f \<Longrightarrow> interior (f o residual f)" |
lemma poly_eval_eval_set_eq:
assumes "closed_fun R g"
assumes "S \<inter> I = S' \<inter> I"
assumes "P \<in> Pring_set R I"
assumes "\<one> \<noteq>\<zero>"
shows "poly_eval R S g P = poly_eval R S' g P" |
lemma list_split:
assumes "n \<le> length bss"
shows "\<exists>b bs. bss = b @ bs \<and> length b = n" |
lemma square_integrable_imp_absolutely_integrable:
assumes f: "f square_integrable S" and S: "S \<in> lmeasurable"
shows "f absolutely_integrable_on S" |
lemma AL_update_idem:
assumes "Assoc_List.lookup ls k = Some v"
shows "Assoc_List.update k v ls = ls" |
lemma finite_if_eq_beyond_finite: "finite S \<Longrightarrow> finite {s. s - S = s' - S}" |
lemma capped_omega_simulation_2:
assumes "s \<le> u"
and "y \<le> u"
and "u * u \<le> u"
and "u * v \<le> v"
and "y * s \<le> s * y"
shows "(s * y)\<^sup>\<omega>\<^sub>v \<le> s\<^sup>\<omega>\<^sub>v" |
lemma Borsuk_maps_homotopic_in_connected_component_eq:
fixes a :: "'a :: euclidean_space"
assumes S: "compact S" "a \<notin> S" "b \<notin> S" and 2: "2 \<le> DIM('a)"
shows "(homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
(\<lambda>x. (x - a) /\<^sub>R norm (x - a))
(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) \<longleftrightarrow>
connected_component (- S) a b)"
(is "?lhs = ?rhs") |
lemma cont_emeasure_spmf: "cont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. emeasure (measure_spmf p))" |
lemma can_coherence:
assumes "par f g" and "can f" and "can g"
shows "f = g" |
lemma establish_convergence_static_properties :
assumes "observable M1"
and "observable M2"
and "minimal M1"
and "minimal M2"
and "inputs M2 = inputs M1"
and "outputs M2 = outputs M1"
and "t \<in> transitions M1"
and "t_source t \<in> reachable_states M1"
and "is_state_cover_assignment M1 V"
and "V (t_source t) @ [(t_input t, t_output t)] \<in> L M2"
and "V ` reachable_states M1 \<subseteq> set T"
and "preserves_divergence M1 M2 (V ` reachable_states M1)"
and "convergence_graph_lookup_invar M1 M2 cg_lookup G"
and "convergence_graph_insert_invar M1 M2 cg_lookup cg_insert"
and "\<And> q1 q2 . q1 \<in> states M1 \<Longrightarrow> q2 \<in> states M1 \<Longrightarrow> q1 \<noteq> q2 \<Longrightarrow> \<exists> io . \<forall> k1 k2 . io \<in> set (dist_fun k1 q1) \<inter> set (dist_fun k2 q2) \<and> distinguishes M1 q1 q2 io"
and "\<And> q . q \<in> reachable_states M1 \<Longrightarrow> set (dist_fun 0 q) \<subseteq> set (after T (V q))"
and "\<And> q k . q \<in> states M1 \<Longrightarrow> finite_tree (dist_fun k q)"
and "L M1 \<inter> set (fst (establish_convergence_static dist_fun M1 V T G cg_insert cg_lookup m t)) = L M2 \<inter> set (fst (establish_convergence_static dist_fun M1 V T G cg_insert cg_lookup m t))"
shows "\<forall> \<gamma> x y . length (\<gamma>@[(x,y)]) \<le> m - size_r M1 \<longrightarrow>
\<gamma> \<in> LS M1 (after_initial M1 (V (t_source t) @ [(t_input t, t_output t)])) \<longrightarrow>
x \<in> inputs M1 \<longrightarrow> y \<in> outputs M1 \<longrightarrow>
L M1 \<inter> ((V ` reachable_states M1) \<union> {\<omega>@\<omega>' | \<omega> \<omega>' . \<omega> \<in> {((V (t_source t)) @ [(t_input t,t_output t)]), (V (t_target t))} \<and> \<omega>' \<in> list.set (prefixes (\<gamma>@[(x,y)]))}) = L M2 \<inter> ((V ` reachable_states M1) \<union> {\<omega>@\<omega>' | \<omega> \<omega>' . \<omega> \<in> {((V (t_source t)) @ [(t_input t,t_output t)]), (V (t_target t))} \<and> \<omega>' \<in> list.set (prefixes (\<gamma>@[(x,y)]))})
\<and> preserves_divergence M1 M2 ((V ` reachable_states M1) \<union> {\<omega>@\<omega>' | \<omega> \<omega>' . \<omega> \<in> {((V (t_source t)) @ [(t_input t,t_output t)]), (V (t_target t))} \<and> \<omega>' \<in> list.set (prefixes (\<gamma>@[(x,y)]))})"
(is "?P1a")
and "preserves_divergence M1 M2 ((V ` reachable_states M1) \<union> {((V (t_source t)) @ [(t_input t,t_output t)]), (V (t_target t))})"
(is "?P1b")
and "convergence_graph_lookup_invar M1 M2 cg_lookup (snd (establish_convergence_static dist_fun M1 V T G cg_insert cg_lookup m t))"
(is "?P2") |
lemma F_ctor_o_fold: "ctor_fold_F s o ctor_F = s o map_pre_F id (ctor_fold_F s)" |
lemma inf_par_distrib2: "d \<parallel> (c\<^sub>0 \<sqinter> c\<^sub>1) = (d \<parallel> c\<^sub>0) \<sqinter> (d \<parallel> c\<^sub>1)" |
lemma guarantees_Int_right_I:
"[| F \<in> Z guarantees X; F \<in> Z guarantees Y |]
==> F \<in> Z guarantees (X \<inter> Y)" |
lemma round_of_int [simp]: "round (of_int n) = n" |
lemma qtname_declclass_simp[simp]: "declclass (q::qtname) = q" |
lemma insort_key_commute:
"x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> insort_key f y o insort_key f x = insort_key f x o insort_key f y" |
lemma widen_strict_to_widen: "C \<prec> D = (C \<preceq> D \<and> C\<noteq>D)" |
lemma mono_assert_fun [simp]:
"mono (assert_fun p)" |
lemma local_lipschitz_constI: "local_lipschitz S T (\<lambda>t x. f t)" |
lemma plus_emeasure:
"a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)" |
lemma timpl_closure_set_Union:
"timpl_closure_set (\<Union>Ms) T = (\<Union>M \<in> Ms. timpl_closure_set M T)" |
lemma sorted_app_l: "sorted cmp (xs@ys) \<Longrightarrow> sorted cmp xs" |
lemma iTILL_iFROM_subset_conv: "([\<dots>n'] \<subseteq> [n\<dots>]) = (n = 0)" |
lemma jumpF_polyL_mult_cancel:
assumes "p'\<noteq>0"
shows "jumpF_polyL (p' * q) (p' * p) a = jumpF_polyL q p a" |
lemma enum_enumerator:
"enum i j = enumerator i j" |
lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
(is "_ \<longleftrightarrow> ?alt") |
lemma dg_SemiCAT_ObjI:
assumes "semicategory \<alpha> \<AA>"
shows "\<AA> \<in>\<^sub>\<circ> dg_SemiCAT \<alpha>\<lparr>Obj\<rparr>" |
lemma Retr_incl:
"\<And>theta. Sretr theta \<subseteq> ZOretrT theta"
(* *)
"\<And>theta. ZOretrT theta \<subseteq> ZOretr theta"
(* *)
"\<And>theta. ZOretrT theta \<subseteq> WretrT theta"
(* *)
"\<And>theta. ZOretr theta \<subseteq> Wretr theta"
(* *)
"\<And>theta. WretrT theta \<subseteq> Wretr theta"
(* *)
"\<And>theta. WretrT theta \<subseteq> RetrT theta" |
lemma EWHILEIT_weaken:
assumes "\<And>x. I x \<Longrightarrow> I' x"
shows "EWHILEIT I' b f x \<le> EWHILEIT I b f x" |
lemma "\<not> OrderedSet Boolean[1] < Set (Boolean[1] :: classes1 type)" |
lemma is_glbD2: "[|S >>| x; S >| u|] ==> u \<sqsubseteq> x" |
lemma pow_morph: "f (x\<^sup>@k) = (f x)\<^sup>@k" |
lemma univ_poly_zero_closed [intro]: "[] \<in> carrier (K[X]\<^bsub>R\<^esub>)" |
lemma substitutes_total:
shows "\<exists>X. substitutes A x M X" |
lemma poly_map_cons:
assumes "a \<in> carrier (R\<^bsup>n\<^esup>)"
shows "poly_map n (f#fs) a = (eval_at_point R a f)#poly_map n fs a" |
lemma matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))" |
lemma sc_hconf_upd_obj: "\<lbrakk> P \<turnstile>sc h \<surd>; h a = Some (Obj C fs); P,h \<turnstile>sc (Obj C fs') \<surd> \<rbrakk> \<Longrightarrow> P \<turnstile>sc h(a\<mapsto>(Obj C fs')) \<surd>" |
theorem "\<And>(y::int) (z::int) (n::int). 3 dvd z \<Longrightarrow> 2 dvd (y::int) \<Longrightarrow>
(\<exists>(x::int). 2*x = y) \<and> (\<exists>(k::int). 3*k = z)" |
lemma inv_into_map_sum:
"inv_into (A <+> B) (map_sum f g) x = map_sum (inv_into A f) (inv_into B g) x"
if "x \<in> f ` A <+> g ` B" "inj_on f A" "inj_on g B" |
lemma length_fresh_rename_ns: "finite G \<Longrightarrow> length (fresh_rename_ns n B insts G) = n" |
lemma "(get_all_matching_src_ips_executable (Iface ''eth0'')
(MatchAnd (MatchNot (Match (Src (IpAddrNetmask (ipv4addr_of_dotdecimal (192,168,0,0)) 24)))) (Match (IIface (Iface ''eth0''))))) =
RangeUnion (WordInterval 0 0xC0A7FFFF) (WordInterval 0xC0A80100 0xFFFFFFFF)" |
lemma cblinfun_left_right_ortho[simp]: \<open>cblinfun_left* o\<^sub>C\<^sub>L cblinfun_right = 0\<close> |
lemma monic_degree_m[simp]: "monic f \<Longrightarrow> degree_m f = degree f" |
lemma IB4: "Br_1 \<phi> \<Longrightarrow> Br_3 \<phi> \<Longrightarrow> Int_4(\<I>\<^sub>B \<phi>)" |
lemma inj_lowerPowers: "inj (lowerPowers b i)" |
lemma hbex_triv_one_point2 [simp]: "(\<exists>x\<^bold>\<in>A. a = x) = (a\<^bold>\<in>A)" |
lemma reduce_basis_cost_N:
assumes "Lg \<ge> nat \<lceil>log (of_rat (4 * \<alpha> / (4 + \<alpha>))) N\<rceil>"
and 0: "Lg > 0"
shows "cost (reduce_basis_cost fs_init) \<le> 49 * m ^ 3 * n * Lg" |
lemma acomp_acomp:
"\<lbrakk>acomp a\<^sub>1 @ acomp a\<^sub>2 @ P \<Turnstile> cfs\<box>;
\<And>cfs. acomp a\<^sub>1 \<Turnstile> cfs\<box> \<Longrightarrow> apred a\<^sub>1 (cfs ! 0) (cfs ! (length cfs - 1));
\<And>cfs. acomp a\<^sub>2 \<Turnstile> cfs\<box> \<Longrightarrow> apred a\<^sub>2 (cfs ! 0) (cfs ! (length cfs - 1))\<rbrakk> \<Longrightarrow>
case cfs ! 0 of (pc, s, stk) \<Rightarrow> pc = 0 \<and> (\<exists>k < length cfs. cfs ! k =
(size (acomp a\<^sub>1 @ acomp a\<^sub>2), s, aval a\<^sub>2 s # aval a\<^sub>1 s # stk))" |
lemma phantom_optionT [simp]: "phantom_optionT x = {}" |
lemma "rot n \<noteq> n" |
lemma rel_gpv''_mono:
assumes "A \<le> A'" "C \<le> C'" "R' \<le> R"
shows "rel_gpv'' A C R \<le> rel_gpv'' A' C' R'" |
lemma GFP_healthy_comp: "\<^bold>\<nu> F = \<^bold>\<nu> (F \<circ> \<H>)" |
lemma singular_subdivision_power_0 [simp]: "(singular_subdivision p ^^ n) 0 = 0" |
lemma thetaZOSeqD_ZOretr:
"thetaZOSeqD \<subseteq> ZOretr (thetaZOSeqD Un ZObis)" |
lemma foldl_LTLOr_prop_entailment:
"S \<Turnstile>\<^sub>P foldl LTLOr i xs = (S \<Turnstile>\<^sub>P i \<or> (\<exists>y \<in> set xs. S \<Turnstile>\<^sub>P y))" |
lemma set_mode_get_shadow_root:
"\<forall>w \<in> set_mode_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_shadow_root_locs ptr'. r h h'))" |
lemma [transfer_rule]:
"((=) ===> pcr_mod_ring) (\<lambda> x. int x mod int (CARD('a :: nontriv))) (of_nat :: nat \<Rightarrow> 'a mod_ring)" |
lemma standard_top_not_bot[intro]:
"standard' C G \<Longrightarrow> :G:\<lbrakk>\<bottom>\<rbrakk> \<noteq> :G:\<lbrakk>\<top>\<rbrakk>" |
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