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lemma 35: \<open>\<turnstile> (p \<rightarrow> q \<rightarrow> r) \<rightarrow> (p \<rightarrow> q) \<rightarrow> p \<rightarrow> r\<close> |
lemma OrdP_cases_disj:
assumes p: "atom p \<sharp> x"
shows "insert (OrdP x) H \<turnstile> x EQ Zero OR Ex p (OrdP (Var p) AND x EQ SUCC (Var p))" |
lemma f'_cong: "(g has_derivative blinfun_apply (f' x)) (at x)" if "x \<in> Y" |
lemma Ate_Refines_LV_VOting:
"PO_refines (ate_ref_rel)
majorities.flv_TS (Ate_TS HOs HOs crds)" |
lemma access_type_safe [simp,intro]: "typeof (s@@l) \<le> ltype l" |
theorem StateParallel_frame_hoare [hoare]:
assumes "vwb_lens a" "vwb_lens b" "a \<bowtie> b" "a \<natural> d\<^sub>1" "b \<natural> d\<^sub>2" "a \<sharp> c\<^sub>1" "b \<sharp> c\<^sub>1" "\<lbrace>c\<^sub>1 \<and> c\<^sub>2\<rbrace>P\<lbrace>d\<^sub>1\<rbrace>\<^sub>u" "\<lbrace>c\<^sub>1 \<and> c\<^sub>2\<rbrace>Q\<lbrace>d\<^sub>2\<rbrace>\<^sub>u"
shows "\<lbrace>c\<^sub>1 \<and> c\<^sub>2\<rbrace>P |a|b|\<^sub>\<sigma> Q\<lbrace>c\<^sub>1 \<and> d\<^sub>1 \<and> d\<^sub>2\<rbrace>\<^sub>u" |
lemma i_set_Plus_closed: "I \<in> i_set \<Longrightarrow> I \<oplus> k \<in> i_set" |
lemma deadlock_state_alt_def_h : "deadlock_state M q = (\<forall> x \<in> inputs M . h M (q,x) = {})" |
lemma all_bex_swap_lemma [iff]:
"(\<forall>x. (\<exists>y\<in>A. x = f y) \<longrightarrow> P x) = (\<forall>y\<in>A. P(f y))" |
lemma kill_short_uwellformed:
assumes "finite (uverts G)" "uwellformed G"
shows "uwellformed (kill_short G k)" |
lemma mset_ran_xfer_pointwise:
assumes "mset_ran a r = mset_ran a' r"
assumes "finite r"
shows "(\<forall>i\<in>r. P (a i)) \<longleftrightarrow> (\<forall>i\<in>r. P (a' i))" |
lemma demorgans2: "-(x \<squnion> y) = -x \<sqinter> -y" |
lemma fresh_minus_atom_set:
fixes S::"atom set"
assumes "finite S"
shows "a \<sharp> S - T \<longleftrightarrow> (a \<notin> T \<longrightarrow> a \<sharp> S)" |
lemma Node_in_tree_sigma:
assumes L: "X \<in> sets (M \<Otimes>\<^sub>M (tree_sigma M \<Otimes>\<^sub>M tree_sigma M))"
shows "{Node l v r | l v r. (v, l, r) \<in> X} \<in> sets (tree_sigma M)" |
lemma infer_v_conj:
assumes "\<Theta> ; \<B> ; GNil \<turnstile> v \<Leftarrow> \<lbrace> z : b | c1 \<rbrace>" and "\<Theta> ; \<B> ; GNil \<turnstile> v \<Leftarrow> \<lbrace> z : b | c2 \<rbrace>"
shows "\<Theta> ; \<B> ; GNil \<turnstile> v \<Leftarrow> \<lbrace> z : b | c1 AND c2 \<rbrace>" |
lemma map_vec_mat_cols: "map (map_vec f) (cols M) = cols ((map_mat f) M)" |
lemma one_side_transitivity:
assumes "P Q OS A B" and
"P Q OS B C"
shows "P Q OS A C" |
lemma FIXME_third_fiddle:
"\<lbrakk> (r \<inter> Y \<times> Y) `` X \<subseteq> X; X \<subseteq> Y; x \<in> X; y \<in> Y - X ; r `` {y} \<inter> X = {} \<rbrakk>
\<Longrightarrow> (r \<inter> (Y - (X - r `` {x})) \<times> (Y - (X - r `` {x}))) `` {y}
= (r \<inter> (Y - X) \<times> (Y - X)) `` {y}" |
lemma floor_has_real_derivative:
fixes f :: "real \<Rightarrow> 'a::{floor_ceiling,order_topology}"
assumes "isCont f x"
and "f x \<notin> \<int>"
shows "((\<lambda>x. floor (f x)) has_real_derivative 0) (at x)" |
lemma True_steps_concD[rule_format]:
"\<forall>p. (True#p,q) : steps (conc L R) w \<longrightarrow>
((\<exists>r. (p,r) : steps L w \<and> q = True#r) \<or>
(\<exists>u a v. w = u@a#v \<and>
(\<exists>r. (p,r) : steps L u \<and> fin L r \<and>
(\<exists>s. (start R,s) : step R a \<and>
(\<exists>t. (s,t) : steps R v \<and> q = False#t)))))" |
lemma l10_2_uniqueness_spec:
assumes "P1 P ReflectL A B" and
"P2 P ReflectL A B"
shows "P1 = P2" |
lemma WHILE_refine_rwof:
assumes "nofail (m \<bind> WHILE c f) \<Longrightarrow> mi \<le> SPEC (\<lambda>s. rwof m c f s \<and> \<not>c s)"
shows "mi \<le> m \<bind> WHILE c f" |
lemma eval_binop_arg2_indep:
"\<not> need_second_arg binop v1 \<Longrightarrow> eval_binop binop v1 x = eval_binop binop v1 y" |
lemma shadow_root_put_get_4 [simp]: "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h' \<Longrightarrow>
preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr getter) h h'" |
lemma filter_map_elem : "t \<in> set (map g (filter f xs)) \<Longrightarrow> \<exists> x \<in> set xs . f x \<and> t = g x" |
lemma set_conv_list [code]:
"set.F g (set xs) = list.F (map g (remdups xs))" |
lemma length_mirror1_aux: "length ps = length (mirror1_aux n ps)" |
lemma sum_list_hd_tl:
fixes xs :: "(_ :: group_add) list"
shows "xs \<noteq> [] \<Longrightarrow> sum_list (tl xs) = (- hd xs) + sum_list xs" |
lemma AbsAxiomCheck:
"OrdinaryObjectsPossiblyConcrete \<longleftrightarrow>
(\<forall> x. ([\<lparr>\<^bold>\<lambda> x . \<^bold>\<box>(\<^bold>\<not>\<lparr>E!, x\<^sup>P\<rparr>), x\<^sup>P\<rparr> in v]
\<longleftrightarrow> (case x of \<alpha>\<nu> y \<Rightarrow> True | _ \<Rightarrow> False)))" |
lemma R26_d [simp]: "pmf (sds R26) d = 1 - pmf (sds R26) a" |
lemma list_sorted_max[simp]:
shows "sorted list \<Longrightarrow> list = (x#xs) \<Longrightarrow> fold max xs x = (last list)" |
lemma len_ge_1: "len ts \<ge> 1" |
lemma subtensor_prod_with_vec:
assumes "order A = 1" "i < hd (dims A)"
shows "subtensor (A \<otimes> B) i = lookup A [i] \<cdot> B" |
lemma a_de_morgan_var_3: "ad (d x + d y) = ad x \<cdot> ad y" |
lemma vsv_vdoubleton:
assumes "a \<noteq> c"
shows "vsv (set {\<langle>a, b\<rangle>, \<langle>c, d\<rangle>})" |
lemma while_denest_5:
"w * ((x \<star> (y * w)) \<star> (x \<star> (y * z))) = w * (((x \<star> y) * w) \<star> ((x \<star> y) * z))" |
lemma subseteq_guards_DynCom:
"\<exists>C'. c=DynCom C' \<and> (\<forall>s. C' s \<subseteq>\<^sub>g C s)" if "c \<subseteq>\<^sub>g DynCom C" |
lemma heap_read_typedI:
"\<lbrakk> heap_read h ad al v; \<And>T. P,h \<turnstile> ad@al : T \<Longrightarrow> P,h \<turnstile> v :\<le> T \<rbrakk> \<Longrightarrow> heap_read_typed P h ad al v" |
lemma LtK_Exp: "LtK X \<noteq> Exp X' Y'" |
lemma bottom_eq_False[simp]: "\<bottom> = False" |
lemma f_shrink_assoc: "xs \<div>\<^sub>f a \<div>\<^sub>f b = xs \<div>\<^sub>f (a * b)" |
lemma "app (fs @ gs) x = app fs (app gs x)" |
lemma inverse_ii [simp]: "inverse quat_ii = -quat_ii" |
lemma tj_stack_incr_disc:
assumes "k < length (tj_stack s)"
and "j < k"
shows "\<delta> s (tj_stack s ! j) > \<delta> s (tj_stack s ! k)" |
lemma fds_eqI_truncate:
assumes "\<And>m. m > 0 \<Longrightarrow> fds_truncate m f = fds_truncate m g"
shows "f = g" |
lemma eigenvalue_root_char_poly: assumes A: "(A :: 'a :: field mat) \<in> carrier_mat n n"
shows "eigenvalue A k \<longleftrightarrow> poly (char_poly A) k = 0" |
lemma interleave_tl: "xs \<in> ys \<otimes> zs \<Longrightarrow> tl xs \<in> tl ys \<otimes> zs \<or> tl xs \<in> ys \<otimes> (tl zs)" |
lemma resolvent_upon_and_partial_saturation :
assumes "redundant P1 S"
assumes "redundant P2 S"
assumes "partial_saturation S A (S \<union> R)"
assumes "C = resolvent_upon P1 P2 A"
shows "redundant C (S \<union> R)" |
lemma ide_char\<^sub>S\<^sub>b\<^sub>C:
shows "ide a \<longleftrightarrow> arr a \<and> C.ide a" |
lemma continuous_at_left_imp_sup_continuous:
fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
assumes "mono f" "\<And>x. continuous (at_left x) f"
shows "sup_continuous f" |
lemma AE_lim_stream:
"AE \<omega> in K.lim_stream x. \<forall>i. snd ((x ## \<omega>) !! i) \<in> DTMC.acc``{snd x} \<and> snd (\<omega> !! i) \<in> J (snd ((x ## \<omega>) !! i)) \<and> fst ((x ## \<omega>) !! i) < fst (\<omega> !! i)"
(is "AE \<omega> in K.lim_stream x. \<forall>i. ?P \<omega> i") |
lemma NSconvergent_NSBseq: "NSconvergent X \<Longrightarrow> NSBseq X" |
lemma ProcProcParModifyReturn:
assumes q: "P \<subseteq> {s. p s = q} \<inter> P'"
\<comment> \<open>@{thm[source] DynProcProcPar} introduces the same constraint as first conjunction in
@{term P'}, so the vcg can simplify it.\<close>
assumes to_prove: "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' (dynCall init p return' c) Q,A"
assumes ret_nrm_modif: "\<forall>s t. t \<in> (Modif (init s))
\<longrightarrow> return' s t = return s t"
assumes ret_abr_modif: "\<forall>s t. t \<in> (ModifAbr (init s))
\<longrightarrow> return' s t = return s t"
assumes modif_clause:
"\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} (Call q) (Modif \<sigma>),(ModifAbr \<sigma>)"
shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (dynCall init p return c) Q,A" |
lemma bounded_linear_prod_right[bounded_linear]: "bounded_linear prod_right" |
lemma mergeSL:
assumes "adm_woL L" shows "adm_wo (mergeSL S L)" |
lemma Iam_calc: "$(Ia)^{m}_n = (\<Sum>j<n. (j+1)/m * (\<Sum>k=j*m..<(j+1)*m. $v.^((k+1)/m)))"
if "m \<noteq> 0" for n m :: nat |
lemma nocp_cp_triv: "nocp Q \<Longrightarrow> cp Q = Q" |
lemma ni_bot:
"ni(bot) = bot" |
lemma sum_Basis_prod_eq:
fixes f::"('a*'b)\<Rightarrow>('a*'b)"
shows "sum f Basis = sum (\<lambda>i. f (i, 0)) Basis + sum (\<lambda>i. f (0, i)) Basis" |
lemma irreflexive_cong:
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> r a b \<longleftrightarrow> r' a b) \<Longrightarrow> irreflexive A r \<longleftrightarrow> irreflexive A r'" |
lemma circline_set_x_axis:
shows "circline_set x_axis = of_complex ` {x. is_real x} \<union> {\<infinity>\<^sub>h}" |
lemma indefinite_integral_continuous_real:
fixes f :: "real \<Rightarrow> 'a::euclidean_space"
assumes "integrable (lebesgue_on {a..b}) f"
shows "continuous_on {a..b} (\<lambda>x. integral\<^sup>L (lebesgue_on {a..x}) f)" |
lemma redT_updIs_insert_Interrupt:
"\<lbrakk> t \<in> redT_updIs is ias; t \<notin> is \<rbrakk> \<Longrightarrow> Interrupt t \<in> set ias" |
lemma concurrent_ops_commute_singleton [intro!]: "concurrent_ops_commute [x]" |
lemma ab_obey_1[PLM]:
"[(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr>) \<^bold>\<rightarrow> ((\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P, F\<rbrace>) \<^bold>\<rightarrow> x\<^sup>P \<^bold>= y\<^sup>P) in v]" |
lemma image_preserves_per:
assumes "A A' Reflect X Y" and
"B B' Reflect X Y"and
"C C' Reflect X Y" and
"Per A B C"
shows "Per A' B' C'" |
lemma continuous_on_open_Collect_neq:
fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
assumes f: "continuous_on S f" and g: "continuous_on S g" and "open S"
shows "open {x\<in>S. f x \<noteq> g x}" |
lemma index_mult_mat_vec[simp]: "i < dim_row A \<Longrightarrow> (A *\<^sub>v v) $ i = row A i \<bullet> v" |
lemma strict_mono_ordinal_of_nat: "strict_mono ordinal_of_nat" |
lemma funas_term_ctxt_apply [simp]:
"funas_term (C\<langle>t\<rangle>) = funas_ctxt C \<union> funas_term t" |
lemma (in monoid_cancel) properfactor_mult_l [simp]:
assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G"
shows "properfactor G (c \<otimes> a) (c \<otimes> b) = properfactor G a b" |
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" |
lemma Inl_Inr_dataspace [simp]:
"[Part UNIV Inl, Part UNIV Inr] \<in> dataspace" |
theorem fusion_lfp_eq:
assumes monoH: "mono H" and monoG: "mono G"
and distF: "dist_over_sup F"
and fgh_comp: "\<And>x. ((F \<circ> G) x = (H \<circ> F) x)"
shows "F (lfp G) = (lfp H)" |
lemma cf_cat_prod_12_of_3_components:
shows "cf_cat_prod_12_of_3 \<AA> \<BB> \<CC>\<lparr>ObjMap\<rparr> =
(\<lambda>A\<in>\<^sub>\<circ>(\<AA> \<times>\<^sub>C\<^sub>3 \<BB> \<times>\<^sub>C\<^sub>3 \<CC>)\<lparr>Obj\<rparr>. [A\<lparr>0\<rparr>, [A\<lparr>1\<^sub>\<nat>\<rparr>, A\<lparr>2\<^sub>\<nat>\<rparr>]\<^sub>\<circ>]\<^sub>\<circ>)"
and "cf_cat_prod_12_of_3 \<AA> \<BB> \<CC>\<lparr>ArrMap\<rparr> =
(\<lambda>F\<in>\<^sub>\<circ>(\<AA> \<times>\<^sub>C\<^sub>3 \<BB> \<times>\<^sub>C\<^sub>3 \<CC>)\<lparr>Arr\<rparr>. [F\<lparr>0\<rparr>, [F\<lparr>1\<^sub>\<nat>\<rparr>, F\<lparr>2\<^sub>\<nat>\<rparr>]\<^sub>\<circ>]\<^sub>\<circ>)"
and [cat_cs_simps]: "cf_cat_prod_12_of_3 \<AA> \<BB> \<CC>\<lparr>HomDom\<rparr> = \<AA> \<times>\<^sub>C\<^sub>3 \<BB> \<times>\<^sub>C\<^sub>3 \<CC>"
and [cat_cs_simps]: "cf_cat_prod_12_of_3 \<AA> \<BB> \<CC>\<lparr>HomCod\<rparr> = \<AA> \<times>\<^sub>C (\<BB> \<times>\<^sub>C \<CC>)" |
lemma inj_on_image_lesspoll_1 [simp]:
assumes "inj_on f A" shows "f ` A \<prec> B \<longleftrightarrow> A \<prec> B" |
lemma bin_code_lcp_concat: assumes "us \<in> lists {u\<^sub>0,u\<^sub>1}" and "vs \<in> lists {u\<^sub>0,u\<^sub>1}" and "\<not> us \<bowtie> vs"
shows "concat us \<cdot> \<alpha> \<and>\<^sub>p concat vs \<cdot> \<alpha> = concat (us \<and>\<^sub>p vs) \<cdot> \<alpha>" |
lemma dverts_mset_sub_dverts: "set_mset (dverts_mset t) \<subseteq> dverts t" |
lemma (in Square_impl)
shows "\<Gamma>\<turnstile>\<lbrace>\<acute>I = 2\<rbrace> \<acute>R :== CALL Square(\<acute>I) \<lbrace>\<acute>R = 4\<rbrace>" |
lemma finite_set [iff]: "finite (set xs)" |
lemma domain_comp: "fst ` set (\<sigma> \<lozenge> \<theta>) = fst ` (set \<sigma> \<union> set \<theta>)" |
lemma strict2[simp]: "f self X invalid = invalid" |
lemma lower_set_imp_not_SN_on:
assumes "s \<in> X" "\<forall>t \<in> X. \<exists>u \<in> X. (t,u) \<in> R" shows "\<not> SN_on R {s}" |
lemma brnL_Int_lt:
assumes n12: "n1 < n2" and n2: "n2 < length cl"
shows
"{brnL cl n1 ..<+ brn (cl!n1)} \<inter> {brnL cl n2 ..<+ brn (cl!n2)} = {}" |
lemma subgroup_inter:
assumes "subgroup A" and "subgroup B"
shows "subgroup (A \<inter> B)" |
lemma (in Ring) nonilp_residue_nilrad:"\<lbrakk>\<not> zeroring R; x \<in> carrier R;
nilpotent (qring R (nilrad R)) (x \<uplus>\<^bsub>R\<^esub> (nilrad R))\<rbrakk> \<Longrightarrow>
x \<uplus>\<^bsub>R\<^esub> (nilrad R) = \<zero>\<^bsub>(qring R (nilrad R))\<^esub>" |
lemma ereal_bot:
fixes x :: ereal
assumes "\<And>B. x \<le> ereal B"
shows "x = - \<infinity>" |
lemma in_Inter_image_nth_conv: "a \<in> \<Inter> (f ` set xs) = (\<forall>i < length xs. a \<in> f (xs!i))" |
lemma uIsInvar_phase_leq_closedPH: "uIsInvar phase_leq_closedPH" |
lemma simplex_furthest_le_exists:
fixes S :: "('a::real_inner) set"
shows "finite S \<Longrightarrow> \<forall>x\<in>(convex hull S). \<exists>y\<in>S. norm (x - a) \<le> norm (y - a)" |
lemma case_bool_If:
"case_bool P Q b = (if b then P else Q)" |
lemma TNil_Lazy_tllist [code]:
"TNil b = Lazy_tllist (\<lambda>_. Inr b)" |
lemma vrat_inverse_closed:
assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
shows "x\<inverse>\<^sub>\<rat> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" |
lemma Thread_neq_sys_xcpts_aux:
"Thread \<noteq> NullPointer"
"Thread \<noteq> ClassCast"
"Thread \<noteq> OutOfMemory"
"Thread \<noteq> ArrayIndexOutOfBounds"
"Thread \<noteq> ArrayStore"
"Thread \<noteq> NegativeArraySize"
"Thread \<noteq> ArithmeticException"
"Thread \<noteq> IllegalMonitorState"
"Thread \<noteq> IllegalThreadState"
"Thread \<noteq> InterruptedException" |
lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y" |
lemma projection_insert_finite:
fixes S :: \<open>'a::chilbert_space set\<close>
assumes a1: "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s" and a2: "finite S"
shows "projection (cspan (insert a S)) u
= projection (cspan {a}) u + projection (cspan S) u" |
lemma disjointI:
assumes "\<And>A B. A \<in> Ss \<Longrightarrow> B \<in> Ss \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
shows "disjoint Ss" |
lemma LV_From_5:
shows "LV (From r n) s \<subseteq> Stars_Append (LV (Star r) s) (\<Union>i\<le>n. LV (From r i) [])" |
lemma indep_and_equiv_implies_ief:
assumes "wb_lens x" "x \<bowtie> y" "x \<approx>\<^sub>L y"
shows "ief_lens x" |
lemma (in Group) rfn_tool16:"\<lbrakk>0 < r; 0 < s; i \<le> (s * r - Suc 0);
G \<guillemotright> f (i div s); (Gp G (f (i div s))) \<triangleright> f (Suc (i div s));
(Gp G (f (i div s))) \<guillemotright> (f (i div s) \<inter> g (s - Suc 0))\<rbrakk> \<Longrightarrow>
(Gp G ((f (Suc (i div s)) \<diamondop>\<^bsub>G\<^esub> (f (i div s) \<inter> g (s - Suc 0))))) \<triangleright>
(f (Suc (i div s)))" |
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