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lemma llist_all2_llist_of_inf_llist [simp]:
"\<not> llist_all2 P (llist_of xs) (inf_llist f)" |
lemma AILiD[rule_format,simp]: "all_in_list p l \<longrightarrow> all_in_list (insertDeny p) l" |
lemma exits_empty [iff]: "exits [] = {}" |
lemma
map_of_zip_upto_length_eq_nth:
assumes "i < length B"
assumes "d = length B"
shows "(map_of (zip [0..<d] B) i) = Some (B ! i)" |
lemma DataValue_DataOverride [simp]:
"((PDataSpace P) = (Data.DataSpace Q)) \<Longrightarrow>
(DataValue (P [D+] Q)) = (map OptionOverride (zip (PDataValue P) (Data.DataValue Q)))" |
lemma pair_in_simplicial_complex_induced:
assumes x: "x < 4" and y: "y < 4"
shows "{x,y} \<in> simplicial_complex_induced_by_monotone_boolean_function 4 bool_fun_threshold_2_3"
(is "?A \<in> simplicial_complex_induced_by_monotone_boolean_function 4 bool_fun_threshold_2_3") |
lemma AxiomP_fresh_iff [simp]: "a \<sharp> AxiomP x \<longleftrightarrow> a \<sharp> x" |
lemma C_eq_if_mr_eq:
"applied_rule_rev C x b = applied_rule_rev C x a \<Longrightarrow> a \<noteq> [] \<Longrightarrow> b \<noteq> [] \<Longrightarrow>
C (list2FWpolicy a) x = C (list2FWpolicy b) x" |
lemma unifiers_insert_ident [simp]:
"unifiers (insert (t, t) E) = unifiers E" |
lemma ltln_weak_strong_stable_words_1:
"w \<Turnstile>\<^sub>n (\<phi> W\<^sub>n \<psi>) \<longleftrightarrow> w \<Turnstile>\<^sub>n \<phi> U\<^sub>n (\<psi> or\<^sub>n (G\<^sub>n \<phi>[\<G>\<F> \<phi> w]\<^sub>\<Pi>\<^sub>1))" (is "?lhs \<longleftrightarrow> ?rhs") |
lemma Iff_E:
"insert A (insert B H) \<turnstile> C \<Longrightarrow> insert (Neg A) (insert (Neg B) H) \<turnstile> C \<Longrightarrow> insert (A IFF B) H \<turnstile> C" |
lemma has_le_has:
assumes h: "has n S"
and nn': "n' \<le> n"
shows "has n' S" |
lemma vfsequence_map_is_arr:
assumes "F : A \<mapsto>\<^bsub>cat_Set \<alpha>\<^esub> B"
shows "vfsequence_map F : vfsequences_on A \<mapsto>\<^bsub>cat_Set \<alpha>\<^esub> vfsequences_on B" |
lemma beta_into_beta_eta_reds: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t" |
lemma gen_model_S5n:
assumes S5n: "S5n M"
shows "S5n (gen_model M w)" |
lemma bounded_linear_mult_right: "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)" |
lemma not_nil_if_in_set:
assumes "x \<in> set xs" shows "xs \<noteq> []" |
lemma is_process8_S: "\<lbrakk> is_process P; s \<in> DIVERGENCES P \<rbrakk> \<Longrightarrow> (s,X) \<in> FAILURES P" |
lemma external_event:
assumes path_Q: "Q \<in> \<P>"
shows "\<exists>d\<in>\<E>. d \<notin> Q" |
lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)" |
lemma simple_rotations_fix_unique:
assumes r:"r \<in> simple_rotations"
shows "r \<noteq> id \<Longrightarrow> r v = v \<Longrightarrow> r w = w \<Longrightarrow> v = w" |
lemma "CoP2 \<^bold>\<not>\<^sup>I \<longrightarrow> TNDm \<^bold>\<not>\<^sup>I" |
lemma sum_sum_distr_fun:
fixes f g :: "'a \<Rightarrow> 'b::dioid_one_zero"
fixes h :: "'a \<Rightarrow> 'a set"
assumes "finite Y"
and "\<And>y. finite (h y)"
shows "\<Sum>((\<lambda>y. \<Sum>(f ` h y) \<cdot> g y) ` Y) = \<Sum>{\<Sum>{f x \<cdot> g y |x. x \<in> (h y)} |y. y \<in> Y}" |
lemma n_bool_lists_correct: "set (n_bool_lists n x) = {xs. length xs = x \<and> count_list xs True = n}" |
lemma (in Ring) csrp_nz_nz:"\<lbrakk>ideal R I; x \<in> carrier (R /\<^sub>r I);
x \<noteq> \<zero>\<^bsub>(R /\<^sub>r I)\<^esub>\<rbrakk> \<Longrightarrow> (csrp_fn R I x) \<noteq> \<zero>" |
lemma lens_quotient_plus_den2: "\<lbrakk> weak_lens x; weak_lens z; x \<bowtie> z; y \<subseteq>\<^sub>L z \<rbrakk> \<Longrightarrow> y /\<^sub>L (x +\<^sub>L z) = (y /\<^sub>L z) ;\<^sub>L snd\<^sub>L " |
lemma [simp]: "rel O Id_on (nodes A) = rel" |
lemma (in TC2) wt_Invoke:
"\<lbrakk> size es = size Ts'; P \<turnstile> C sees M: Ts\<rightarrow>T = m in D; P \<turnstile> Ts' [\<le>] Ts \<rbrakk>
\<Longrightarrow> \<turnstile> [Invoke M (size es)],[] [::] [ty\<^sub>i' (rev Ts' @ Class C # ST) E A, ty\<^sub>i' (T#ST) E A]" |
lemma butlast_strict_prefix_eq_butlast:
assumes "length s = length t" and "strict_prefix (butlast s) t"
shows "strict_prefix (butlast s) t \<longleftrightarrow> (butlast s) = (butlast t)" |
lemma shiftl_of_0 [simp]:
\<open>a << 0 = a\<close> |
lemma lsl_trans: "\<nu> \<circ> \<nu> \<le> \<nu>" |
lemma FOREACHoci_rule':
assumes FIN: "finite S"
assumes I0: "I S \<sigma>0"
assumes IP:
"\<And>x it \<sigma>. \<lbrakk> c \<sigma>; x\<in>it; it\<subseteq>S; I it \<sigma>; \<forall>y\<in>it - {x}. R x y;
\<forall>y\<in>S - it. R y x \<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it-{x}))"
assumes II1: "\<And>\<sigma>. \<lbrakk>I {} \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"
assumes II2: "\<And>it \<sigma>. \<lbrakk> it\<subseteq>S; I it \<sigma>; \<not>c \<sigma>;
\<forall>x\<in>it. \<forall>y\<in>S - it. R y x \<rbrakk> \<Longrightarrow> P \<sigma>"
shows "FOREACHoci R I S c f \<sigma>0 \<le> SPEC P" |
lemma not_Ici_le_Icc[simp]: "\<not> {l\<le>\<^sub>a..} \<subseteq> {l'\<le>\<^sub>a..\<le>\<^sub>ah'}" |
lemma braun_ssa_CFG_SSA_Transformed:
"CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry defs uses defs' uses' phis' var" |
lemma hcomplex_diff_eq_eq [simp]: "x - y = z \<longleftrightarrow> x = z + y"
for x y z :: hcomplex |
lemma get_curr_win_low_equal: "low_equal s1 s2 \<Longrightarrow>
(fst (fst (get_curr_win () s1))) = (fst (fst (get_curr_win () s2)))" |
lemma LFP_lowerbound:
assumes "x \<in> carrier L" "f x \<sqsubseteq> x"
shows "LFP f \<sqsubseteq> x" |
lemma None_eq_join_option: "None = join_option x \<longleftrightarrow> x = None \<or> x = Some None" |
lemma subseq_append_le_same_iff: "subseq (xs @ ys) ys \<longleftrightarrow> xs = []" |
lemma givenBy_o_subset: "givenBy (w o v) <= givenBy v" |
lemma finite_subcls1: "finite subcls1" |
lemma compact_sequence_with_limit:
fixes f :: "nat \<Rightarrow> 'a::heine_borel"
shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> compact (insert l (range f))" |
lemma run_parametric [transfer_rule]:
"(rel_dds A B ===> A ===> rel_prod B (rel_dds A B)) run run" |
lemma lossless_parallel_oracle [simp]:
"lossless_spmf (parallel_oracle s12 xy) \<longleftrightarrow>
(\<forall>x. xy = Inl x \<longrightarrow> lossless_spmf (left (fst s12) x)) \<and>
(\<forall>y. xy = Inr y \<longrightarrow> lossless_spmf (right (snd s12) y))" |
lemma per3_preserves_bet2_aux:
assumes "Col PO A C" and
"A \<noteq> C'" and
"Bet A B' C'" and
"PO \<noteq> A" and
"PO \<noteq> B'" and
"PO \<noteq> C'" and
"Per PO B' B" and
"Per PO C' C" and
"Col A B C" and
"Col PO A C'"
shows "Bet A B C" |
lemma lookup_eq_Some_iff:
assumes invar: "invar_trie ((Trie vo kvs) :: ('key, 'val) trie)"
shows "lookup_trie (Trie vo kvs) ks = Some v \<longleftrightarrow>
(ks = [] \<and> vo = Some v) \<or>
(\<exists>k t ks'. ks = k # ks' \<and>
(k, t) \<in> set kvs \<and> lookup_trie t ks' = Some v)" |
lemma cIsInvar_paperIDs_geq_subPH: "cIsInvar paperIDs_geq_subPH" |
lemma watch_label: "label_eq l (watch e t) = (fst (shd t) = String.implode l)" |
lemma (in mut_m) strong_tricolour[intro]:
notes fun_upd_apply[simp]
shows
"\<lbrace> mark_object_invL
\<^bold>\<and> mut_get_roots.mark_object_invL m
\<^bold>\<and> mut_store_del.mark_object_invL m
\<^bold>\<and> mut_store_ins.mark_object_invL m
\<^bold>\<and> LSTP (fA_rel_inv \<^bold>\<and> fM_rel_inv \<^bold>\<and> handshake_phase_inv \<^bold>\<and> mutators_phase_inv \<^bold>\<and> strong_tricolour_inv \<^bold>\<and> sys_phase_inv \<^bold>\<and> valid_refs_inv) \<rbrace>
mutator m
\<lbrace> LSTP strong_tricolour_inv \<rbrace>" |
lemma leadsTo_Un_distrib:
"F \<in> (A \<union> B) leadsTo C = (F \<in> A leadsTo C & F \<in> B leadsTo C)" |
lemma field_differentiable_diff_const [simp,derivative_intros]:
"(-)c field_differentiable F" |
lemma new_tv_Fun2[simp]:
"new_tv n (t1 =-> t2) = (new_tv n t1 \<and> new_tv n t2)" |
lemma ln_sum_bigo_ln: "(\<lambda>x::real. ln (x + c)) \<in> O(ln)" |
lemma subcls_widen_methd [rule_format (no_asm)]:
"[|G\<turnstile>T'\<preceq>C T; wf_prog wf_mb G|] ==>
\<forall>D rT b. method (G,T) sig = Some (D,rT ,b) -->
(\<exists>D' rT' b'. method (G,T') sig = Some (D',rT',b') \<and> G\<turnstile>D'\<preceq>C D \<and> G\<turnstile>rT'\<preceq>rT)" |
lemma WT_converter_parallel_converter2 [WT_intro]:
assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv1 \<surd>"
and "\<I>1',\<I>2' \<turnstile>\<^sub>C conv2 \<surd>"
shows "\<I>1 \<oplus>\<^sub>\<I> \<I>1',\<I>2 \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C parallel_converter2 conv1 conv2 \<surd>" |
lemma vpsubsetE:
assumes "A \<subset>\<^sub>\<circ> B"
obtains x where "A \<subseteq>\<^sub>\<circ> B" and "x \<notin>\<^sub>\<circ> A" and "x \<in>\<^sub>\<circ> B" |
lemma pp_sup_p [simp]:
"--(x \<squnion> -x) = top" |
lemma lexord_cancel_right: "(u \<cdot> z, v \<cdot> w) \<in> lexord r \<Longrightarrow> \<not> u \<bowtie> v \<Longrightarrow> (u,v) \<in> lexord r" |
lemma irred_terms_and_reduced_subst:
assumes "f = (\<lambda>t. (trm_rep t S))"
assumes "\<eta> = (map_subst f \<sigma>)"
assumes "all_trms_irreducible (subst_set E \<sigma>) f"
assumes "I = int_clset S"
assumes "equivalent_on \<sigma> \<eta> (vars_of_cl (cl_ecl C)) I"
assumes "lower_on \<eta> \<sigma> (vars_of_cl (cl_ecl C))"
assumes "E = (trms_ecl C)"
assumes "\<forall>x \<in> S. \<forall>y. (y \<in> trms_ecl x \<longrightarrow> dom_trm y (cl_ecl x))"
assumes "C \<in> S"
assumes "fo_interpretation I"
shows "all_trms_irreducible (subst_set E \<eta> ) f" |
theorem has_small_limits:
assumes "small (UNIV :: 'i set)"
shows "has_limits (undefined :: 'i)" |
lemma eval_psi_ineq_aux_mono:
assumes "psi n = x" "psi m = x" "psi n \<le> 3 / 2 * ln 2 * n" "n \<le> m"
shows "psi m \<le> 3 / 2 * ln 2 * m" |
lemma transpose_map_map:
"transpose (map (map f) xs) = map (map f) (transpose xs)" |
lemma items_3_finite[intro!, simp]: "finite (items_3 A p s)" |
lemma inj_on_lift:
assumes P: "part I0 P" and Q: "part I0 Q" and PQ: "finer P Q"
and F: "inj_on F P" and FP: "part J0 (F ` P)" and emp: "{} \<notin> F ` P"
shows "inj_on (lift P F) Q" |
lemma in_graphD: "(k, v) \<in> graph m \<Longrightarrow> m k = Some v" |
lemma ani_star_induct:
"ani(y) \<le> ani(x * y) \<Longrightarrow> ani(y) \<le> ani(x\<^sup>\<star> * y)" |
lemma basis_subset_eq:
assumes "basis B\<^sub>1"
assumes "basis B\<^sub>2"
assumes "B\<^sub>1 \<subseteq> B\<^sub>2"
shows "B\<^sub>1 = B\<^sub>2" |
lemma cmod_square:
shows "(cmod z)\<^sup>2 = Re (z * cnj z)" |
lemma sum_symmetric:
assumes "y = y\<^sup>T"
shows "sum (x\<^sup>T \<sqinter> y) = sum (x \<sqinter> y)" |
lemma adopt_node_removes_first_child:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h'"
assumes "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r node # children"
shows "h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children" |
lemma takeWhile_not_last:
"distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs" |
lemma cat_singleton_qm_fst_def[simp]:
"(\<Prod>\<^sub>Ci\<in>\<^sub>\<circ>set {0}. (i = 0 ? \<AA> : \<BB>)) = (\<Prod>\<^sub>Ci\<in>\<^sub>\<circ>set {0}. \<AA>)" |
lemma symKey_neq_priEK: "K \<in> symKeys \<Longrightarrow> K \<noteq> priEK A" |
lemma Group_RSpan_single :
assumes "m \<in> M"
shows "Group (RSpan [m])" |
lemma effect_returnI [effect_intros]:
"h = h' \<Longrightarrow> effect (return x) h h' x" |
lemma sums_logderiv_zeta:
assumes "Re s > 1"
shows "(\<lambda>p. if prime p then of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums
-(deriv zeta s / zeta s)" (is "?f sums _") |
lemma eventually_diff_zero_imp_eq:
fixes f g :: "real \<Rightarrow> real"
assumes "eventually (\<lambda>x. f x - g x = 0) at_top"
shows "eventually (\<lambda>x. f x = g x) at_top" |
lemma hcomp_in_hhomE [elim]:
assumes "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow> c\<guillemotright>"
and "\<lbrakk> arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>; src \<mu> = a; trg \<nu> = c \<rbrakk> \<Longrightarrow> T"
shows T |
lemma "(i::int) mod 1 = 0" |
lemma subset_step:
fixes p:: "real poly"
fixes qs1 qs2 :: "real poly list"
fixes signs1 signs2 :: "rat list list"
assumes csa1: "set (characterize_consistent_signs_at_roots_copr p qs1) \<subseteq> set (signs1)"
assumes csa2: "set (characterize_consistent_signs_at_roots_copr p qs2) \<subseteq> set (signs2)"
shows "set (characterize_consistent_signs_at_roots_copr p
(qs1 @ qs2))
\<subseteq> set (signs_smash signs1 signs2)" |
lemma ofsm_table_subset:
assumes "i \<le> j"
shows "ofsm_table M f j q \<subseteq> ofsm_table M f i q" |
lemma generaliseRefl': "PROP Pure.prop (PROP P \<Longrightarrow> PROP P)" |
lemma ccApprox_ttree_restr[simp]: "ccApprox (ttree_restr S t) = cc_restr S (ccApprox t)" |
lemma reach_window_run_tj: "reach_window args t0 sub rho (i, ti, si, j, tj, sj) \<Longrightarrow>
reaches_on (w_run_t args) t0 (map fst rho) tj" |
lemma xfer_waits:
assumes "sim_rel s cs"
shows "is_WAIT (tts cs t) \<longleftrightarrow> is_WAIT (tts s t)" |
lemma l8_14_2_1b_bis:
assumes "A B Perp C D" and
"Col X A B" and
"Col X C D"
shows "X PerpAt A B C D" |
lemma lm068:
assumes "runiq (f^-1)" "A \<inter> Range f = {}"
shows "converse ` { f \<union> {(x, y)} | y . y \<in> A } \<subseteq> runiqs" |
lemma lift_nth: "i<length xs \<Longrightarrow> map (lift Q) xs ! i = lift Q (xs! i)" |
lemma (in sub_rel_of) sp_from_old_verts_imp_sp_in_old :
assumes "extends g e g'"
assumes "v1 \<in> Graph.vertices g"
assumes "v2 \<in> Graph.vertices g"
assumes "subpath g' v1 es v2 subs"
shows "subpath g v1 es v2 subs" |
lemma array_shrink_get [simp]:
"\<lbrakk> i < s; s \<le> array_length a \<rbrakk> \<Longrightarrow> array_get (array_shrink a s) i = array_get a i" |
lemma list_nth_append0:
assumes "i < length x"
shows "x ! i = (x @ z) ! i" |
lemma xml1many2elements_gen_mono [partial_function_mono]:
fixes p1 :: "xml \<Rightarrow> ('b \<Rightarrow> (string +\<^sub>\<bottom> 'c)) \<Rightarrow> string +\<^sub>\<bottom> 'd"
assumes p1: "\<And>y. mono_sum_bot (p1 y)"
"\<And>y. mono_sum_bot (p3 y)"
"\<And>y. mono_sum_bot (p4 y)"
shows "mono_sum_bot (\<lambda>g. xml1many2elements_gen t (\<lambda>y. p1 y g) p2 (\<lambda>y. p3 y g) (\<lambda>y. p4 y g) f x)" |
lemma ass_function_0':
assumes r: "ass_function ass"
shows "(ass x div x = 0) = (x=0)" |
lemma C_eq_until_separated:
"DenyAll\<in>set(policy2list p) \<Longrightarrow> all_in_list(policy2list p) l \<Longrightarrow> allNetsDistinct(policy2list p) \<Longrightarrow>
Cp (list2FWpolicy (separate (sort (removeShadowRules2 (remdups (rm_MT_rules Cp
(insertDeny (removeShadowRules1 (policy2list p)))))) l))) =
Cp p" |
lemma norm_triangle_mono:
"norm a \<le> r \<Longrightarrow> norm b \<le> s \<Longrightarrow> norm (a + b) \<le> r + s" |
lemma ivl_limpt_diff:
assumes "finite s" "a < b" "(x::real) \<in> {a..b} - s"
shows "x islimpt {a..b} - s" |
lemma post_assumption_above_one:
"q \<in> assumption \<Longrightarrow> post 1 \<le> post (q ^ o)" |
lemma step_4_push_small_size_ok_3_aux: "\<lbrakk>
4 \<le> remaining_steps (States dir big small);
size_new small + remaining_steps (States dir big small) + 2 \<le> 3 * size_new big
\<rbrakk> \<Longrightarrow> Suc (size_new small) + (remaining_steps (States dir big small) - 4) + 2 \<le> 3 * size_new big" |
lemma sublens'_prop1:
assumes "vwb_lens X" "X \<subseteq>\<^sub>L' Y"
shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> s\<^sub>1 (get\<^bsub>Y\<^esub> s\<^sub>2)) s\<^sub>3 = put\<^bsub>Y\<^esub> s\<^sub>1 (get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> s\<^sub>2 s\<^sub>3))" |
lemma monadic_nfoldli_simp[simp]:
"monadic_nfoldli [] c f s = RETURN s"
"monadic_nfoldli (x#ls) c f s = do {
b\<leftarrow>c s;
if b then f x s \<bind> monadic_nfoldli ls c f else RETURN s
}" |