UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
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 }"

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> \" 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 "\1,\2 \<turnstile>\<^sub>C conv1 \<surd>" and "\1',\2' \<turnstile>\<^sub>C conv2 \<surd>" shows "\1 \<oplus>\<^sub>\ \1',\2 \<oplus>\<^sub>\ \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 }"