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

Statement: stringlengths 7 24.3k
lemma Suc_length_remove1: "x \<in> set xs \<Longrightarrow> Suc (length (remove1 x xs)) = length xs"
lemma fold_conv_fold_keys: "RBT_Set2.fold f rbt b = List.fold f (RBT_Set2.keys rbt) b"
lemma D\<^sub>i: "\<^bold>\<forall>x.\<^bold>\<exists>j. ID j \<^bold>\<and> x\<cdot>j \<cong> x"
lemma trans_S_point: "\<And> x y z. (x, y) \<in> S \<Longrightarrow> (y, z) \<in> S \<Longrightarrow> (x, z) \<in> S"
lemma (in finite_boolean_algebra) UNIV_card: "card (UNIV::'a set) = card (Pow \<J>)"
lemma neg_Idef: "\<forall>A. \<^bold>\<not>A \<^bold>\<approx> \<I>(\<^bold>\<midarrow>A) \<^bold>\<or> Q"
lemma bits_div_by_0 [simp]: \<open>a div 0 = 0\<close>
lemma while_sup_one_left_unfold: "1 \<le> x \<Longrightarrow> x * (x \<star> y) = x \<star> y"
lemma proj4_eq_flag: "\<And>dip rt. dip\<in>kD(rt) \<Longrightarrow> \<pi>\<^sub>4(the (rt dip)) = the (flag rt dip)"
lemma L4_2c: assumes run: "ipath gba.E \<sigma>" and "\<mu> R\<^sub>r \<eta> \<in> old (\<sigma> 0)" shows "\<forall>i. \<eta> \<in> old (\<sigma> i) \<or> (\<exists>j<i. \<mu> \<in> old (\<sigma> j))"
lemma filtermap_eq_append: assumes "filtermap pred func tr = al1 @ al2" shows "\<exists> tr1 tr2. tr = tr1 @ tr2 \<and> filtermap pred func tr1 = al1 \<and> filtermap pred func tr2 = al2"
lemma lbisimulation_this: "lbisimulation R A B"
lemma (in kat) "test p \<Longrightarrow> test q \<Longrightarrow> p \<cdot> x \<cdot> y \<le> x \<cdot> y \<cdot> q \<Longrightarrow>(\<exists>r. test r \<and> p \<cdot> x \<le> x \<cdot> r \<and> r \<cdot> y \<le> y \<cdot> q)"
lemma abs_conv_abscissa_prod_le: assumes "\<And>x. x \<in> A \<Longrightarrow> abs_conv_abscissa (f x :: 'a :: dirichlet_series fds) \<le> d" shows "abs_conv_abscissa (prod f A) \<le> d"
lemma sleq_refl: "sleq x x"
lemma [simp]: "pc\<^sub>1 + size(compE\<^sub>2 e\<^sub>1) \<le> pc\<^sub>2 \<Longrightarrow> pcs(compxE\<^sub>2 e\<^sub>1 pc\<^sub>1 d\<^sub>1) \<inter> pcs(compxE\<^sub>2 e\<^sub>2 pc\<^sub>2 d\<^sub>2) = {}"
lemma analz_Un: "analz G \<union> analz H \<subseteq> analz (G \<union> H)"
lemma "\<not> p2 n"
lemma ideal_generated_pair_exists_UNIV: shows "(ideal_generated {a,b} = ideal_generated {1}) = (\<exists>p q. pa+qb = 1)" (is "?lhs = ?rhs")
lemma (in Order) fTo_conditional_Un_Chain_mem2:" \<lbrakk>C \<in> carrier (fTo D); Chain (Iod (fTo D) {S \<in> carrier fTo D. C \<subseteq> S}) Ca; Ca = {}\<rbrakk> \<Longrightarrow> C \<in> upper_bounds (Iod (fTo D) {S \<in> carrier (fTo D). C \<subseteq> S}) Ca"
lemma simpleDefs_phiDefs_var_disjoint: assumes "v \<in> phiDefs g n" "n \<in> set (\<alpha>n g)" shows "var g v \<notin> oldDefs g n"
lemma subst_psubst: "\<lbrakk> closed_env \<rho>; FV v = {} \<rbrakk> \<Longrightarrow> subst x v (psubst ((x, EVar x) # \<rho>) e) = psubst ((x, v) # \<rho>) e"
lemma sseq_imp_seq: shows "sseq t u \<Longrightarrow> seq t u"
lemma E_nonneg_fun: fixes f::"'a\<Rightarrow>real" shows "(\<forall>x\<in>set_pmf X. 0\<le>f x) \<Longrightarrow> 0 \<le> E (map_pmf f X)"
lemma (in Group) Qg_i:"G \<triangleright> N \<Longrightarrow> \<forall>x \<in> set_rcs G N. c_top G N (c_iop G N x) x = N"
lemma \<tau>Exec_1_dt_preserves_correct_state: assumes wf: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" and exec: "\<tau>Exec_1_dt P t \<sigma> \<sigma>'" shows "\<Phi> \|- t:\<sigma> [ok] \<Longrightarrow> \<Phi> \|- t:\<sigma>' [ok]"
lemma CK_nf_pos_neg_disjoint: assumes "CK_nf_pos f" assumes "CK_nf_neg g" shows "f \<noteq> g"
lemma uncurry0_add_app_tag: "uncurry0 (RETURN c) = uncurry0 (RETURN$c)"
lemma totatives_subset: "totatives n \<subseteq> {0<..n}"
lemma SpecAnnoNoAbrupt: "\<lbrakk>P \<subseteq> {s. \<exists> Z. s\<in>P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> t \<in> Q)}; \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) (c Z) (Q' Z),{}; \<forall>Z. c Z = c undefined \<rbrakk> \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (specAnno P' c Q' (\<lambda>s. {})) Q,A"
lemma keys_of_delete [rewrite]: "keys_of (delete_map x M) = keys_of M - {x}"
lemma "(Node (3::int) Leaf Leaf) \<otimes> (Node (1::int) Leaf Leaf) = (Node 4 Leaf Leaf)"
lemma append_eq_symm: "t2 @ t1 \<sim> t1 @ t2"
lemma Diff_triv_mset: "M \<inter># N = {#} \<Longrightarrow> M - N = M"
lemma Lp_infinity_zero_space: assumes "p > (0::ennreal)" shows "zero_space\<^sub>N (\<LL> p M) = {f \<in> borel_measurable M. AE x in M. f x = 0}"
lemma in_hhomI [intro, simp]: assumes "arr \<mu>" and "src \<mu> = a" and "trg \<mu> = b" shows "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"
lemma analz_insert_Pan [simp]: "analz (insert (Pan A) H) = insert (Pan A) (analz H)"
lemma not_equiv_funI: assumes "\<And>c\<^sub>1 c\<^sub>2 n. c\<^sub>1 > 0 \<Longrightarrow> c\<^sub>2 > 0 \<Longrightarrow> \<exists>m>n. c\<^sub>1 * f m < g m \<or> c\<^sub>2 * g m < f m" shows "\<not> f \<cong> g"
lemma Cod_dom [simp]: assumes "arr f" shows "Cod (dom f) = Dom f"
lemma local_lipschitz_therm_dyn: assumes "0 < (a::real)" shows "local_lipschitz UNIV UNIV (\<lambda>t::real. f a L)"
lemma first_t_fusion [simp]: "last x = first y \<Longrightarrow> first (t_fusion x y) = first x"
lemma map_getOneIp_distinct: assumes distinct: "distinct xs" and disjoint: "(\<forall>x1 \<in> set xs. \<forall>x2 \<in> set xs. x1 \<noteq> x2 \<longrightarrow> wordinterval_to_set x1 \<inter> wordinterval_to_set x2 = {})" and notempty: "\<forall>x \<in> set xs. \<not> wordinterval_empty x" shows "distinct (map getOneIp xs)"
lemma lower_triangular_mult: assumes "lower_triangular_mat X" "lower_triangular_mat Y" shows "lower_triangular_mat (X ** Y)"
lemma min_plus_max: shows "(min a b) + (max a b) = a + b"
lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x"
lemma DomainHierarchyNG_offending_set: "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar = DomainHierarchyNG_offending_set"
lemma impl_of_delete [code abstract]: "impl_of (delete k al) = AList.delete_aux k (impl_of al)"
lemma (in ShareRep_impl) ShareRep_modifies: shows "\<forall>\<sigma>. \<Gamma>\<turnstile>{\<sigma>} PROC ShareRep (\<acute>nodeslist, \<acute>p) {t. t may_only_modify_globals \<sigma> in [rep]}"
lemma hermitean_det_zero_trace_zero: assumes "mat_det A = 0" and "mat_trace A = (0::complex)" and "hermitean A" shows "A = mat_zero"
lemma delete_index_ge_length: "n \<ge> length xs \<Longrightarrow> delete_index n xs = xs"
lemma ennreal_mult_eq_top_iff: fixes a b :: ennreal shows "a * b = top \<longleftrightarrow> (a = top \<and> b \<noteq> 0) \<or> (b = top \<and> a \<noteq> 0)"
lemma pair_subst_ident[intro]: "(fv t \<union> fv t') \<inter> subst_domain \<theta> = {} \<Longrightarrow> (t,t') \<cdot>\<^sub>p \<theta> = (t,t')"
lemma foldl_map_append_is_some_if: assumes "b x = Some y \<or> (\<exists>m \<in> set ms. m x = Some y)" and "\<forall>m' \<in> set ms. m' x = Some y \<or> m' x = None" shows "foldl (++) b ms x = Some y"
lemma triangle_set_graph_edge_ss_bound: fixes G :: "ugraph" and Gnew :: "ugraph" assumes "uwellformed G" "finite (uverts G)" "uedges Gnew \<subseteq> uedges G" "uverts Gnew = uverts G" shows "card (triangle_set G) \<ge> card (triangle_set Gnew)"
lemma n_omega_mult: "n(x\<^sup>\<omega> * y) = n(x\<^sup>\<omega>)"
lemma assign_local_skip: "\<lbrace>\<lambda>\<sigma>. exec_stop \<sigma> \<and> P \<sigma> \<rbrace> upd :==\<^sub>L rhs \<lbrace>\<lambda>r \<sigma>. exec_stop \<sigma> \<and> P \<sigma> \<rbrace>"
lemma mkeps_flat: assumes "nullable(r)" shows "flat (mkeps r) = []"
lemma finite_gpv_lift_spmf [simp]: "finite_gpv (lift_spmf p)"
lemma assumes "lockstep_backward_simulation step1 step2 match" shows "plus_backward_simulation step1 step2 match"
lemma ub_exp_nonneg: "real_of_float (ub_exp prec x) \<ge> 0"
lemma in_trancl_closure_iff_in_trancl_fun: "(a,b) \<in> (set TI)\<^sup>+ \<longleftrightarrow> in_trancl TI a b" (is "?A TI a b \<longleftrightarrow> ?B TI a b")
lemma matchCases[consumes 1, case_names cMatch]: fixes a :: name and b :: name and P :: pi and Rs :: residual and F :: "name \<Rightarrow> name \<Rightarrow> bool" assumes "[a\<frown>b]P \<longmapsto> Rs" and "\<lbrakk>P \<longmapsto> Rs; a = b\<rbrakk> \<Longrightarrow> F a a" shows "F a b"
lemma ground_aux_simps[simp]: "ground_aux zer S = True" "ground_aux (Var k) S = (if atom k \<in> S then True else False)" "ground_aux (suc t) S = (ground_aux t S)" "ground_aux (pls t u) S = (ground_aux t S \<and> ground_aux u S)" "ground_aux (tms t u) S = (ground_aux t S \<and> ground_aux u S)"
lemma lambda_predicates_3_2[axiom]: "[[(\<^bold>\<lambda>\<^sup>2 (\<lambda> x y . \<lparr>F, x\<^sup>P, y\<^sup>P\<rparr>)) \<^bold>= F]]"
lemma Un_set_offending_flows_bound_minus_subseteq: assumes wfG: "wf_graph \<lparr> nodes = V, edges = E \<rparr>" and Foffending: "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E\<rparr> nP) \<subseteq> X" shows "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E - E'\<rparr> nP) \<subseteq> X - E'"
lemma add_root[simp]: "z \<cdot> w \<in> z* \<longleftrightarrow> w \<in> z*"
lemma ltln_Release_alterdef: "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> \<longleftrightarrow> w \<Turnstile>\<^sub>n (G\<^sub>n \<psi>) or\<^sub>n (\<psi> U\<^sub>n (\<phi> and\<^sub>n \<psi>))"
theorem encode_problem_serializable_sound: assumes "is_valid_problem_strips \<Pi>" and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "is_parallel_solution_for_problem \<Pi> (\<Phi>\<inverse> \<Pi> \<A> t)" and "\<forall>k < length (\<Phi>\<inverse> \<Pi> \<A> t). are_all_operators_non_interfering ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"
lemma sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_ii_b: assumes "is_valid_problem_sas_plus \<Psi>" shows "finite (bounded_solution_set_strips' (\<phi> \<Psi>) k)"
lemma (in wide_subsemicategory) wide_subsemicategory_axioms'[smc_cs_intros]: assumes "\<alpha>' = \<alpha>" and "\<BB>' = \<BB>" shows "\<BB>' \<subseteq>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>'\<^esub> \<CC>"
lemma covib:"ov^-1 O b \<subseteq> b \<union> m \<union> ov \<union> f^-1 \<union> d^-1"
lemma nemp_exp_pos[intro]: "w \<noteq> \<epsilon> \<Longrightarrow> r\<^sup>@k = w \<Longrightarrow> k \<noteq> 0"
lemma PO_m2_inv2_trans [iff]: "{m2_inv1_auth} trans m2 {> m2_inv1_auth}"
lemma equal_pow_resI''': assumes "n > 0" assumes "a \<in> nonzero Q\<^sub>p" assumes "b \<in> nonzero Q\<^sub>p" assumes "c \<in> nonzero Q\<^sub>p" assumes "pow_res n (c \<otimes> a) = pow_res n (c \<otimes> b)" shows "pow_res n a = pow_res n b"
lemma smaller_compatible_core: assumes "y \<succeq> x" shows "x ## \|y\|"
lemma NonUniformExecutionBase: fixes cfg assumes Cfg: "initial cfg" "nonUniform cfg" shows "execution trans sends start [cfg] [] \<and> nonUniform (last [cfg]) \<and> (\<exists> cfgList' msgList'. nonUniform (last cfgList') \<and> prefixList [cfg] cfgList' \<and> prefixList [] msgList' \<and> (execution trans sends start cfgList' msgList') \<and> (\<exists> msg'. execution.minimalEnabled [cfg] [] msg' \<and> msg' \<in> set msgList'))"
theorem hoaret_complete: "\<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> \<turnstile>\<^sub>t {P}c{Q}"
lemma eq_scheduler_refl[intro]: "eq_scheduler sc sc"
lemma scene_space_compats [simp]: "pairwise (##\<^sub>S) (set Vars)"
lemma nonneg_Reals_cases: assumes "x \<in> \<real>\<^sub>\<ge>\<^sub>0" obtains r where "x = of_real r" "r \<ge> 0"
lemma Card_gcard [iff]: "Card (gcard X)"
lemma wf_greater_bounded[simp, intro!]: "wf (greater_bounded N)"
lemma div_reals [simp]: assumes "is_real a" and "is_real b" shows "is_real (a / b)"
lemma phi_coset_eq_self: assumes "a \<in> G // K" shows "\<phi> a \<cdot>\| K = a"
lemma zipRT_identity: "Done\<cdot>ID \<diamondop> r = r"
lemma rel_spmf_simps: "rel_spmf R p q \<longleftrightarrow> (\<exists>pq. (\<forall>(x, y)\<in>set_spmf pq. R x y) \<and> map_spmf fst pq = p \<and> map_spmf snd pq = q)"
lemma surj_exception_of_option [simp]: "surj exception_of_option"
lemma square_subset: "[\| square R S T U; T \<le> T' \|] ==> square R S T' U"
lemma N2_enabled_at_b: "\<turnstile> pc2 = #b \<longrightarrow> Enabled (<N2>_(x,y,sem,pc1,pc2))"
lemma inext_mono2_infin_fin: " \<lbrakk> n \<in> I; n \<noteq> Max I \<or> infinite I \<rbrakk> \<Longrightarrow> n < inext n I"
lemma psubst_ode: assumes good_interp:"is_interp I" shows "ODE_sem I ODE = ODE_sem (PFadjoint I \<sigma>) ODE"
lemma InvariantImpliedLiteralsAndFormulaFalseThenFormulaAndDecisionsAreNotSatisfiable: fixes M :: LiteralTrail and F :: Formula assumes "InvariantImpliedLiterals F M" and "formulaFalse F (elements M)" shows "\<not> satisfiable (F @ val2form (decisions M))"
lemma nth_imp_genPrefix: "length xs <= length ys \<Longrightarrow> (\<forall>i. i < length xs \<longrightarrow> (xs ! i, ys ! i) \<in> r) \<Longrightarrow> (xs, ys) \<in> genPrefix r"
lemma sim_refl: "E,E: s \<sqsubseteq>\<^sub>id s"
lemma a_redu_NotL_elim: assumes a: "NotL <a>.M x \<longrightarrow>\<^sub>a R" shows "\<exists>M'. R = NotL <a>.M' x \<and> M\<longrightarrow>\<^sub>aM'"
lemma SD_implies_BSD : "(SD \<V> Tr\<^bsub>ES\<^esub>) \<Longrightarrow> BSD \<V> Tr\<^bsub>ES\<^esub> "
lemma drinks_0_rejects: "\<not> (fst a = STR ''select'' \<and> length (snd a) = 1) \<Longrightarrow> (possible_steps drinks 0 r (fst a) (snd a)) = {\|\|}"
lemma eval_subst: "eval \<phi> = Yes \<Longrightarrow> eval (subst \<phi> m) = Yes" "eval \<phi> = No \<Longrightarrow> eval (subst \<phi> m) = No"
lemma lNact_notActive: fixes c t n k assumes "k\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>" and "k<n" shows "\<not>\<parallel>c\<parallel>\<^bsub>t k\<^esub>"

lemma Suc_length_remove1: "x \<in> set xs \<Longrightarrow> Suc (length (remove1 x xs)) = length xs"

lemma fold_conv_fold_keys: "RBT_Set2.fold f rbt b = List.fold f (RBT_Set2.keys rbt) b"

lemma D\<^sub>i: "\<^bold>\<forall>x.\<^bold>\<exists>j. ID j \<^bold>\<and> x\<cdot>j \<cong> x"

lemma trans_S_point: "\<And> x y z. (x, y) \<in> S \<Longrightarrow> (y, z) \<in> S \<Longrightarrow> (x, z) \<in> S"

lemma (in finite_boolean_algebra) UNIV_card: "card (UNIV::'a set) = card (Pow \<J>)"

lemma neg_Idef: "\<forall>A. \<^bold>\<not>A \<^bold>\<approx> \<I>(\<^bold>\<midarrow>A) \<^bold>\<or> Q"

lemma bits_div_by_0 [simp]: \<open>a div 0 = 0\<close>

lemma while_sup_one_left_unfold: "1 \<le> x \<Longrightarrow> x * (x \<star> y) = x \<star> y"

lemma proj4_eq_flag: "\<And>dip rt. dip\<in>kD(rt) \<Longrightarrow> \<pi>\<^sub>4(the (rt dip)) = the (flag rt dip)"

lemma L4_2c: assumes run: "ipath gba.E \<sigma>" and "\<mu> R\<^sub>r \<eta> \<in> old (\<sigma> 0)" shows "\<forall>i. \<eta> \<in> old (\<sigma> i) \<or> (\<exists>j<i. \<mu> \<in> old (\<sigma> j))"

lemma filtermap_eq_append: assumes "filtermap pred func tr = al1 @ al2" shows "\<exists> tr1 tr2. tr = tr1 @ tr2 \<and> filtermap pred func tr1 = al1 \<and> filtermap pred func tr2 = al2"

lemma lbisimulation_this: "lbisimulation R A B"

lemma (in kat) "test p \<Longrightarrow> test q \<Longrightarrow> p \<cdot> x \<cdot> y \<le> x \<cdot> y \<cdot> q \<Longrightarrow>(\<exists>r. test r \<and> p \<cdot> x \<le> x \<cdot> r \<and> r \<cdot> y \<le> y \<cdot> q)"

lemma abs_conv_abscissa_prod_le: assumes "\<And>x. x \<in> A \<Longrightarrow> abs_conv_abscissa (f x :: 'a :: dirichlet_series fds) \<le> d" shows "abs_conv_abscissa (prod f A) \<le> d"

lemma sleq_refl: "sleq x x"

lemma [simp]: "pc\<^sub>1 + size(compE\<^sub>2 e\<^sub>1) \<le> pc\<^sub>2 \<Longrightarrow> pcs(compxE\<^sub>2 e\<^sub>1 pc\<^sub>1 d\<^sub>1) \<inter> pcs(compxE\<^sub>2 e\<^sub>2 pc\<^sub>2 d\<^sub>2) = {}"

lemma analz_Un: "analz G \<union> analz H \<subseteq> analz (G \<union> H)"

lemma "\<not> p2 n"

lemma ideal_generated_pair_exists_UNIV: shows "(ideal_generated {a,b} = ideal_generated {1}) = (\<exists>p q. p*a+q*b = 1)" (is "?lhs = ?rhs")

lemma (in Order) fTo_conditional_Un_Chain_mem2:" \<lbrakk>C \<in> carrier (fTo D); Chain (Iod (fTo D) {S \<in> carrier fTo D. C \<subseteq> S}) Ca; Ca = {}\<rbrakk> \<Longrightarrow> C \<in> upper_bounds (Iod (fTo D) {S \<in> carrier (fTo D). C \<subseteq> S}) Ca"

lemma simpleDefs_phiDefs_var_disjoint: assumes "v \<in> phiDefs g n" "n \<in> set (\<alpha>n g)" shows "var g v \<notin> oldDefs g n"

lemma subst_psubst: "\<lbrakk> closed_env \<rho>; FV v = {} \<rbrakk> \<Longrightarrow> subst x v (psubst ((x, EVar x) # \<rho>) e) = psubst ((x, v) # \<rho>) e"

lemma sseq_imp_seq: shows "sseq t u \<Longrightarrow> seq t u"

lemma E_nonneg_fun: fixes f::"'a\<Rightarrow>real" shows "(\<forall>x\<in>set_pmf X. 0\<le>f x) \<Longrightarrow> 0 \<le> E (map_pmf f X)"

lemma (in Group) Qg_i:"G \<triangleright> N \<Longrightarrow> \<forall>x \<in> set_rcs G N. c_top G N (c_iop G N x) x = N"

lemma \<tau>Exec_1_dt_preserves_correct_state: assumes wf: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" and exec: "\<tau>Exec_1_dt P t \<sigma> \<sigma>'" shows "\<Phi> |- t:\<sigma> [ok] \<Longrightarrow> \<Phi> |- t:\<sigma>' [ok]"

lemma CK_nf_pos_neg_disjoint: assumes "CK_nf_pos f" assumes "CK_nf_neg g" shows "f \<noteq> g"

lemma uncurry0_add_app_tag: "uncurry0 (RETURN c) = uncurry0 (RETURN$c)"

lemma totatives_subset: "totatives n \<subseteq> {0<..n}"

lemma SpecAnnoNoAbrupt: "\<lbrakk>P \<subseteq> {s. \<exists> Z. s\<in>P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> t \<in> Q)}; \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) (c Z) (Q' Z),{}; \<forall>Z. c Z = c undefined \<rbrakk> \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (specAnno P' c Q' (\<lambda>s. {})) Q,A"

lemma keys_of_delete [rewrite]: "keys_of (delete_map x M) = keys_of M - {x}"

lemma "(Node (3::int) Leaf Leaf) \<otimes> (Node (1::int) Leaf Leaf) = (Node 4 Leaf Leaf)"

lemma append_eq_symm: "t2 @ t1 \<sim> t1 @ t2"

lemma Diff_triv_mset: "M \<inter># N = {#} \<Longrightarrow> M - N = M"

lemma Lp_infinity_zero_space: assumes "p > (0::ennreal)" shows "zero_space\<^sub>N (\<LL> p M) = {f \<in> borel_measurable M. AE x in M. f x = 0}"

lemma in_hhomI [intro, simp]: assumes "arr \<mu>" and "src \<mu> = a" and "trg \<mu> = b" shows "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"

lemma analz_insert_Pan [simp]: "analz (insert (Pan A) H) = insert (Pan A) (analz H)"

lemma not_equiv_funI: assumes "\<And>c\<^sub>1 c\<^sub>2 n. c\<^sub>1 > 0 \<Longrightarrow> c\<^sub>2 > 0 \<Longrightarrow> \<exists>m>n. c\<^sub>1 * f m < g m \<or> c\<^sub>2 * g m < f m" shows "\<not> f \<cong> g"

lemma Cod_dom [simp]: assumes "arr f" shows "Cod (dom f) = Dom f"

lemma local_lipschitz_therm_dyn: assumes "0 < (a::real)" shows "local_lipschitz UNIV UNIV (\<lambda>t::real. f a L)"

lemma first_t_fusion [simp]: "last x = first y \<Longrightarrow> first (t_fusion x y) = first x"

lemma map_getOneIp_distinct: assumes distinct: "distinct xs" and disjoint: "(\<forall>x1 \<in> set xs. \<forall>x2 \<in> set xs. x1 \<noteq> x2 \<longrightarrow> wordinterval_to_set x1 \<inter> wordinterval_to_set x2 = {})" and notempty: "\<forall>x \<in> set xs. \<not> wordinterval_empty x" shows "distinct (map getOneIp xs)"

lemma lower_triangular_mult: assumes "lower_triangular_mat X" "lower_triangular_mat Y" shows "lower_triangular_mat (X ** Y)"

lemma min_plus_max: shows "(min a b) + (max a b) = a + b"

lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x"

lemma DomainHierarchyNG_offending_set: "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar = DomainHierarchyNG_offending_set"

lemma impl_of_delete [code abstract]: "impl_of (delete k al) = AList.delete_aux k (impl_of al)"

lemma (in ShareRep_impl) ShareRep_modifies: shows "\<forall>\<sigma>. \<Gamma>\<turnstile>{\<sigma>} PROC ShareRep (\<acute>nodeslist, \<acute>p) {t. t may_only_modify_globals \<sigma> in [rep]}"

lemma hermitean_det_zero_trace_zero: assumes "mat_det A = 0" and "mat_trace A = (0::complex)" and "hermitean A" shows "A = mat_zero"

lemma delete_index_ge_length: "n \<ge> length xs \<Longrightarrow> delete_index n xs = xs"

lemma ennreal_mult_eq_top_iff: fixes a b :: ennreal shows "a * b = top \<longleftrightarrow> (a = top \<and> b \<noteq> 0) \<or> (b = top \<and> a \<noteq> 0)"

lemma pair_subst_ident[intro]: "(fv t \<union> fv t') \<inter> subst_domain \<theta> = {} \<Longrightarrow> (t,t') \<cdot>\<^sub>p \<theta> = (t,t')"

lemma foldl_map_append_is_some_if: assumes "b x = Some y \<or> (\<exists>m \<in> set ms. m x = Some y)" and "\<forall>m' \<in> set ms. m' x = Some y \<or> m' x = None" shows "foldl (++) b ms x = Some y"

lemma triangle_set_graph_edge_ss_bound: fixes G :: "ugraph" and Gnew :: "ugraph" assumes "uwellformed G" "finite (uverts G)" "uedges Gnew \<subseteq> uedges G" "uverts Gnew = uverts G" shows "card (triangle_set G) \<ge> card (triangle_set Gnew)"

lemma n_omega_mult: "n(x\<^sup>\<omega> * y) = n(x\<^sup>\<omega>)"

lemma assign_local_skip: "\<lbrace>\<lambda>\<sigma>. exec_stop \<sigma> \<and> P \<sigma> \<rbrace> upd :==\<^sub>L rhs \<lbrace>\<lambda>r \<sigma>. exec_stop \<sigma> \<and> P \<sigma> \<rbrace>"

lemma mkeps_flat: assumes "nullable(r)" shows "flat (mkeps r) = []"

lemma finite_gpv_lift_spmf [simp]: "finite_gpv (lift_spmf p)"

lemma assumes "lockstep_backward_simulation step1 step2 match" shows "plus_backward_simulation step1 step2 match"

lemma ub_exp_nonneg: "real_of_float (ub_exp prec x) \<ge> 0"

lemma in_trancl_closure_iff_in_trancl_fun: "(a,b) \<in> (set TI)\<^sup>+ \<longleftrightarrow> in_trancl TI a b" (is "?A TI a b \<longleftrightarrow> ?B TI a b")

lemma matchCases[consumes 1, case_names cMatch]: fixes a :: name and b :: name and P :: pi and Rs :: residual and F :: "name \<Rightarrow> name \<Rightarrow> bool" assumes "[a\<frown>b]P \<longmapsto> Rs" and "\<lbrakk>P \<longmapsto> Rs; a = b\<rbrakk> \<Longrightarrow> F a a" shows "F a b"

lemma ground_aux_simps[simp]: "ground_aux zer S = True" "ground_aux (Var k) S = (if atom k \<in> S then True else False)" "ground_aux (suc t) S = (ground_aux t S)" "ground_aux (pls t u) S = (ground_aux t S \<and> ground_aux u S)" "ground_aux (tms t u) S = (ground_aux t S \<and> ground_aux u S)"

lemma lambda_predicates_3_2[axiom]: "[[(\<^bold>\<lambda>\<^sup>2 (\<lambda> x y . \<lparr>F, x\<^sup>P, y\<^sup>P\<rparr>)) \<^bold>= F]]"

lemma Un_set_offending_flows_bound_minus_subseteq: assumes wfG: "wf_graph \<lparr> nodes = V, edges = E \<rparr>" and Foffending: "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E\<rparr> nP) \<subseteq> X" shows "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E - E'\<rparr> nP) \<subseteq> X - E'"

lemma add_root[simp]: "z \<cdot> w \<in> z* \<longleftrightarrow> w \<in> z*"

lemma ltln_Release_alterdef: "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> \<longleftrightarrow> w \<Turnstile>\<^sub>n (G\<^sub>n \<psi>) or\<^sub>n (\<psi> U\<^sub>n (\<phi> and\<^sub>n \<psi>))"

theorem encode_problem_serializable_sound: assumes "is_valid_problem_strips \<Pi>" and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "is_parallel_solution_for_problem \<Pi> (\<Phi>\<inverse> \<Pi> \<A> t)" and "\<forall>k < length (\<Phi>\<inverse> \<Pi> \<A> t). are_all_operators_non_interfering ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"

lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"

lemma sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_ii_b: assumes "is_valid_problem_sas_plus \<Psi>" shows "finite (bounded_solution_set_strips' (\<phi> \<Psi>) k)"

lemma (in wide_subsemicategory) wide_subsemicategory_axioms'[smc_cs_intros]: assumes "\<alpha>' = \<alpha>" and "\<BB>' = \<BB>" shows "\<BB>' \<subseteq>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>'\<^esub> \<CC>"

lemma covib:"ov^-1 O b \<subseteq> b \<union> m \<union> ov \<union> f^-1 \<union> d^-1"

lemma nemp_exp_pos[intro]: "w \<noteq> \<epsilon> \<Longrightarrow> r\<^sup>@k = w \<Longrightarrow> k \<noteq> 0"

lemma PO_m2_inv2_trans [iff]: "{m2_inv1_auth} trans m2 {> m2_inv1_auth}"

lemma equal_pow_resI''': assumes "n > 0" assumes "a \<in> nonzero Q\<^sub>p" assumes "b \<in> nonzero Q\<^sub>p" assumes "c \<in> nonzero Q\<^sub>p" assumes "pow_res n (c \<otimes> a) = pow_res n (c \<otimes> b)" shows "pow_res n a = pow_res n b"

lemma smaller_compatible_core: assumes "y \<succeq> x" shows "x ## |y|"

lemma NonUniformExecutionBase: fixes cfg assumes Cfg: "initial cfg" "nonUniform cfg" shows "execution trans sends start [cfg] [] \<and> nonUniform (last [cfg]) \<and> (\<exists> cfgList' msgList'. nonUniform (last cfgList') \<and> prefixList [cfg] cfgList' \<and> prefixList [] msgList' \<and> (execution trans sends start cfgList' msgList') \<and> (\<exists> msg'. execution.minimalEnabled [cfg] [] msg' \<and> msg' \<in> set msgList'))"

theorem hoaret_complete: "\<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> \<turnstile>\<^sub>t {P}c{Q}"

lemma eq_scheduler_refl[intro]: "eq_scheduler sc sc"

lemma scene_space_compats [simp]: "pairwise (##\<^sub>S) (set Vars)"

lemma nonneg_Reals_cases: assumes "x \<in> \<real>\<^sub>\<ge>\<^sub>0" obtains r where "x = of_real r" "r \<ge> 0"

lemma Card_gcard [iff]: "Card (gcard X)"

lemma wf_greater_bounded[simp, intro!]: "wf (greater_bounded N)"

lemma div_reals [simp]: assumes "is_real a" and "is_real b" shows "is_real (a / b)"

lemma phi_coset_eq_self: assumes "a \<in> G // K" shows "\<phi> a \<cdot>| K = a"

lemma zipRT_identity: "Done\<cdot>ID \<diamondop> r = r"

lemma rel_spmf_simps: "rel_spmf R p q \<longleftrightarrow> (\<exists>pq. (\<forall>(x, y)\<in>set_spmf pq. R x y) \<and> map_spmf fst pq = p \<and> map_spmf snd pq = q)"

lemma surj_exception_of_option [simp]: "surj exception_of_option"

lemma square_subset: "[| square R S T U; T \<le> T' |] ==> square R S T' U"

lemma N2_enabled_at_b: "\<turnstile> pc2 = #b \<longrightarrow> Enabled (<N2>_(x,y,sem,pc1,pc2))"

lemma inext_mono2_infin_fin: " \<lbrakk> n \<in> I; n \<noteq> Max I \<or> infinite I \<rbrakk> \<Longrightarrow> n < inext n I"

lemma psubst_ode: assumes good_interp:"is_interp I" shows "ODE_sem I ODE = ODE_sem (PFadjoint I \<sigma>) ODE"

lemma InvariantImpliedLiteralsAndFormulaFalseThenFormulaAndDecisionsAreNotSatisfiable: fixes M :: LiteralTrail and F :: Formula assumes "InvariantImpliedLiterals F M" and "formulaFalse F (elements M)" shows "\<not> satisfiable (F @ val2form (decisions M))"

lemma nth_imp_genPrefix: "length xs <= length ys \<Longrightarrow> (\<forall>i. i < length xs \<longrightarrow> (xs ! i, ys ! i) \<in> r) \<Longrightarrow> (xs, ys) \<in> genPrefix r"

lemma sim_refl: "E,E: s \<sqsubseteq>\<^sub>id s"

lemma a_redu_NotL_elim: assumes a: "NotL <a>.M x \<longrightarrow>\<^sub>a R" shows "\<exists>M'. R = NotL <a>.M' x \<and> M\<longrightarrow>\<^sub>aM'"

lemma SD_implies_BSD : "(SD \<V> Tr\<^bsub>ES\<^esub>) \<Longrightarrow> BSD \<V> Tr\<^bsub>ES\<^esub> "

lemma drinks_0_rejects: "\<not> (fst a = STR ''select'' \<and> length (snd a) = 1) \<Longrightarrow> (possible_steps drinks 0 r (fst a) (snd a)) = {||}"

lemma eval_subst: "eval \<phi> = Yes \<Longrightarrow> eval (subst \<phi> m) = Yes" "eval \<phi> = No \<Longrightarrow> eval (subst \<phi> m) = No"

lemma lNact_notActive: fixes c t n k assumes "k\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>" and "k<n" shows "\<not>\<parallel>c\<parallel>\<^bsub>t k\<^esub>"