Datasets:

UDACA

/

Split-Isa-AF-MMA

like 0

lemma point_to_polys_affine_alg_set: assumes "as \<in> carrier (R\<^bsup>n\<^esup>)" shows "affine_alg_set R n (set (point_to_polys as)) = {as}"

lemma inv_renaming_cancel_r_list[simp]: "is_renaming_list rs \<Longrightarrow> rs \<odot>s map inv_renaming rs = replicate (length rs) id_subst"

lemma cexpr_subst_val_aux_eq_cexpr_subst: "cexpr_subst_val_aux x e v = cexpr_subst x (CVal v) e"

lemma echelon_form_JNF_Hermite_of_row_i': fixes A::"int mat" assumes "A \<in> carrier_mat m n" assumes eA: "echelon_form_JNF A" and "i<m" and "1 < m" and "1 < n" (*Required from the mod_type restrictions*) shows "echelon_form_JNF (Hermite_of_row_i A i)"

lemma right_inverse_linear: assumes lf: "linear scale scale f" and gf: "f \<circ> g = id" shows "linear scale scale g"

lemma unit_disc_fix_conjugate_comp_moebius [simp]: assumes "unit_disc_fix M" shows "unit_disc_fix_f (conjugate \<circ> moebius_pt M)"

lemma grid_mono: assumes "j \<le> n" shows "t j \<le> t n"

lemma elem_exists_count_min: "\<exists> i \<in>{..<dim_vec v}. v $ i = x \<Longrightarrow> count_vec v x \<ge> 1"

lemma subst2_simps[simp]: "subst s2 X = Q (LC lx)" "subst s2 Y = Q (LC lx)" "subst s2 (imp X Y) = imp (subst s2 X) (subst s2 Y)"

lemma ordinal_oLog_monoR: "x \<le> y \<Longrightarrow> oLog b x \<le> oLog b y"

lemma compl_structD1: assumes "compl_struct compl" and "dickson_grading d" and "sps \<noteq> []" and "set sps \<subseteq> set ps" shows "dgrad_p_set_le d (fst ` (set (fst (compl gs bs (ps -- sps) sps data)))) (args_to_set (gs, bs, ps))"

lemma "check_eqv (Abs_idx (1, 0)) (FEq_Plus 0 0 1) FFalse"

lemma zero_change_imp_all_preconds_submap: fixes s s' assumes "(vars_change as vs s = [])" "(sat_precond_as s as)" "(ListMem b as)" "(fmrestrict_set vs s = fmrestrict_set vs s')" shows "(fmrestrict_set vs (fst b) \<subseteq>\<^sub>f fmrestrict_set vs s')"

lemma inf_sup_distrib2_1: "((y :: 'a :: distrib_lattice) \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (y \<sqinter> x)"

lemma is_rewritable_alt_spp: assumes "0 \<notin> set bs" shows "is_rewritable bs p u = is_rewritable_spp (map spp_of bs) (spp_of p) u"

lemma compute_plus_up[code]: "plus_up p x y = - plus_down p (-x) (-y)"

lemma step_open_isCOMact: assumes "step s a = (ou,s')" and "open s \<noteq> open s'" shows "\<not> isCOMact a \<and> \<not> (\<exists> ua. isuPost ua \<and> a = Uact ua)"

lemma connected_nest: fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set" assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)" and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m" shows "connected(\<Inter> (range S))"

lemma substt [simp]: \<open>to_tm (FOL_Fitting.substt t s v) = sub_term v (to_tm s) (to_tm t)\<close> \<open>to_tm_list (FOL_Fitting.substts l s v) = sub_list v (to_tm s) (to_tm_list l)\<close>

lemma hom_dvd_hom[simp]: "hom x dvd hom y \<longleftrightarrow> x dvd y"

lemma A_fv: "\<And>A. \<A> e = \<lfloor>A\<rfloor> \<Longrightarrow> A \<subseteq> fv e" and "\<And>A. \<A>s es = \<lfloor>A\<rfloor> \<Longrightarrow> A \<subseteq> fvs es"

lemma pair_qbs_fst: assumes "qbs_space Y \<noteq> {}" shows "map_qbs fst (X \<Otimes>\<^sub>Q Y) = X"

lemma lfp_exp_sound: assumes fR: "t R \<tturnstile> R" and sR: "sound R" shows "sound (lfp_exp t)"

lemma simple_const_inter_block_size: "(\<And> bl. bl \<in># \<B> \<Longrightarrow> \<m> < card bl) \<Longrightarrow> simple_design \<V> \<B>"

lemma countable_or_card_of: assumes "countable A" shows "(finite A \<and> |A| <o |UNIV::nat set| ) \<or> (infinite A \<and> |A| =o |UNIV::nat set| )"

lemma wf_chain_append: "wf_chain \<Gamma> (rs1@rs2) \<longleftrightarrow> wf_chain \<Gamma> rs1 \<and> wf_chain \<Gamma> rs2"

lemma pr_pri_agrk_parts [rule_format]: "(evs, S, A, U) \<in> protocol \<Longrightarrow> Pri_AgrK x \<notin> U \<longrightarrow> Pri_AgrK x \<notin> parts (A \<union> spies evs)"

lemma min_int_poly_rat_code_unfold [code_unfold]: "min_int_poly = poly_rat"

lemma conj_refine_left: "(Q \<Rightarrow> P) \<sqsubseteq> R \<Longrightarrow> P \<sqsubseteq> (Q \<and> R)"

lemma empty_list_valid_merge: "(\<forall>(v,e) \<in> set []. set v \<inter> dlverts t1 = {} \<and> v \<noteq> [] \<and> e \<notin> darcs t1 \<union> {e1})"

lemma induced_hom_Abs_freelist_conv_sum_list: "ss\<in>lists S \<Longrightarrow> F (\<lceil>FreeGroup S|Abs_freelist ss|Q\<rceil>) = (\<Sum>s\<leftarrow>ss. f s)"

lemma transpose_mat_of_rows: "(mat_of_rows n vs)\<^sup>T = mat_of_cols n vs"

lemma cat_GRPH_CId_app[cat_GRPH_simps]: assumes "digraph \<alpha> \<CC>" shows "cat_GRPH \<alpha>\<lparr>CId\<rparr>\<lparr>\<CC>\<rparr> = dghm_id \<CC>"

lemma op_ntcf_ntcf_const[cat_op_simps]: "op_ntcf (ntcf_const \<JJ> \<CC> f) = ntcf_const (op_cat \<JJ>) (op_cat \<CC>) f"

lemma iprev_cut_le_conv: "n \<le> t \<Longrightarrow> iprev n (I \<down>\<le> t) = iprev n I"

lemma heap_is_wellformed_children_disc_nodes: "heap_is_wellformed h \<Longrightarrow> node_ptr |\<in>| node_ptr_kinds h \<Longrightarrow> \<not>(\<exists>parent \<in> fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r) \<Longrightarrow> (\<exists>document_ptr \<in> fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r)"

lemma Zorns_po_lemma_nonempty: assumes po: "Partial_order r" and u: "\<And>C. \<lbrakk>C \<in> Chains r; C\<noteq>{}\<rbrakk> \<Longrightarrow> \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r" and "r \<noteq> {}" shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"

lemma left_add_zero_mat[simp]: "(A :: 'a :: monoid_add mat) \<in> carrier_mat nr nc \<Longrightarrow> 0\<^sub>m nr nc + A = A"

lemma lookup_except_when: "lookup (except p S) = (\<lambda>t. lookup p t when t \<notin> S)"

lemma maybe_counterexample2: "\<lbrakk>a = Just\<cdot>x; b = Just\<cdot>y; k\<cdot>x = Nothing; k\<cdot>y = Just\<cdot>z\<rbrakk> \<Longrightarrow> fplus\<cdot>a\<cdot>b \<bind> k \<noteq> fplus\<cdot>(a \<bind> k)\<cdot>(b \<bind> k)"

lemma verify_plan_correct: "verify_plan problem \<pi>s = Inr () \<longleftrightarrow> ast_problem.well_formed problem \<and> ast_problem.valid_plan problem \<pi>s"

theorem wls_vsubst_Op_simp[simp]: assumes "wlsInp delta inp" and "wlsBinp delta binp" shows "((Op delta inp binp) #[y1 // y]_ys) = Op delta (inp %[y1 // y]_ys) (binp %%[y1 // y]_ys)"

lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"

lemma starlike_imp_connected: fixes S :: "'a::real_normed_vector set" shows "starlike S \<Longrightarrow> connected S"

lemma matches_cong: "\<forall>x\<in>fv \<phi>. v!x = v'!x \<Longrightarrow> matches v \<phi> e = matches v' \<phi> e"

lemma condition_4_part_3: fixes A::"'a::{field}^'columns::{mod_type}^'rows::{mod_type}" and k::nat defines ia:"ia\<equiv>(if \<forall>m. is_zero_row_upt_k m k A then 0 else to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1)" defines B:"B\<equiv>(snd (Gauss_Jordan_column_k (ia,A) k))" assumes rref: "reduced_row_echelon_form_upt_k A k" and not_zero_i_suc_k: "\<not> is_zero_row_upt_k i (Suc k) B" and i_not_j: "i \<noteq> j" and not_zero_m: " \<not> is_zero_row_upt_k m k A" and zero_below_greatest: "\<forall>m\<ge>(GREATEST n. \<not> is_zero_row_upt_k n k A) + 1. A $ m $ from_nat k = 0" shows "A $ j $ (LEAST n. A $ i $ n \<noteq> 0) = 0"

lemma BPd'_BCh'_mechanism_domain: shows "mechanism_domain BPd' BCh'"

lemma I_cyclic: assumes "is_dvd a" and "hd_coeff a = 1" and "i mod divisor a = j mod divisor a" shows "I\<^sub>Z a (i#e) = I\<^sub>Z a (j#e)"

lemma semialg_val_strict_ineq_set_is_semialg': assumes "f \<in> carrier (SA k)" shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) < C}"

theorem GodIsEssential: "\<lfloor>\<^bold>\<forall>x. G x \<^bold>\<rightarrow> ((\<E> \<down>\<^sub>1G) x)\<rfloor>"

lemma the_NF_steps: assumes "(a, b) \<in> A\<^sup>*" shows "the_NF A a = the_NF A b"

lemma allNeededINChannelsTestL2p3: "allNeededINChannels level2 {data1, data10, data11}"

lemma L_bdd_above[simp, intro]: "bdd_above ((\<lambda>p. L p v s) ` X)"

lemma subsetClosed'[simp]: fixes p :: "name prm" and xvec :: "name list" and P :: "'a::fs_name" shows "(set (p \<bullet> xvec) \<subseteq> supp (p \<bullet> P)) = (set xvec \<subseteq> supp P)"

lemma lzip_inf_llist_llist_of [simp]: "lzip (inf_llist f) (llist_of xs) = llist_of (zip (map f [0..<length xs]) xs)"

lemma invariant_steps: "list_all P as" if "steps (a # as)" "P a"

lemma sym_lens_compl [simp]: "sym_lens a \<Longrightarrow> sym_lens (-\<^sub>L a)"

lemma analz_trans: "X \<in> analz G \<Longrightarrow> G \<subseteq> analz H \<Longrightarrow> X \<in> analz H"

lemma nstd_case3: "\<forall>rs n. c \<noteq> Oc\<up>(Suc rs) @ Bk\<up>(n) \<Longrightarrow> NSTD (trpl_code (a, b, c))"

lemma eval_monom_pos: assumes "basis_wf basis" "fst monom > 0" shows "eventually (\<lambda>x. eval_monom monom basis x > 0) at_top"

lemma NSBseqD2: "NSBseq X \<Longrightarrow> ( *f* X) N \<in> HFinite"

lemma funpow_shift1_1: "(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) = Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"

lemma in_associates_Hermite_of: fixes A::"'a::{bezout_ring_div,normalization_semidom,unique_euclidean_ring}^'cols::{mod_type}^'rows::{mod_type}" assumes a: "ass_function ass" and r: "res_function res" and b: "is_bezout_ext bezout" and i: "\<not> is_zero_row i (Hermite_of A ass res bezout)" shows "Hermite_of A ass res bezout $ i $ (LEAST n. Hermite_of A ass res bezout $ i $ n \<noteq> 0) \<in> range ass"

lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"

lemma fun_pair_wf\<^sub>t\<^sub>r\<^sub>m: "wf\<^sub>t\<^sub>r\<^sub>m t \<Longrightarrow> wf\<^sub>t\<^sub>r\<^sub>m t' \<Longrightarrow> wf\<^sub>t\<^sub>r\<^sub>m (pair (t,t'))"

lemma finite_numbers[simp,intro]: "finite [n]"

lemma vimage_is_vempty[iff]: "r `\<^sub>\<circ> A = 0 \<longleftrightarrow> vdisjnt (\<D>\<^sub>\<circ> r) A"

lemma n_preserve1_var: "n x \<cdot> y \<le> n x \<cdot> y \<cdot> n x \<Longrightarrow> n x \<cdot> (n x \<cdot> y + t x \<cdot> z)\<^sup>\<dagger> \<le> (n x \<cdot> y)\<^sup>\<dagger> \<cdot> n x"

lemma cl_max': assumes c: "pcp C" assumes sc: "subset_closed C" shows "F \<triangleright> pcp_lim C S \<in> C \<Longrightarrow> F \<in> pcp_lim C S" "F \<triangleright> G \<triangleright> pcp_lim C S \<in> C \<Longrightarrow> F \<in> pcp_lim C S \<and> G \<in> pcp_lim C S"

lemma period_rev_conv [reversal_rule]: "period (rev w) n \<longleftrightarrow> period w n"

lemma disj2: assumes disj_x_y: "disj x y s" assumes disj_x_z: "disj x z s" assumes unreach_l_x: "\<not> s\<turnstile> l reachable_from x" shows "disj x y (s\<langle>l:=z\<rangle>)"

lemma pfp_inv: "pfp f x = Some y \<Longrightarrow> (\<And>x. P x \<Longrightarrow> P(f x)) \<Longrightarrow> P x \<Longrightarrow> P y"

lemma oth_class_taut_3_b[PLM]: "[(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> (\<psi> \<^bold>& \<phi>) in v]"

lemma length_hd_le_concat: assumes "as \<noteq> []" shows "length (hd as) \<le> length (concat as)"

lemma open_Collect_positive: fixes f :: "'a::topological_space \<Rightarrow> real" assumes f: "continuous_on s f" shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"

lemma binop_known_addrs: assumes ok: "start_heap_ok" shows "binop bop v1 v2 = \<lfloor>Inl v\<rfloor> \<Longrightarrow> ka_Val v \<subseteq> ka_Val v1 \<union> ka_Val v2 \<union> set start_addrs" and "binop bop v1 v2 = \<lfloor>Inr a\<rfloor> \<Longrightarrow> a \<in> ka_Val v1 \<union> ka_Val v2 \<union> set start_addrs"

lemma match_subst_closed: assumes "match pat t = Some env" "closed_except rhs (frees pat)" "closed t" shows "closed (subst rhs env)"

lemma sum_upto_moebius_times_floor_linear: "sum_upto (\<lambda>n. moebius_mu n * \<lfloor>x / real n\<rfloor>) x = (if x \<ge> 1 then 1 else 0)"

lemma factor_dvd_f_0: assumes "factor dvd f" shows "pl.Mp factor \<noteq> 0"

lemma subst_sym[sym]: "\<lbrakk>s1 \<doteq> s2\<rbrakk> \<Longrightarrow> s2 \<doteq> s1"

lemma get_shadow_root_is_l_get_shadow_root [instances]: "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"

lemma convert_eval: "peval P a = ppeval (convert P) a v"

lemma lists_succ_snoc: "lists_succ (xss @ [xs]) = lists_succ xss o list_succ xs"

lemma aboveS_notIn: "a \<notin> aboveS r a"

lemma finite_range: "finite (range index)"

theorem hta_prod'_correct: assumes TA: "hashedTa H1" "hashedTa H2" assumes HI: "hta_has_idx_s H1" "hta_has_idx_sf H2" shows "ta_lang (hta_\<alpha> (hta_prod' H1 H2)) = ta_lang (hta_\<alpha> H1) \<inter> ta_lang (hta_\<alpha> H2)" "hashedTa (hta_prod' H1 H2)"

lemma OUren: "ORadmit ODE \<Longrightarrow> ODE_sem I (OUrename x y ODE) \<nu> = RSadj x y (ODE_sem I ODE (RSadj x y \<nu>))"

lemma mono_Ndet2: "P \<sqsubseteq> Q \<Longrightarrow> (\<forall> s. s \<notin> D (P \<sqinter> S) \<longrightarrow> Ra (P \<sqinter> S) s = Ra (Q \<sqinter> S) s)"

lemma bij_betw_add[simp]: "bij_betw ((\<oplus>\<^sub>a) a) A B \<longleftrightarrow> (\<oplus>\<^sub>a) a ` A = B"

lemma qbs_prob_integral_add: assumes "qbs_integrable (s::'a qbs_prob_space) f" and "qbs_integrable s g" shows "qbs_prob_integral s (\<lambda>x. f x + g x) = qbs_prob_integral s f + qbs_prob_integral s g"

lemma list_it_alt: "list_it s = map_iterator_dom (map.iteratei s)"

lemma bigo_const3: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)" for c :: "'a::linordered_field"

lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> l)"

lemma instInp: assumes \<tau>: "\<tau> \<in> ptrm (Suc 0)" and [simp]: "t \<in> trm" and [simp]: "FvarsT t = Variable ` {(Suc 0)..n}" shows "instInp \<tau> t \<in> ptrm n"

lemma add_block_rep_number_in: assumes "x \<in> b" shows "(add_block b) rep x = \<B> rep x + 1"

lemma edka_complexity_refine: "edka_complexity \<le> \<Down>Id edka"

lemma (in \<Z>) M\<alpha>_Rel_arrow_rl_is_cat_Rel_iso_arr'[cat_Rel_par_set_cs_intros]: assumes "A \<in>\<^sub>\<circ> cat_Rel \<alpha>\<lparr>Obj\<rparr>" and "B \<in>\<^sub>\<circ> cat_Rel \<alpha>\<lparr>Obj\<rparr>" and "C \<in>\<^sub>\<circ> cat_Rel \<alpha>\<lparr>Obj\<rparr>" and "A' = A \<times>\<^sub>\<circ> (B \<times>\<^sub>\<circ> C)" and "B' = (A \<times>\<^sub>\<circ> B) \<times>\<^sub>\<circ> C" and "\<CC>' = cat_Rel \<alpha>" shows "M\<alpha>_Rel_arrow_rl A B C : A' \<mapsto>\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<CC>'\<^esub> B'"

lemma carrier_single[simp]: "carrier (single y) = {y}"

lemma mset_list_remove1[simp]: "mset (list_remove1 x l) = mset l - {#x#}"

lemma "\<Gamma>\<turnstile> \<lbrace>\<acute>N = 5\<rbrace> \<acute>N :== 2 * \<acute>N \<lbrace>\<acute>N = 10\<rbrace>"