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

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lemma prv_scnj2_imp_cnj: "\<phi> \<in> fmla \<Longrightarrow> \<psi> \<in> fmla \<Longrightarrow> prv (imp (scnj {\<phi>,\<psi>}) (cnj \<phi> \<psi>))"
lemma sfoldl_op'_strict[simp]: "op'\<cdot>pat\<cdot>(sfoldl\<cdot>(op'\<cdot>pat)\<cdot>(us, \<bottom>)\<cdot>xs)\<cdot>x = \<bottom>"
lemma WildOr: "Wildcard (Or A B), \<Delta> \<equiv> Or (Wildcard A) (Wildcard B)"
lemma convex_rel_interior_closure: fixes S :: "'n::euclidean_space set" assumes "convex S" shows "rel_interior (closure S) = rel_interior S"
lemma val_Zp_dist_infty: assumes "a \<in> carrier Zp" assumes "b \<in> carrier Zp" assumes "val_Zp_dist a b = \<infinity>" shows "a = b"
lemma lowest_tops_lowest: "lowest_tops es = Some a \<Longrightarrow> e \<in> set es \<Longrightarrow> ifex_ordered e \<Longrightarrow> v \<in> ifex_var_set e \<Longrightarrow> a \<le> v"
lemma dbm_entry_dbm_min: assumes "dbm_entry_val u (Some c1) (Some c2) (min a b)" shows "dbm_entry_val u (Some c1) (Some c2) b"
lemma union_mono_aux: "A \<subseteq> B \<Longrightarrow> A \<union> C \<subseteq> B \<union> C"
lemma upd_hd_next: assumes p_ps: "List p next (p#ps)" shows "List (next p) (next(p := q)) ps"
lemma sys_block_size_subset: "sys_block_sizes \<subseteq> \<K>"
theorem Zorn's_Lemma: assumes r: "\<And>c. c \<in> chains S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S" and aS: "a \<in> S" shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> z = y"
lemma vrestr_comp: assumes "g \<in> measurable M M" shows "(f o g)--` A = g--` (f--` A)"
lemma subc_sub_closed_var [simp]: \<open>new c p \<Longrightarrow> closed (Suc m) p \<Longrightarrow> subc c (Var m) (subst p (App c []) m) = p\<close>
lemma (in prob_space) indep_sets_finite: assumes I: "I \<noteq> {}" "finite I" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" shows "indep_sets F I \<longleftrightarrow> (\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j)))"
lemma r_hd [simp]: "eval r_hd [e] \<down>= e_hd e"
lemma fixes s :: "'a :: {nat_power_normed_field,banach,euclidean_space}" assumes "s \<bullet> 1 > 1" shows euler_product_fds_zeta: "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - 1 / nat_power p s) else 1) \<longlonglongrightarrow> eval_fds fds_zeta s" (is ?th1) and eval_fds_zeta_nonzero: "eval_fds fds_zeta s \<noteq> 0"
lemma le_oexp': assumes "Ord \<alpha>" "1 < \<alpha>" "Ord \<beta>" shows "\<beta> \<le> \<alpha>\<up>\<beta>"
lemma closed_Collect_eq: fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "closed {x. f x = g x}"
lemma bool11: "(True \<longrightarrow> P) = P"
lemma less_eq_sup_top: "-x \<le> -y \<longleftrightarrow> --x \<squnion> -y = top"
lemma f_nth_eq_map_f_nth: "\<lbrakk>as \<noteq> []; length as \<ge> n\<rbrakk> \<Longrightarrow> f (as ! (n - Suc 0)) = map f as ! (n - Suc 0)"
lemma mult_distrib_inverse' [simp]: "(a * b) / a = b"
lemma Fix2: "\<lbrakk>Monotone \<phi>; \<phi> (FIX \<phi>) (s, t,\<beta>)\<rbrakk> \<Longrightarrow> FIX \<phi> (s,t,\<beta>)"
lemma assumes "x > 0" shows erf_remainder_integral_nonneg: "erf_remainder_integral n x \<ge> 0" and erf_remainder_integral_bound: "erf_remainder_integral n x \<le> exp (-x\<^sup>2) / x ^ (2*n+1)"
lemma maybe_and_not_true: "(x \<and>? y \<noteq> true) = (x \<noteq> true \<or> y \<noteq> true)"
lemma insert_ops_exist: assumes "rga_ops xs" shows "\<exists>ys. set xs = set ys \<and> insert_ops ys"
lemma Lnode\<^sub>i_height: "height_up\<^sub>i (Lnode\<^sub>i k xs) = height (Leaf xs)"
lemma mat_tensor_prod_2_prelim [simp]: assumes "state 1 v" and "state 1 w" shows "v \<Otimes> w = mat_of_cols_list 4 [[v $$ (0,0) * w $$ (0,0), v $$ (0,0) * w $$ (1,0), v $$ (1,0) * w $$ (0,0), v $$ (1,0) * w $$ (1,0)]]"
lemma op_ntcf_components[cat_op_simps]: shows "op_ntcf \<NN>\<lparr>NTMap\<rparr> = \<NN>\<lparr>NTMap\<rparr>" and "op_ntcf \<NN>\<lparr>NTDom\<rparr> = op_cf (\<NN>\<lparr>NTCod\<rparr>)" and "op_ntcf \<NN>\<lparr>NTCod\<rparr> = op_cf (\<NN>\<lparr>NTDom\<rparr>)" and "op_ntcf \<NN>\<lparr>NTDGDom\<rparr> = op_cat (\<NN>\<lparr>NTDGDom\<rparr>)" and "op_ntcf \<NN>\<lparr>NTDGCod\<rparr> = op_cat (\<NN>\<lparr>NTDGCod\<rparr>)"
lemma mpoly_degree_mult_eq: fixes p q :: "'a :: idom mpoly" assumes "p \<noteq> 0" "q \<noteq> 0" shows "MPoly_Type.degree (p * q) x = MPoly_Type.degree p x + MPoly_Type.degree q x"
lemma k_frontier_is_node[intro, simp]: "list_all k_isNode (k_frontier a)"
lemma (in prob_space) AE_in_set_eq_1: assumes A[measurable]: "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
lemma one_point: assumes "mwb_lens x" "x \<sharp> v" shows "(\<exists> x \<bullet> P \<and> var x =\<^sub>u v) = P\<lbrakk>v/x\<rbrakk>"
lemma hm_rel_id_conv: "hm_rel id_assn id_assn = hm_rel_np" \<comment> \<open>Used for generic algorithms: Unfold with this, then let decl-impl compose with \<open>map_rel\<close> again.\<close>
lemma (in Module) ex_extension:"\<lbrakk>R module N; free_generator R M H; f \<in> H \<rightarrow> carrier N; H1 \<subseteq> H; h \<in> H - H1; (H1, g) \<in> fsps R M N f H\<rbrakk> \<Longrightarrow> \<exists>k. ((H1 \<union> {h}), k) \<in> fsps R M N f H"
lemma cross7_basis_zero: " u=0 \<Longrightarrow> (u \<times>\<^sub>7 axis 1 1 = 0) \<and> (u \<times>\<^sub>7 axis 2 1 = 0) \<and> (u \<times>\<^sub>7 axis 3 1 = 0) \<and> (u \<times>\<^sub>7 axis 4 1 = 0) \<and> (u \<times>\<^sub>7 axis 5 1 = 0 ) \<and> (u \<times>\<^sub>7 axis 6 1 = 0 ) \<and> (u \<times>\<^sub>7 axis 7 1 = 0) "
lemma fix_asset_price: shows "\<exists>x Mkt2. x \<notin> stocks Mkt \<and> coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr"
lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
lemma Ratreal_code[code]: "Ratreal = real_of_u \<circ> mau_of_rat"
theorem sim_first: assumes H: "s \<approx> t" shows "first s = first t"
lemma transfer_rfoldl[refine_transfer]: assumes "\<And>s x. RETURN (f s x) \<le> F s x" shows "RETURN (foldl f s l) \<le> rfoldl F s l"
lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q"
lemma self_finance_updated_ind: assumes "portfolio pf" and "\<forall>n w. prices Mkt asset n w \<noteq> 0" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = cls_val_process Mkt pf (Suc n) w + (prices Mkt asset (Suc n) w / (prices Mkt asset n w)) * (val_process Mkt (self_finance Mkt v pf asset) n w - val_process Mkt pf n w)"
lemma monotone_extreme_imp_extreme_bound: "extreme_bound A (\<sqsubseteq>) (f ` I) (f e)"
lemma residuated_strict: "residuated f \<Longrightarrow> f \<bottom> = \<bottom>"
lemma inner_eucl_of_list: fixes i::"'a::executable_euclidean_space" assumes i: "i \<in> Basis" assumes l: "length xs = DIM('a)" shows "i \<bullet> eucl_of_list xs = xs ! (index Basis_list i)"
lemma while_denest_3: "(x \<star> w) \<star> x\<^sup>\<omega> = (x \<star> w)\<^sup>\<omega>"
theorem constr_fp_uniq: assumes "constr F E" "mono F" "\<Sqinter> (range E) = C" shows "(C \<and> \<mu> F) = (C \<and> \<nu> F)"
lemma msubst_lit [usubst]: "\<guillemotleft>x\<guillemotright>\<lbrakk>x\<rightarrow>v\<rbrakk> = v"
lemma Max_le_Min_imp_singleton: " \<lbrakk> finite A; A \<noteq> {}; Max A \<le> Min A \<rbrakk> \<Longrightarrow> A = {Min A}"
lemma single_valued_monom_rel': \<open>IS_LEFT_UNIQUE monom_rel\<close>
lemma correctCompositionDiffLevelsA82: "correctCompositionDiffLevels sA82"
lemma map_of_ty_k': "\<lbrakk>distinct (map (\<lambda>(cl, var, ty). var) cl_var_ty_list); distinct (map (\<lambda>(y, cl, var, var', v). var') y_cl_var_var'_v_list); x' \<notin> (\<lambda>(y, cl, var, var', v). x_var var') ` set y_cl_var_var'_v_list; map fst y_cl_var_var'_v_list = map fst y_ty_list; map (\<lambda>(cl, var, ty). ty) cl_var_ty_list = map snd y_ty_list; map (\<lambda>(y, cl, var, u). var) y_cl_var_var'_v_list = map (\<lambda>(cl, var, ty). var) cl_var_ty_list; map_of (map (\<lambda>(cl, var, y). (x_var var, y)) cl_var_ty_list) (x_var var) = Some ty_var\<rbrakk> \<Longrightarrow> (if x_var (case_option var (case_x (\<lambda>var'. var') var) (map_of (map (\<lambda>(y, cl, var, var', v). (x_var var, x_var var')) y_cl_var_var'_v_list) (x_var var))) = x' then Some ty_x_m else (\<Gamma> ++ map_of (map (\<lambda>((y, cl, var, var', v), y', y). (x_var var', y)) (zip y_cl_var_var'_v_list y_ty_list))) (x_var (case_option var (case_x (\<lambda>var'. var') var) (map_of (map (\<lambda>(y, cl, var, var', v). (x_var var, x_var var')) y_cl_var_var'_v_list) (x_var var))))) = Some ty_var"
lemma ii_squared [simp]: "quat_ii\<^sup>2 = -1"
lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
lemma OrdNotEqP_self_contra: "{x NEQ x} \<turnstile> Fls"
lemma ins_root_order: assumes "root_order k t" shows "root_order_up\<^sub>i k (ins k x t)"
lemma unite_wf_env: fixes e v w defines "e' \<equiv> unite v w e" assumes pre: "pre_dfss v e" and w: "w \<in> successors v" "w \<notin> vsuccs e v" "w \<in> visited e" "w \<notin> explored e" shows "wf_env e'"
lemma monad_state_alt_envT' [locale_witness]: "monad_state_alt return (bind :: ('a, 'm) bind) (get :: ('s, 'm) get) put alt \<Longrightarrow> monad_state_alt return (bind :: ('a, ('r, 'm) envT) bind) (get :: ('s, ('r, 'm) envT) get) put alt"
lemma d_delta_ln_upper_15: "x > 0 \<Longrightarrow> ((\<lambda>x. ln_upper_15 x - ln x) has_field_derivative diff_delta_ln_upper_15 x) (at x)"
lemma B'[simp]:"x \<noteq> bot \<Longrightarrow> x \<noteq> null \<Longrightarrow> (snd \<lceil>\<lceil>Rep_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>) \<noteq> bot"
lemma upper_real_interval[simp]: "upper (real_interval x) = upper x"
lemma us_mono: assumes "i < j" and "j < sum_list y" shows "\<^bold>u y i <\<^sub>v \<^bold>u y j"
lemma assumes "f \<in> X \<rightarrow>\<^sub>Q \<real>\<^sub>Q" and "\<alpha> \<in> qbs_Mx X" shows "(\<lambda>x. 2 * f (\<alpha> x) + (f (\<alpha> x))^2) \<in> \<real>\<^sub>Q \<rightarrow>\<^sub>Q \<real>\<^sub>Q"
lemma p_subid_interr: "(x \<cdot> z \<cdot> 1\<^sub>\<pi>) \<parallel> (y \<cdot> z \<cdot> 1\<^sub>\<pi>) = (x \<parallel> y) \<cdot> z \<cdot> 1\<^sub>\<pi>"
lemma upd_cond_alt: "upd_cond Q \<pi> u v' \<longleftrightarrow> (v',u) \<in> edges g \<and> v'\<notin>S (A Q \<pi>) \<and> enat (w {v',u}) < Q v'"
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = add_mset a A"
lemma Neg_Imp_I [intro!]: "H \<turnstile> A \<Longrightarrow> insert B H \<turnstile> Fls \<Longrightarrow> H \<turnstile> Neg (A IMP B)"
lemma pt_unit_inst: shows "pt TYPE(unit) TYPE('x)"
lemma usedBy_bop [unrest]: "\<lbrakk> x \<natural> u; x \<natural> v \<rbrakk> \<Longrightarrow> x \<natural> bop f u v"
lemma Arccos_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Im(z + \<i> * csqrt(1 - z\<^sup>2))"
lemma o_subst: "subst \<pi>1 o subst \<pi>2 = subst (subst \<pi>1 o \<pi>2)"
lemma path_connected_arc_complement: fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space" assumes "arc \<gamma>" "2 \<le> DIM('a)" shows "path_connected(- path_image \<gamma>)"
lemma [simp]: "\<alpha> [] = (\<lambda>_. 0)"
lemma share_all_until_volatile_write_update_sb: assumes congr: "\<And>S. share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S" shows "\<And>\<S> i. \<lbrakk>i < length ts; ts!i = (p,is,\<theta>,sb,\<D>,\<O>,\<R>)\<rbrakk> \<Longrightarrow> share_all_until_volatile_write ts \<S> = share_all_until_volatile_write (ts[i := (p', is',\<theta>', sb', \<D>', \<O>',\<R>')]) \<S>"
lemma norm_ket_k_ge_K: "k \<ge> K \<Longrightarrow> inner_prod (ket_k k) (ket_k k) = 0"
lemma trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_finite[simp]: "finite (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p x)"
lemma closure_complete: assumes "lang r = lang s" shows "EX bs. closure as ([({r},{s})],[]) = Some([],bs)" (is ?C)
theorem wls_psubst_Op_simp[simp]: assumes "wlsInp delta inp" and "wlsBinp delta binp" and "wlsEnv rho" shows "((Op delta inp binp) #[rho]) = Op delta (inp %[rho]) (binp %%[rho])"
lemma rsum_append: "rsum (D1 @ D2) f = rsum D1 f + rsum D2 f"
lemma inv_atI[intro]: assumes "\<And> i. i < n \<Longrightarrow> p A i m" shows "inv_at p A m"
lemma Gromov_product_isometry: assumes "isometry_on UNIV f" shows "Gromov_product_at (f x) (f y) (f z) = Gromov_product_at x y z"
lemma Initial_Label: "CTXT 0 [] outp P \<Longrightarrow> P"
lemma reduce_implyP: \<open>reduce (ps \<^bold>\<leadsto>\<^sub>! q) = (map reduce ps \<^bold>\<leadsto>\<^sub>! reduce q)\<close>
lemma nat_diff_left_cancel_eq1: "\<lbrakk> k - m = k - (n::nat); m < k \<rbrakk> \<Longrightarrow> m = n"
lemma ac_fract: assumes "c \<in> carrier Q\<^sub>p" assumes "a \<in> nonzero Z\<^sub>p" assumes "b \<in> nonzero Z\<^sub>p" assumes "c = frac a b" shows "angular_component c = (ac_Zp a)\<otimes>\<^bsub>Z\<^sub>p\<^esub> inv \<^bsub>Z\<^sub>p\<^esub>(ac_Zp b)"
lemma analytic_at: "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
lemma inc_above_dec_above_iff: "inc_above b i (dec_above b i x) = x \<longleftrightarrow> x < b \<or> b + i \<le> x"
lemma array_map_conv_foldl_array_set: assumes len: "array_length A = array_length a" shows "array_map f a = foldl (\<lambda>A (k, v). array_set A k (f k v)) A (assoc_list_of_array a)"
lemma real_less_ereal_iff: "real_of_ereal y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
lemma rejects_termination: "observe_execution drinks 0 <> [(STR ''select'', [Str ''coke'']), (STR ''rejects'', [Num 50]), (STR ''coin'', [Num 50])] = [[]]"
lemma all_edges_between_subset: "all_edges_between X Y G \<subseteq> X\<times>Y"
lemma sdnets_noteq: "onlyTwoNets a \<Longrightarrow> onlyTwoNets aa \<Longrightarrow> first_bothNet a \<noteq> first_bothNet aa \<Longrightarrow> \<not> member DenyAll a \<Longrightarrow> \<not> member DenyAll aa \<Longrightarrow> sdnets a \<noteq> sdnets aa"
lemma bisimScopeExtSym: fixes x :: name and Q :: "('a, 'b, 'c) psi" and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>" and "x \<sharp> Q" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> (\<lparr>\<nu>x\<rparr>P) \<parallel> Q"
lemma "0 \<le> (n :: nat)"
lemma lconf_hext: "\<lbrakk> P,h \<turnstile> l (:\<le>)\<^sub>w E; h \<unlhd> h' \<rbrakk> \<Longrightarrow> P,h' \<turnstile> l (:\<le>)\<^sub>w E"
lemma is_fieldD: "is_field m \<Longrightarrow> \<exists> declC f. m=(declC,fdecl f)"
lemma set_iterator_rule_P: "\<lbrakk> set_iterator it S0; I S0 \<sigma>0; !!S \<sigma> x. \<lbrakk> c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0 \<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>); !!\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>; !!\<sigma> S. S \<subseteq> S0 \<Longrightarrow> S \<noteq> {} \<Longrightarrow> \<not> c \<sigma> \<Longrightarrow> I S \<sigma> \<Longrightarrow> P \<sigma> \<rbrakk> \<Longrightarrow> P (it c f \<sigma>0)"
lemma valid_dbm_non_empty_diag: assumes "valid_dbm M" "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<forall> k \<le> n. M k k \<ge> \<one>"
lemma PhiWhilePOp_Monotone:"Monotone (PhiWhilePOp A b \<Phi>)"

lemma prv_scnj2_imp_cnj: "\<phi> \<in> fmla \<Longrightarrow> \<psi> \<in> fmla \<Longrightarrow> prv (imp (scnj {\<phi>,\<psi>}) (cnj \<phi> \<psi>))"

lemma sfoldl_op'_strict[simp]: "op'\<cdot>pat\<cdot>(sfoldl\<cdot>(op'\<cdot>pat)\<cdot>(us, \<bottom>)\<cdot>xs)\<cdot>x = \<bottom>"

lemma WildOr: "Wildcard (Or A B), \<Delta> \<equiv> Or (Wildcard A) (Wildcard B)"

lemma convex_rel_interior_closure: fixes S :: "'n::euclidean_space set" assumes "convex S" shows "rel_interior (closure S) = rel_interior S"

lemma val_Zp_dist_infty: assumes "a \<in> carrier Zp" assumes "b \<in> carrier Zp" assumes "val_Zp_dist a b = \<infinity>" shows "a = b"

lemma lowest_tops_lowest: "lowest_tops es = Some a \<Longrightarrow> e \<in> set es \<Longrightarrow> ifex_ordered e \<Longrightarrow> v \<in> ifex_var_set e \<Longrightarrow> a \<le> v"

lemma dbm_entry_dbm_min: assumes "dbm_entry_val u (Some c1) (Some c2) (min a b)" shows "dbm_entry_val u (Some c1) (Some c2) b"

lemma union_mono_aux: "A \<subseteq> B \<Longrightarrow> A \<union> C \<subseteq> B \<union> C"

lemma upd_hd_next: assumes p_ps: "List p next (p#ps)" shows "List (next p) (next(p := q)) ps"

lemma sys_block_size_subset: "sys_block_sizes \<subseteq> \<K>"

theorem Zorn's_Lemma: assumes r: "\<And>c. c \<in> chains S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S" and aS: "a \<in> S" shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> z = y"

lemma vrestr_comp: assumes "g \<in> measurable M M" shows "(f o g)--` A = g--` (f--` A)"

lemma subc_sub_closed_var [simp]: \<open>new c p \<Longrightarrow> closed (Suc m) p \<Longrightarrow> subc c (Var m) (subst p (App c []) m) = p\<close>

lemma (in prob_space) indep_sets_finite: assumes I: "I \<noteq> {}" "finite I" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" shows "indep_sets F I \<longleftrightarrow> (\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j)))"

lemma r_hd [simp]: "eval r_hd [e] \<down>= e_hd e"

lemma fixes s :: "'a :: {nat_power_normed_field,banach,euclidean_space}" assumes "s \<bullet> 1 > 1" shows euler_product_fds_zeta: "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - 1 / nat_power p s) else 1) \<longlonglongrightarrow> eval_fds fds_zeta s" (is ?th1) and eval_fds_zeta_nonzero: "eval_fds fds_zeta s \<noteq> 0"

lemma le_oexp': assumes "Ord \<alpha>" "1 < \<alpha>" "Ord \<beta>" shows "\<beta> \<le> \<alpha>\<up>\<beta>"

lemma closed_Collect_eq: fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "closed {x. f x = g x}"

lemma bool11: "(True \<longrightarrow> P) = P"

lemma less_eq_sup_top: "-x \<le> -y \<longleftrightarrow> --x \<squnion> -y = top"

lemma f_nth_eq_map_f_nth: "\<lbrakk>as \<noteq> []; length as \<ge> n\<rbrakk> \<Longrightarrow> f (as ! (n - Suc 0)) = map f as ! (n - Suc 0)"

lemma mult_distrib_inverse' [simp]: "(a * b) / a = b"

lemma Fix2: "\<lbrakk>Monotone \<phi>; \<phi> (FIX \<phi>) (s, t,\<beta>)\<rbrakk> \<Longrightarrow> FIX \<phi> (s,t,\<beta>)"

lemma assumes "x > 0" shows erf_remainder_integral_nonneg: "erf_remainder_integral n x \<ge> 0" and erf_remainder_integral_bound: "erf_remainder_integral n x \<le> exp (-x\<^sup>2) / x ^ (2*n+1)"

lemma maybe_and_not_true: "(x \<and>? y \<noteq> true) = (x \<noteq> true \<or> y \<noteq> true)"

lemma insert_ops_exist: assumes "rga_ops xs" shows "\<exists>ys. set xs = set ys \<and> insert_ops ys"

lemma Lnode\<^sub>i_height: "height_up\<^sub>i (Lnode\<^sub>i k xs) = height (Leaf xs)"

lemma mat_tensor_prod_2_prelim [simp]: assumes "state 1 v" and "state 1 w" shows "v \<Otimes> w = mat_of_cols_list 4 [[v $$ (0,0) * w $$ (0,0), v $$ (0,0) * w $$ (1,0), v $$ (1,0) * w $$ (0,0), v $$ (1,0) * w $$ (1,0)]]"

lemma op_ntcf_components[cat_op_simps]: shows "op_ntcf \<NN>\<lparr>NTMap\<rparr> = \<NN>\<lparr>NTMap\<rparr>" and "op_ntcf \<NN>\<lparr>NTDom\<rparr> = op_cf (\<NN>\<lparr>NTCod\<rparr>)" and "op_ntcf \<NN>\<lparr>NTCod\<rparr> = op_cf (\<NN>\<lparr>NTDom\<rparr>)" and "op_ntcf \<NN>\<lparr>NTDGDom\<rparr> = op_cat (\<NN>\<lparr>NTDGDom\<rparr>)" and "op_ntcf \<NN>\<lparr>NTDGCod\<rparr> = op_cat (\<NN>\<lparr>NTDGCod\<rparr>)"

lemma mpoly_degree_mult_eq: fixes p q :: "'a :: idom mpoly" assumes "p \<noteq> 0" "q \<noteq> 0" shows "MPoly_Type.degree (p * q) x = MPoly_Type.degree p x + MPoly_Type.degree q x"

lemma k_frontier_is_node[intro, simp]: "list_all k_isNode (k_frontier a)"

lemma (in prob_space) AE_in_set_eq_1: assumes A[measurable]: "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"

lemma one_point: assumes "mwb_lens x" "x \<sharp> v" shows "(\<exists> x \<bullet> P \<and> var x =\<^sub>u v) = P\<lbrakk>v/x\<rbrakk>"

lemma hm_rel_id_conv: "hm_rel id_assn id_assn = hm_rel_np" \<comment> \<open>Used for generic algorithms: Unfold with this, then let decl-impl compose with \<open>map_rel\<close> again.\<close>

lemma (in Module) ex_extension:"\<lbrakk>R module N; free_generator R M H; f \<in> H \<rightarrow> carrier N; H1 \<subseteq> H; h \<in> H - H1; (H1, g) \<in> fsps R M N f H\<rbrakk> \<Longrightarrow> \<exists>k. ((H1 \<union> {h}), k) \<in> fsps R M N f H"

lemma cross7_basis_zero: " u=0 \<Longrightarrow> (u \<times>\<^sub>7 axis 1 1 = 0) \<and> (u \<times>\<^sub>7 axis 2 1 = 0) \<and> (u \<times>\<^sub>7 axis 3 1 = 0) \<and> (u \<times>\<^sub>7 axis 4 1 = 0) \<and> (u \<times>\<^sub>7 axis 5 1 = 0 ) \<and> (u \<times>\<^sub>7 axis 6 1 = 0 ) \<and> (u \<times>\<^sub>7 axis 7 1 = 0) "

lemma fix_asset_price: shows "\<exists>x Mkt2. x \<notin> stocks Mkt \<and> coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr"

lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"

lemma Ratreal_code[code]: "Ratreal = real_of_u \<circ> mau_of_rat"

theorem sim_first: assumes H: "s \<approx> t" shows "first s = first t"

lemma transfer_rfoldl[refine_transfer]: assumes "\<And>s x. RETURN (f s x) \<le> F s x" shows "RETURN (foldl f s l) \<le> rfoldl F s l"

lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q"

lemma self_finance_updated_ind: assumes "portfolio pf" and "\<forall>n w. prices Mkt asset n w \<noteq> 0" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = cls_val_process Mkt pf (Suc n) w + (prices Mkt asset (Suc n) w / (prices Mkt asset n w)) * (val_process Mkt (self_finance Mkt v pf asset) n w - val_process Mkt pf n w)"

lemma monotone_extreme_imp_extreme_bound: "extreme_bound A (\<sqsubseteq>) (f ` I) (f e)"

lemma residuated_strict: "residuated f \<Longrightarrow> f \<bottom> = \<bottom>"

lemma inner_eucl_of_list: fixes i::"'a::executable_euclidean_space" assumes i: "i \<in> Basis" assumes l: "length xs = DIM('a)" shows "i \<bullet> eucl_of_list xs = xs ! (index Basis_list i)"

lemma while_denest_3: "(x \<star> w) \<star> x\<^sup>\<omega> = (x \<star> w)\<^sup>\<omega>"

theorem constr_fp_uniq: assumes "constr F E" "mono F" "\<Sqinter> (range E) = C" shows "(C \<and> \<mu> F) = (C \<and> \<nu> F)"

lemma msubst_lit [usubst]: "\<guillemotleft>x\<guillemotright>\<lbrakk>x\<rightarrow>v\<rbrakk> = v"

lemma Max_le_Min_imp_singleton: " \<lbrakk> finite A; A \<noteq> {}; Max A \<le> Min A \<rbrakk> \<Longrightarrow> A = {Min A}"

lemma single_valued_monom_rel': \<open>IS_LEFT_UNIQUE monom_rel\<close>

lemma correctCompositionDiffLevelsA82: "correctCompositionDiffLevels sA82"

lemma map_of_ty_k': "\<lbrakk>distinct (map (\<lambda>(cl, var, ty). var) cl_var_ty_list); distinct (map (\<lambda>(y, cl, var, var', v). var') y_cl_var_var'_v_list); x' \<notin> (\<lambda>(y, cl, var, var', v). x_var var') ` set y_cl_var_var'_v_list; map fst y_cl_var_var'_v_list = map fst y_ty_list; map (\<lambda>(cl, var, ty). ty) cl_var_ty_list = map snd y_ty_list; map (\<lambda>(y, cl, var, u). var) y_cl_var_var'_v_list = map (\<lambda>(cl, var, ty). var) cl_var_ty_list; map_of (map (\<lambda>(cl, var, y). (x_var var, y)) cl_var_ty_list) (x_var var) = Some ty_var\<rbrakk> \<Longrightarrow> (if x_var (case_option var (case_x (\<lambda>var'. var') var) (map_of (map (\<lambda>(y, cl, var, var', v). (x_var var, x_var var')) y_cl_var_var'_v_list) (x_var var))) = x' then Some ty_x_m else (\<Gamma> ++ map_of (map (\<lambda>((y, cl, var, var', v), y', y). (x_var var', y)) (zip y_cl_var_var'_v_list y_ty_list))) (x_var (case_option var (case_x (\<lambda>var'. var') var) (map_of (map (\<lambda>(y, cl, var, var', v). (x_var var, x_var var')) y_cl_var_var'_v_list) (x_var var))))) = Some ty_var"

lemma ii_squared [simp]: "quat_ii\<^sup>2 = -1"

lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"

lemma OrdNotEqP_self_contra: "{x NEQ x} \<turnstile> Fls"

lemma ins_root_order: assumes "root_order k t" shows "root_order_up\<^sub>i k (ins k x t)"

lemma unite_wf_env: fixes e v w defines "e' \<equiv> unite v w e" assumes pre: "pre_dfss v e" and w: "w \<in> successors v" "w \<notin> vsuccs e v" "w \<in> visited e" "w \<notin> explored e" shows "wf_env e'"

lemma monad_state_alt_envT' [locale_witness]: "monad_state_alt return (bind :: ('a, 'm) bind) (get :: ('s, 'm) get) put alt \<Longrightarrow> monad_state_alt return (bind :: ('a, ('r, 'm) envT) bind) (get :: ('s, ('r, 'm) envT) get) put alt"

lemma d_delta_ln_upper_15: "x > 0 \<Longrightarrow> ((\<lambda>x. ln_upper_15 x - ln x) has_field_derivative diff_delta_ln_upper_15 x) (at x)"

lemma B'[simp]:"x \<noteq> bot \<Longrightarrow> x \<noteq> null \<Longrightarrow> (snd \<lceil>\<lceil>Rep_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>) \<noteq> bot"

lemma upper_real_interval[simp]: "upper (real_interval x) = upper x"

lemma us_mono: assumes "i < j" and "j < sum_list y" shows "\<^bold>u y i <\<^sub>v \<^bold>u y j"

lemma assumes "f \<in> X \<rightarrow>\<^sub>Q \<real>\<^sub>Q" and "\<alpha> \<in> qbs_Mx X" shows "(\<lambda>x. 2 * f (\<alpha> x) + (f (\<alpha> x))^2) \<in> \<real>\<^sub>Q \<rightarrow>\<^sub>Q \<real>\<^sub>Q"

lemma p_subid_interr: "(x \<cdot> z \<cdot> 1\<^sub>\<pi>) \<parallel> (y \<cdot> z \<cdot> 1\<^sub>\<pi>) = (x \<parallel> y) \<cdot> z \<cdot> 1\<^sub>\<pi>"

lemma upd_cond_alt: "upd_cond Q \<pi> u v' \<longleftrightarrow> (v',u) \<in> edges g \<and> v'\<notin>S (A Q \<pi>) \<and> enat (w {v',u}) < Q v'"

lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = add_mset a A"

lemma Neg_Imp_I [intro!]: "H \<turnstile> A \<Longrightarrow> insert B H \<turnstile> Fls \<Longrightarrow> H \<turnstile> Neg (A IMP B)"

lemma pt_unit_inst: shows "pt TYPE(unit) TYPE('x)"

lemma usedBy_bop [unrest]: "\<lbrakk> x \<natural> u; x \<natural> v \<rbrakk> \<Longrightarrow> x \<natural> bop f u v"

lemma Arccos_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Im(z + \<i> * csqrt(1 - z\<^sup>2))"

lemma o_subst: "subst \<pi>1 o subst \<pi>2 = subst (subst \<pi>1 o \<pi>2)"

lemma path_connected_arc_complement: fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space" assumes "arc \<gamma>" "2 \<le> DIM('a)" shows "path_connected(- path_image \<gamma>)"

lemma [simp]: "\<alpha> [] = (\<lambda>_. 0)"

lemma share_all_until_volatile_write_update_sb: assumes congr: "\<And>S. share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S" shows "\<And>\<S> i. \<lbrakk>i < length ts; ts!i = (p,is,\<theta>,sb,\<D>,\<O>,\<R>)\<rbrakk> \<Longrightarrow> share_all_until_volatile_write ts \<S> = share_all_until_volatile_write (ts[i := (p', is',\<theta>', sb', \<D>', \<O>',\<R>')]) \<S>"

lemma norm_ket_k_ge_K: "k \<ge> K \<Longrightarrow> inner_prod (ket_k k) (ket_k k) = 0"

lemma trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_finite[simp]: "finite (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p x)"

lemma closure_complete: assumes "lang r = lang s" shows "EX bs. closure as ([({r},{s})],[]) = Some([],bs)" (is ?C)

theorem wls_psubst_Op_simp[simp]: assumes "wlsInp delta inp" and "wlsBinp delta binp" and "wlsEnv rho" shows "((Op delta inp binp) #[rho]) = Op delta (inp %[rho]) (binp %%[rho])"

lemma rsum_append: "rsum (D1 @ D2) f = rsum D1 f + rsum D2 f"

lemma inv_atI[intro]: assumes "\<And> i. i < n \<Longrightarrow> p A i m" shows "inv_at p A m"

lemma Gromov_product_isometry: assumes "isometry_on UNIV f" shows "Gromov_product_at (f x) (f y) (f z) = Gromov_product_at x y z"

lemma Initial_Label: "CTXT 0 [] outp P \<Longrightarrow> P"

lemma reduce_implyP: \<open>reduce (ps \<^bold>\<leadsto>\<^sub>! q) = (map reduce ps \<^bold>\<leadsto>\<^sub>! reduce q)\<close>

lemma nat_diff_left_cancel_eq1: "\<lbrakk> k - m = k - (n::nat); m < k \<rbrakk> \<Longrightarrow> m = n"

lemma ac_fract: assumes "c \<in> carrier Q\<^sub>p" assumes "a \<in> nonzero Z\<^sub>p" assumes "b \<in> nonzero Z\<^sub>p" assumes "c = frac a b" shows "angular_component c = (ac_Zp a)\<otimes>\<^bsub>Z\<^sub>p\<^esub> inv \<^bsub>Z\<^sub>p\<^esub>(ac_Zp b)"

lemma analytic_at: "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"

lemma inc_above_dec_above_iff: "inc_above b i (dec_above b i x) = x \<longleftrightarrow> x < b \<or> b + i \<le> x"

lemma array_map_conv_foldl_array_set: assumes len: "array_length A = array_length a" shows "array_map f a = foldl (\<lambda>A (k, v). array_set A k (f k v)) A (assoc_list_of_array a)"

lemma real_less_ereal_iff: "real_of_ereal y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"

lemma rejects_termination: "observe_execution drinks 0 <> [(STR ''select'', [Str ''coke'']), (STR ''rejects'', [Num 50]), (STR ''coin'', [Num 50])] = [[]]"

lemma all_edges_between_subset: "all_edges_between X Y G \<subseteq> X\<times>Y"

lemma sdnets_noteq: "onlyTwoNets a \<Longrightarrow> onlyTwoNets aa \<Longrightarrow> first_bothNet a \<noteq> first_bothNet aa \<Longrightarrow> \<not> member DenyAll a \<Longrightarrow> \<not> member DenyAll aa \<Longrightarrow> sdnets a \<noteq> sdnets aa"

lemma bisimScopeExtSym: fixes x :: name and Q :: "('a, 'b, 'c) psi" and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>" and "x \<sharp> Q" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> (\<lparr>\<nu>x\<rparr>P) \<parallel> Q"

lemma "0 \<le> (n :: nat)"

lemma lconf_hext: "\<lbrakk> P,h \<turnstile> l (:\<le>)\<^sub>w E; h \<unlhd> h' \<rbrakk> \<Longrightarrow> P,h' \<turnstile> l (:\<le>)\<^sub>w E"

lemma is_fieldD: "is_field m \<Longrightarrow> \<exists> declC f. m=(declC,fdecl f)"

lemma set_iterator_rule_P: "\<lbrakk> set_iterator it S0; I S0 \<sigma>0; !!S \<sigma> x. \<lbrakk> c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0 \<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>); !!\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>; !!\<sigma> S. S \<subseteq> S0 \<Longrightarrow> S \<noteq> {} \<Longrightarrow> \<not> c \<sigma> \<Longrightarrow> I S \<sigma> \<Longrightarrow> P \<sigma> \<rbrakk> \<Longrightarrow> P (it c f \<sigma>0)"

lemma valid_dbm_non_empty_diag: assumes "valid_dbm M" "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<forall> k \<le> n. M k k \<ge> \<one>"

lemma PhiWhilePOp_Monotone:"Monotone (PhiWhilePOp A b \<Phi>)"