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

Statement: stringlengths 7 24.3k
lemma prime_field_finite_field_ops_integer: "prime_field_gen (finite_field_ops_integer (integer_of_int p)) mod_ring_rel_integer p"
lemma measurable_restrict[measurable (raw)]: assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
lemma string_unop_type_det: "string_unop_type op \<tau> \<sigma>\<^sub>1 \<Longrightarrow> string_unop_type op \<tau> \<sigma>\<^sub>2 \<Longrightarrow> \<sigma>\<^sub>1 = \<sigma>\<^sub>2"
lemma wf_darcs_normalize1: "wf_darcs t1 \<Longrightarrow> wf_darcs (normalize1 t1)"
lemma same_values_fo_int: assumes "value_is_compatible_with_structure f" shows "fo_interpretation (same_values f)"
lemma nn_integral_nat_function: fixes f :: "'a \<Rightarrow> nat" assumes "f \<in> measurable M (count_space UNIV)" shows "(\<integral>\<^sup>+x. of_nat (f x) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"
lemma neg_meas_setD1 : assumes "neg_meas_set E" shows "E \<in> sets M"
lemma is_expansion_lift: assumes "basis_wf (b # basis)" "is_expansion C basis" shows "is_expansion (lift_ms C) (b # basis)"
lemma bounded_disc_\<P>\<^sub>1': "bounded ((\<lambda>x. ((\<P>\<^sub>1 x ^^ m) v) s) ` X)"
lemma rolling_rule_ltr: "fix\<cdot>(g oo f) \<sqsubseteq> g\<cdot>(fix\<cdot>(f oo g))"
lemma fixes xs :: "'a list" and ys :: "'b list" shows "foo (x # xs) ys = (case ys of [] \<Rightarrow> 0 \| _ # _ \<Rightarrow> foo ([] :: 'a list) ([] :: 'b list))"
lemma de_morgan_orthogonal_complement_inter: fixes A B::"'a::chilbert_space set" assumes a1: \<open>closed_csubspace A\<close> and a2: \<open>closed_csubspace B\<close> shows \<open>orthogonal_complement (A \<inter> B) = orthogonal_complement A +\<^sub>M orthogonal_complement B\<close>
lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
lemma fst_initmissing_netgmap_default_toy_init_netlift: "fst (initmissing (netgmap sr s)) = default toy_init (netlift sr s)"
lemma non_speculative_readI [intro?]: "(\<And>ttas s' t x ta x' m' i ad al v v'. \<lbrakk> s -\<triangleright>ttas\<rightarrow>* s'; non_speculative P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))); thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>; t \<turnstile> (x, shr s') -ta\<rightarrow> (x', m'); actions_ok s' t ta; i < length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>; non_speculative P (w_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)); \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v); v' \<in> w_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas) @ take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) (ad, al) \<rbrakk> \<Longrightarrow> \<exists>ta' x'' m''. t \<turnstile> (x, shr s') -ta'\<rightarrow> (x'', m'') \<and> actions_ok s' t ta' \<and> i < length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> \<and> take i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> = take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<and> \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v') \<and> length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> \<le> max n (length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) \<Longrightarrow> non_speculative_read n s vs"
theorem wls_subst_vsubst_trans: assumes "wls s X" and "wls (asSort ys) Y" and "fresh ys y1 X" shows "((X #[y1 // y]_ys) #[Y / y1]_ys) = (X #[Y / y]_ys)"
lemma lad_on_substitutes_on_irc_on: assumes "f_range_on A f" assumes "substitutes_on A f" assumes "lad_on A f" shows "irc_on A f"
lemma r_contains_prim: "prim_recfn 1 (r_contains xs)"
lemma skl1_step4_refines_a0i_commit_skip_r: "{R0sk1iar \<inter> UNIV\<times>skl1_inv6} a0i_commit A B \<langle>Ni, NonceF (Rb$nr), Exp gnx (NonceF (Rb$ny))\<rangle> \<union> Id, skl1_step4 Rb A B Ni gnx {>R0sk1iar}"
lemma triv_spref: "s \<noteq> \<epsilon> \<Longrightarrow> r <p r \<cdot> s"
lemma (in encoding_wrt_barbs) enc_weakly_reflects_barbs_iff_source_target_rel: shows "enc_weakly_reflects_barbs = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_weakly_reflects_barbs Rel (STCalWB SWB TWB))"
lemma mpoly_map_vars_mult [simp]: "mpoly_map_vars f (p * q) = mpoly_map_vars f p * mpoly_map_vars f q"
lemma ennreal_real_conv_ennreal_of_enat: "ennreal (real n) = ennreal_of_enat n"
lemma size_subF: "!!theta. size (subF theta A) = size (A::formula)"
lemma steps_r_alt: "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto>* \<langle>l'', R''\<rangle> \<Longrightarrow> A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle> \<Longrightarrow> A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l'', R''\<rangle>"
lemma Ana_abs_aux1: fixes \<delta>::"(('fun,'atom,'sets,'lbl) prot_fun, nat, ('fun,'atom,'sets,'lbl) prot_var) gsubst" and \<alpha>::"nat \<Rightarrow> 'sets set" assumes "Ana\<^sub>f f = (K,T)" shows "(K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>\<alpha>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<alpha> = K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (\<lambda>n. \<delta> n \<cdot>\<^sub>\<alpha> \<alpha>)"
lemma vfield_iff: "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r \<longleftrightarrow> (\<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<or> \<langle>b, a\<rangle> \<in>\<^sub>\<circ> r)"
lemma has_fresh_z2: fixes t::"'b::fs" shows "\<exists>z c. atom z \<sharp> t \<and> \<tau> = \<lbrace> z : b_of \<tau> \| c \<rbrace>"
lemma add_block_wf: "incidence_system (\<V> \<union> b) (add_block b)"
lemma subset_dgrad_set_zero: "F \<subseteq> dgrad_set (\<lambda>_. 0) m"
lemma map_eq_appendE: assumes "map f ls = fl@fl'" obtains l l' where "ls=l@l'" and "map f l=fl" and "map f l' = fl'"
lemma Inv_in_Hom [simp]: assumes "Can t" shows "Inv t \<in> Hom (Cod t) (Dom t)"
lemma p_ge_0: "0 < int p"
lemma setdist_pos_le [simp]: "0 \<le> setdist s t"
lemma "P (x::('a, 'b) T3)"
lemma concat_index_split_sound_fst_arg: "i < length (concat xs) \<Longrightarrow> fst (concat_index_split (0, i) xs) < length xs"
lemma sumset_iterated_empty: "r>0 \<Longrightarrow> sumset_iterated {} r = {}"
lemma mult_cancel: assumes "trans s" and "irrefl s" shows "(X + Z, Y + Z) \<in> mult s \<longleftrightarrow> (X, Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
lemma list_match_lel_lel: assumes "c1 @ qs # c2 = c1' @ qs' # c2'" obtains (left) c21' where "c1 = c1' @ qs' # c21'" "c2' = c21' @ qs # c2" \| (center) "c1' = c1" "qs' = qs" "c2' = c2" \| (right) c21 where "c1' = c1 @ qs # c21" "c2 = c21 @ qs' # c2'"
lemma keys_monom_multE: assumes "v \<in> keys (monom_mult c t p)" obtains u where "u \<in> keys p" and "v = t \<oplus> u"
lemma G'_steps_V_all: "list_all V xs" if "G'.steps xs" "V (hd xs)"
lemma scale_shift_ms_aux_conv_mslmap: "scale_shift_ms_aux x = mslmap (map_prod ((*) (fst x)) ((+) (snd x)))"
lemma BIT_no_paid: "\<forall>((free,paid),_) \<in> (BIT_step s q). paid=[]"
lemma sumCases[consumes 1, case_names cSum1 cSum2]: fixes P :: ccs and Q :: ccs and \<alpha> :: act and R :: ccs assumes "P \<oplus> Q \<longmapsto>\<alpha> \<prec> R" and "\<And>P'. P \<longmapsto>\<alpha> \<prec> P' \<Longrightarrow> Prop P'" and "\<And>Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> Prop Q'" shows "Prop R"
lemma Integer\<^sub>n\<^sub>u\<^sub>l\<^sub>l_defined : "\<delta> Integer\<^sub>n\<^sub>u\<^sub>l\<^sub>l = true"
lemma distinct_scaled_inl: "distinct (map ((*\<^sub>R) 2) (ccw.selsort 0 (inl (snd X))))"
lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
lemma nfv_And[simp]: "nfv (And \<phi> \<psi>) = max (nfv \<phi>) (nfv \<psi>)"
lemma has_notempty_1: assumes Sne: "S \<noteq> {}" shows "has 1 S"
lemma (in Ring) LSM_sub_M:"\<lbrakk>R module M; T \<subseteq> carrier M\<rbrakk> \<Longrightarrow> (LSM\<^bsub>R\<^esub> M T) \<subseteq> carrier M"
lemma equal_div_mod:assumes "((j::nat) div a) = (i div a)" and "(j mod a) = (i mod a)" shows "j = i"
lemma ordLeq_cmax: assumes r: "Card_order r" and s: "Card_order s" shows "r \<le>o cmax r s \<and> s \<le>o cmax r s"
lemma cmap_insort_empty_cong: assumes "xs = ys" and "\<And>x. x \<in>\<in> xs \<Longrightarrow> ff x = gg x" and x: "ff x = []" shows "cmap ff (insort_key f x xs) = cmap gg ys"
theorem (in xvalid_program_progress) concurrent_direct_execution_simulates_store_buffer_execution_empty: assumes exec_sb: "(ts\<^sub>s\<^sub>b,m\<^sub>s\<^sub>b,x) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts\<^sub>s\<^sub>b',m\<^sub>s\<^sub>b',x')" assumes init: "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b" assumes valid: "valid ts\<^sub>s\<^sub>b" (* FIXME: move into initial ?) assumes empty: "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>s\<^sub>b' \<longrightarrow> ts\<^sub>s\<^sub>b'!i=(p,is,xs,sb,\<D>,\<O>,\<R>)\<longrightarrow> sb=[]" assumes sim: "(ts\<^sub>s\<^sub>b,m\<^sub>s\<^sub>b,\<S>\<^sub>s\<^sub>b) \<sim> (ts,m,\<S>)" assumes safe: "safe_reach_direct safe_free_flowing (ts,m,\<S>)" shows "\<exists>ts' \<S>'. (ts,m,\<S>) \<Rightarrow>\<^sub>d\<^sup> (ts',m\<^sub>s\<^sub>b',\<S>') \<and> ts\<^sub>s\<^sub>b' \<sim>\<^sub>d ts'"
lemma OclReject_invalid[simp,code_unfold]:"invalid->reject\<^sub>S\<^sub>e\<^sub>t(a \| P a) = invalid"
lemma ngbai_gi_refine[autoref_rules]: fixes S :: "('statei \<times> 'state) set" assumes "SIDE_GEN_ALGO (is_bounded_hashcode S seq bhc)" assumes "SIDE_GEN_ALGO (is_valid_def_hm_size TYPE('statei) hms)" assumes "GEN_OP seq HOL.eq (S \<rightarrow> S \<rightarrow> bool_rel)" shows "(NGBA_Algorithms.ngbai_gi seq bhc hms, ngba_g) \<in> \<langle>L, S\<rangle> ngbai_ngba_rel \<rightarrow> \<langle>unit_rel, S\<rangle> g_impl_rel_ext"
lemma code_GS_rec_am_arr_opt[code]: "opt_policy_gs'' (vec_to_bfun v) = vec_to_fun ((snd (foldl (\<lambda>(v, d) s. let (am, m) = least_max_arg_max_enum (\<lambda>a. r (s, a) + l * (\<Sum>s' \<in> UNIV. pmf (K (s,a)) s' * v $ s')) (A s) in (vec_upd v s m, vec_upd d s am)) (v, (\<chi> s. (least_enum (\<lambda>a. a \<in> A s)))) (sorted_list_of_set UNIV))))"
lemma fmdrop_set_single[simp]: "fmdrop_set {a} m = fmdrop a m"
lemma nat_of_natural_of_nat_inverse [simp]: "nat_of_natural (natural_of_nat n) = n"
lemma GR': \<open>\<turnstile> \<^bold>\<not> p\<langle>\<^bold>#n/0\<rangle> \<^bold>\<longrightarrow> q \<Longrightarrow> max_var p < n \<Longrightarrow> max_var q < n \<Longrightarrow> \<turnstile> \<^bold>\<not> \<^bold>\<forall>p \<^bold>\<longrightarrow> q\<close>
lemma rebalance_middle_tree_height: assumes "height t = height sub" and "case rs of (rsub,rsep) # list \<Rightarrow> height rsub = height t \| [] \<Rightarrow> True" shows "height (rebalance_middle_tree k ls sub sep rs t) = height (Node (ls@(sub,sep)#rs) t)"
lemma tick_count_strict_sub: assumes \<open>dilating f sub r\<close> shows \<open>tick_count_strict sub c n = tick_count_strict r c (f n)\<close>
lemma ord_spmfI: "\<lbrakk> \<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> ord x y; map_spmf fst pq = p; map_spmf snd pq = q \<rbrakk> \<Longrightarrow> p \<sqsubseteq> q"
lemma one_over_fun_closed: assumes "f \<in> carrier (SA n)" shows "one_over_fun n f \<in> carrier (SA n)"
lemma pnet_maintains_dom: assumes "(s, a, s') \<in> trans (pnet np p)" shows "net_ips s = net_ips s'"
lemma baEN2[elim]: fixes c::'id and t::"nat \<Rightarrow> cnf" and t'::"nat \<Rightarrow> 'cmp" and n::nat assumes nAct: "\<nexists>i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" and al: "eval c t t' n (ba \<phi>)" shows "\<phi> (t' n)"
lemma mset_le_subtract_right: "(A::'a multiset)+B \<subseteq># X \<Longrightarrow> A \<subseteq># X-B \<and> B\<subseteq>#X"
lemma poly_ring_add_mono: assumes "n \<le> m" assumes "A \<in> carrier (R[\<X>\<^bsub>n\<^esub>])" assumes "B \<in> carrier (R[\<X>\<^bsub>n\<^esub>])" shows "A \<oplus>\<^bsub>R[\<X>\<^bsub>n\<^esub>]\<^esub> B = A \<oplus>\<^bsub>coord_ring R m\<^esub> B"
lemma "x > y \<Longrightarrow> x - y \<noteq> (0::int)"
lemma bind_left_fail_SE[simp] : "(x \<leftarrow> fail\<^sub>S\<^sub>E; P x) = fail\<^sub>S\<^sub>E"
lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
lemma A81_DAcc_level1: "DAcc level1 sA81 = {sA91, sA92}"
lemma used_appIR: "X \<in> used evs \<Longrightarrow> X \<in> used (evs @ evs')"
lemma int_prime_not_sqr : assumes "is_prime p" shows "int p \<noteq> n\<^sup>2"
lemma spair_sigs_alt_spp: "spair_sigs p q = spair_sigs_spp (spp_of p) (spp_of q)"
lemma word_mod_def [code]: "a mod b = word_of_int (uint a mod uint b)"
lemma sup_co_test: "co_test x \<Longrightarrow> co_test y \<Longrightarrow> co_test (x \<squnion> y)"
lemma stutter_equiv_sym [sym]: "\<sigma> \<approx> \<tau> \<Longrightarrow> \<tau> \<approx> \<sigma>"
lemma phull_closed_lin_red: assumes "phull B \<subseteq> phull A" and "p \<in> phull A" and "lin_red B p q" shows "q \<in> phull A"
lemma heap_upds_ok_invariant: "invariant step heap_upds_ok_conf"
lemma [simp]: "A \<sqsubseteq> A"
lemma homeomorphic_frontiers: fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" assumes "S homeomorphic T" "closed S" "closed T" "interior S = {} \<longleftrightarrow> interior T = {}" shows "(frontier S) homeomorphic (frontier T)"
lemma preamble_maximal_io_paths : assumes "is_preamble P M q" and "observable M" and "path P (initial P) p" and "target (initial P) p = q" shows "\<nexists>io' . io' \<noteq> [] \<and> p_io p @ io' \<in> L P"
lemma genempty_substs: "genempty Q \<Longrightarrow> length xs = length ys \<Longrightarrow> genempty (Q[xs \<^bold>\<rightarrow>\<^sup>* ys])"
lemma up_dist_sup: "\<up>(x \<squnion> y) = \<up>x \<inter> \<up>y"
lemma init_prefixE [elim]: "prefix f n = prefix g n \<Longrightarrow> f \<triangleright> n = g \<triangleright> n"
lemma dtree_leaf_child_empty: "leaf r \<Longrightarrow> {(x,e). (if e \<in> out_arcs T r then x = to_dtree_aux (head T e) else False)} = {}"
lemma proc_ord2_set: "P \<sqsubseteq> Q \<Longrightarrow> {(s, X). s \<notin> D P \<and> (s, X) \<in> F P} = {(s, X). s \<notin> D P \<and> (s, X) \<in> F Q}"
lemma map_nth_drop: "i < length xs \<Longrightarrow> map f (nth_drop i xs) = nth_drop i (map f xs)"
lemma is_processT6_S1: "\<lbrakk> tick \<notin> X;(s @ [tick], {}) \<in> F P \<rbrakk> \<Longrightarrow> (s::'a event list, X) \<in> F P"
lemma Sup_trans_least: "\<lbrakk> \<forall>t\<in>S. le_utrans t u; \<And>P. unitary P \<Longrightarrow> unitary (u P) \<rbrakk> \<Longrightarrow> le_utrans (Sup_trans S) u"
lemma lagrange_interpolation_poly: assumes dist: "distinct (map fst xs_ys)" and p: "p = lagrange_interpolation_poly xs_ys" shows "\<And> x y. (x,y) \<in> set xs_ys \<Longrightarrow> poly p x = y"
lemma BIT_b2: assumes A: "x \<noteq> y" "init \<in> {[x,y],[y,x]}" "v \<in> lang (seq [Atom x, Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) v = 0.75 * (length v - 1) - 0.5" (is ?T) and "config'_rand BIT (type0 init x y) v = (type0 init y x)" (is ?C)
lemma liftLM_comm: "liftL_trm (liftM_trm t n) m = liftM_trm (liftL_trm t m) n" "liftL_cmd (liftM_cmd c n) m = liftM_cmd (liftL_cmd c m) n"
lemma observable_path_unique : assumes "observable M" and "path M q p" and "path M q p'" and "p_io p = p_io p'" shows "p = p'"
lemma in_nv2T: "x \<in> nv2T T \<longleftrightarrow> T = Var x"
lemma empty_object_not_conflicting_objects [elim!]: "(sep_heap y) obj_id = None \<Longrightarrow> not_conflicting_objects x y obj_id"
lemma Union_included_in_supp: fixes S::"('a::fs set)" assumes fin: "finite S" shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"
lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
lemma cf_comp_is_tiny_functor[cat_small_cs_intros]: assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>C\<^sub>F \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<CC>"

lemma prime_field_finite_field_ops_integer: "prime_field_gen (finite_field_ops_integer (integer_of_int p)) mod_ring_rel_integer p"

lemma measurable_restrict[measurable (raw)]: assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"

lemma string_unop_type_det: "string_unop_type op \<tau> \<sigma>\<^sub>1 \<Longrightarrow> string_unop_type op \<tau> \<sigma>\<^sub>2 \<Longrightarrow> \<sigma>\<^sub>1 = \<sigma>\<^sub>2"

lemma wf_darcs_normalize1: "wf_darcs t1 \<Longrightarrow> wf_darcs (normalize1 t1)"

lemma same_values_fo_int: assumes "value_is_compatible_with_structure f" shows "fo_interpretation (same_values f)"

lemma nn_integral_nat_function: fixes f :: "'a \<Rightarrow> nat" assumes "f \<in> measurable M (count_space UNIV)" shows "(\<integral>\<^sup>+x. of_nat (f x) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"

lemma neg_meas_setD1 : assumes "neg_meas_set E" shows "E \<in> sets M"

lemma is_expansion_lift: assumes "basis_wf (b # basis)" "is_expansion C basis" shows "is_expansion (lift_ms C) (b # basis)"

lemma bounded_disc_\<P>\<^sub>1': "bounded ((\<lambda>x. ((\<P>\<^sub>1 x ^^ m) v) s) ` X)"

lemma rolling_rule_ltr: "fix\<cdot>(g oo f) \<sqsubseteq> g\<cdot>(fix\<cdot>(f oo g))"

lemma fixes xs :: "'a list" and ys :: "'b list" shows "foo (x # xs) ys = (case ys of [] \<Rightarrow> 0 | _ # _ \<Rightarrow> foo ([] :: 'a list) ([] :: 'b list))"

lemma de_morgan_orthogonal_complement_inter: fixes A B::"'a::chilbert_space set" assumes a1: \<open>closed_csubspace A\<close> and a2: \<open>closed_csubspace B\<close> shows \<open>orthogonal_complement (A \<inter> B) = orthogonal_complement A +\<^sub>M orthogonal_complement B\<close>

lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"

lemma fst_initmissing_netgmap_default_toy_init_netlift: "fst (initmissing (netgmap sr s)) = default toy_init (netlift sr s)"

lemma non_speculative_readI [intro?]: "(\<And>ttas s' t x ta x' m' i ad al v v'. \<lbrakk> s -\<triangleright>ttas\<rightarrow>* s'; non_speculative P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))); thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>; t \<turnstile> (x, shr s') -ta\<rightarrow> (x', m'); actions_ok s' t ta; i < length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>; non_speculative P (w_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)); \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v); v' \<in> w_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas) @ take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) (ad, al) \<rbrakk> \<Longrightarrow> \<exists>ta' x'' m''. t \<turnstile> (x, shr s') -ta'\<rightarrow> (x'', m'') \<and> actions_ok s' t ta' \<and> i < length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> \<and> take i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> = take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<and> \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v') \<and> length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> \<le> max n (length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) \<Longrightarrow> non_speculative_read n s vs"

theorem wls_subst_vsubst_trans: assumes "wls s X" and "wls (asSort ys) Y" and "fresh ys y1 X" shows "((X #[y1 // y]_ys) #[Y / y1]_ys) = (X #[Y / y]_ys)"

lemma lad_on_substitutes_on_irc_on: assumes "f_range_on A f" assumes "substitutes_on A f" assumes "lad_on A f" shows "irc_on A f"

lemma r_contains_prim: "prim_recfn 1 (r_contains xs)"

lemma skl1_step4_refines_a0i_commit_skip_r: "{R0sk1iar \<inter> UNIV\<times>skl1_inv6} a0i_commit A B \<langle>Ni, NonceF (Rb$nr), Exp gnx (NonceF (Rb$ny))\<rangle> \<union> Id, skl1_step4 Rb A B Ni gnx {>R0sk1iar}"

lemma triv_spref: "s \<noteq> \<epsilon> \<Longrightarrow> r <p r \<cdot> s"

lemma (in encoding_wrt_barbs) enc_weakly_reflects_barbs_iff_source_target_rel: shows "enc_weakly_reflects_barbs = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_weakly_reflects_barbs Rel (STCalWB SWB TWB))"

lemma mpoly_map_vars_mult [simp]: "mpoly_map_vars f (p * q) = mpoly_map_vars f p * mpoly_map_vars f q"

lemma ennreal_real_conv_ennreal_of_enat: "ennreal (real n) = ennreal_of_enat n"

lemma size_subF: "!!theta. size (subF theta A) = size (A::formula)"

lemma steps_r_alt: "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto>* \<langle>l'', R''\<rangle> \<Longrightarrow> A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle> \<Longrightarrow> A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l'', R''\<rangle>"

lemma Ana_abs_aux1: fixes \<delta>::"(('fun,'atom,'sets,'lbl) prot_fun, nat, ('fun,'atom,'sets,'lbl) prot_var) gsubst" and \<alpha>::"nat \<Rightarrow> 'sets set" assumes "Ana\<^sub>f f = (K,T)" shows "(K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>\<alpha>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<alpha> = K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (\<lambda>n. \<delta> n \<cdot>\<^sub>\<alpha> \<alpha>)"

lemma vfield_iff: "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r \<longleftrightarrow> (\<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<or> \<langle>b, a\<rangle> \<in>\<^sub>\<circ> r)"

lemma has_fresh_z2: fixes t::"'b::fs" shows "\<exists>z c. atom z \<sharp> t \<and> \<tau> = \<lbrace> z : b_of \<tau> | c \<rbrace>"

lemma add_block_wf: "incidence_system (\<V> \<union> b) (add_block b)"

lemma subset_dgrad_set_zero: "F \<subseteq> dgrad_set (\<lambda>_. 0) m"

lemma map_eq_appendE: assumes "map f ls = fl@fl'" obtains l l' where "ls=l@l'" and "map f l=fl" and "map f l' = fl'"

lemma Inv_in_Hom [simp]: assumes "Can t" shows "Inv t \<in> Hom (Cod t) (Dom t)"

lemma p_ge_0: "0 < int p"

lemma setdist_pos_le [simp]: "0 \<le> setdist s t"

lemma "P (x::('a, 'b) T3)"

lemma concat_index_split_sound_fst_arg: "i < length (concat xs) \<Longrightarrow> fst (concat_index_split (0, i) xs) < length xs"

lemma sumset_iterated_empty: "r>0 \<Longrightarrow> sumset_iterated {} r = {}"

lemma mult_cancel: assumes "trans s" and "irrefl s" shows "(X + Z, Y + Z) \<in> mult s \<longleftrightarrow> (X, Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")

lemma list_match_lel_lel: assumes "c1 @ qs # c2 = c1' @ qs' # c2'" obtains (left) c21' where "c1 = c1' @ qs' # c21'" "c2' = c21' @ qs # c2" | (center) "c1' = c1" "qs' = qs" "c2' = c2" | (right) c21 where "c1' = c1 @ qs # c21" "c2 = c21 @ qs' # c2'"

lemma keys_monom_multE: assumes "v \<in> keys (monom_mult c t p)" obtains u where "u \<in> keys p" and "v = t \<oplus> u"

lemma G'_steps_V_all: "list_all V xs" if "G'.steps xs" "V (hd xs)"

lemma scale_shift_ms_aux_conv_mslmap: "scale_shift_ms_aux x = mslmap (map_prod ((*) (fst x)) ((+) (snd x)))"

lemma BIT_no_paid: "\<forall>((free,paid),_) \<in> (BIT_step s q). paid=[]"

lemma sumCases[consumes 1, case_names cSum1 cSum2]: fixes P :: ccs and Q :: ccs and \<alpha> :: act and R :: ccs assumes "P \<oplus> Q \<longmapsto>\<alpha> \<prec> R" and "\<And>P'. P \<longmapsto>\<alpha> \<prec> P' \<Longrightarrow> Prop P'" and "\<And>Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> Prop Q'" shows "Prop R"

lemma Integer\<^sub>n\<^sub>u\<^sub>l\<^sub>l_defined : "\<delta> Integer\<^sub>n\<^sub>u\<^sub>l\<^sub>l = true"

lemma distinct_scaled_inl: "distinct (map ((*\<^sub>R) 2) (ccw.selsort 0 (inl (snd X))))"

lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"

lemma nfv_And[simp]: "nfv (And \<phi> \<psi>) = max (nfv \<phi>) (nfv \<psi>)"

lemma has_notempty_1: assumes Sne: "S \<noteq> {}" shows "has 1 S"

lemma (in Ring) LSM_sub_M:"\<lbrakk>R module M; T \<subseteq> carrier M\<rbrakk> \<Longrightarrow> (LSM\<^bsub>R\<^esub> M T) \<subseteq> carrier M"

lemma equal_div_mod:assumes "((j::nat) div a) = (i div a)" and "(j mod a) = (i mod a)" shows "j = i"

lemma ordLeq_cmax: assumes r: "Card_order r" and s: "Card_order s" shows "r \<le>o cmax r s \<and> s \<le>o cmax r s"

lemma cmap_insort_empty_cong: assumes "xs = ys" and "\<And>x. x \<in>\<in> xs \<Longrightarrow> ff x = gg x" and x: "ff x = []" shows "cmap ff (insort_key f x xs) = cmap gg ys"

theorem (in xvalid_program_progress) concurrent_direct_execution_simulates_store_buffer_execution_empty: assumes exec_sb: "(ts\<^sub>s\<^sub>b,m\<^sub>s\<^sub>b,x) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts\<^sub>s\<^sub>b',m\<^sub>s\<^sub>b',x')" assumes init: "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b" assumes valid: "valid ts\<^sub>s\<^sub>b" (* FIXME: move into initial ?*) assumes empty: "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>s\<^sub>b' \<longrightarrow> ts\<^sub>s\<^sub>b'!i=(p,is,xs,sb,\<D>,\<O>,\<R>)\<longrightarrow> sb=[]" assumes sim: "(ts\<^sub>s\<^sub>b,m\<^sub>s\<^sub>b,\<S>\<^sub>s\<^sub>b) \<sim> (ts,m,\<S>)" assumes safe: "safe_reach_direct safe_free_flowing (ts,m,\<S>)" shows "\<exists>ts' \<S>'. (ts,m,\<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts',m\<^sub>s\<^sub>b',\<S>') \<and> ts\<^sub>s\<^sub>b' \<sim>\<^sub>d ts'"

lemma OclReject_invalid[simp,code_unfold]:"invalid->reject\<^sub>S\<^sub>e\<^sub>t(a | P a) = invalid"

lemma ngbai_gi_refine[autoref_rules]: fixes S :: "('statei \<times> 'state) set" assumes "SIDE_GEN_ALGO (is_bounded_hashcode S seq bhc)" assumes "SIDE_GEN_ALGO (is_valid_def_hm_size TYPE('statei) hms)" assumes "GEN_OP seq HOL.eq (S \<rightarrow> S \<rightarrow> bool_rel)" shows "(NGBA_Algorithms.ngbai_gi seq bhc hms, ngba_g) \<in> \<langle>L, S\<rangle> ngbai_ngba_rel \<rightarrow> \<langle>unit_rel, S\<rangle> g_impl_rel_ext"

lemma code_GS_rec_am_arr_opt[code]: "opt_policy_gs'' (vec_to_bfun v) = vec_to_fun ((snd (foldl (\<lambda>(v, d) s. let (am, m) = least_max_arg_max_enum (\<lambda>a. r (s, a) + l * (\<Sum>s' \<in> UNIV. pmf (K (s,a)) s' * v $ s')) (A s) in (vec_upd v s m, vec_upd d s am)) (v, (\<chi> s. (least_enum (\<lambda>a. a \<in> A s)))) (sorted_list_of_set UNIV))))"

lemma fmdrop_set_single[simp]: "fmdrop_set {a} m = fmdrop a m"

lemma nat_of_natural_of_nat_inverse [simp]: "nat_of_natural (natural_of_nat n) = n"

lemma GR': \<open>\<turnstile> \<^bold>\<not> p\<langle>\<^bold>#n/0\<rangle> \<^bold>\<longrightarrow> q \<Longrightarrow> max_var p < n \<Longrightarrow> max_var q < n \<Longrightarrow> \<turnstile> \<^bold>\<not> \<^bold>\<forall>p \<^bold>\<longrightarrow> q\<close>

lemma rebalance_middle_tree_height: assumes "height t = height sub" and "case rs of (rsub,rsep) # list \<Rightarrow> height rsub = height t | [] \<Rightarrow> True" shows "height (rebalance_middle_tree k ls sub sep rs t) = height (Node (ls@(sub,sep)#rs) t)"

lemma tick_count_strict_sub: assumes \<open>dilating f sub r\<close> shows \<open>tick_count_strict sub c n = tick_count_strict r c (f n)\<close>

lemma ord_spmfI: "\<lbrakk> \<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> ord x y; map_spmf fst pq = p; map_spmf snd pq = q \<rbrakk> \<Longrightarrow> p \<sqsubseteq> q"

lemma one_over_fun_closed: assumes "f \<in> carrier (SA n)" shows "one_over_fun n f \<in> carrier (SA n)"

lemma pnet_maintains_dom: assumes "(s, a, s') \<in> trans (pnet np p)" shows "net_ips s = net_ips s'"

lemma baEN2[elim]: fixes c::'id and t::"nat \<Rightarrow> cnf" and t'::"nat \<Rightarrow> 'cmp" and n::nat assumes nAct: "\<nexists>i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" and al: "eval c t t' n (ba \<phi>)" shows "\<phi> (t' n)"

lemma mset_le_subtract_right: "(A::'a multiset)+B \<subseteq># X \<Longrightarrow> A \<subseteq># X-B \<and> B\<subseteq>#X"

lemma poly_ring_add_mono: assumes "n \<le> m" assumes "A \<in> carrier (R[\<X>\<^bsub>n\<^esub>])" assumes "B \<in> carrier (R[\<X>\<^bsub>n\<^esub>])" shows "A \<oplus>\<^bsub>R[\<X>\<^bsub>n\<^esub>]\<^esub> B = A \<oplus>\<^bsub>coord_ring R m\<^esub> B"

lemma "x > y \<Longrightarrow> x - y \<noteq> (0::int)"

lemma bind_left_fail_SE[simp] : "(x \<leftarrow> fail\<^sub>S\<^sub>E; P x) = fail\<^sub>S\<^sub>E"

lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"

lemma A81_DAcc_level1: "DAcc level1 sA81 = {sA91, sA92}"

lemma used_appIR: "X \<in> used evs \<Longrightarrow> X \<in> used (evs @ evs')"

lemma int_prime_not_sqr : assumes "is_prime p" shows "int p \<noteq> n\<^sup>2"

lemma spair_sigs_alt_spp: "spair_sigs p q = spair_sigs_spp (spp_of p) (spp_of q)"

lemma word_mod_def [code]: "a mod b = word_of_int (uint a mod uint b)"

lemma sup_co_test: "co_test x \<Longrightarrow> co_test y \<Longrightarrow> co_test (x \<squnion> y)"

lemma stutter_equiv_sym [sym]: "\<sigma> \<approx> \<tau> \<Longrightarrow> \<tau> \<approx> \<sigma>"

lemma phull_closed_lin_red: assumes "phull B \<subseteq> phull A" and "p \<in> phull A" and "lin_red B p q" shows "q \<in> phull A"

lemma heap_upds_ok_invariant: "invariant step heap_upds_ok_conf"

lemma [simp]: "A \<sqsubseteq> A"

lemma homeomorphic_frontiers: fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" assumes "S homeomorphic T" "closed S" "closed T" "interior S = {} \<longleftrightarrow> interior T = {}" shows "(frontier S) homeomorphic (frontier T)"

lemma preamble_maximal_io_paths : assumes "is_preamble P M q" and "observable M" and "path P (initial P) p" and "target (initial P) p = q" shows "\<nexists>io' . io' \<noteq> [] \<and> p_io p @ io' \<in> L P"

lemma genempty_substs: "genempty Q \<Longrightarrow> length xs = length ys \<Longrightarrow> genempty (Q[xs \<^bold>\<rightarrow>\<^sup>* ys])"

lemma up_dist_sup: "\<up>(x \<squnion> y) = \<up>x \<inter> \<up>y"

lemma init_prefixE [elim]: "prefix f n = prefix g n \<Longrightarrow> f \<triangleright> n = g \<triangleright> n"

lemma dtree_leaf_child_empty: "leaf r \<Longrightarrow> {(x,e). (if e \<in> out_arcs T r then x = to_dtree_aux (head T e) else False)} = {}"

lemma proc_ord2_set: "P \<sqsubseteq> Q \<Longrightarrow> {(s, X). s \<notin> D P \<and> (s, X) \<in> F P} = {(s, X). s \<notin> D P \<and> (s, X) \<in> F Q}"

lemma map_nth_drop: "i < length xs \<Longrightarrow> map f (nth_drop i xs) = nth_drop i (map f xs)"

lemma is_processT6_S1: "\<lbrakk> tick \<notin> X;(s @ [tick], {}) \<in> F P \<rbrakk> \<Longrightarrow> (s::'a event list, X) \<in> F P"

lemma Sup_trans_least: "\<lbrakk> \<forall>t\<in>S. le_utrans t u; \<And>P. unitary P \<Longrightarrow> unitary (u P) \<rbrakk> \<Longrightarrow> le_utrans (Sup_trans S) u"

lemma lagrange_interpolation_poly: assumes dist: "distinct (map fst xs_ys)" and p: "p = lagrange_interpolation_poly xs_ys" shows "\<And> x y. (x,y) \<in> set xs_ys \<Longrightarrow> poly p x = y"

lemma BIT_b2: assumes A: "x \<noteq> y" "init \<in> {[x,y],[y,x]}" "v \<in> lang (seq [Atom x, Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) v = 0.75 * (length v - 1) - 0.5" (is ?T) and "config'_rand BIT (type0 init x y) v = (type0 init y x)" (is ?C)

lemma liftLM_comm: "liftL_trm (liftM_trm t n) m = liftM_trm (liftL_trm t m) n" "liftL_cmd (liftM_cmd c n) m = liftM_cmd (liftL_cmd c m) n"

lemma observable_path_unique : assumes "observable M" and "path M q p" and "path M q p'" and "p_io p = p_io p'" shows "p = p'"

lemma in_nv2T: "x \<in> nv2T T \<longleftrightarrow> T = Var x"

lemma empty_object_not_conflicting_objects [elim!]: "(sep_heap y) obj_id = None \<Longrightarrow> not_conflicting_objects x y obj_id"

lemma Union_included_in_supp: fixes S::"('a::fs set)" assumes fin: "finite S" shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"

lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"

lemma cf_comp_is_tiny_functor[cat_small_cs_intros]: assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>C\<^sub>F \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<CC>"