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

Statement: stringlengths 7 24.3k
lemma sub_qbs_Mx[simp]: "qbs_Mx (sub_qbs X U) = {f \<in> UNIV \<rightarrow> qbs_space X \<inter> U. f \<in> qbs_Mx X}"
lemma car_is_alg: "is_algebraic R n (carrier (R\<^bsup>n\<^esup>))"
lemma (in is_cat_finite_obj_prod) cat_fin_obj_prod_index_vfinite: "vfinite I"
lemma SKIP_preserves [iff]: "SKIP \<in> preserves v"
lemma SeqStTermP_cong: "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'; H \<turnstile> s EQ s'; H \<turnstile> k EQ k'\<rbrakk> \<Longrightarrow> H \<turnstile> SeqStTermP v i t u s k IFF SeqStTermP v i t' u' s' k'"
lemma op_cf_cf_cn_comp[cat_op_simps]: "op_cf (\<GG> \<^sub>C\<^sub>F\<circ> \<FF>) = op_cf \<GG> \<^sub>C\<^sub>F\<circ> op_cf \<FF>"
lemma Inf_lim: fixes a :: "'a::{complete_linorder,linorder_topology}" assumes "\<And>n. b n \<in> s" and "b \<longlonglongrightarrow> a" shows "Inf s \<le> a"
lemma lift0: "(liftPoly i j 0) = 0"
lemma vfinite_vcard_vinsert_nin[simp]: assumes "vfinite A" and "a \<notin>\<^sub>\<circ> A" shows "vcard (vinsert a A) = csucc (vcard A)"
lemma refines_adm_aux: "option.admissible (\<lambda>xa. \<forall>h a aa b. xa h = Some (a, aa, b) \<longrightarrow> execute (t x) h = Some (a, aa, b))"
lemma in_rem_implicit_pres_\<delta>: "op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> rem_implicit_pres op \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops prob))"
lemma spaceN_sum [simp]: assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> space\<^sub>N N" shows "(\<Sum>i\<in>I. x i) \<in> space\<^sub>N N"
lemma vsubset_vinsert: "A \<subseteq>\<^sub>\<circ> vinsert x B \<longleftrightarrow> (if x \<in>\<^sub>\<circ> A then A -\<^sub>\<circ> set {x} \<subseteq>\<^sub>\<circ> B else A \<subseteq>\<^sub>\<circ> B)"
lemma tm_semi_id_gt0_halts': shows "\<lbrace>\<lambda>tap. \<exists>l. tap = ([], [Oc, Oc] @ Bk\<up> l)\<rbrace> tm_semi_id_gt0 \<lbrace>\<lambda>tap. \<exists>l. tap = ([], [Oc, Oc] @ Bk\<up> l)\<rbrace>"
lemma fps_lr_inverse_fps_X_plus1: "fps_left_inverse (1 + fps_X) (1::'a::ring_1) = Abs_fps (\<lambda>n. (-1)^n)" "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)"
lemma aial_empty_impl: "(aial_empty,RETURN op_list_empty) \<in> \<langle>ial_rel1 maxsize\<rangle>nres_rel"
lemma hnorm_mult_ineq: "\<And>x y::'a::real_normed_algebra star. hnorm (x * y) \<le> hnorm x * hnorm y"
lemma const_finfun: "(\<lambda>x. a) \<in> finfun"
theorem ikkbz_sub_dverts_optimal: assumes "\<exists>x. fwd_sub root (dverts t) x" shows "\<exists>zs. fwd_sub root (dverts (ikkbz_sub t)) zs \<and> (\<forall>as. fwd_sub root (dverts t) as \<longrightarrow> cost (rev zs) \<le> cost (rev as))"
lemma effect_swapI [effect_intros]: assumes "i < length h a" "h' = update a i x h" "r = get h a ! i" shows "effect (swap i x a) h h' r"
lemma notInKeys_notInVars : "x \<notin> keys m \<Longrightarrow> x \<notin> vars(MPoly_Type.monom m a)"
lemma doesnt_read_or_modify_doesnt_modify: "doesnt_read_or_modify c x \<Longrightarrow> doesnt_modify c x"
lemma LESS_SUC[simp]: "LESS ord 0 (SUC ord idx)" "LESS ord (Suc l) (SUC ord idx) = LESS ord l idx" "ord \<noteq> ord' \<Longrightarrow> LESS ord l (SUC ord' idx) = LESS ord l idx" "LESS ord l idx \<Longrightarrow> LESS ord l (SUC ord' idx)"
lemma UNIV_eqvt [eqvt]: shows "p \<bullet> UNIV = UNIV"
lemma expands_to_imp_exp_ln_eq: assumes "(l expands_to L) bs" "eventually (\<lambda>x. l x \<le> f x) at_top" "trimmed_pos L" "basis_wf bs" shows "eventually (\<lambda>x. exp (ln (f x)) = f x) at_top"
lemma cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
lemma anonymousI[intro]: "(\<And>P f x y. \<lbrakk> profile A Is P; bij_betw f Is Is; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> (x \<^bsub>(scf P)\<^esub>\<preceq> y) = (x \<^bsub>(scf (P \<circ> f))\<^esub>\<preceq> y)) \<Longrightarrow> anonymous scf A Is"
lemma eliminate_aux_vars_of_int_poly: "eliminate_aux_vars (map_mpoly (of_int :: _ \<Rightarrow> 'a :: {comm_ring_1,ring_char_0}) mp) (of_int_poly \<circ> qs) is = of_int_poly (eliminate_aux_vars mp qs is)"
lemma (in order_topology) increasing_tendsto: assumes bdd: "eventually (\<lambda>n. f n \<le> l) F" and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F" shows "(f \<longlongrightarrow> l) F"
lemma not_exact_match_in_doubt_deny_approx_match: "matcher_agree_on_exact_matches \<gamma> \<beta> \<Longrightarrow> a = Accept \<or> a = Reject \<or> a = Drop \<Longrightarrow> \<not> Semantics.matches \<gamma> m p \<Longrightarrow> ((a = Drop \<or> a = Reject) \<and> Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p) \<or> \<not> Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p"
lemma exec_rename_to_exec: assumes \<Gamma>: "\<forall>p bdy. \<Gamma> p = Some bdy \<longrightarrow> \<Gamma>' (h p) = Some (rename h bdy)" assumes exec: "\<Gamma>'\<turnstile>\<langle>rc,s\<rangle> \<Rightarrow> t" shows "\<And>c. rename h c = rc\<Longrightarrow> \<exists>t'. \<Gamma>\<turnstile>\<langle>c,s\<rangle> \<Rightarrow> t' \<and> (t'=Stuck \<or> t'=t)"
lemma check_in_heap_returns_result [simp]: "ptr \|\<in>\| object_ptr_kinds h \<Longrightarrow> h \<turnstile> (check_in_heap ptr \<bind> f) \<rightarrow>\<^sub>r x = h \<turnstile> f () \<rightarrow>\<^sub>r x"
lemma single_LeadsTo_I: "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"
lemma trans_constraint_creats_LEQ_only: assumes "transf_constraint x \<noteq> []" shows "(\<forall>x \<in> set (transf_constraint x). \<exists>a b. x = LEQ a b)"
lemma backward_terminating_path_root_iff: "(\<exists>r . backward_terminating_path_root r x) \<longleftrightarrow> backward_terminating_path x"
lemma exp_cf_cat_cf_id_cat: assumes "category \<alpha> \<CC>" and "category \<alpha> \<DD>" shows "exp_cf_cat \<alpha> (cf_id \<CC>) \<DD> = cf_id (cat_FUNCT \<alpha> \<DD> \<CC>)"
lemma monad_commute_envT [locale_witness]: assumes "monad_commute return bind" shows "monad_commute return_env bind_env"
lemma partition_on_eq_quotient: assumes P: "partition_on A P" shows "A // {(x, y). \<exists>p \<in> P. x \<in> p \<and> y \<in> p} = P"
lemma Cons_not_Nil [iff]: "x # xs ~= Nil"
lemma dependent_inj_imageD: assumes d: "m2.dependent (f ` s)" and i: "inj_on f (m1.span s)" shows "m1.dependent s"
lemma tree_map_compose: "map_tree g (map_tree f t) = map_tree (g \<circ> f) t"
lemma expr_sem_pair_vars: "expr_sem \<sigma> <Var x, Var y> = return_val <\|\<sigma> x, \<sigma> y\|>"
lemma root_hash_T_simps [simp]: "root_hash_T rha (T\<^sub>m x) = T\<^sub>h (root_hash_F rha (root_hash_T rha) x)"
lemma fds_mangoldt': "fds mangoldt = fds_zeta * fds_deriv (fds moebius_mu)"
lemma OrdNotEqP_subst [simp]: "(OrdNotEqP x y)(i::=t) = OrdNotEqP (subst i t x) (subst i t y)"
lemma JFcol_bd: "\<forall>(j1 :: 'a JF1) (j2 :: 'a JF2). \|JF1col n j1\| <o bd_F \<and> \|JF2col n j2\| <o bd_F"
lemma permI: assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a" shows "f \<in> perm"
lemma computable_rec_inseparable_conv: "computable A \<longleftrightarrow> \<not> rec_inseparable A (- A)"
lemma bigtheta_f: obtains a where "a > A" "f \<in> \<Theta>(\<lambda>x. x powr p *(1 + integral (\<lambda>u. g' u / u powr (p + 1)) a x))"
lemma genby_uminus_genset_subset: "uminus ` S \<subseteq> \<langle>S\<rangle>"
lemma "matches\<cdot>[::]\<cdot>[::] = [:0:]"
lemma ad_agr_list_mono: "X \<subseteq> Y \<Longrightarrow> ad_agr_list Y ys xs \<Longrightarrow> ad_agr_list X ys xs"
lemma fds_nth_norm [simp]: "fds_nth (fds_norm f) n = norm (fds_nth f n)"
lemma set_of_delete_root: assumes "t = MKT n l r h" and "avl t" and "is_ord t" shows "set_of (delete_root t) = (set_of t) - {n}"
lemma lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
lemma orth_at__ncop1: assumes "U \<noteq> X" and "X OrthAt A B C U V" shows "\<not> Coplanar A B C U"
theorem compositionality_BSIA: "\<lbrakk> BSD \<V>1 Tr\<^bsub>ES1\<^esub>; BSD \<V>2 Tr\<^bsub>ES2\<^esub>; BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>; BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>; (\<rho>1 \<V>1) \<subseteq> (\<rho> \<V>) \<inter> E\<^bsub>ES1\<^esub>; (\<rho>2 \<V>2) \<subseteq> (\<rho> \<V>) \<inter> E\<^bsub>ES2\<^esub> \<rbrakk> \<Longrightarrow> BSIA \<rho> \<V> (Tr\<^bsub>(ES1 \<parallel> ES2)\<^esub>)"
lemma WL_eq_imp: "eq_imp (\<lambda>(_::unit) s. (ghost_honorary_grey (s p), W (s p))) (WL p)"
lemma foldr_scene_union_add_tail: "\<lbrakk> pairwise (##\<^sub>S) (set xs); \<forall> x\<in>set xs. x ##\<^sub>S b \<rbrakk> \<Longrightarrow> \<Squnion>\<^sub>S xs \<squnion>\<^sub>S b = foldr (\<squnion>\<^sub>S) xs b"
lemma iso_tuple_UNIV_I: "x \<in> UNIV \<equiv> True"
lemma gmctxt_cl_mono_funas: assumes "\<F> \<subseteq> \<G>" shows "gmctxt_cl \<F> \<R> \<subseteq> gmctxt_cl \<G> \<R>"
lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
lemma \<Delta>\<^sub>\<epsilon>_impl [code]: "\<Delta>\<^sub>\<epsilon> (TA \<Delta>\<^sub>A \<Delta>\<^sub>A\<^sub>\<epsilon>) (TA \<Delta>\<^sub>B \<Delta>\<^sub>B\<^sub>\<epsilon>) = the (\<Delta>\<^sub>\<epsilon>_impl \<Delta>\<^sub>A \<Delta>\<^sub>A\<^sub>\<epsilon> \<Delta>\<^sub>B \<Delta>\<^sub>B\<^sub>\<epsilon>)"
lemma (in finite_measure) finite_measure_mono_AE: assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M" shows "measure M A \<le> measure M B"
lemma invoke_pure [simp]: "pure (invoke [] ptr args) h"
lemma r2f_inf_pres: "inf_pres \<F>"
lemma forball_contra1:"\<lbrakk>\<forall>y\<in>A. P x y \<longrightarrow> Q y; \<forall>y\<in>A. \<not> Q y\<rbrakk> \<Longrightarrow> \<forall>y\<in>A. \<not> P x y"
lemma these_tm_ntcfs_iff: (not simp) "\<NN> \<in>\<^sub>\<circ> these_tm_ntcfs \<alpha> \<AA> \<BB> \<FF> \<GG> \<longleftrightarrow> \<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
lemma measurable_stream_space2: assumes f_snth: "\<And>n. (\<lambda>x. f x !! n) \<in> measurable N M" shows "f \<in> measurable N (stream_space M)"
lemma sumlist_as_finsum: assumes "set xs \<subseteq> carrier G" and "distinct xs" shows "sumlist xs = (\<Oplus>x\<in>set xs. x)"
lemma fixes a b::"'a::linorder_topology" assumes "continuous_on {a .. b} f" "a < b" shows continuous_on_Icc_at_rightD: "(f \<longlongrightarrow> f a) (at_right a)" and continuous_on_Icc_at_leftD: "(f \<longlongrightarrow> f b) (at_left b)"
lemma negligible_real_ivlI: fixes a b::real assumes "a \<ge> b" shows "negligible {a .. b}"
lemma Subset_E: "H \<turnstile> t SUBS u \<Longrightarrow> H \<turnstile> a IN t \<Longrightarrow> insert (a IN u) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"
lemma distinguishing_formula_eqvt [simp]: assumes "\<not> (P =\<cdot> Q)" shows "p \<bullet> distinguishing_formula P Q = distinguishing_formula (p \<bullet> P) (p \<bullet> Q)"
lemma tagged_division_of_union_self: assumes "p tagged_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
lemma (in ring_hom_ring) induced_ring_hom: assumes "subring K R" shows "ring_hom_ring (R \<lparr> carrier := K \<rparr>) S h"
lemma lunit_ide_eq: assumes "ide f" shows "\<l>[f] = \<lparr>Chn = \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Dom = \<lparr>Leg0 = Leg0 (Dom f) \<cdot> \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Leg1 = \<p>\<^sub>1[C.cod (Leg1 (Dom f)), Leg1 (Dom f)]\<rparr>, Cod = Cod f\<rparr>"
lemma actPres: fixes P :: ccs and Q :: ccs and \<alpha> :: act assumes "P \<approx> Q" shows "\<alpha>.(P) \<approx> \<alpha>.(Q)"
lemma project_act_Restrict_subset_project_act: "project_act h (Restrict C act) \<subseteq> project_act h act"
lemma fetch_aft_erase_b_b: "fetch (mopup_a n @ shift mopup_b (2 * n)) (2n + 3) Bk = (R, Suc (2 n + 3))"
lemma image_prod_mset_multiplicity: "prod_mset (image_mset f M) = prod (\<lambda>x. f x ^ count M x) (set_mset M)"
lemma algebraic_ii [simp]: "algebraic \<i>"
lemma lang_nderiv: "lang (nderiv a r) = Deriv a (lang r)"
lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
lemma eSwapAbs_simp3[simp]: "eSwapAbs MOD zs z1 z2 ERR = ERR"
lemma trivial_relative_homology_group_gen: assumes "continuous_map X (subtopology X S) f" "homotopic_with (\<lambda>h. True) (subtopology X S) (subtopology X S) f id" "homotopic_with (\<lambda>k. True) X X f id" shows "trivial_group(relative_homology_group p X S)"
lemma mult_ge_prts: fixes a b :: "'a::lattice_ring" assumes "a1 \<le> a" and "a \<le> a2" and "b1 \<le> b" and "b \<le> b2" shows "a * b \<ge> nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
lemma jmm_allocate_Eps: "(SOME ha. ha \<in> jmm_allocate h hT) = (h', a') \<Longrightarrow> jmm_allocate h hT \<noteq> {} \<longrightarrow> (h', a') \<in> jmm_allocate h hT"
lemma triv_suf: "u \<le>s v \<cdot> u"
lemma lcnj[simp,intro!]: "set \<phi>s \<subseteq> fmla \<Longrightarrow> lcnj \<phi>s \<in> fmla"
lemma cn_cf_ObjMap_CId[cat_cn_cs_simps]: assumes "\<FF> : \<AA> \<^sub>C\<mapsto>\<mapsto>\<^bsub>\<alpha>\<^esub> \<BB>" and "c \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>\<AA>\<lparr>CId\<rparr>\<lparr>c\<rparr>\<rparr> = \<BB>\<lparr>CId\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>c\<rparr>\<rparr>"
lemma radius_diameter_singleton_eq: assumes "card V = 1" shows "radius = diameter"
lemma i_drop_commute: "s \<Up> a \<Up> b = s \<Up> b \<Up> a"
lemma composable_tm_twice_compile_tm [simp]: "composable_tm (twice_compile_tm, 0)"
lemma finite_infinity: "is_finite a \<Longrightarrow> \<not> is_infinity a"
lemma is_Proj_complement[simp]: fixes P :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a\<close> assumes a1: "is_Proj P" shows "is_Proj (id_cblinfun - P)"
lemma sublocale_sas_plus_finite_domain_representation_ii: fixes \<Psi>::"('v,'d) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v \<in> set ((\<Psi>)\<^sub>\<V>\<^sub>+). (\<R>\<^sub>+ \<Psi> v) \<noteq> {}" and "\<forall>op \<in> set ((\<Psi>)\<^sub>\<O>\<^sub>+). is_valid_operator_sas_plus \<Psi> op" and "dom ((\<Psi>)\<^sub>I\<^sub>+) = set ((\<Psi>)\<^sub>\<V>\<^sub>+)" and "\<forall>v \<in> dom ((\<Psi>)\<^sub>I\<^sub>+). the (((\<Psi>)\<^sub>I\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v" and "dom ((\<Psi>)\<^sub>G\<^sub>+) \<subseteq> set ((\<Psi>)\<^sub>\<V>\<^sub>+)" and "\<forall>v \<in> dom ((\<Psi>)\<^sub>G\<^sub>+). the (((\<Psi>)\<^sub>G\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v"
lemma orthogonal_real_eq: "RV_inner.orthogonal = real_inner_class.orthogonal"
lemma OclIsEmpty_null[simp,code_unfold]:"(null->isEmpty\<^sub>B\<^sub>a\<^sub>g()) = true"
lemma (in aGroup) aSum_zero:"a \<in> carrier A \<Longrightarrow> aSum A n \<zero> = \<zero>"

lemma sub_qbs_Mx[simp]: "qbs_Mx (sub_qbs X U) = {f \<in> UNIV \<rightarrow> qbs_space X \<inter> U. f \<in> qbs_Mx X}"

lemma car_is_alg: "is_algebraic R n (carrier (R\<^bsup>n\<^esup>))"

lemma (in is_cat_finite_obj_prod) cat_fin_obj_prod_index_vfinite: "vfinite I"

lemma SKIP_preserves [iff]: "SKIP \<in> preserves v"

lemma SeqStTermP_cong: "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'; H \<turnstile> s EQ s'; H \<turnstile> k EQ k'\<rbrakk> \<Longrightarrow> H \<turnstile> SeqStTermP v i t u s k IFF SeqStTermP v i t' u' s' k'"

lemma op_cf_cf_cn_comp[cat_op_simps]: "op_cf (\<GG> \<^sub>C\<^sub>F\<circ> \<FF>) = op_cf \<GG> \<^sub>C\<^sub>F\<circ> op_cf \<FF>"

lemma Inf_lim: fixes a :: "'a::{complete_linorder,linorder_topology}" assumes "\<And>n. b n \<in> s" and "b \<longlonglongrightarrow> a" shows "Inf s \<le> a"

lemma lift0: "(liftPoly i j 0) = 0"

lemma vfinite_vcard_vinsert_nin[simp]: assumes "vfinite A" and "a \<notin>\<^sub>\<circ> A" shows "vcard (vinsert a A) = csucc (vcard A)"

lemma refines_adm_aux: "option.admissible (\<lambda>xa. \<forall>h a aa b. xa h = Some (a, aa, b) \<longrightarrow> execute (t x) h = Some (a, aa, b))"

lemma in_rem_implicit_pres_\<delta>: "op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> rem_implicit_pres op \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops prob))"

lemma spaceN_sum [simp]: assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> space\<^sub>N N" shows "(\<Sum>i\<in>I. x i) \<in> space\<^sub>N N"

lemma vsubset_vinsert: "A \<subseteq>\<^sub>\<circ> vinsert x B \<longleftrightarrow> (if x \<in>\<^sub>\<circ> A then A -\<^sub>\<circ> set {x} \<subseteq>\<^sub>\<circ> B else A \<subseteq>\<^sub>\<circ> B)"

lemma tm_semi_id_gt0_halts': shows "\<lbrace>\<lambda>tap. \<exists>l. tap = ([], [Oc, Oc] @ Bk\<up> l)\<rbrace> tm_semi_id_gt0 \<lbrace>\<lambda>tap. \<exists>l. tap = ([], [Oc, Oc] @ Bk\<up> l)\<rbrace>"

lemma fps_lr_inverse_fps_X_plus1: "fps_left_inverse (1 + fps_X) (1::'a::ring_1) = Abs_fps (\<lambda>n. (-1)^n)" "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)"

lemma aial_empty_impl: "(aial_empty,RETURN op_list_empty) \<in> \<langle>ial_rel1 maxsize\<rangle>nres_rel"

lemma hnorm_mult_ineq: "\<And>x y::'a::real_normed_algebra star. hnorm (x * y) \<le> hnorm x * hnorm y"

lemma const_finfun: "(\<lambda>x. a) \<in> finfun"

theorem ikkbz_sub_dverts_optimal: assumes "\<exists>x. fwd_sub root (dverts t) x" shows "\<exists>zs. fwd_sub root (dverts (ikkbz_sub t)) zs \<and> (\<forall>as. fwd_sub root (dverts t) as \<longrightarrow> cost (rev zs) \<le> cost (rev as))"

lemma effect_swapI [effect_intros]: assumes "i < length h a" "h' = update a i x h" "r = get h a ! i" shows "effect (swap i x a) h h' r"

lemma notInKeys_notInVars : "x \<notin> keys m \<Longrightarrow> x \<notin> vars(MPoly_Type.monom m a)"

lemma doesnt_read_or_modify_doesnt_modify: "doesnt_read_or_modify c x \<Longrightarrow> doesnt_modify c x"

lemma LESS_SUC[simp]: "LESS ord 0 (SUC ord idx)" "LESS ord (Suc l) (SUC ord idx) = LESS ord l idx" "ord \<noteq> ord' \<Longrightarrow> LESS ord l (SUC ord' idx) = LESS ord l idx" "LESS ord l idx \<Longrightarrow> LESS ord l (SUC ord' idx)"

lemma UNIV_eqvt [eqvt]: shows "p \<bullet> UNIV = UNIV"

lemma expands_to_imp_exp_ln_eq: assumes "(l expands_to L) bs" "eventually (\<lambda>x. l x \<le> f x) at_top" "trimmed_pos L" "basis_wf bs" shows "eventually (\<lambda>x. exp (ln (f x)) = f x) at_top"

lemma cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"

lemma anonymousI[intro]: "(\<And>P f x y. \<lbrakk> profile A Is P; bij_betw f Is Is; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> (x \<^bsub>(scf P)\<^esub>\<preceq> y) = (x \<^bsub>(scf (P \<circ> f))\<^esub>\<preceq> y)) \<Longrightarrow> anonymous scf A Is"

lemma eliminate_aux_vars_of_int_poly: "eliminate_aux_vars (map_mpoly (of_int :: _ \<Rightarrow> 'a :: {comm_ring_1,ring_char_0}) mp) (of_int_poly \<circ> qs) is = of_int_poly (eliminate_aux_vars mp qs is)"

lemma (in order_topology) increasing_tendsto: assumes bdd: "eventually (\<lambda>n. f n \<le> l) F" and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F" shows "(f \<longlongrightarrow> l) F"

lemma not_exact_match_in_doubt_deny_approx_match: "matcher_agree_on_exact_matches \<gamma> \<beta> \<Longrightarrow> a = Accept \<or> a = Reject \<or> a = Drop \<Longrightarrow> \<not> Semantics.matches \<gamma> m p \<Longrightarrow> ((a = Drop \<or> a = Reject) \<and> Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p) \<or> \<not> Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p"

lemma exec_rename_to_exec: assumes \<Gamma>: "\<forall>p bdy. \<Gamma> p = Some bdy \<longrightarrow> \<Gamma>' (h p) = Some (rename h bdy)" assumes exec: "\<Gamma>'\<turnstile>\<langle>rc,s\<rangle> \<Rightarrow> t" shows "\<And>c. rename h c = rc\<Longrightarrow> \<exists>t'. \<Gamma>\<turnstile>\<langle>c,s\<rangle> \<Rightarrow> t' \<and> (t'=Stuck \<or> t'=t)"

lemma check_in_heap_returns_result [simp]: "ptr |\<in>| object_ptr_kinds h \<Longrightarrow> h \<turnstile> (check_in_heap ptr \<bind> f) \<rightarrow>\<^sub>r x = h \<turnstile> f () \<rightarrow>\<^sub>r x"

lemma single_LeadsTo_I: "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"

lemma trans_constraint_creats_LEQ_only: assumes "transf_constraint x \<noteq> []" shows "(\<forall>x \<in> set (transf_constraint x). \<exists>a b. x = LEQ a b)"

lemma backward_terminating_path_root_iff: "(\<exists>r . backward_terminating_path_root r x) \<longleftrightarrow> backward_terminating_path x"

lemma exp_cf_cat_cf_id_cat: assumes "category \<alpha> \<CC>" and "category \<alpha> \<DD>" shows "exp_cf_cat \<alpha> (cf_id \<CC>) \<DD> = cf_id (cat_FUNCT \<alpha> \<DD> \<CC>)"

lemma monad_commute_envT [locale_witness]: assumes "monad_commute return bind" shows "monad_commute return_env bind_env"

lemma partition_on_eq_quotient: assumes P: "partition_on A P" shows "A // {(x, y). \<exists>p \<in> P. x \<in> p \<and> y \<in> p} = P"

lemma Cons_not_Nil [iff]: "x # xs ~= Nil"

lemma dependent_inj_imageD: assumes d: "m2.dependent (f ` s)" and i: "inj_on f (m1.span s)" shows "m1.dependent s"

lemma tree_map_compose: "map_tree g (map_tree f t) = map_tree (g \<circ> f) t"

lemma expr_sem_pair_vars: "expr_sem \<sigma> <Var x, Var y> = return_val <|\<sigma> x, \<sigma> y|>"

lemma root_hash_T_simps [simp]: "root_hash_T rha (T\<^sub>m x) = T\<^sub>h (root_hash_F rha (root_hash_T rha) x)"

lemma fds_mangoldt': "fds mangoldt = fds_zeta * fds_deriv (fds moebius_mu)"

lemma OrdNotEqP_subst [simp]: "(OrdNotEqP x y)(i::=t) = OrdNotEqP (subst i t x) (subst i t y)"

lemma JFcol_bd: "\<forall>(j1 :: 'a JF1) (j2 :: 'a JF2). |JF1col n j1| <o bd_F \<and> |JF2col n j2| <o bd_F"

lemma permI: assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a" shows "f \<in> perm"

lemma computable_rec_inseparable_conv: "computable A \<longleftrightarrow> \<not> rec_inseparable A (- A)"

lemma bigtheta_f: obtains a where "a > A" "f \<in> \<Theta>(\<lambda>x. x powr p *(1 + integral (\<lambda>u. g' u / u powr (p + 1)) a x))"

lemma genby_uminus_genset_subset: "uminus ` S \<subseteq> \<langle>S\<rangle>"

lemma "matches\<cdot>[::]\<cdot>[::] = [:0:]"

lemma ad_agr_list_mono: "X \<subseteq> Y \<Longrightarrow> ad_agr_list Y ys xs \<Longrightarrow> ad_agr_list X ys xs"

lemma fds_nth_norm [simp]: "fds_nth (fds_norm f) n = norm (fds_nth f n)"

lemma set_of_delete_root: assumes "t = MKT n l r h" and "avl t" and "is_ord t" shows "set_of (delete_root t) = (set_of t) - {n}"

lemma lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"

lemma orth_at__ncop1: assumes "U \<noteq> X" and "X OrthAt A B C U V" shows "\<not> Coplanar A B C U"

theorem compositionality_BSIA: "\<lbrakk> BSD \<V>1 Tr\<^bsub>ES1\<^esub>; BSD \<V>2 Tr\<^bsub>ES2\<^esub>; BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>; BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>; (\<rho>1 \<V>1) \<subseteq> (\<rho> \<V>) \<inter> E\<^bsub>ES1\<^esub>; (\<rho>2 \<V>2) \<subseteq> (\<rho> \<V>) \<inter> E\<^bsub>ES2\<^esub> \<rbrakk> \<Longrightarrow> BSIA \<rho> \<V> (Tr\<^bsub>(ES1 \<parallel> ES2)\<^esub>)"

lemma WL_eq_imp: "eq_imp (\<lambda>(_::unit) s. (ghost_honorary_grey (s p), W (s p))) (WL p)"

lemma foldr_scene_union_add_tail: "\<lbrakk> pairwise (##\<^sub>S) (set xs); \<forall> x\<in>set xs. x ##\<^sub>S b \<rbrakk> \<Longrightarrow> \<Squnion>\<^sub>S xs \<squnion>\<^sub>S b = foldr (\<squnion>\<^sub>S) xs b"

lemma iso_tuple_UNIV_I: "x \<in> UNIV \<equiv> True"

lemma gmctxt_cl_mono_funas: assumes "\<F> \<subseteq> \<G>" shows "gmctxt_cl \<F> \<R> \<subseteq> gmctxt_cl \<G> \<R>"

lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"

lemma \<Delta>\<^sub>\<epsilon>_impl [code]: "\<Delta>\<^sub>\<epsilon> (TA \<Delta>\<^sub>A \<Delta>\<^sub>A\<^sub>\<epsilon>) (TA \<Delta>\<^sub>B \<Delta>\<^sub>B\<^sub>\<epsilon>) = the (\<Delta>\<^sub>\<epsilon>_impl \<Delta>\<^sub>A \<Delta>\<^sub>A\<^sub>\<epsilon> \<Delta>\<^sub>B \<Delta>\<^sub>B\<^sub>\<epsilon>)"

lemma (in finite_measure) finite_measure_mono_AE: assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M" shows "measure M A \<le> measure M B"

lemma invoke_pure [simp]: "pure (invoke [] ptr args) h"

lemma r2f_inf_pres: "inf_pres \<F>"

lemma forball_contra1:"\<lbrakk>\<forall>y\<in>A. P x y \<longrightarrow> Q y; \<forall>y\<in>A. \<not> Q y\<rbrakk> \<Longrightarrow> \<forall>y\<in>A. \<not> P x y"

lemma these_tm_ntcfs_iff: (*not simp*) "\<NN> \<in>\<^sub>\<circ> these_tm_ntcfs \<alpha> \<AA> \<BB> \<FF> \<GG> \<longleftrightarrow> \<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"

lemma measurable_stream_space2: assumes f_snth: "\<And>n. (\<lambda>x. f x !! n) \<in> measurable N M" shows "f \<in> measurable N (stream_space M)"

lemma sumlist_as_finsum: assumes "set xs \<subseteq> carrier G" and "distinct xs" shows "sumlist xs = (\<Oplus>x\<in>set xs. x)"

lemma fixes a b::"'a::linorder_topology" assumes "continuous_on {a .. b} f" "a < b" shows continuous_on_Icc_at_rightD: "(f \<longlongrightarrow> f a) (at_right a)" and continuous_on_Icc_at_leftD: "(f \<longlongrightarrow> f b) (at_left b)"

lemma negligible_real_ivlI: fixes a b::real assumes "a \<ge> b" shows "negligible {a .. b}"

lemma Subset_E: "H \<turnstile> t SUBS u \<Longrightarrow> H \<turnstile> a IN t \<Longrightarrow> insert (a IN u) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"

lemma distinguishing_formula_eqvt [simp]: assumes "\<not> (P =\<cdot> Q)" shows "p \<bullet> distinguishing_formula P Q = distinguishing_formula (p \<bullet> P) (p \<bullet> Q)"

lemma tagged_division_of_union_self: assumes "p tagged_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"

lemma (in ring_hom_ring) induced_ring_hom: assumes "subring K R" shows "ring_hom_ring (R \<lparr> carrier := K \<rparr>) S h"

lemma lunit_ide_eq: assumes "ide f" shows "\<l>[f] = \<lparr>Chn = \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Dom = \<lparr>Leg0 = Leg0 (Dom f) \<cdot> \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Leg1 = \<p>\<^sub>1[C.cod (Leg1 (Dom f)), Leg1 (Dom f)]\<rparr>, Cod = Cod f\<rparr>"

lemma actPres: fixes P :: ccs and Q :: ccs and \<alpha> :: act assumes "P \<approx> Q" shows "\<alpha>.(P) \<approx> \<alpha>.(Q)"

lemma project_act_Restrict_subset_project_act: "project_act h (Restrict C act) \<subseteq> project_act h act"

lemma fetch_aft_erase_b_b: "fetch (mopup_a n @ shift mopup_b (2 * n)) (2*n + 3) Bk = (R, Suc (2 * n + 3))"

lemma image_prod_mset_multiplicity: "prod_mset (image_mset f M) = prod (\<lambda>x. f x ^ count M x) (set_mset M)"

lemma algebraic_ii [simp]: "algebraic \<i>"

lemma lang_nderiv: "lang (nderiv a r) = Deriv a (lang r)"

lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"

lemma eSwapAbs_simp3[simp]: "eSwapAbs MOD zs z1 z2 ERR = ERR"

lemma trivial_relative_homology_group_gen: assumes "continuous_map X (subtopology X S) f" "homotopic_with (\<lambda>h. True) (subtopology X S) (subtopology X S) f id" "homotopic_with (\<lambda>k. True) X X f id" shows "trivial_group(relative_homology_group p X S)"

lemma mult_ge_prts: fixes a b :: "'a::lattice_ring" assumes "a1 \<le> a" and "a \<le> a2" and "b1 \<le> b" and "b \<le> b2" shows "a * b \<ge> nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"

lemma jmm_allocate_Eps: "(SOME ha. ha \<in> jmm_allocate h hT) = (h', a') \<Longrightarrow> jmm_allocate h hT \<noteq> {} \<longrightarrow> (h', a') \<in> jmm_allocate h hT"

lemma triv_suf: "u \<le>s v \<cdot> u"

lemma lcnj[simp,intro!]: "set \<phi>s \<subseteq> fmla \<Longrightarrow> lcnj \<phi>s \<in> fmla"

lemma cn_cf_ObjMap_CId[cat_cn_cs_simps]: assumes "\<FF> : \<AA> \<^sub>C\<mapsto>\<mapsto>\<^bsub>\<alpha>\<^esub> \<BB>" and "c \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>\<AA>\<lparr>CId\<rparr>\<lparr>c\<rparr>\<rparr> = \<BB>\<lparr>CId\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>c\<rparr>\<rparr>"

lemma radius_diameter_singleton_eq: assumes "card V = 1" shows "radius = diameter"

lemma i_drop_commute: "s \<Up> a \<Up> b = s \<Up> b \<Up> a"

lemma composable_tm_twice_compile_tm [simp]: "composable_tm (twice_compile_tm, 0)"

lemma finite_infinity: "is_finite a \<Longrightarrow> \<not> is_infinity a"

lemma is_Proj_complement[simp]: fixes P :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a\<close> assumes a1: "is_Proj P" shows "is_Proj (id_cblinfun - P)"

lemma sublocale_sas_plus_finite_domain_representation_ii: fixes \<Psi>::"('v,'d) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v \<in> set ((\<Psi>)\<^sub>\<V>\<^sub>+). (\<R>\<^sub>+ \<Psi> v) \<noteq> {}" and "\<forall>op \<in> set ((\<Psi>)\<^sub>\<O>\<^sub>+). is_valid_operator_sas_plus \<Psi> op" and "dom ((\<Psi>)\<^sub>I\<^sub>+) = set ((\<Psi>)\<^sub>\<V>\<^sub>+)" and "\<forall>v \<in> dom ((\<Psi>)\<^sub>I\<^sub>+). the (((\<Psi>)\<^sub>I\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v" and "dom ((\<Psi>)\<^sub>G\<^sub>+) \<subseteq> set ((\<Psi>)\<^sub>\<V>\<^sub>+)" and "\<forall>v \<in> dom ((\<Psi>)\<^sub>G\<^sub>+). the (((\<Psi>)\<^sub>G\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v"

lemma orthogonal_real_eq: "RV_inner.orthogonal = real_inner_class.orthogonal"

lemma OclIsEmpty_null[simp,code_unfold]:"(null->isEmpty\<^sub>B\<^sub>a\<^sub>g()) = true"

lemma (in aGroup) aSum_zero:"a \<in> carrier A \<Longrightarrow> aSum A n \<zero> = \<zero>"