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

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lemma fincomp_mono_neutral_cong: assumes [simp]: "finite B" "finite A" and *: "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>" "\<And>i. i \<in> A - B \<Longrightarrow> g i = \<one>" and gh: "\<And>x. x \<in> A \<inter> B \<Longrightarrow> g x = h x" and g: "g \<in> A \<rightarrow> M" and h: "h \<in> B \<rightarrow> M" shows "fincomp g A = fincomp h B"
lemma ground_term_of_gterm [simp]: "ground (term_of_gterm s)"
lemma pres_exec: assumes "(q,\<tau>) \<in> set (step p (ss!p))" and "\<forall>n \<in> set (ss!p). p > n" and "ss!p \<in> A" and "p < n" shows "\<tau> \<in> A "
lemma DC_refines_AC: "(a \<Sqinter> b) \<sqsubseteq> (a \<Squnion> b)"
lemma mult_scalar_zero_right [simp]: "p \<odot> 0 = 0"
lemma dense_lattice_char_2: "(\<forall>x y . x \<sqinter> y = bot \<longrightarrow> x = bot \<or> y = bot) \<longleftrightarrow> (\<forall>x . regular x \<longrightarrow> x = bot \<or> x = top)"
lemma sublinear_wp_Bind: "\<lbrakk> \<And>s. sublinear (wp (a (f s))) \<rbrakk> \<Longrightarrow> sublinear (wp (Bind f a))"
lemma perzeta_conv_hurwitz_zeta_multiplication: fixes k :: nat and a :: int and s :: complex assumes "k > 0" "s \<noteq> 1" shows "k powr s * perzeta (a / k) s = (\<Sum>n=1..k. exp (2 * pi * n * a / k * \<i>) * hurwitz_zeta (n / k) s)" (is "?lhs s = ?rhs s")
lemma paths_withoutI: assumes "xs \<in> paths t" assumes "x \<notin> set xs" shows "xs \<in> paths (without x t)"
lemma Stop_Sim: "Stop \<preceq>S Stop"
theorem i_Exec_Stream_causal: " input1 \<Down> n = input2 \<Down> n \<Longrightarrow> (i_Exec_Comp_Stream trans_fun input1 c) \<Down> n = (i_Exec_Comp_Stream trans_fun input2 c) \<Down> n"
lemma adds_antisym: assumes "s adds t" "t adds s" shows "s = t"
lemma less_eq_mask_iff_take_bit_eq_self: \<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close> for w :: \<open>'a::len word\<close>
lemma is_poincare_line_mk_circline: assumes "(A, B, C, D) \<in> hermitean_nonzero" shows "is_poincare_line (mk_circline A B C D) \<longleftrightarrow> (cmod B)\<^sup>2 > (cmod A)\<^sup>2 \<and> A = D"
lemma Si_neg: assumes "T \<ge> 0" shows "Si (- T) = - Si T"
lemma (in information_space) fixes X :: "'a \<Rightarrow> 'b" assumes X[measurable]: "distributed M MX X f" and nn: "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> f x" shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq)
lemma run_argsD: "run \<A> s t \<Longrightarrow> length (gargs s) = length (gargs t) \<and> (\<forall> i < length (gargs t). run \<A> (gargs s ! i) (gargs t ! i))"
lemma shows integrable_I0i_1_div_plus_square: "interval_lebesgue_integrable lborel 0 \<infinity> (\<lambda>x. 1 / (1 + x^2))" and LBINT_I0i_1_div_plus_square: "LBINT x=0..\<infinity>. 1 / (1 + x^2) = pi / 2"
lemma opp_add: assumes p\<^sub>1: "on_curve a b p\<^sub>1" and p\<^sub>2: "on_curve a b p\<^sub>2" shows "opp (add a p\<^sub>1 p\<^sub>2) = add a (opp p\<^sub>1) (opp p\<^sub>2)"
lemma T_ins_tree_simple_bound: "T_ins_tree t ts \<le> length ts + 1"
lemma card_suc_ordLess_imp_ordLeq: assumes ORD: "Card_order r" "Card_order r'" "card_order r'" and LESS: "r <o card_suc r'" shows "r \<le>o r'"
lemma graph_homomorphism_composes[intro]: assumes "graph_homomorphism a b x" "graph_homomorphism b c y" shows "graph_homomorphism a c (x O y)"
lemma tree_edges_finite[simp, intro!]: "finite (tree_edges s)"
lemma expl_cond_expect_prop_sets2: assumes "disc_fct Y" and "point_measurable (fct_gen_subalgebra M N Y) (space N) Y" and "D = {w\<in> space M. Y w \<in> space N \<and> (P (expl_cond_expect M Y X w))}" shows "D\<in> sets (fct_gen_subalgebra M N Y)"
lemma \<open>push_bit (Suc (Suc (Suc 0))) (Suc 0) = 8\<close>
lemma typ_of1_imp_has_typ1: "typ_of1 Ts t = Some ty \<Longrightarrow> has_typ1 Ts t ty"
lemma singelton_trancl [simp]: "{a}\<^sup>+ = {a}"
lemma pminus_Nil: "-- [] = []"
lemma some_gcd_ff_list_smult: "a \<noteq> 0 \<Longrightarrow> some_gcd_ff_list (map (() a) xs) =dff a some_gcd_ff_list xs"
lemma ofail_NF_wp [wp]: "ovalidNF (\<lambda>_. False) ofail Q"
lemma transpose_add: "A \<in> carrier_mat nr nc \<Longrightarrow> B \<in> carrier_mat nr nc \<Longrightarrow> transpose_mat (A + B) = transpose_mat A + transpose_mat B"
lemma eq_option_refine[sepref_fr_rules]: assumes "CONSTRAINT is_pure A" shows "(uncurry eq,uncurry (RETURN oo dflt_option_eq)) \<in> (dflt_option_assn dflt A)\<^sup>k *\<^sub>a (dflt_option_assn dflt A)\<^sup>k \<rightarrow>\<^sub>a bool_assn"
lemma dflt_cmp_2inv[simp]: "dflt_cmp (comp2le cmp) (comp2lt cmp) = cmp"
lemma (in nf_invar) C_ne_max_dist: assumes "C\<noteq>{}" shows "d \<le> max_dist src"
lemma NEG_\<G>: "NEG \<subseteq> \<G>"
lemma sccs_finished: "\<Union>(sccs s) \<subseteq> dom (finished s)"
lemma lift_profile_add_mset [simp]: "lift_profile (add_mset X A) = add_mset (lift_applist X) (lift_profile A)"
lemma(in UP_domain) pow_sum: assumes "p \<in> carrier P" assumes "q \<in> carrier P" assumes "degree q < degree p" shows "degree ((p \<oplus>\<^bsub>P\<^esub> q )[^]\<^bsub>P\<^esub>n) = (degree p)*n"
lemma iso3 [simp]: "d (d x \<cdot> U) = d x "
lemma inference_closed_sets_are_saturated: assumes "inference_closed S" assumes "\<forall>x \<in> S. (finite (cl_ecl x))" shows "clause_saturated S"
lemma meval_sched_snocI: "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns\<^esub> (cms\<^sub>1'', mem\<^sub>1'') \<and> (cms\<^sub>1'', mem\<^sub>1'') \<leadsto>\<^bsub>n\<^esub> (cms\<^sub>1', mem\<^sub>1') \<Longrightarrow> (cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns@[n]\<^esub> (cms\<^sub>1', mem\<^sub>1')"
lemma vertical_chop_assoc2: "(v=v1--v2) \<and> (v1=v3--v4) \<longrightarrow> (\<exists>v'. (v=v3--v') \<and> (v'=v4--v2))"
lemma INF_eq_minf: "(INF i\<in>I. f i::ereal) \<noteq> -\<infinity> \<longleftrightarrow> (\<exists>b>-\<infinity>. \<forall>i\<in>I. b \<le> f i)"
lemma meet_iso: "x \<le> y \<Longrightarrow> x \<cdot> z \<le> y \<cdot> z"
lemma list_gcd_greatest: "(\<And> x. x \<in> set xs \<Longrightarrow> y dvd x) \<Longrightarrow> y dvd (list_gcd xs)"
lemma matrix_le_norm_mono: assumes "0 \<le> (blinfun_to_matrix C)" and "(blinfun_to_matrix C) \<le> (blinfun_to_matrix D)" shows "norm C \<le> norm D"
lemma Show_nat_not_empty: \<open>(Show\<^sub>n\<^sub>a\<^sub>t n) \<noteq> []\<close>
lemma ce_rel_lm_25: "r \<in> ce_rels \<Longrightarrow> r^-1 \<in> ce_rels"
lemma multpw_map: assumes "\<And>x y. x \<in># X \<Longrightarrow> y \<in># Y \<Longrightarrow> (x, y) \<in> ns \<Longrightarrow> (f x, g y) \<in> ns'" and "(X, Y) \<in> multpw ns" shows "(image_mset f X, image_mset g Y) \<in> multpw ns'"
lemma single_valued_inter2: "single_valued R \<Longrightarrow> single_valued (S\<inter>R)"
lemma remove_child_children_subset: assumes "h \<turnstile> remove_child parent child \<rightarrow>\<^sub>h h'" and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'" and known_ptrs: "known_ptrs h" and type_wf: "type_wf h" shows "set children' \<subseteq> set children"
lemma finite_subint1: "finite (subint1 G)"
lemma assumes "bv2int 0 = 0" and "bv2int 1 = 1" and "bv2int 2 = 2" and "bv2int 3 = 3" and "\<forall>x::2 word. bv2int x > 0" shows "\<forall>i::int. i < 0 \<longrightarrow> (\<forall>x::2 word. bv2int x > i)"
lemma case_prod_preserve [quot_preserve]: assumes q1: "Quotient3 R1 Abs1 Rep1" and q2: "Quotient3 R2 Abs2 Rep2" and q3: "Quotient3 R3 Abs3 Rep3" shows "((Abs1 ---> Abs2 ---> Rep3) ---> map_prod Rep1 Rep2 ---> Abs3) case_prod = case_prod"
lemma unity_root_eq_1_iff_int: fixes k :: nat and n :: int assumes "k > 0" shows "unity_root k n = 1 \<longleftrightarrow> k dvd n"
lemma decr_grading_p_monomial: "decr_grading_p d n (monomial c v) = monomial c (decr_grading_term d n v)"
lemma smcf_comp_is_ff_semifunctor[smcf_cs_intros]: assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<CC>"
lemma iex_disj_distrib: " (\<diamond> t I. P t \<or> Q t) = ((\<diamond> t I. P t) \<or> (\<diamond> t I. Q t))"
lemma wt[simp]: "M.wt T \<Longrightarrow> wt T"
lemma iMin_Un[rule_format]: " \<lbrakk> A \<noteq> {}; B \<noteq> {} \<rbrakk> \<Longrightarrow> iMin (A \<union> B) = min (iMin A) (iMin B)"
lemma tauActTauChain: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto>\<tau> \<prec> P'" shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
lemma or_foldl_sem:"List.member \<Gamma> \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I (foldr Or \<Gamma> FF)"
lemma acyclic_on_empty[simp]: "acyclic_on {} r"
lemma end_smult_distrib_right : assumes "range T \<subseteq> V" shows "(a+b) \<cdot>\<cdot> T = a \<cdot>\<cdot> T + b \<cdot>\<cdot> T"
lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
lemma cong_gen_cong: "cong (gen_cong R)"
lemma fst_K0[simp]: "map_pmf fst (K0 p S0) = S0"
lemma matrix_equation_main_step: fixes p:: "real poly" fixes qs:: "real poly list" fixes I:: "nat list" fixes signs:: "rat list list" assumes nonzero: "p\<noteq>0" assumes distinct_signs: "distinct signs" assumes all_info: "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set(signs)" assumes welldefined: "list_constr I (length qs)" assumes pairwise_rel_prime_1: "\<forall>q. ((List.member qs q) \<longrightarrow> (coprime p q))" shows "(vec_of_list (mtx_row signs I) \<bullet> (construct_lhs_vector p qs signs)) = construct_NofI p (retrieve_polys qs I)"
lemma FullInit: "\<turnstile> FullInit \<longrightarrow> BInit inp qc out"
lemma lemma_3_5_i_4: "(d4 a b = 1) = (a = b)"
lemma nths_Cons: "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"
lemma timpl_closure_no_Abs_eq: assumes "t \<in> timpl_closure s TI" and "\<forall>f \<in> funs_term t. \<not>is_Abs f" shows "t = s"
lemma e_app_intro[intro]: "\<lbrakk> VFun f \<in> E e1 \<rho>; v2 \<in> E e2 \<rho>; (v2',v3') \<in> fset f; v2' \<sqsubseteq> v2; v3 \<sqsubseteq> v3'\<rbrakk> \<Longrightarrow> v3 \<in> E (EApp e1 e2) \<rho>"
lemma map_bset_permute: "p \<bullet> B = map_bset (permute p) B"
lemma lookup_map2_val_neutr: assumes "\<And>k x. f k x 0 = x" and "\<And>k x. f k 0 x = x" shows "lookup (map2_val_neutr f xs ys) k = f k (lookup xs k) (lookup ys k)"
lemma induced_automorphism_chamber_map: "chamber C \<Longrightarrow> chamber (\<s>`C)"
lemma strictly_precedes_alt_def2: \<open>{ \<rho>. \<forall>n::nat. (run_tick_count \<rho> K\<^sub>2 n) \<le> (run_tick_count_strictly \<rho> K\<^sub>1 n) } = { \<rho>. (\<not>hamlet ((Rep_run \<rho>) 0 K\<^sub>2)) \<and> (\<forall>n::nat. (run_tick_count \<rho> K\<^sub>2 (Suc n)) \<le> (run_tick_count \<rho> K\<^sub>1 n)) }\<close> (is \<open>?P = ?P'\<close>)
lemma count_nodes_eq_0_iff [simp]: "count_nodes t = 0 \<longleftrightarrow> t = Leaf"
lemma arcs_fset_id: "fset (Abs_fset (arcs T)) = arcs T"
lemma group_law_group: "group_law inverse neutral mult"
lemma gcd_0_right [simp]: "gcd a 0 = normalize a"
lemma (in imap) concurrent_create_delete_independent_technical: assumes "i \<in> is" and "xs prefix of j" and "(i, Create i e) \<in> set (node_deliver_messages xs)" and "(ir, Delete is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Create i e) (ir, Delete is e)"
lemma fold_delta_a_init_a_mrexps[simp]: "fold delta_a w (init_a s) \<in> UNIV \<times> mrexps s"
lemma eval_fds_integral_has_field_derivative: fixes s :: "'a :: dirichlet_series" assumes "ereal (s \<bullet> 1) > abs_conv_abscissa f" assumes "fds_nth f 1 = 0" shows "(eval_fds (fds_integral c f) has_field_derivative eval_fds f s) (at s)"
lemma LIMSEQ_cong: assumes "f \<longlonglongrightarrow> x" "\<forall>\<^sup>\<infinity>n. f n = g n" shows "g \<longlonglongrightarrow> x"
lemma gpv_stop_map' [simp]: "gpv_stop (map_gpv' f g h gpv) = map_gpv' (map_option f) g (map_option h) (gpv_stop gpv)"
lemma dia_lowerB: "\<lbrakk> u \<in> verts G; v \<in> verts G \<rbrakk> \<Longrightarrow> diameter w \<ge> \<mu> w u v"
lemma fls_nth_compose_power: assumes "d > 0" shows "fls_nth (fls_compose_power f d) n = (if int d dvd n then fls_nth f (n div int d) else 0)"
lemma pos_neg_kauff_mat: "kauff_mat ((basic [over]) \<circ> (basic [under])) = kauff_mat ((basic [vert,vert]) \<circ> (basic [vert,vert])) "
lemma subst_fmla_commute [simp]: "atom j \<sharp> A \<Longrightarrow> (A(i::=t))(j::=u) = A(i ::= subst j u t)"
lemma le_inf_eq_inf_transp[intro, trans]: assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2" "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>3"
lemma GroupH : "Group H"
lemma get_object_ptr_simp1 [simp]: "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr object h) = Some object"
lemma differentiable_cmult_left_iff [simp]: fixes c::"'a::real_normed_field" shows "(\<lambda>t. c * q t) differentiable at t \<longleftrightarrow> c = 0 \<or> (\<lambda>t. q t) differentiable at t" (is "?lhs = ?rhs")
lemma norm_power_diff: fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}" assumes "norm z \<le> 1" "norm w \<le> 1" shows "norm (z^m - w^m) \<le> m * norm (z - w)"
lemma fA_rel_inv[intro]: notes fun_upd_apply[simp] shows "\<lbrace> LSTP (fA_rel_inv \<^bold>\<and> tso_store_inv) \<rbrace> sys \<lbrace> LSTP fA_rel_inv \<rbrace>"
lemma r_not_updated_stays_the_same: "r \<notin> fst ` set U \<Longrightarrow> apply_updates U c d $ r = d $ r"
lemma correctCompositionDiffLevelsS11opt: "correctCompositionDiffLevels sS11opt"
lemma prv_ldsj_imp_remdups: assumes "set \<phi>s \<subseteq> fmla" shows "prv (imp (ldsj \<phi>s) (ldsj (remdups \<phi>s)))"
lemma bifunctor_proj_fst_ObjMap_app[cat_cs_simps]: assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>C \<BB>)\<lparr>Obj\<rparr>" shows "(\<SS>\<^bsub>\<AA>,\<BB>\<^esub>(-,b)\<^sub>C\<^sub>F)\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<SS>\<lparr>ObjMap\<rparr>\<lparr>a, b\<rparr>\<^sub>\<bullet>"

lemma fincomp_mono_neutral_cong: assumes [simp]: "finite B" "finite A" and *: "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>" "\<And>i. i \<in> A - B \<Longrightarrow> g i = \<one>" and gh: "\<And>x. x \<in> A \<inter> B \<Longrightarrow> g x = h x" and g: "g \<in> A \<rightarrow> M" and h: "h \<in> B \<rightarrow> M" shows "fincomp g A = fincomp h B"

lemma ground_term_of_gterm [simp]: "ground (term_of_gterm s)"

lemma pres_exec: assumes "(q,\<tau>) \<in> set (step p (ss!p))" and "\<forall>n \<in> set (ss!p). p > n" and "ss!p \<in> A" and "p < n" shows "\<tau> \<in> A "

lemma DC_refines_AC: "(a \<Sqinter> b) \<sqsubseteq> (a \<Squnion> b)"

lemma mult_scalar_zero_right [simp]: "p \<odot> 0 = 0"

lemma dense_lattice_char_2: "(\<forall>x y . x \<sqinter> y = bot \<longrightarrow> x = bot \<or> y = bot) \<longleftrightarrow> (\<forall>x . regular x \<longrightarrow> x = bot \<or> x = top)"

lemma sublinear_wp_Bind: "\<lbrakk> \<And>s. sublinear (wp (a (f s))) \<rbrakk> \<Longrightarrow> sublinear (wp (Bind f a))"

lemma perzeta_conv_hurwitz_zeta_multiplication: fixes k :: nat and a :: int and s :: complex assumes "k > 0" "s \<noteq> 1" shows "k powr s * perzeta (a / k) s = (\<Sum>n=1..k. exp (2 * pi * n * a / k * \<i>) * hurwitz_zeta (n / k) s)" (is "?lhs s = ?rhs s")

lemma paths_withoutI: assumes "xs \<in> paths t" assumes "x \<notin> set xs" shows "xs \<in> paths (without x t)"

lemma Stop_Sim: "Stop \<preceq>S Stop"

theorem i_Exec_Stream_causal: " input1 \<Down> n = input2 \<Down> n \<Longrightarrow> (i_Exec_Comp_Stream trans_fun input1 c) \<Down> n = (i_Exec_Comp_Stream trans_fun input2 c) \<Down> n"

lemma adds_antisym: assumes "s adds t" "t adds s" shows "s = t"

lemma less_eq_mask_iff_take_bit_eq_self: \<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close> for w :: \<open>'a::len word\<close>

lemma is_poincare_line_mk_circline: assumes "(A, B, C, D) \<in> hermitean_nonzero" shows "is_poincare_line (mk_circline A B C D) \<longleftrightarrow> (cmod B)\<^sup>2 > (cmod A)\<^sup>2 \<and> A = D"

lemma Si_neg: assumes "T \<ge> 0" shows "Si (- T) = - Si T"

lemma (in information_space) fixes X :: "'a \<Rightarrow> 'b" assumes X[measurable]: "distributed M MX X f" and nn: "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> f x" shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq)

lemma run_argsD: "run \<A> s t \<Longrightarrow> length (gargs s) = length (gargs t) \<and> (\<forall> i < length (gargs t). run \<A> (gargs s ! i) (gargs t ! i))"

lemma shows integrable_I0i_1_div_plus_square: "interval_lebesgue_integrable lborel 0 \<infinity> (\<lambda>x. 1 / (1 + x^2))" and LBINT_I0i_1_div_plus_square: "LBINT x=0..\<infinity>. 1 / (1 + x^2) = pi / 2"

lemma opp_add: assumes p\<^sub>1: "on_curve a b p\<^sub>1" and p\<^sub>2: "on_curve a b p\<^sub>2" shows "opp (add a p\<^sub>1 p\<^sub>2) = add a (opp p\<^sub>1) (opp p\<^sub>2)"

lemma T_ins_tree_simple_bound: "T_ins_tree t ts \<le> length ts + 1"

lemma card_suc_ordLess_imp_ordLeq: assumes ORD: "Card_order r" "Card_order r'" "card_order r'" and LESS: "r <o card_suc r'" shows "r \<le>o r'"

lemma graph_homomorphism_composes[intro]: assumes "graph_homomorphism a b x" "graph_homomorphism b c y" shows "graph_homomorphism a c (x O y)"

lemma tree_edges_finite[simp, intro!]: "finite (tree_edges s)"

lemma expl_cond_expect_prop_sets2: assumes "disc_fct Y" and "point_measurable (fct_gen_subalgebra M N Y) (space N) Y" and "D = {w\<in> space M. Y w \<in> space N \<and> (P (expl_cond_expect M Y X w))}" shows "D\<in> sets (fct_gen_subalgebra M N Y)"

lemma \<open>push_bit (Suc (Suc (Suc 0))) (Suc 0) = 8\<close>

lemma typ_of1_imp_has_typ1: "typ_of1 Ts t = Some ty \<Longrightarrow> has_typ1 Ts t ty"

lemma singelton_trancl [simp]: "{a}\<^sup>+ = {a}"

lemma pminus_Nil: "-- [] = []"

lemma some_gcd_ff_list_smult: "a \<noteq> 0 \<Longrightarrow> some_gcd_ff_list (map ((*) a) xs) =dff a * some_gcd_ff_list xs"

lemma ofail_NF_wp [wp]: "ovalidNF (\<lambda>_. False) ofail Q"

lemma transpose_add: "A \<in> carrier_mat nr nc \<Longrightarrow> B \<in> carrier_mat nr nc \<Longrightarrow> transpose_mat (A + B) = transpose_mat A + transpose_mat B"

lemma eq_option_refine[sepref_fr_rules]: assumes "CONSTRAINT is_pure A" shows "(uncurry eq,uncurry (RETURN oo dflt_option_eq)) \<in> (dflt_option_assn dflt A)\<^sup>k *\<^sub>a (dflt_option_assn dflt A)\<^sup>k \<rightarrow>\<^sub>a bool_assn"

lemma dflt_cmp_2inv[simp]: "dflt_cmp (comp2le cmp) (comp2lt cmp) = cmp"

lemma (in nf_invar) C_ne_max_dist: assumes "C\<noteq>{}" shows "d \<le> max_dist src"

lemma NEG_\<G>: "NEG \<subseteq> \<G>"

lemma sccs_finished: "\<Union>(sccs s) \<subseteq> dom (finished s)"

lemma lift_profile_add_mset [simp]: "lift_profile (add_mset X A) = add_mset (lift_applist X) (lift_profile A)"

lemma(in UP_domain) pow_sum: assumes "p \<in> carrier P" assumes "q \<in> carrier P" assumes "degree q < degree p" shows "degree ((p \<oplus>\<^bsub>P\<^esub> q )[^]\<^bsub>P\<^esub>n) = (degree p)*n"

lemma iso3 [simp]: "d (d x \<cdot> U) = d x "

lemma inference_closed_sets_are_saturated: assumes "inference_closed S" assumes "\<forall>x \<in> S. (finite (cl_ecl x))" shows "clause_saturated S"

lemma meval_sched_snocI: "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns\<^esub> (cms\<^sub>1'', mem\<^sub>1'') \<and> (cms\<^sub>1'', mem\<^sub>1'') \<leadsto>\<^bsub>n\<^esub> (cms\<^sub>1', mem\<^sub>1') \<Longrightarrow> (cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns@[n]\<^esub> (cms\<^sub>1', mem\<^sub>1')"

lemma vertical_chop_assoc2: "(v=v1--v2) \<and> (v1=v3--v4) \<longrightarrow> (\<exists>v'. (v=v3--v') \<and> (v'=v4--v2))"

lemma INF_eq_minf: "(INF i\<in>I. f i::ereal) \<noteq> -\<infinity> \<longleftrightarrow> (\<exists>b>-\<infinity>. \<forall>i\<in>I. b \<le> f i)"

lemma meet_iso: "x \<le> y \<Longrightarrow> x \<cdot> z \<le> y \<cdot> z"

lemma list_gcd_greatest: "(\<And> x. x \<in> set xs \<Longrightarrow> y dvd x) \<Longrightarrow> y dvd (list_gcd xs)"

lemma matrix_le_norm_mono: assumes "0 \<le> (blinfun_to_matrix C)" and "(blinfun_to_matrix C) \<le> (blinfun_to_matrix D)" shows "norm C \<le> norm D"

lemma Show_nat_not_empty: \<open>(Show\<^sub>n\<^sub>a\<^sub>t n) \<noteq> []\<close>

lemma ce_rel_lm_25: "r \<in> ce_rels \<Longrightarrow> r^-1 \<in> ce_rels"

lemma multpw_map: assumes "\<And>x y. x \<in># X \<Longrightarrow> y \<in># Y \<Longrightarrow> (x, y) \<in> ns \<Longrightarrow> (f x, g y) \<in> ns'" and "(X, Y) \<in> multpw ns" shows "(image_mset f X, image_mset g Y) \<in> multpw ns'"

lemma single_valued_inter2: "single_valued R \<Longrightarrow> single_valued (S\<inter>R)"

lemma remove_child_children_subset: assumes "h \<turnstile> remove_child parent child \<rightarrow>\<^sub>h h'" and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'" and known_ptrs: "known_ptrs h" and type_wf: "type_wf h" shows "set children' \<subseteq> set children"

lemma finite_subint1: "finite (subint1 G)"

lemma assumes "bv2int 0 = 0" and "bv2int 1 = 1" and "bv2int 2 = 2" and "bv2int 3 = 3" and "\<forall>x::2 word. bv2int x > 0" shows "\<forall>i::int. i < 0 \<longrightarrow> (\<forall>x::2 word. bv2int x > i)"

lemma case_prod_preserve [quot_preserve]: assumes q1: "Quotient3 R1 Abs1 Rep1" and q2: "Quotient3 R2 Abs2 Rep2" and q3: "Quotient3 R3 Abs3 Rep3" shows "((Abs1 ---> Abs2 ---> Rep3) ---> map_prod Rep1 Rep2 ---> Abs3) case_prod = case_prod"

lemma unity_root_eq_1_iff_int: fixes k :: nat and n :: int assumes "k > 0" shows "unity_root k n = 1 \<longleftrightarrow> k dvd n"

lemma decr_grading_p_monomial: "decr_grading_p d n (monomial c v) = monomial c (decr_grading_term d n v)"

lemma smcf_comp_is_ff_semifunctor[smcf_cs_intros]: assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<CC>"

lemma iex_disj_distrib: " (\<diamond> t I. P t \<or> Q t) = ((\<diamond> t I. P t) \<or> (\<diamond> t I. Q t))"

lemma wt[simp]: "M.wt T \<Longrightarrow> wt T"

lemma iMin_Un[rule_format]: " \<lbrakk> A \<noteq> {}; B \<noteq> {} \<rbrakk> \<Longrightarrow> iMin (A \<union> B) = min (iMin A) (iMin B)"

lemma tauActTauChain: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto>\<tau> \<prec> P'" shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"

lemma or_foldl_sem:"List.member \<Gamma> \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I (foldr Or \<Gamma> FF)"

lemma acyclic_on_empty[simp]: "acyclic_on {} r"

lemma end_smult_distrib_right : assumes "range T \<subseteq> V" shows "(a+b) \<cdot>\<cdot> T = a \<cdot>\<cdot> T + b \<cdot>\<cdot> T"

lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"

lemma cong_gen_cong: "cong (gen_cong R)"

lemma fst_K0[simp]: "map_pmf fst (K0 p S0) = S0"

lemma matrix_equation_main_step: fixes p:: "real poly" fixes qs:: "real poly list" fixes I:: "nat list" fixes signs:: "rat list list" assumes nonzero: "p\<noteq>0" assumes distinct_signs: "distinct signs" assumes all_info: "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set(signs)" assumes welldefined: "list_constr I (length qs)" assumes pairwise_rel_prime_1: "\<forall>q. ((List.member qs q) \<longrightarrow> (coprime p q))" shows "(vec_of_list (mtx_row signs I) \<bullet> (construct_lhs_vector p qs signs)) = construct_NofI p (retrieve_polys qs I)"

lemma FullInit: "\<turnstile> FullInit \<longrightarrow> BInit inp qc out"

lemma lemma_3_5_i_4: "(d4 a b = 1) = (a = b)"

lemma nths_Cons: "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"

lemma timpl_closure_no_Abs_eq: assumes "t \<in> timpl_closure s TI" and "\<forall>f \<in> funs_term t. \<not>is_Abs f" shows "t = s"

lemma e_app_intro[intro]: "\<lbrakk> VFun f \<in> E e1 \<rho>; v2 \<in> E e2 \<rho>; (v2',v3') \<in> fset f; v2' \<sqsubseteq> v2; v3 \<sqsubseteq> v3'\<rbrakk> \<Longrightarrow> v3 \<in> E (EApp e1 e2) \<rho>"

lemma map_bset_permute: "p \<bullet> B = map_bset (permute p) B"

lemma lookup_map2_val_neutr: assumes "\<And>k x. f k x 0 = x" and "\<And>k x. f k 0 x = x" shows "lookup (map2_val_neutr f xs ys) k = f k (lookup xs k) (lookup ys k)"

lemma induced_automorphism_chamber_map: "chamber C \<Longrightarrow> chamber (\<s>`C)"

lemma strictly_precedes_alt_def2: \<open>{ \<rho>. \<forall>n::nat. (run_tick_count \<rho> K\<^sub>2 n) \<le> (run_tick_count_strictly \<rho> K\<^sub>1 n) } = { \<rho>. (\<not>hamlet ((Rep_run \<rho>) 0 K\<^sub>2)) \<and> (\<forall>n::nat. (run_tick_count \<rho> K\<^sub>2 (Suc n)) \<le> (run_tick_count \<rho> K\<^sub>1 n)) }\<close> (is \<open>?P = ?P'\<close>)

lemma count_nodes_eq_0_iff [simp]: "count_nodes t = 0 \<longleftrightarrow> t = Leaf"

lemma arcs_fset_id: "fset (Abs_fset (arcs T)) = arcs T"

lemma group_law_group: "group_law inverse neutral mult"

lemma gcd_0_right [simp]: "gcd a 0 = normalize a"

lemma (in imap) concurrent_create_delete_independent_technical: assumes "i \<in> is" and "xs prefix of j" and "(i, Create i e) \<in> set (node_deliver_messages xs)" and "(ir, Delete is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Create i e) (ir, Delete is e)"

lemma fold_delta_a_init_a_mrexps[simp]: "fold delta_a w (init_a s) \<in> UNIV \<times> mrexps s"

lemma eval_fds_integral_has_field_derivative: fixes s :: "'a :: dirichlet_series" assumes "ereal (s \<bullet> 1) > abs_conv_abscissa f" assumes "fds_nth f 1 = 0" shows "(eval_fds (fds_integral c f) has_field_derivative eval_fds f s) (at s)"

lemma LIMSEQ_cong: assumes "f \<longlonglongrightarrow> x" "\<forall>\<^sup>\<infinity>n. f n = g n" shows "g \<longlonglongrightarrow> x"

lemma gpv_stop_map' [simp]: "gpv_stop (map_gpv' f g h gpv) = map_gpv' (map_option f) g (map_option h) (gpv_stop gpv)"

lemma dia_lowerB: "\<lbrakk> u \<in> verts G; v \<in> verts G \<rbrakk> \<Longrightarrow> diameter w \<ge> \<mu> w u v"

lemma fls_nth_compose_power: assumes "d > 0" shows "fls_nth (fls_compose_power f d) n = (if int d dvd n then fls_nth f (n div int d) else 0)"

lemma pos_neg_kauff_mat: "kauff_mat ((basic [over]) \<circ> (basic [under])) = kauff_mat ((basic [vert,vert]) \<circ> (basic [vert,vert])) "

lemma subst_fmla_commute [simp]: "atom j \<sharp> A \<Longrightarrow> (A(i::=t))(j::=u) = A(i ::= subst j u t)"

lemma le_inf_eq_inf_transp[intro, trans]: assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2" "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>3"

lemma GroupH : "Group H"

lemma get_object_ptr_simp1 [simp]: "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr object h) = Some object"

lemma differentiable_cmult_left_iff [simp]: fixes c::"'a::real_normed_field" shows "(\<lambda>t. c * q t) differentiable at t \<longleftrightarrow> c = 0 \<or> (\<lambda>t. q t) differentiable at t" (is "?lhs = ?rhs")

lemma norm_power_diff: fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}" assumes "norm z \<le> 1" "norm w \<le> 1" shows "norm (z^m - w^m) \<le> m * norm (z - w)"

lemma fA_rel_inv[intro]: notes fun_upd_apply[simp] shows "\<lbrace> LSTP (fA_rel_inv \<^bold>\<and> tso_store_inv) \<rbrace> sys \<lbrace> LSTP fA_rel_inv \<rbrace>"

lemma r_not_updated_stays_the_same: "r \<notin> fst ` set U \<Longrightarrow> apply_updates U c d $ r = d $ r"

lemma correctCompositionDiffLevelsS11opt: "correctCompositionDiffLevels sS11opt"

lemma prv_ldsj_imp_remdups: assumes "set \<phi>s \<subseteq> fmla" shows "prv (imp (ldsj \<phi>s) (ldsj (remdups \<phi>s)))"

lemma bifunctor_proj_fst_ObjMap_app[cat_cs_simps]: assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>C \<BB>)\<lparr>Obj\<rparr>" shows "(\<SS>\<^bsub>\<AA>,\<BB>\<^esub>(-,b)\<^sub>C\<^sub>F)\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<SS>\<lparr>ObjMap\<rparr>\<lparr>a, b\<rparr>\<^sub>\<bullet>"