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

Statement: stringlengths 7 24.3k
lemma run_limit_two_connectedI: assumes A: "ipath E r" assumes B: "a \<in> limit r" "b\<in>limit r" shows "(a,b)\<in>E\<^sup>+"
lemma \<delta>_symbolsI: "(q \<rightarrow> f qs)\<in>\<delta> \<Longrightarrow> f\<in>\<delta>_symbols \<delta>"
lemma S_ne: "U\<^sub>M \<noteq> {}"
lemma mult_two_impl2[dest]: assumes "a * 2 = 1 + 2 * b" shows "(a::enat) = \<infinity> \<and> b=\<infinity>"
theorem shift_safe: assumes "\<forall>ii<i. \<not>sublist_at s t ii" "t!(i+j) \<noteq> s!j" and [simp]: "j < length s" and matches: "\<forall>jj<j. t!(i+jj) = s!jj" defines assignment: "i' \<equiv> i + (j - \<ff> s j + 1)" shows "\<forall>ii<i'. \<not>sublist_at s t ii"
lemma eq_key_imp_eq_value: "v1 = v2" if "distinct (map fst xs)" "(k, v1) \<in> set xs" "(k, v2) \<in> set xs"
lemma JFcol_minimal: "\<lbrakk>\<And>a. F1set1 (dtor1 a) \<subseteq> K1 a; \<And>b. F2set1 (dtor2 b) \<subseteq> K2 b; \<And>a a'. a' \<in> F1set2 (dtor1 a) \<Longrightarrow> K1 a' \<subseteq> K1 a; \<And>a b'. b' \<in> F1set3 (dtor1 a) \<Longrightarrow> K2 b' \<subseteq> K1 a; \<And>b a'. a' \<in> F2set2 (dtor2 b) \<Longrightarrow> K1 a' \<subseteq> K2 b; \<And>b b'. b' \<in> F2set3 (dtor2 b) \<Longrightarrow> K2 b' \<subseteq> K2 b\<rbrakk> \<Longrightarrow> \<forall>a b. JF1col n a \<subseteq> K1 a \<and> JF2col n b \<subseteq> K2 b"
lemma inverse_subgroupD: assumes sub: "subgroup (inverse ` H) G (\<cdot>) \<one>" and inv: "H \<subseteq> Units" shows "subgroup H G (\<cdot>) \<one>"
lemma (in gcd_condition_monoid) gcd_closed [simp]: assumes "a \<in> carrier G" "b \<in> carrier G" shows "somegcd G a b \<in> carrier G"
lemma psubt_bot_impl[simp]: "set (psubt_bot_impl R) = psubt_lhs_bot (set R)"
lemma count_empty [simp]: "count {#} a = 0"
lemma dist_stereographic_finite: assumes "stereographic M1 = of_complex m1" and "stereographic M2 = of_complex m2" shows "dist_riemann_sphere' M1 M2 = 2 * cmod (m1 - m2) / (sqrt (1 + (cmod m1)\<^sup>2) * sqrt (1 + (cmod m2)\<^sup>2))"
lemma addition_assoc:"\<lbrakk>assoc_bpp A f; x \<in> addition_set f A; y \<in> addition_set f A; z \<in> addition_set f A\<rbrakk> \<Longrightarrow> (x \<^sub>f+ y) \<^sub>f+ z = x \<^sub>f+ (y \<^sub>f+ z)"
lemma inj_f[intro!, simp]: "inj_on f (nodes A)"
lemma "Integer[1] \<squnion> (Real[?] :: classes1 type) = Real[?]"
lemma "y \<cdot> x \<cdot> x \<cdot> x \<cdot> y \<le>p y \<cdot> x \<cdot> x \<cdot> y \<cdot> y \<cdot> x \<Longrightarrow> x \<cdot> y = y \<cdot> x"
lemma unique_KC1: "[\|Says C B \<lbrace>Crypt KC1 X, Crypt EK \<lbrace>Key KC1, Y\<rbrace>\<rbrace> \<in> set evs; Says C B' \<lbrace>Crypt KC1 X', Crypt EK' \<lbrace>Key KC1, Y'\<rbrace>\<rbrace> \<in> set evs; C \<notin> bad; evs \<in> set_cr\|] ==> B'=B \<and> Y'=Y"
lemma conversep_restrict_relp [simp]: "(R \<upharpoonleft> P \<otimes> Q)\<inverse>\<inverse> = R\<inverse>\<inverse> \<upharpoonleft> Q \<otimes> P"
lemma monad_state_writerT' [locale_witness]: "monad_state return (bind :: ('a \<times> 'w list, 'm) bind) (get :: ('s, 'm) get) put \<Longrightarrow> monad_state return (bind :: ('a, ('w, 'a, 'm) writerT) bind) (get :: ('s, ('w, 'a, 'm) writerT) get) put"
lemma subst_support_comp_split: fixes \<theta> \<delta> \<I>::"('a,'b) subst" assumes "(\<theta> \<circ>\<^sub>s \<delta>) supports \<I>" shows "subst_domain \<theta> \<inter> range_vars \<theta> = {} \<Longrightarrow> \<theta> supports \<I>" and "subst_domain \<theta> \<inter> subst_domain \<delta> = {} \<Longrightarrow> \<delta> supports \<I>"
lemma replicate_count: "count (mset (replicate n x)) x = n"
lemma word_div_lt_eq_0: "x < y \<Longrightarrow> x div y = 0" for x :: "'a :: len word"
lemma stable_imp_take_bit_eq: \<open>take_bit n a = (if even a then 0 else 2 ^ n - 1)\<close> if \<open>a div 2 = a\<close>
lemma (in Worder) segment_unique1:"\<lbrakk>a \<in> carrier D; b \<in> carrier D; a \<prec> b\<rbrakk> \<Longrightarrow> \<not> ord_equiv (Iod D (segment D b)) (Iod D (segment D a))"
lemma id_fixespointwise: "fixespointwise id A"
lemma proj_poly_syz_eq_zero_iff: "proj_poly_syz n p = 0 \<longleftrightarrow> (component_of_term ` keys p \<subseteq> {0..<n})"
lemma [intro!]: "v \<in> set abs_ast_variable_section \<Longrightarrow> x < length (snd (snd (astDom ! v))) \<Longrightarrow> x \<in> set (the (abs_range_map v))"
lemma isos_compose [intro]: assumes "iso f" and "iso f'" and "seq f' f" shows "iso (f' \<cdot> f)"
lemma lemma_2: assumes "desargues_config A B C A' B' C' M N P R" and "incid A (line B' C') \<or> incid C' (line A B)" and "incid C (line A' B') \<or> incid B' (line A C)" and "incid B (line A' C') \<or> incid A' (line B C)" shows "col M N P \<or> triangle_circumscribes_triangle A B C A' B' C' \<or> triangle_circumscribes_triangle A' B' C' A B C"
lemma le_mask_iff_lt_2n: "n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))"
lemma may_lock_upd_locks_conv [simp]: "lock_actions_ok l t Ls \<Longrightarrow> may_lock (upd_locks l t Ls) t = may_lock l t"
lemma atoms_of_covers': shows "\<Union> (atoms_of Xs) = \<Union> Xs"
lemma inR2': "\<Gamma> \<Rightarrow> F, G, H, \<Delta> \<down> n \<Longrightarrow> \<Gamma> \<Rightarrow> G, H, F, \<Delta> \<down> n"
lemma invar_step_n: "invar (states :: 'a states) \<Longrightarrow> invar ((step^^n) states)"
lemma add_conj_le: shows "z + cnj z \<le> 2 * cmod z"
lemma "(\<lambda>b. b<maxsize, X) \<in> A \<rightarrow> bool_rel"
theorem substT_liftT [simp]: "k \<le> k' \<Longrightarrow> k' < k + n \<Longrightarrow> (\<up>\<^sub>\<tau> n k T)[k' \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau> = \<up>\<^sub>\<tau> (n - 1) k T"
lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"
lemma polymap0 : "polymap 0 = 0"
lemma Domain_Restrict [simp]: "Domain (Restrict A r) = A \<inter> Domain r"
lemma SCallRedsNonStatic: "\<lbrakk> P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,False\<rangle>; P \<turnstile> C sees M,NonStatic:Ts\<rightarrow>T = m in D \<rbrakk> \<Longrightarrow> P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,s\<^sub>2,False\<rangle>"
lemma upper_asymptotic_density_union: "upper_asymptotic_density (A \<union> B) \<le> upper_asymptotic_density A + upper_asymptotic_density B"
lemma coplanar_perm_22: assumes "Coplanar A B C D" shows "Coplanar D C A B"
lemma sorted_repeat_poly_list_rel_with0_wrtl_Cons_iff: \<open>((ys, n) # p, a) \<in> sorted_repeat_poly_list_rel_with0_wrt S term_poly_list_rel \<longleftrightarrow> (p, remove1_mset (mset ys, n) a) \<in> sorted_repeat_poly_list_rel_with0_wrt S term_poly_list_rel \<and> (mset ys, n) \<in># a \<and> (\<forall>x \<in> set p. S ys (fst x)) \<and> sorted_wrt (rel2p var_order_rel) ys \<and> distinct ys\<close>
lemma singleton_abelian_group [simp]: "comm_group (singleton_group a)"
lemma sigma_lemma: "primerec rg (Suc (length xs)) \<Longrightarrow> rec_exec (rec_sigma rg) (xs @ [x]) = Sigma (rec_exec rg) (xs @ [x])"
theorem DC_eq [simp]: shows "Decrypt K (Crypt K X) = X"
lemma Inl_cont_H[simp]: "Inl -` (cont (H n)) = Inl -` (cont (pick n))"
lemma not_ephemeralD_pos_period': assumes C: "C \<in> UNIV // communicating" shows "not_ephemeral C \<longleftrightarrow> 0 < period C"
lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. \<Pi>' j\<in>J. UNIV) = UNIV"
lemma theorem_13: assumes phi: "\<phi> \<in> tf_mformula" "\<phi> \<in> \<A>" and sub: "POS \<subseteq> ACC_mf \<phi>" "ACC_cf_mf \<phi> = {}" shows "cs \<phi> > k powr (4 / 7 * sqrt k)"
lemma types_agree_imp_e_typing: assumes "types_agree t v" shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : ([] _> [t])"
lemma min_union_comm: "A \<union>\<^sub>m B = B \<union>\<^sub>m A"
lemma dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S>_at_pre_nullstrict [simp] : "(null).boss@pre = invalid"
lemma final_subdivFace': "\<And>f u n g. minGraphProps g \<Longrightarrow> pre_subdivFace' g f r u n ovs \<Longrightarrow> f \<in> \<F> g \<Longrightarrow> (ovs = [] \<longrightarrow> n=0 \<and> u = last(verticesFrom f r)) \<Longrightarrow> \<exists>f' \<in> set(finals(subdivFace' g f u n ovs)) - set(finals g). (f\<^bsup>-1\<^esup> \<bullet> r,r) \<in> \<E> f' \<and> \|vertices f'\| = n + \|ovs\| + (if r=u then 1 else \|between (vertices f) r u\| + 2)"
lemma lang_rderiv_lderivs[simp]: "\<lbrakk>wf n r; wf_word n w; a \<in> \<Sigma> n\<rbrakk> \<Longrightarrow> lang n (rderiv a (lderivs w r)) = lang n (lderivs w (rderiv a r))"
lemma is_cnf_encode_initial_state: assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (\<Phi>\<^sub>I \<Pi>)"
lemma avl_delete_root: assumes "avl t" and "t \<noteq> ET" shows "avl(delete_root t)"
lemma SmallerMultipleStepsWithLimit: fixes k A limit assumes "\<forall> n \<ge> limit . (A (Suc n)) < (A n)" shows "\<forall> n \<ge> limit . (A (n + k)) \<le> (A n) - k"
lemma domConc: "x \<in> dom (C (list2FWpolicy b)) \<Longrightarrow> b \<noteq> [] \<Longrightarrow> x \<in> dom (C (list2FWpolicy (a @ b)))"
lemma lunit_eqI: assumes "ide f" and "\<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright>" and "trg f \<star> \<mu> = (\<i>[trg f] \<star> f) \<cdot> (inv \<a>[trg f, trg f, f])" shows "\<mu> = \<l>[f]"
lemma close_preserved: fixes \<Gamma>::"Ctxt" assumes asm: "((ftv T) - (ftv \<Gamma>)) \<sharp>* \<theta>" shows "\<theta><close \<Gamma> T> = close (\<theta><\<Gamma>>) (\<theta><T>)"
lemma finfun_Diag_const_code [code]: "($K$ b, K$ c$) = (K$ (b, c))" "($K$ b, finfun_update_code g a c$) = finfun_update_code ($K$ b, g$) a (b, c)"
lemma strict_mono_limit_ordinal: "strict_mono f \<Longrightarrow> limit_ordinal (oLimit f)"
lemma invariantD: assumes "invariant ds P" shows "P ds (fp_cop_F ds)"
lemma difference_list_set : assumes "finite_tree t1" shows "List.set (difference_list t1 t2) = (set t1 - set t2)"
lemma col_space_eq_row_space_transpose: shows "col_space A = row_space A\<^sup>T"
lemma lattice_of_altdef_lincomb: assumes "set fs \<subseteq> carrier_vec n" shows "lattice_of fs = {y. \<exists>f. lincomb (of_int \<circ> f) (set fs) = y}"
lemma sum_intro: "(\<forall> w. O w z \<longleftrightarrow> (O w x \<or> O w y)) \<Longrightarrow> x \<oplus> y = z"
lemma dvd_imp_degree: \<open>degree x \<le> degree y\<close> if \<open>x dvd y\<close> \<open>x \<noteq> 0\<close> \<open>y \<noteq> 0\<close> for x y :: \<open>'a::{comm_semiring_1,semiring_no_zero_divisors} poly\<close>
lemma GEN_OP_EQ_Id: "GEN_OP (=) (=) (Id\<rightarrow>Id\<rightarrow>bool_rel)"
lemma M1_dim: "M1 \<in> carrier_mat K K"
lemma is_semialg_function_tupleI: assumes "\<And> f. f \<in> set fs \<Longrightarrow> is_semialg_function n f" shows "is_semialg_function_tuple n fs"
lemma quasinorm_sum_limit: "\<exists>f1 f2. (\<forall>n. f = f1 n + f2 n) \<and> (\<lambda>n. eNorm N1 (f1 n) + eNorm N2 (f2 n)) \<longlonglongrightarrow> eNorm (N1 +\<^sub>N N2) f"
lemma lcoeff_nonzero_deg: assumes deg: "deg R p \<noteq> 0" and R: "p \<in> carrier P" shows "coeff P p (deg R p) \<noteq> \<zero>"
lemma point_in_S_polar_is_tangent: assumes "p \<in> S" and "q \<in> S" and "proj2_incident q (polar p)" shows "q = p"
theorem unification: assumes fin: "finite L" assumes uni: "unifier\<^sub>l\<^sub>s \<sigma> L" shows "\<exists>\<theta>. mgu\<^sub>l\<^sub>s \<theta> L"
lemma vector_up_closed: "vector x \<Longrightarrow> up_closed x"
lemma (in UP_univ_prop) Eval_monom: "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"
lemma var_Monotone: "Monotone (\<lambda> \<Phi> . (\<lambda> (s,t) .(b,\<Phi>,s,t):var))"
lemma is_element_ptr_ref [simp]: "is_element_ptr (element_ptr.Ref n)"
lemma sturm_squarefree'_adjacent_root_propagate_left: assumes "p \<noteq> 0" assumes "i < length (sturm_squarefree' (p :: real poly)) - 1" assumes "poly (sturm_squarefree' p ! i) x = 0" and "poly (sturm_squarefree' p ! (i + 1)) x = 0" shows "\<forall>j\<le>i+1. poly (sturm_squarefree' p ! j) x = 0"
lemma poly_add_zero': assumes "set p \<subseteq> carrier R" shows "poly_add p [] = normalize p" and "poly_add [] p = normalize p"
lemma cancel_comm_semigroup_ow_transfer[transfer_rule]: assumes [transfer_rule]: "bi_unique A" "right_total A" shows "(rel_set A ===> (A ===> A ===> A) ===> (A ===> A ===> A) ===> (=)) cancel_comm_semigroup_ow cancel_comm_semigroup_ow"
lemma funpow_dist1_eq_funcset: assumes "y \<in> orbit f x" "x \<in> A" "f \<in> A \<rightarrow> A" "\<And>y. y \<in> A \<Longrightarrow> f y = g y" shows "funpow_dist1 f x y = funpow_dist1 g x y"
lemma limsup_INF_SUP: "limsup f = (INF n. SUP m\<in>{n..}. f m)"
lemma lin_dense_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> t < y)"
theorem ln_lower_1_eq: "0<x \<Longrightarrow> ln_lower_1 x = (x - 1)/x"
lemma sol_dim_in_sat_dual: "x \<in> sat_dual A c \<Longrightarrow> dim_vec x = dim_row A"
lemma solution_CH[simp]: "solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)"
lemma orthogonal_complement_of_cspan: \<open>orthogonal_complement A = orthogonal_complement (cspan A)\<close>
lemma map_of_aset_insert [rewrite]: "unique_keys_set (S \<union> {(k, v)}) \<Longrightarrow> map_of_aset (S \<union> {(k, v)}) = (map_of_aset S) {k \<rightarrow> v}"
lemma has_bochner_integral_indicator: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> has_bochner_integral M (\<lambda>x. indicator A x \<^sub>R c) (measure M A \<^sub>R c)"
lemma Max_lessThan: "0 < n \<Longrightarrow> Max {..<n} = n - Suc 0"
lemma proofTree_def2: "proofTree = ( % x . bounded x & founded subs (SATAxiom o sequent) x)"
lemma restricted_graph_memI: assumes "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)" assumes "take n a \<in> S" assumes "g (take n a) = a!n" shows "a \<in> restricted_graph n g S"
lemma satC_iff_set: "satC \<xi> c \<longleftrightarrow> (\<exists> l \<in> set c. satL \<xi> l)"
lemma fact_ge_self: "fact n \<ge> n"
lemma compare_height_alt: "RBT_Impl.compare_height sx s t tx = compare_height sx s t tx"
lemma "STR ''-92134039538802366542421159375273829975'' = createSInt 128 45648483135649456465465452123894894554654654654654646999465"

lemma run_limit_two_connectedI: assumes A: "ipath E r" assumes B: "a \<in> limit r" "b\<in>limit r" shows "(a,b)\<in>E\<^sup>+"

lemma \<delta>_symbolsI: "(q \<rightarrow> f qs)\<in>\<delta> \<Longrightarrow> f\<in>\<delta>_symbols \<delta>"

lemma S_ne: "U\<^sub>M \<noteq> {}"

lemma mult_two_impl2[dest]: assumes "a * 2 = 1 + 2 * b" shows "(a::enat) = \<infinity> \<and> b=\<infinity>"

theorem shift_safe: assumes "\<forall>ii<i. \<not>sublist_at s t ii" "t!(i+j) \<noteq> s!j" and [simp]: "j < length s" and matches: "\<forall>jj<j. t!(i+jj) = s!jj" defines assignment: "i' \<equiv> i + (j - \<ff> s j + 1)" shows "\<forall>ii<i'. \<not>sublist_at s t ii"

lemma eq_key_imp_eq_value: "v1 = v2" if "distinct (map fst xs)" "(k, v1) \<in> set xs" "(k, v2) \<in> set xs"

lemma JFcol_minimal: "\<lbrakk>\<And>a. F1set1 (dtor1 a) \<subseteq> K1 a; \<And>b. F2set1 (dtor2 b) \<subseteq> K2 b; \<And>a a'. a' \<in> F1set2 (dtor1 a) \<Longrightarrow> K1 a' \<subseteq> K1 a; \<And>a b'. b' \<in> F1set3 (dtor1 a) \<Longrightarrow> K2 b' \<subseteq> K1 a; \<And>b a'. a' \<in> F2set2 (dtor2 b) \<Longrightarrow> K1 a' \<subseteq> K2 b; \<And>b b'. b' \<in> F2set3 (dtor2 b) \<Longrightarrow> K2 b' \<subseteq> K2 b\<rbrakk> \<Longrightarrow> \<forall>a b. JF1col n a \<subseteq> K1 a \<and> JF2col n b \<subseteq> K2 b"

lemma inverse_subgroupD: assumes sub: "subgroup (inverse ` H) G (\<cdot>) \<one>" and inv: "H \<subseteq> Units" shows "subgroup H G (\<cdot>) \<one>"

lemma (in gcd_condition_monoid) gcd_closed [simp]: assumes "a \<in> carrier G" "b \<in> carrier G" shows "somegcd G a b \<in> carrier G"

lemma psubt_bot_impl[simp]: "set (psubt_bot_impl R) = psubt_lhs_bot (set R)"

lemma count_empty [simp]: "count {#} a = 0"

lemma dist_stereographic_finite: assumes "stereographic M1 = of_complex m1" and "stereographic M2 = of_complex m2" shows "dist_riemann_sphere' M1 M2 = 2 * cmod (m1 - m2) / (sqrt (1 + (cmod m1)\<^sup>2) * sqrt (1 + (cmod m2)\<^sup>2))"

lemma addition_assoc:"\<lbrakk>assoc_bpp A f; x \<in> addition_set f A; y \<in> addition_set f A; z \<in> addition_set f A\<rbrakk> \<Longrightarrow> (x \<^sub>f+ y) \<^sub>f+ z = x \<^sub>f+ (y \<^sub>f+ z)"

lemma inj_f[intro!, simp]: "inj_on f (nodes A)"

lemma "Integer[1] \<squnion> (Real[?] :: classes1 type) = Real[?]"

lemma "y \<cdot> x \<cdot> x \<cdot> x \<cdot> y \<le>p y \<cdot> x \<cdot> x \<cdot> y \<cdot> y \<cdot> x \<Longrightarrow> x \<cdot> y = y \<cdot> x"

lemma unique_KC1: "[|Says C B \<lbrace>Crypt KC1 X, Crypt EK \<lbrace>Key KC1, Y\<rbrace>\<rbrace> \<in> set evs; Says C B' \<lbrace>Crypt KC1 X', Crypt EK' \<lbrace>Key KC1, Y'\<rbrace>\<rbrace> \<in> set evs; C \<notin> bad; evs \<in> set_cr|] ==> B'=B \<and> Y'=Y"

lemma conversep_restrict_relp [simp]: "(R \<upharpoonleft> P \<otimes> Q)\<inverse>\<inverse> = R\<inverse>\<inverse> \<upharpoonleft> Q \<otimes> P"

lemma monad_state_writerT' [locale_witness]: "monad_state return (bind :: ('a \<times> 'w list, 'm) bind) (get :: ('s, 'm) get) put \<Longrightarrow> monad_state return (bind :: ('a, ('w, 'a, 'm) writerT) bind) (get :: ('s, ('w, 'a, 'm) writerT) get) put"

lemma subst_support_comp_split: fixes \<theta> \<delta> \::"('a,'b) subst" assumes "(\<theta> \<circ>\<^sub>s \<delta>) supports \" shows "subst_domain \<theta> \<inter> range_vars \<theta> = {} \<Longrightarrow> \<theta> supports \" and "subst_domain \<theta> \<inter> subst_domain \<delta> = {} \<Longrightarrow> \<delta> supports \"

lemma replicate_count: "count (mset (replicate n x)) x = n"

lemma word_div_lt_eq_0: "x < y \<Longrightarrow> x div y = 0" for x :: "'a :: len word"

lemma stable_imp_take_bit_eq: \<open>take_bit n a = (if even a then 0 else 2 ^ n - 1)\<close> if \<open>a div 2 = a\<close>

lemma (in Worder) segment_unique1:"\<lbrakk>a \<in> carrier D; b \<in> carrier D; a \<prec> b\<rbrakk> \<Longrightarrow> \<not> ord_equiv (Iod D (segment D b)) (Iod D (segment D a))"

lemma id_fixespointwise: "fixespointwise id A"

lemma proj_poly_syz_eq_zero_iff: "proj_poly_syz n p = 0 \<longleftrightarrow> (component_of_term ` keys p \<subseteq> {0..<n})"

lemma [intro!]: "v \<in> set abs_ast_variable_section \<Longrightarrow> x < length (snd (snd (astDom ! v))) \<Longrightarrow> x \<in> set (the (abs_range_map v))"

lemma isos_compose [intro]: assumes "iso f" and "iso f'" and "seq f' f" shows "iso (f' \<cdot> f)"

lemma lemma_2: assumes "desargues_config A B C A' B' C' M N P R" and "incid A (line B' C') \<or> incid C' (line A B)" and "incid C (line A' B') \<or> incid B' (line A C)" and "incid B (line A' C') \<or> incid A' (line B C)" shows "col M N P \<or> triangle_circumscribes_triangle A B C A' B' C' \<or> triangle_circumscribes_triangle A' B' C' A B C"

lemma le_mask_iff_lt_2n: "n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))"

lemma may_lock_upd_locks_conv [simp]: "lock_actions_ok l t Ls \<Longrightarrow> may_lock (upd_locks l t Ls) t = may_lock l t"

lemma atoms_of_covers': shows "\<Union> (atoms_of Xs) = \<Union> Xs"

lemma inR2': "\<Gamma> \<Rightarrow> F, G, H, \<Delta> \<down> n \<Longrightarrow> \<Gamma> \<Rightarrow> G, H, F, \<Delta> \<down> n"

lemma invar_step_n: "invar (states :: 'a states) \<Longrightarrow> invar ((step^^n) states)"

lemma add_conj_le: shows "z + cnj z \<le> 2 * cmod z"

lemma "(\<lambda>b. b<maxsize, X) \<in> A \<rightarrow> bool_rel"

theorem substT_liftT [simp]: "k \<le> k' \<Longrightarrow> k' < k + n \<Longrightarrow> (\<up>\<^sub>\<tau> n k T)[k' \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau> = \<up>\<^sub>\<tau> (n - 1) k T"

lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"

lemma polymap0 : "polymap 0 = 0"

lemma Domain_Restrict [simp]: "Domain (Restrict A r) = A \<inter> Domain r"

lemma SCallRedsNonStatic: "\<lbrakk> P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,False\<rangle>; P \<turnstile> C sees M,NonStatic:Ts\<rightarrow>T = m in D \<rbrakk> \<Longrightarrow> P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,s\<^sub>2,False\<rangle>"

lemma upper_asymptotic_density_union: "upper_asymptotic_density (A \<union> B) \<le> upper_asymptotic_density A + upper_asymptotic_density B"

lemma coplanar_perm_22: assumes "Coplanar A B C D" shows "Coplanar D C A B"

lemma sorted_repeat_poly_list_rel_with0_wrtl_Cons_iff: \<open>((ys, n) # p, a) \<in> sorted_repeat_poly_list_rel_with0_wrt S term_poly_list_rel \<longleftrightarrow> (p, remove1_mset (mset ys, n) a) \<in> sorted_repeat_poly_list_rel_with0_wrt S term_poly_list_rel \<and> (mset ys, n) \<in># a \<and> (\<forall>x \<in> set p. S ys (fst x)) \<and> sorted_wrt (rel2p var_order_rel) ys \<and> distinct ys\<close>

lemma singleton_abelian_group [simp]: "comm_group (singleton_group a)"

lemma sigma_lemma: "primerec rg (Suc (length xs)) \<Longrightarrow> rec_exec (rec_sigma rg) (xs @ [x]) = Sigma (rec_exec rg) (xs @ [x])"

theorem DC_eq [simp]: shows "Decrypt K (Crypt K X) = X"

lemma Inl_cont_H[simp]: "Inl -` (cont (H n)) = Inl -` (cont (pick n))"

lemma not_ephemeralD_pos_period': assumes C: "C \<in> UNIV // communicating" shows "not_ephemeral C \<longleftrightarrow> 0 < period C"

lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. \<Pi>' j\<in>J. UNIV) = UNIV"

lemma theorem_13: assumes phi: "\<phi> \<in> tf_mformula" "\<phi> \<in> \<A>" and sub: "POS \<subseteq> ACC_mf \<phi>" "ACC_cf_mf \<phi> = {}" shows "cs \<phi> > k powr (4 / 7 * sqrt k)"

lemma types_agree_imp_e_typing: assumes "types_agree t v" shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : ([] _> [t])"

lemma min_union_comm: "A \<union>\<^sub>m B = B \<union>\<^sub>m A"

lemma dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\\<O>\<S>\<S>_at_pre_nullstrict [simp] : "(null).boss@pre = invalid"

lemma final_subdivFace': "\<And>f u n g. minGraphProps g \<Longrightarrow> pre_subdivFace' g f r u n ovs \<Longrightarrow> f \<in> \<F> g \<Longrightarrow> (ovs = [] \<longrightarrow> n=0 \<and> u = last(verticesFrom f r)) \<Longrightarrow> \<exists>f' \<in> set(finals(subdivFace' g f u n ovs)) - set(finals g). (f\<^bsup>-1\<^esup> \<bullet> r,r) \<in> \<E> f' \<and> |vertices f'| = n + |ovs| + (if r=u then 1 else |between (vertices f) r u| + 2)"

lemma lang_rderiv_lderivs[simp]: "\<lbrakk>wf n r; wf_word n w; a \<in> \<Sigma> n\<rbrakk> \<Longrightarrow> lang n (rderiv a (lderivs w r)) = lang n (lderivs w (rderiv a r))"

lemma is_cnf_encode_initial_state: assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (\<Phi>\<^sub>I \<Pi>)"

lemma avl_delete_root: assumes "avl t" and "t \<noteq> ET" shows "avl(delete_root t)"

lemma SmallerMultipleStepsWithLimit: fixes k A limit assumes "\<forall> n \<ge> limit . (A (Suc n)) < (A n)" shows "\<forall> n \<ge> limit . (A (n + k)) \<le> (A n) - k"

lemma domConc: "x \<in> dom (C (list2FWpolicy b)) \<Longrightarrow> b \<noteq> [] \<Longrightarrow> x \<in> dom (C (list2FWpolicy (a @ b)))"

lemma lunit_eqI: assumes "ide f" and "\<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright>" and "trg f \<star> \<mu> = (\[trg f] \<star> f) \<cdot> (inv \<a>[trg f, trg f, f])" shows "\<mu> = \<l>[f]"

lemma close_preserved: fixes \<Gamma>::"Ctxt" assumes asm: "((ftv T) - (ftv \<Gamma>)) \<sharp>* \<theta>" shows "\<theta><close \<Gamma> T> = close (\<theta><\<Gamma>>) (\<theta><T>)"

lemma finfun_Diag_const_code [code]: "($K$ b, K$ c$) = (K$ (b, c))" "($K$ b, finfun_update_code g a c$) = finfun_update_code ($K$ b, g$) a (b, c)"

lemma strict_mono_limit_ordinal: "strict_mono f \<Longrightarrow> limit_ordinal (oLimit f)"

lemma invariantD: assumes "invariant ds P" shows "P ds (fp_cop_F ds)"

lemma difference_list_set : assumes "finite_tree t1" shows "List.set (difference_list t1 t2) = (set t1 - set t2)"

lemma col_space_eq_row_space_transpose: shows "col_space A = row_space A\<^sup>T"

lemma lattice_of_altdef_lincomb: assumes "set fs \<subseteq> carrier_vec n" shows "lattice_of fs = {y. \<exists>f. lincomb (of_int \<circ> f) (set fs) = y}"

lemma sum_intro: "(\<forall> w. O w z \<longleftrightarrow> (O w x \<or> O w y)) \<Longrightarrow> x \<oplus> y = z"

lemma dvd_imp_degree: \<open>degree x \<le> degree y\<close> if \<open>x dvd y\<close> \<open>x \<noteq> 0\<close> \<open>y \<noteq> 0\<close> for x y :: \<open>'a::{comm_semiring_1,semiring_no_zero_divisors} poly\<close>

lemma GEN_OP_EQ_Id: "GEN_OP (=) (=) (Id\<rightarrow>Id\<rightarrow>bool_rel)"

lemma M1_dim: "M1 \<in> carrier_mat K K"

lemma is_semialg_function_tupleI: assumes "\<And> f. f \<in> set fs \<Longrightarrow> is_semialg_function n f" shows "is_semialg_function_tuple n fs"

lemma quasinorm_sum_limit: "\<exists>f1 f2. (\<forall>n. f = f1 n + f2 n) \<and> (\<lambda>n. eNorm N1 (f1 n) + eNorm N2 (f2 n)) \<longlonglongrightarrow> eNorm (N1 +\<^sub>N N2) f"

lemma lcoeff_nonzero_deg: assumes deg: "deg R p \<noteq> 0" and R: "p \<in> carrier P" shows "coeff P p (deg R p) \<noteq> \<zero>"

lemma point_in_S_polar_is_tangent: assumes "p \<in> S" and "q \<in> S" and "proj2_incident q (polar p)" shows "q = p"

theorem unification: assumes fin: "finite L" assumes uni: "unifier\<^sub>l\<^sub>s \<sigma> L" shows "\<exists>\<theta>. mgu\<^sub>l\<^sub>s \<theta> L"

lemma vector_up_closed: "vector x \<Longrightarrow> up_closed x"

lemma (in UP_univ_prop) Eval_monom: "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"

lemma var_Monotone: "Monotone (\<lambda> \<Phi> . (\<lambda> (s,t) .(b,\<Phi>,s,t):var))"

lemma is_element_ptr_ref [simp]: "is_element_ptr (element_ptr.Ref n)"

lemma sturm_squarefree'_adjacent_root_propagate_left: assumes "p \<noteq> 0" assumes "i < length (sturm_squarefree' (p :: real poly)) - 1" assumes "poly (sturm_squarefree' p ! i) x = 0" and "poly (sturm_squarefree' p ! (i + 1)) x = 0" shows "\<forall>j\<le>i+1. poly (sturm_squarefree' p ! j) x = 0"

lemma poly_add_zero': assumes "set p \<subseteq> carrier R" shows "poly_add p [] = normalize p" and "poly_add [] p = normalize p"

lemma cancel_comm_semigroup_ow_transfer[transfer_rule]: assumes [transfer_rule]: "bi_unique A" "right_total A" shows "(rel_set A ===> (A ===> A ===> A) ===> (A ===> A ===> A) ===> (=)) cancel_comm_semigroup_ow cancel_comm_semigroup_ow"

lemma funpow_dist1_eq_funcset: assumes "y \<in> orbit f x" "x \<in> A" "f \<in> A \<rightarrow> A" "\<And>y. y \<in> A \<Longrightarrow> f y = g y" shows "funpow_dist1 f x y = funpow_dist1 g x y"

lemma limsup_INF_SUP: "limsup f = (INF n. SUP m\<in>{n..}. f m)"

lemma lin_dense_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall>t. l < t \<and> t t \<notin> U) \<and> l < x \<and> x t < x \<longrightarrow> (\<forall>y. l < y \<and> y t < y)"

theorem ln_lower_1_eq: "0<x \<Longrightarrow> ln_lower_1 x = (x - 1)/x"

lemma sol_dim_in_sat_dual: "x \<in> sat_dual A c \<Longrightarrow> dim_vec x = dim_row A"

lemma solution_CH[simp]: "solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)"

lemma orthogonal_complement_of_cspan: \<open>orthogonal_complement A = orthogonal_complement (cspan A)\<close>

lemma map_of_aset_insert [rewrite]: "unique_keys_set (S \<union> {(k, v)}) \<Longrightarrow> map_of_aset (S \<union> {(k, v)}) = (map_of_aset S) {k \<rightarrow> v}"

lemma has_bochner_integral_indicator: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> has_bochner_integral M (\<lambda>x. indicator A x *\<^sub>R c) (measure M A *\<^sub>R c)"

lemma Max_lessThan: "0 < n \<Longrightarrow> Max {..<n} = n - Suc 0"

lemma proofTree_def2: "proofTree = ( % x . bounded x & founded subs (SATAxiom o sequent) x)"

lemma restricted_graph_memI: assumes "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)" assumes "take n a \<in> S" assumes "g (take n a) = a!n" shows "a \<in> restricted_graph n g S"

lemma satC_iff_set: "satC \<xi> c \<longleftrightarrow> (\<exists> l \<in> set c. satL \<xi> l)"

lemma fact_ge_self: "fact n \<ge> n"

lemma compare_height_alt: "RBT_Impl.compare_height sx s t tx = compare_height sx s t tx"

lemma "STR ''-92134039538802366542421159375273829975'' = createSInt 128 45648483135649456465465452123894894554654654654654646999465"