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

Statement: stringlengths 7 24.3k
lemma inv2_rbt_joinR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<ge> bheight r \<Longrightarrow> inv2 (rbt_joinR l a b r)"
lemma poincare_map_has_derivative_step: assumes Deriv: "(poincare_map P has_derivative blinfun_apply D) (at (flow0 x0 h))" assumes ret: "returns_to P x0" assumes cont: "continuous (at x0 within S) (return_time P)" assumes less: "0 \<le> h" "h < return_time P x0" assumes cP: "closed P" and x0: "x0 \<in> S" shows "((\<lambda>x. poincare_map P x) has_derivative (D o\<^sub>L Dflow x0 h)) (at x0 within S)"
lemma has_derivative_Blinfun: assumes "(f has_derivative f') F" shows "(f has_derivative Blinfun f') F"
lemma a_sum_le_U_sum: "wf ot \<Longrightarrow> acost_sum ot \<le> U_sum ot"
lemma op_cf_comma_is_functor: assumes "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" shows "op_cf_comma \<GG> \<HH> : op_cat (\<GG> \<^sub>C\<^sub>F\<down>\<^sub>C\<^sub>F \<HH>) \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> (op_cf \<HH>) \<^sub>C\<^sub>F\<down>\<^sub>C\<^sub>F (op_cf \<GG>)"
lemma wf_interp_for_formula_any_Inr: "wf_interp_for_formula (w, Inr P # I) \<phi> \<Longrightarrow> \<forall>P \<subseteq> {0 .. length w - 1}. wf_interp_for_formula (w, Inr P # I) \<phi>"
lemma ind_is_open_S [iff]: "ind_is_open S"
lemma set_compre_subset: "B \<subseteq> A \<Longrightarrow> {x \<in> B. P x} \<subseteq> {x \<in> A. P x}"
lemma ik\<^sub>s\<^sub>t_assignment_rhs\<^sub>s\<^sub>t_wfrestrictedvars_subset: "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>s\<^sub>t A) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t A"
lemma "\<lbrakk>test p; test q\<rbrakk> \<Longrightarrow> p \<cdot> x = p \<cdot> x \<cdot> q \<longleftrightarrow> p \<cdot> x \<le> x \<cdot> q"
lemma ord_strict_pI: assumes "lookup p v = 0" and "lookup q v \<noteq> 0" and "\<And>u. v \<prec>\<^sub>t u \<Longrightarrow> lookup p u = lookup q u" shows "p \<prec>\<^sub>p q"
lemma swap_triple: assumes "a \<noteq> b" and "c \<noteq> b" assumes "sort_of a = sort_of b" "sort_of b = sort_of c" shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
lemma (in Group) compseries_is_tW_cmpser:"\<lbrakk>0 < r; compseries G r f\<rbrakk> \<Longrightarrow> tW_cmpser G r f"
lemma monom_eval_add: assumes "closed_fun R g" shows "monom_eval R (add_mset x M) g = (g x) \<otimes> (monom_eval R M g)"
lemma pre_post_left_sub_dist: "-p\<squnion>-q\<stileturn>-r \<le> -p\<stileturn>-r"
lemma tso_lock_invL[intro]: "\<lbrace> tso_lock_invL \<rbrace> mutator m"
lemma box_an_an_same: "\|an(x)]an(x) = 1"
lemma an_neq_inf:"an n \<noteq> \<infinity>"
lemma good_mapI: assumes surj_h: "surj h" and prem: "!! x x' y y'. h(x,y) = h(x',y') ==> x=x'" shows "good_map h"
lemma pivot_unsat_core_id: "\<lbrakk>\<triangle> (\<T> s); x\<^sub>i \<in> lvars (\<T> s); x\<^sub>j \<in> rvars_of_lvar (\<T> s) x\<^sub>i\<rbrakk> \<Longrightarrow> \<U>\<^sub>c (pivot x\<^sub>i x\<^sub>j s) = \<U>\<^sub>c s"
lemma not_convr [simp]: "(\<not> p)\<^sup>- = (\<not> p\<^sup>-)"
theorem (* Detailed proof *) fixes f:: "'a::complete_lattice \<Rightarrow> 'a" assumes CONT: "cont f" shows "lfp f = (SUP i. (f^^i) bot)"
lemma lfilter_id_conv: "lfilter P xs = xs \<longleftrightarrow> (\<forall>x\<in>lset xs. P x)" (is "?lhs \<longleftrightarrow> ?rhs")
lemma preal_sup: assumes le: "\<And>X. X \<in> P \<Longrightarrow> X \<le> Y" and "P \<noteq> {}" shows "cut (\<Union>X \<in> P. Rep_preal(X))"
lemma [trans] : "P' \<subseteq> P \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P c Q,A \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P' c Q,A"
lemma vector_to_cblinfun_apply_one_dim[simp]: shows "vector_to_cblinfun \<phi> \<^sub>V \<gamma> = one_dim_iso \<gamma> \<^sub>C \<phi>"
lemma typ_of1_decr_gen': "typ_of1 (Ts@Ts') (decr (length Ts) t) = tyo\<Longrightarrow> \<not> loose_bvar1 t (length Ts) \<Longrightarrow> typ_of1 (Ts@[T]@Ts') t = tyo"
lemma Accessible_right_lang_eq [simp]: "q \<in> accessible \<Longrightarrow> Accessible.right_lang q = right_lang q"
lemma repeatable_both_both_nxt: assumes "t' \<otimes>\<otimes> t = t'" shows "t' \<otimes>\<otimes> t'' \<otimes>\<otimes> t = t' \<otimes>\<otimes> t''"
lemma projection_on_union: "l \<upharpoonleft> Y = [] \<Longrightarrow> l \<upharpoonleft> (X \<union> Y) = l \<upharpoonleft> X"
lemma lappend_is_LNil_conv [iff]: "(s @@ t = LNil) = (s = LNil \<and> t = LNil)"
lemma distinct_RS2[rule_format,simp]: "distinct p \<longrightarrow> distinct (removeShadowRules2 p)"
lemma Failures_implies_Traces: " \<lbrakk>is_process P; (s, X) \<in> FAILURES P\<rbrakk> \<Longrightarrow> s \<in> TRACES P"
lemma push_push_aux: "peval (push_param P (Suc m)) (push a n) = peval (push_param P m) a"
lemma zero_notin_vpair: "0 \<notin> elts \<langle>x,y\<rangle>"
lemma concat_pp_less: assumes "concat (take k ws) <p concat (take n ws)" shows "k < n"
lemma Inter_eqvt [eqvt]: shows "p \<bullet> \<Inter>S = \<Inter>(p \<bullet> S)"
lemma pell_valuation_solution_pos_nat: fixes z :: "nat \<times> nat" assumes "solution z" shows "pell_valuation z > 0"
lemma parts_trans: "\<lbrakk> X\<in> parts G; G \<subseteq> parts H \<rbrakk> \<Longrightarrow> X\<in> parts H"
lemma modifies_lhsv'_gen: assumes "lhsv\<pi> \<pi> \<subseteq> vs" assumes "lhsv' c \<subseteq> vs" assumes "\<pi>: (c,s) \<Rightarrow> t" shows "modifies vs t s"
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a)" for a :: "'a::real_normed_vector"
lemma uset_laws [simp]: "x \<in>\<^sub>u {}\<^sub>u = false" "x \<in>\<^sub>u {m..n}\<^sub>u = (m \<le>\<^sub>u x \<and> x \<le>\<^sub>u n)"
lemma wellformed_items_Scan: "wellformed_items I \<Longrightarrow> T \<subseteq> \<X> k \<Longrightarrow> wellformed_items (Scan T k I)"
lemma real_greaterThanLessThan_minus_infinity_eq: "real_of_ereal ` {-\<infinity><..<N::ereal} = (if N = \<infinity> then UNIV else if N = -\<infinity> then {} else {..<real_of_ereal N})"
lemma reduce'_semantics: assumes \<open>static q\<close> shows \<open>(M, w \<Turnstile>\<^sub>! [p]\<^sub>! q) = (M, w \<Turnstile>\<^sub>! reduce' p q)\<close>
lemma gexp_max_input_In: "max_input (In v l) = AExp.max_input (V v)"
lemma if_doesnt_read: "x \<notin> bexp_vars e \<Longrightarrow> \<C>_vars x \<inter> bexp_vars e = {} \<Longrightarrow> doesnt_read_or_modify (If e c\<^sub>1 c\<^sub>2 \<otimes> annos) x"
lemma (in dup) shows "s<a := i>\<cdot>x = s\<cdot>x"
lemma containsI: "x \<in> A ==> contains A x"
lemma holdsIstate_paperIDs_geq_subPH: "holdsIstate paperIDs_geq_subPH"
lemma OUTfromChCorrect_data22: "OUTfromChCorrect data22"
lemma ts_ord_add_isol: "n x \<sqsubseteq> n y \<Longrightarrow> n z \<oplus> n x \<sqsubseteq> n z \<oplus> n y"
lemma psubst_lcnj[simp]: "set \<phi>s \<subseteq> fmla \<Longrightarrow> snd ` (set txs) \<subseteq> var \<Longrightarrow> fst ` (set txs) \<subseteq> trm \<Longrightarrow> distinct (map snd txs) \<Longrightarrow> psubst (lcnj \<phi>s) txs = lcnj (map (\<lambda>\<phi>. psubst \<phi> txs) \<phi>s)"
lemma coeff_mult_C[simp]: "coeff (a * C x) n = coeff a n * x"
lemma closed_UNIV [simp]: "closed UNIV f"
lemma dual_inv2 [simp]: "\<partial> \<circ> \<partial>\<^sup>- = id"
lemma product_language_state[simp]: "LS (product A B) (q1,q2) = LS A q1 \<inter> LS B q2"
lemma acquired_reads_no_pending_write_in: "x \<in> acquired_reads False xs A \<Longrightarrow> x \<in> acquired_reads False xs B"
lemma step2: "gdpr_scenario' \<rightarrow>\<^sub>n gdpr_scenario''"
lemma i_Exec_Comp_Stream_Acc_Output__Init__eq_Msg_iSince_conv: " \<lbrakk> 0 < k; m \<noteq> \<NoMsg>; s = (output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun (input \<odot>\<^sub>i k) c) \<rbrakk> \<Longrightarrow> ((i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c) t = m) = (s t2 = \<NoMsg>. t2 \<S> t1 [Suc (t * k)\<dots>,k - Suc 0]. s t1 = m)"
lemma reachFrom_step_induct[consumes 1, case_names Refl Step]: assumes s: "reachFrom s s'" and refl: "P s" and step: "\<And>s' a ou s''. reachFrom s s' \<Longrightarrow> P s' \<Longrightarrow> step s' a = (ou, s'') \<Longrightarrow> P s''" shows "P s'"
lemma omega_induct[consumes 1, case_names 0 succ]: assumes "n \<in>\<^sub>\<circ> \<omega>" and "P 0" and "\<And>n. \<lbrakk> n \<in>\<^sub>\<circ> \<omega>; P n \<rbrakk> \<Longrightarrow> P (succ n)" shows "P n"
lemma subst_ev_var_flip[simp]: fixes e::e and y::x and x::x assumes "atom y \<sharp> e" shows "(y \<leftrightarrow> x) \<bullet> e = e [x::=V_var y]\<^sub>e\<^sub>v"
lemma permutes_iff: "p permutes {..<n} \<Longrightarrow> i < n \<Longrightarrow> j < n \<Longrightarrow> p i = p j \<longleftrightarrow> i = j"
lemma between_distinct_r12: "distinct vs \<Longrightarrow> ram1 \<noteq> ram2 \<Longrightarrow> distinct (ram1 # between vs ram1 ram2 @ [ram2])"
lemma less_add_order_eq_0: assumes "a+^k = 0" "k < add_order a" shows "k = 0"
lemma resInputFreeTrans[dest]: fixes x :: name fixes a :: name and y :: name and P :: pi and \<alpha> :: freeRes and P' :: pi assumes "<\<nu>x>a<y>.P \<longmapsto>\<alpha> \<prec> P'" shows False
theorem preservation: assumes v: "v \<in> E e \<rho>" and rr: "e \<longrightarrow>* e'" shows "v \<in> E e' \<rho>"
lemma preserves_arr: shows "A.arr T \<Longrightarrow> B.arr (map T)"
lemma iter: "\<forall>a. (iter g (g a) n) = (g (iter g a n))"
lemma appGetStatic[simp]: "app\<^sub>i (Getstatic C F D,P,pc,mxs,T\<^sub>r,s) = (\<exists> vT ST LT. s = (ST, LT) \<and> length ST < mxs \<and> P \<turnstile> C sees F,Static:vT in D)"
lemma root_primitive_part [simp]: fixes p :: "'a :: {semiring_gcd, semiring_no_zero_divisors} poly" shows "poly (primitive_part p) x = 0 \<longleftrightarrow> poly p x = 0"
lemma contour_integral_reversepath: assumes "valid_path g" shows "contour_integral (reversepath g) f = - (contour_integral g f)"
lemma degree_rebase_poly_le: "degree (#p) \<le> degree p"
lemma negative_len_shortest: "length xs = n \<Longrightarrow> len m i i xs < 0 \<Longrightarrow> \<exists> j ys. distinct (j # ys) \<and> len m j j ys < 0 \<and> j \<in> set (i # xs) \<and> set ys \<subseteq> set xs"
lemma strictly_between_implies_angle_eq_pi: assumes "between (A, C) B" assumes "A \<noteq> B" "B \<noteq> C" shows "angle A B C = pi"
lemma (in weak_partial_order) greatest_Lower_cong_l: assumes "x .= x'" and "x \<in> carrier L" "x' \<in> carrier L" shows "greatest L x (Lower L A) = greatest L x' (Lower L A)"
lemma [transfer_rule]: "(pcr_fmap (=) (=) ===> pcr_linear_poly) (\<lambda> f x. case f x of None \<Rightarrow> 0 \| Some x \<Rightarrow> x) LinearPoly"
lemma items_le_fix_D: assumes items_le_fix: "items_le k I = I" assumes x_dom: "x \<in> I" shows "item_end x \<le> k"
lemma lossless_complete_game: assumes lossless_init: "\<forall> h w. lossless_spmf (init h w)" and lossless_response: "\<forall> r w e. lossless_spmf (response r w e)" shows "lossless_spmf (completeness_game h w e)"
lemma out_subseqs1_set_as_out_subseqs2_set: "out_subseqs1_set ((i,v) # prevs) = { f head v } \<union> out_subseqs1_set prevs \<union> out_subseqs2_set v [0..<i]"
lemma row_echelon_form_berlekamp_resulting_mat: "row_echelon_form (berlekamp_resulting_mat u)"
lemma tp_correct': assumes layout: "ly = layout_of ap" and compile: "tp = tm_of ap" and crsp: "crsp ly (0, lm) (Suc 0, l, r) ires" and abc_halt: "abc_steps_l (0, lm) ap stp = (length ap, am)" shows "\<exists> stp k. steps (Suc 0, l, r) (tp, 0) stp = (start_of ly (length ap), Bk # Bk # ires, <am> @ Bk\<up>k)"
lemma inter_guards_sym: "\<And>c. (c1 \<inter>\<^sub>g c2) = Some c \<Longrightarrow> (c2 \<inter>\<^sub>g c1) = Some c"
lemma(in padic_integers) val_Zp_not_equal_ord_plus_minus: assumes "x \<in> carrier Zp" assumes "y \<in> carrier Zp" assumes "val_Zp x \<noteq> (val_Zp y)" shows "val_Zp (x \<ominus> y) = val_Zp (x \<oplus> y)"
lemma distinct_edgesI: assumes "distinct p" shows "distinct (walk_edges p)"
lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
lemma normalize_src_ports_preserves_normalized_not_has_disc_negated: assumes n: "normalized_nnf_match m" and nodisc: "\<not> has_disc_negated disc2 False m" and disc2_noProt: "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated is_Src_Ports False m" shows "m'\<in> set (normalize_src_ports m) \<Longrightarrow> \<not> has_disc_negated disc2 False m'"
lemma [code_unfold]: "((x::OclAny) \<doteq> y) = StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y"
lemma T_eq: "\<Gamma> \<turnstile> t : T \<Longrightarrow> T = T' \<Longrightarrow> \<Gamma> \<turnstile> t : T'"
lemma inext_nth_iNext_Suc: "(\<circle> t (I \<rightarrow> n) I. P t) = P (I \<rightarrow> Suc n)"
lemma "rec_list nil cons [] = nil"
lemma lift_simulation_eval: "L1.eval s1 s1' \<Longrightarrow> match i s1 s2 \<Longrightarrow> \<exists>i' s2'. L2.eval s2 s2' \<and> match i' s1' s2'"
lemma assumes "<P> c <Q>" assumes "(h,as)\<Turnstile>P" shows hoare_triple_success: "success c h" and hoare_triple_effect: "\<exists>h' r. effect c h h' r \<and> (h',new_addrs h as h')\<Turnstile>Q r"
lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)"
lemma distinct_merge_distinct[simp]: "\<lbrakk>sorted xs; distinct xs; sorted ys; distinct ys\<rbrakk> \<Longrightarrow> distinct (merge_distinct xs ys)"
lemma BinOpRedsVal: assumes e\<^sub>1_steps: "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>" and e\<^sub>2_steps: "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>" and op: "binop(bop,v\<^sub>1,v\<^sub>2) = Some v" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2, s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2\<rangle>" (<)(is "(?x, ?z) \<in> (red P)\<^sup>*")
lemma infinite_hyp_changes_not_Lim: assumes "f \<in> U" and "\<forall>n. \<exists>m\<^sub>1>n. \<exists>m\<^sub>2>n. s (f \<triangleright> m\<^sub>1) \<noteq> s (f \<triangleright> m\<^sub>2)" shows "\<not> learn_lim \<psi> U s"
lemma adj_to_edge_set_card_lim: assumes "e \<in> edge_set" shows "card e > 0 \<and> card e \<le> 2"
lemma lF_char: assumes "C.ide f" shows "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright>" and "F (trg\<^sub>C f) \<star>\<^sub>D lF f = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]" and "\<exists>!\<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> F (trg\<^sub>C f) \<star>\<^sub>D \<mu> = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"

lemma inv2_rbt_joinR: "inv2 l \<Longrightarrow> inv2 r \<Longrightarrow> bheight l \<ge> bheight r \<Longrightarrow> inv2 (rbt_joinR l a b r)"

lemma poincare_map_has_derivative_step: assumes Deriv: "(poincare_map P has_derivative blinfun_apply D) (at (flow0 x0 h))" assumes ret: "returns_to P x0" assumes cont: "continuous (at x0 within S) (return_time P)" assumes less: "0 \<le> h" "h < return_time P x0" assumes cP: "closed P" and x0: "x0 \<in> S" shows "((\<lambda>x. poincare_map P x) has_derivative (D o\<^sub>L Dflow x0 h)) (at x0 within S)"

lemma has_derivative_Blinfun: assumes "(f has_derivative f') F" shows "(f has_derivative Blinfun f') F"

lemma a_sum_le_U_sum: "wf ot \<Longrightarrow> acost_sum ot \<le> U_sum ot"

lemma op_cf_comma_is_functor: assumes "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" shows "op_cf_comma \<GG> \<HH> : op_cat (\<GG> \<^sub>C\<^sub>F\<down>\<^sub>C\<^sub>F \<HH>) \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> (op_cf \<HH>) \<^sub>C\<^sub>F\<down>\<^sub>C\<^sub>F (op_cf \<GG>)"

lemma wf_interp_for_formula_any_Inr: "wf_interp_for_formula (w, Inr P # I) \<phi> \<Longrightarrow> \<forall>P \<subseteq> {0 .. length w - 1}. wf_interp_for_formula (w, Inr P # I) \<phi>"

lemma ind_is_open_S [iff]: "ind_is_open S"

lemma set_compre_subset: "B \<subseteq> A \<Longrightarrow> {x \<in> B. P x} \<subseteq> {x \<in> A. P x}"

lemma ik\<^sub>s\<^sub>t_assignment_rhs\<^sub>s\<^sub>t_wfrestrictedvars_subset: "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>s\<^sub>t A) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t A"

lemma "\<lbrakk>test p; test q\<rbrakk> \<Longrightarrow> p \<cdot> x = p \<cdot> x \<cdot> q \<longleftrightarrow> p \<cdot> x \<le> x \<cdot> q"

lemma ord_strict_pI: assumes "lookup p v = 0" and "lookup q v \<noteq> 0" and "\<And>u. v \<prec>\<^sub>t u \<Longrightarrow> lookup p u = lookup q u" shows "p \<prec>\<^sub>p q"

lemma swap_triple: assumes "a \<noteq> b" and "c \<noteq> b" assumes "sort_of a = sort_of b" "sort_of b = sort_of c" shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"

lemma (in Group) compseries_is_tW_cmpser:"\<lbrakk>0 < r; compseries G r f\<rbrakk> \<Longrightarrow> tW_cmpser G r f"

lemma monom_eval_add: assumes "closed_fun R g" shows "monom_eval R (add_mset x M) g = (g x) \<otimes> (monom_eval R M g)"

lemma pre_post_left_sub_dist: "-p\<squnion>-q\<stileturn>-r \<le> -p\<stileturn>-r"

lemma tso_lock_invL[intro]: "\<lbrace> tso_lock_invL \<rbrace> mutator m"

lemma box_an_an_same: "|an(x)]an(x) = 1"

lemma an_neq_inf:"an n \<noteq> \<infinity>"

lemma good_mapI: assumes surj_h: "surj h" and prem: "!! x x' y y'. h(x,y) = h(x',y') ==> x=x'" shows "good_map h"

lemma pivot_unsat_core_id: "\<lbrakk>\<triangle> (\<T> s); x\<^sub>i \<in> lvars (\<T> s); x\<^sub>j \<in> rvars_of_lvar (\<T> s) x\<^sub>i\<rbrakk> \<Longrightarrow> \\<^sub>c (pivot x\<^sub>i x\<^sub>j s) = \\<^sub>c s"

lemma not_convr [simp]: "(\<not> p)\<^sup>- = (\<not> p\<^sup>-)"

theorem (* Detailed proof *) fixes f:: "'a::complete_lattice \<Rightarrow> 'a" assumes CONT: "cont f" shows "lfp f = (SUP i. (f^^i) bot)"

lemma lfilter_id_conv: "lfilter P xs = xs \<longleftrightarrow> (\<forall>x\<in>lset xs. P x)" (is "?lhs \<longleftrightarrow> ?rhs")

lemma preal_sup: assumes le: "\<And>X. X \<in> P \<Longrightarrow> X \<le> Y" and "P \<noteq> {}" shows "cut (\<Union>X \<in> P. Rep_preal(X))"

lemma [trans] : "P' \<subseteq> P \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P c Q,A \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P' c Q,A"

lemma vector_to_cblinfun_apply_one_dim[simp]: shows "vector_to_cblinfun \<phi> *\<^sub>V \<gamma> = one_dim_iso \<gamma> *\<^sub>C \<phi>"

lemma typ_of1_decr_gen': "typ_of1 (Ts@Ts') (decr (length Ts) t) = tyo\<Longrightarrow> \<not> loose_bvar1 t (length Ts) \<Longrightarrow> typ_of1 (Ts@[T]@Ts') t = tyo"

lemma Accessible_right_lang_eq [simp]: "q \<in> accessible \<Longrightarrow> Accessible.right_lang q = right_lang q"

lemma repeatable_both_both_nxt: assumes "t' \<otimes>\<otimes> t = t'" shows "t' \<otimes>\<otimes> t'' \<otimes>\<otimes> t = t' \<otimes>\<otimes> t''"

lemma projection_on_union: "l \<upharpoonleft> Y = [] \<Longrightarrow> l \<upharpoonleft> (X \<union> Y) = l \<upharpoonleft> X"

lemma lappend_is_LNil_conv [iff]: "(s @@ t = LNil) = (s = LNil \<and> t = LNil)"

lemma distinct_RS2[rule_format,simp]: "distinct p \<longrightarrow> distinct (removeShadowRules2 p)"

lemma Failures_implies_Traces: " \<lbrakk>is_process P; (s, X) \<in> FAILURES P\<rbrakk> \<Longrightarrow> s \<in> TRACES P"

lemma push_push_aux: "peval (push_param P (Suc m)) (push a n) = peval (push_param P m) a"

lemma zero_notin_vpair: "0 \<notin> elts \<langle>x,y\<rangle>"

lemma concat_pp_less: assumes "concat (take k ws) <p concat (take n ws)" shows "k < n"

lemma Inter_eqvt [eqvt]: shows "p \<bullet> \<Inter>S = \<Inter>(p \<bullet> S)"

lemma pell_valuation_solution_pos_nat: fixes z :: "nat \<times> nat" assumes "solution z" shows "pell_valuation z > 0"

lemma parts_trans: "\<lbrakk> X\<in> parts G; G \<subseteq> parts H \<rbrakk> \<Longrightarrow> X\<in> parts H"

lemma modifies_lhsv'_gen: assumes "lhsv\<pi> \<pi> \<subseteq> vs" assumes "lhsv' c \<subseteq> vs" assumes "\<pi>: (c,s) \<Rightarrow> t" shows "modifies vs t s"

lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a)" for a :: "'a::real_normed_vector"

lemma uset_laws [simp]: "x \<in>\<^sub>u {}\<^sub>u = false" "x \<in>\<^sub>u {m..n}\<^sub>u = (m \<le>\<^sub>u x \<and> x \<le>\<^sub>u n)"

lemma wellformed_items_Scan: "wellformed_items I \<Longrightarrow> T \<subseteq> \<X> k \<Longrightarrow> wellformed_items (Scan T k I)"

lemma real_greaterThanLessThan_minus_infinity_eq: "real_of_ereal ` {-\<infinity><..<N::ereal} = (if N = \<infinity> then UNIV else if N = -\<infinity> then {} else {..<real_of_ereal N})"

lemma reduce'_semantics: assumes \<open>static q\<close> shows \<open>(M, w \<Turnstile>\<^sub>! [p]\<^sub>! q) = (M, w \<Turnstile>\<^sub>! reduce' p q)\<close>

lemma gexp_max_input_In: "max_input (In v l) = AExp.max_input (V v)"

lemma if_doesnt_read: "x \<notin> bexp_vars e \<Longrightarrow> \<C>_vars x \<inter> bexp_vars e = {} \<Longrightarrow> doesnt_read_or_modify (If e c\<^sub>1 c\<^sub>2 \<otimes> annos) x"

lemma (in dup) shows "s<a := i>\<cdot>x = s\<cdot>x"

lemma containsI: "x \<in> A ==> contains A x"

lemma holdsIstate_paperIDs_geq_subPH: "holdsIstate paperIDs_geq_subPH"

lemma OUTfromChCorrect_data22: "OUTfromChCorrect data22"

lemma ts_ord_add_isol: "n x \<sqsubseteq> n y \<Longrightarrow> n z \<oplus> n x \<sqsubseteq> n z \<oplus> n y"

lemma psubst_lcnj[simp]: "set \<phi>s \<subseteq> fmla \<Longrightarrow> snd ` (set txs) \<subseteq> var \<Longrightarrow> fst ` (set txs) \<subseteq> trm \<Longrightarrow> distinct (map snd txs) \<Longrightarrow> psubst (lcnj \<phi>s) txs = lcnj (map (\<lambda>\<phi>. psubst \<phi> txs) \<phi>s)"

lemma coeff_mult_C[simp]: "coeff (a * C x) n = coeff a n * x"

lemma closed_UNIV [simp]: "closed UNIV f"

lemma dual_inv2 [simp]: "\<partial> \<circ> \<partial>\<^sup>- = id"

lemma product_language_state[simp]: "LS (product A B) (q1,q2) = LS A q1 \<inter> LS B q2"

lemma acquired_reads_no_pending_write_in: "x \<in> acquired_reads False xs A \<Longrightarrow> x \<in> acquired_reads False xs B"

lemma step2: "gdpr_scenario' \<rightarrow>\<^sub>n gdpr_scenario''"

lemma i_Exec_Comp_Stream_Acc_Output__Init__eq_Msg_iSince_conv: " \<lbrakk> 0 < k; m \<noteq> \<NoMsg>; s = (output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun (input \<odot>\<^sub>i k) c) \<rbrakk> \<Longrightarrow> ((i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c) t = m) = (s t2 = \<NoMsg>. t2 \<S> t1 [Suc (t * k)\<dots>,k - Suc 0]. s t1 = m)"

lemma reachFrom_step_induct[consumes 1, case_names Refl Step]: assumes s: "reachFrom s s'" and refl: "P s" and step: "\<And>s' a ou s''. reachFrom s s' \<Longrightarrow> P s' \<Longrightarrow> step s' a = (ou, s'') \<Longrightarrow> P s''" shows "P s'"

lemma omega_induct[consumes 1, case_names 0 succ]: assumes "n \<in>\<^sub>\<circ> \<omega>" and "P 0" and "\<And>n. \<lbrakk> n \<in>\<^sub>\<circ> \<omega>; P n \<rbrakk> \<Longrightarrow> P (succ n)" shows "P n"

lemma subst_ev_var_flip[simp]: fixes e::e and y::x and x::x assumes "atom y \<sharp> e" shows "(y \<leftrightarrow> x) \<bullet> e = e [x::=V_var y]\<^sub>e\<^sub>v"

lemma permutes_iff: "p permutes {..<n} \<Longrightarrow> i < n \<Longrightarrow> j < n \<Longrightarrow> p i = p j \<longleftrightarrow> i = j"

lemma between_distinct_r12: "distinct vs \<Longrightarrow> ram1 \<noteq> ram2 \<Longrightarrow> distinct (ram1 # between vs ram1 ram2 @ [ram2])"

lemma less_add_order_eq_0: assumes "a+^k = 0" "k < add_order a" shows "k = 0"

lemma resInputFreeTrans[dest]: fixes x :: name fixes a :: name and y :: name and P :: pi and \<alpha> :: freeRes and P' :: pi assumes "<\<nu>x>a<y>.P \<longmapsto>\<alpha> \<prec> P'" shows False

theorem preservation: assumes v: "v \<in> E e \<rho>" and rr: "e \<longrightarrow>* e'" shows "v \<in> E e' \<rho>"

lemma preserves_arr: shows "A.arr T \<Longrightarrow> B.arr (map T)"

lemma iter: "\<forall>a. (iter g (g a) n) = (g (iter g a n))"

lemma appGetStatic[simp]: "app\<^sub>i (Getstatic C F D,P,pc,mxs,T\<^sub>r,s) = (\<exists> vT ST LT. s = (ST, LT) \<and> length ST < mxs \<and> P \<turnstile> C sees F,Static:vT in D)"

lemma root_primitive_part [simp]: fixes p :: "'a :: {semiring_gcd, semiring_no_zero_divisors} poly" shows "poly (primitive_part p) x = 0 \<longleftrightarrow> poly p x = 0"

lemma contour_integral_reversepath: assumes "valid_path g" shows "contour_integral (reversepath g) f = - (contour_integral g f)"

lemma degree_rebase_poly_le: "degree (#p) \<le> degree p"

lemma negative_len_shortest: "length xs = n \<Longrightarrow> len m i i xs < 0 \<Longrightarrow> \<exists> j ys. distinct (j # ys) \<and> len m j j ys < 0 \<and> j \<in> set (i # xs) \<and> set ys \<subseteq> set xs"

lemma strictly_between_implies_angle_eq_pi: assumes "between (A, C) B" assumes "A \<noteq> B" "B \<noteq> C" shows "angle A B C = pi"

lemma (in weak_partial_order) greatest_Lower_cong_l: assumes "x .= x'" and "x \<in> carrier L" "x' \<in> carrier L" shows "greatest L x (Lower L A) = greatest L x' (Lower L A)"

lemma [transfer_rule]: "(pcr_fmap (=) (=) ===> pcr_linear_poly) (\<lambda> f x. case f x of None \<Rightarrow> 0 | Some x \<Rightarrow> x) LinearPoly"

lemma items_le_fix_D: assumes items_le_fix: "items_le k I = I" assumes x_dom: "x \<in> I" shows "item_end x \<le> k"

lemma lossless_complete_game: assumes lossless_init: "\<forall> h w. lossless_spmf (init h w)" and lossless_response: "\<forall> r w e. lossless_spmf (response r w e)" shows "lossless_spmf (completeness_game h w e)"

lemma out_subseqs1_set_as_out_subseqs2_set: "out_subseqs1_set ((i,v) # prevs) = { f head v } \<union> out_subseqs1_set prevs \<union> out_subseqs2_set v [0..<i]"

lemma row_echelon_form_berlekamp_resulting_mat: "row_echelon_form (berlekamp_resulting_mat u)"

lemma tp_correct': assumes layout: "ly = layout_of ap" and compile: "tp = tm_of ap" and crsp: "crsp ly (0, lm) (Suc 0, l, r) ires" and abc_halt: "abc_steps_l (0, lm) ap stp = (length ap, am)" shows "\<exists> stp k. steps (Suc 0, l, r) (tp, 0) stp = (start_of ly (length ap), Bk # Bk # ires, <am> @ Bk\<up>k)"

lemma inter_guards_sym: "\<And>c. (c1 \<inter>\<^sub>g c2) = Some c \<Longrightarrow> (c2 \<inter>\<^sub>g c1) = Some c"

lemma(in padic_integers) val_Zp_not_equal_ord_plus_minus: assumes "x \<in> carrier Zp" assumes "y \<in> carrier Zp" assumes "val_Zp x \<noteq> (val_Zp y)" shows "val_Zp (x \<ominus> y) = val_Zp (x \<oplus> y)"

lemma distinct_edgesI: assumes "distinct p" shows "distinct (walk_edges p)"

lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)"

lemma normalize_src_ports_preserves_normalized_not_has_disc_negated: assumes n: "normalized_nnf_match m" and nodisc: "\<not> has_disc_negated disc2 False m" and disc2_noProt: "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated is_Src_Ports False m" shows "m'\<in> set (normalize_src_ports m) \<Longrightarrow> \<not> has_disc_negated disc2 False m'"

lemma [code_unfold]: "((x::OclAny) \<doteq> y) = StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y"

lemma T_eq: "\<Gamma> \<turnstile> t : T \<Longrightarrow> T = T' \<Longrightarrow> \<Gamma> \<turnstile> t : T'"

lemma inext_nth_iNext_Suc: "(\<circle> t (I \<rightarrow> n) I. P t) = P (I \<rightarrow> Suc n)"

lemma "rec_list nil cons [] = nil"

lemma lift_simulation_eval: "L1.eval s1 s1' \<Longrightarrow> match i s1 s2 \<Longrightarrow> \<exists>i' s2'. L2.eval s2 s2' \<and> match i' s1' s2'"

lemma assumes " c <Q>" assumes "(h,as)\<Turnstile>P" shows hoare_triple_success: "success c h" and hoare_triple_effect: "\<exists>h' r. effect c h h' r \<and> (h',new_addrs h as h')\<Turnstile>Q r"

lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)"

lemma distinct_merge_distinct[simp]: "\<lbrakk>sorted xs; distinct xs; sorted ys; distinct ys\<rbrakk> \<Longrightarrow> distinct (merge_distinct xs ys)"

lemma BinOpRedsVal: assumes e\<^sub>1_steps: "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>" and e\<^sub>2_steps: "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>" and op: "binop(bop,v\<^sub>1,v\<^sub>2) = Some v" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2, s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2\<rangle>" (*<*)(is "(?x, ?z) \<in> (red P)\<^sup>*")

lemma infinite_hyp_changes_not_Lim: assumes "f \<in> U" and "\<forall>n. \<exists>m\<^sub>1>n. \<exists>m\<^sub>2>n. s (f \<triangleright> m\<^sub>1) \<noteq> s (f \<triangleright> m\<^sub>2)" shows "\<not> learn_lim \<psi> U s"

lemma adj_to_edge_set_card_lim: assumes "e \<in> edge_set" shows "card e > 0 \<and> card e \<le> 2"

lemma lF_char: assumes "C.ide f" shows "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright>" and "F (trg\<^sub>C f) \<star>\<^sub>D lF f = (\[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]" and "\<exists>!\<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> F (trg\<^sub>C f) \<star>\<^sub>D \<mu> = (\[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"