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

Statement: stringlengths 7 24.3k
lemma next_preserves_inv_implications_nonneg: assumes "next s" and "holds inv_implications_nonneg s" and "holds inv_imp_plus_work_nonneg s" shows "nxt (holds inv_implications_nonneg) s"
lemma card_Int_conv: " \<lbrakk> finite A; finite B \<rbrakk> \<Longrightarrow> card (A \<inter> B) = card A + card B - card (A \<union> B)"
lemma prv_imp_neg_imp_cnjL: assumes "\<phi> \<in> fmla" "\<phi>1 \<in> fmla" "\<phi>2 \<in> fmla" and "prv (imp \<phi> (neg \<phi>1))" shows "prv (imp \<phi> (neg (cnj \<phi>1 \<phi>2)))"
lemma sig_red_zeroE: assumes "sig_red_zero sing_reg F r" obtains s where "(sig_red sing_reg (\<preceq>) F)\<^sup>\<^sup> r s" and "rep_list s = 0"
lemma load_word_mem_10_low_equal: assumes a1: "low_equal s1 s2" shows "load_word_mem s1 address 10 = load_word_mem s2 address 10"
lemma selector_pushout: assumes "valid_selector Rs selector" "selector G'' = Some (R,f)" defines "G \<equiv> graph_of G''" assumes "graph G" defines "g \<equiv> extend (fst G'') R f" defines "G' \<equiv> LG (on_triple g `` edges (snd R) \<union> (snd G'')) {0..<max (fst G'') (nextMax (Range g))}" shows "pushout_step (t:: 'x itself) R G G'"
lemma path_connectedin_absolute [simp]: "path_connectedin (subtopology X S) S \<longleftrightarrow> path_connectedin X S"
lemma check_poly_eq: fixes p :: "('v :: linorder,'a :: poly_carrier)poly" assumes chk: "check_poly_eq p q" shows "p =p q"
lemma Arg2pi_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg2pi (z * of_real r) = Arg2pi z"
lemma states_closed: assumes "s \<in> states" assumes "(s, t) \<in> acc_on (- {error, ok})" shows "t \<in> states"
lemma gsfw_join_valid: "gsfw_valid f1 \<Longrightarrow> gsfw_valid f2 \<Longrightarrow> gsfw_valid (generalized_fw_join f1 f2)"
theorem fw_shortest_path_up_to: "D m i j k = fw m n k i j" if "cyc_free_subs n {0..k} m" "i \<le> n" "j \<le> n" "k \<le> n"
lemma sound_opr_alt: "sound_opr opr f = ((\<forall>s. s \<Turnstile>\<^sub>= (precondition opr) \<longrightarrow> (\<exists>s'. f s = (Some s') \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set(dels (effect opr)) \<and> s \<Turnstile>\<^sub>= atm \<longrightarrow> s' \<Turnstile>\<^sub>= atm) \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set (adds (effect opr)) \<and> s \<Turnstile>\<^sub>= Not atm \<longrightarrow> s' \<Turnstile>\<^sub>= Not atm) \<and> (\<forall>atm. atm \<in> set(adds (effect opr)) \<longrightarrow> s' \<Turnstile>\<^sub>= atm) \<and> (\<forall>fmla. fmla \<in> set (dels (effect opr)) \<and> fmla \<notin> set(adds (effect opr)) \<longrightarrow> s' \<Turnstile>\<^sub>= (Not fmla)) \<and> (\<forall>a b. s \<Turnstile>\<^sub>= Atom (Eq a b) \<longrightarrow> s' \<Turnstile>\<^sub>= Atom (Eq a b)) \<and> (\<forall>a b. s \<Turnstile>\<^sub>= Not (Atom (Eq a b)) \<longrightarrow> s' \<Turnstile>\<^sub>= Not (Atom (Eq a b))) )) \<and> (\<forall>fmla\<in>set(adds (effect opr)). is_predAtom fmla))"
lemma subtree_rank_ge_if_mdeg_le1: "\<lbrakk>is_subtree (Node r {\|(t1,e1)\|}) t; max_deg (Node r {\|(t1,e1)\|}) \<le> 1; v \<noteq> r; v \<in> dverts t; \<exists>y \<in> set v. \<not>(\<exists>x'\<in>set (Dtree.root t1). x' \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y) \<and> (\<exists>x\<in>set r. x \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y)\<rbrakk> \<Longrightarrow> rank (rev (Dtree.root t1)) \<le> rank (rev v)"
lemma subst_liftn: "i \<le> n + k \<and> k \<le> i \<Longrightarrow> (liftn (Suc n) s k)[t/i] = liftn n s k"
lemma coeff_mult_degree_sum: "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)"
lemma fixes s :: complex assumes s: "Re s > 1" shows abs_summable_Dirichlet_L: "summable (\<lambda>n. norm (\<chi> n * of_nat n powr -s))" and summable_Dirichlet_L: "summable (\<lambda>n. \<chi> n * of_nat n powr -s)" and sums_Dirichlet_L: "(\<lambda>n. \<chi> n * n powr -s) sums Dirichlet_L n \<chi> s" and Dirichlet_L_conv_eval_fds_weak: "Dirichlet_L n \<chi> s = eval_fds (fds \<chi>) s"
lemma mult_div_mono_left: fixes c::real assumes nnc: "0 \<le> c" and nzc: "c \<noteq> 0" and inv: "a \<le> inverse c * b" shows "c * a \<le> b"
lemma if_Red1_mthr_imp_if_Red1_mthr': assumes lok: "if_lock_oks1 (locks s) (thr s)" and elo: "ts_ok (init_fin_lift (\<lambda>t exexs h. el_loc_ok1 exexs)) (thr s) (shr s)" and Red: "Red1_mthr.if.redT uf P s tta s'" shows "Red1_mthr.if.redT (\<not> uf) P s tta s'"
lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
lemma st_vec_nonneg[simp]: "st_vec x $ i \<ge> 0"
lemma change_respecting_doesnt_modify: assumes cr: "change_respecting (cms, mem) (cms', mem') X g" assumes eval: "(cms, mem) \<leadsto> (cms', mem')" assumes domf: "dom f = X" assumes x_in_dom: "x \<in> dom (g f)" assumes noread: "doesnt_read (fst cms) x" shows "mem x = mem' x"
lemma COND_terms_hf: assumes "hf_valid ainfo uinfo hf nxt" "terms_hf hf \<subseteq> analz ik" "no_oracle ainfo uinfo" shows "\<exists>hfs. hf \<in> set hfs \<and> (\<exists>uinfo' . (ainfo, hfs) \<in> (auth_seg2 uinfo'))"
lemma round_unique: "of_int y > x - 1/2 \<Longrightarrow> of_int y \<le> x + 1/2 \<Longrightarrow> round x = y"
lemma (in jozsa) jozsa_algo_result [simp]: shows "jozsa_algo = \<psi>\<^sub>3"
lemma qfa_enq_correct: "list (enq x (qfa l)) = (list (qfa l)) @ [x]"
lemma monad_state_prob_stateT [locale_witness]: "monad_state_prob return_state bind_state get_state put_state sample_state"
lemma not_diff_distrib: \<open>NOT (a - b) = NOT a + b\<close>
theorem SF1: assumes h1: "\|~ P \<and> [N]_v \<longrightarrow> \<circle>P \<or> \<circle>Q" and h2: "\|~ P \<and> \<langle>N \<and> A\<rangle>_v \<longrightarrow> \<circle>Q" and h3: "\<turnstile> \<box>P \<and> \<box>[N]_v \<and> \<box>F \<longrightarrow> \<diamond>Enabled \<langle>A\<rangle>_v" and h4: "\|~ P \<and> Unchanged v \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>[N]_v \<and> SF(A)_v \<and> \<box>F \<longrightarrow> (P \<leadsto> Q)"
lemma emb_step_equiv': "emb_step t s \<longleftrightarrow> (\<exists>p. p \<noteq> [] \<and> emb_step_at' p t = s) \<and> t \<noteq> s"
lemma hcomplex_of_hypreal_pow: "\<And>x. hcomplex_of_hypreal (x ^ n) = hcomplex_of_hypreal x ^ n"
lemma compact_trapC: shows "compact trapC"
lemma nabla_sum_expand [simp]: "\|x\<rangle> \<nabla> (x + y) + \|y\<rangle> \<nabla> (x + y) = \<nabla> (x + y)"
lemma f_join_singleton_if: " xs \<Join>\<^sub>f {n} = (if n < length xs then [xs ! n] else [])"
lemma mem_card1_singleton: "\<lbrakk> u \<in> U; card U = 1 \<rbrakk> \<Longrightarrow> U = {u}"
lemma costBIT_1y: assumes "x\<noteq>y" "x : {x0,y0}" "y\<in>{x0,y0}" shows "E (type1 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s y \<bind> (\<lambda>(a, is'). return_pmf (real (t\<^sub>p (fst s) y a))))) = 3/4"
lemma effect_apply_eqvt' [eqvt]: "p \<bullet> \<langle>f\<rangle>P = \<langle>p \<bullet> f\<rangle>(p \<bullet> P)"
lemma HEndPhase0_blocksRead: assumes act: "HEndPhase0 s s' p" shows "\<exists>d. blocksRead s p d \<noteq> {}"
lemma mset_le_single_iff[iff]: "{#x#} \<le> {#y#} \<longleftrightarrow> x \<le> y" for x y :: "'a::order"
lemma consistent_imp_nonneg_constant: assumes "consistent x t T B p incr" assumes "t < T" shows "B \<ge> 0"
lemma p_inf_pp_pp [simp]: "-(--x \<sqinter> --y) = -(x \<sqinter> y)"
lemma iso_Euclidean_complements_lemma1: assumes S: "closedin (Euclidean_space m) S" and cmf: "continuous_map(subtopology (Euclidean_space m) S) (Euclidean_space n) f" obtains g where "continuous_map (Euclidean_space m) (Euclidean_space n) g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
lemma fall2_imp_alw_index: assumes 0: "\<And> \<pi> \<pi>'. wfp AP' \<pi> \<and> wfp AP' \<pi>' \<longrightarrow> \<phi> [] [\<pi>,\<pi>'] = f (stateOf \<pi>) (stateOf \<pi>') \<and> \<psi> [] [\<pi>,\<pi>'] = g (stateOf \<pi>) (stateOf \<pi>')" shows "fall2 (\<lambda> \<pi>' \<pi> \<pi>l. imp (alw (\<phi> \<pi>l)) (alw (\<psi> \<pi>l)) (\<pi>l @ [\<pi>,\<pi>'])) [] \<longleftrightarrow> (\<forall> \<pi> \<pi>'. wfp AP' \<pi> \<and> wfp AP' \<pi>' \<and> stateOf \<pi> = s0 \<and> stateOf \<pi>' = s0 \<longrightarrow> (\<forall> i. (\<forall> j \<le> i. f (fst (\<pi> !! j)) (fst (\<pi>' !! j))) \<longrightarrow> g (fst (\<pi> !! i)) (fst (\<pi>' !! i))) )" (is "?L \<longleftrightarrow> ?R")
lemma fetch_wcode_adjust_tm[simp]: "fetch wcode_adjust_tm (Suc 0) Bk = (WO, 1)" "fetch wcode_adjust_tm (Suc 0) Oc = (R, 2)" "fetch wcode_adjust_tm (Suc (Suc 0)) Oc = (R, 3)" "fetch wcode_adjust_tm (Suc (Suc (Suc 0))) Oc = (R, 4)" "fetch wcode_adjust_tm (Suc (Suc (Suc 0))) Bk = (R, 3)" "fetch wcode_adjust_tm 4 Bk = (L, 8)" "fetch wcode_adjust_tm 4 Oc = (L, 5)" "fetch wcode_adjust_tm 5 Oc = (WB, 5)" "fetch wcode_adjust_tm 5 Bk = (L, 6)" "fetch wcode_adjust_tm 6 Oc = (R, 7)" "fetch wcode_adjust_tm 6 Bk = (L, 6)" "fetch wcode_adjust_tm 7 Bk = (WO, 2)" "fetch wcode_adjust_tm 8 Bk = (L, 9)" "fetch wcode_adjust_tm 8 Oc = (WB, 8)" "fetch wcode_adjust_tm 9 Oc = (L, 10)" "fetch wcode_adjust_tm 9 Bk = (L, 9)" "fetch wcode_adjust_tm 10 Bk = (L, 11)" "fetch wcode_adjust_tm 10 Oc = (L, 10)" "fetch wcode_adjust_tm 11 Oc = (L, 11)" "fetch wcode_adjust_tm 11 Bk = (R, 0)"
theorem Completeness: "I \<noteq> {} \<Longrightarrow> finite I \<Longrightarrow> Kripke {s :: ('s :: state). \<exists> i \<in> I. (i \<rightarrow>\<^sub>i* s)} (I :: ('s :: state)set) \<turnstile> EF s \<Longrightarrow> \<exists> (A :: ('s :: state) attree). \<turnstile> A \<and> attack A = (I,s)"
lemma inorder_baliR: "inorder(baliR l a r) = inorder l @ a # inorder r"
lemma diagonalize_is_idempotent: shows "D o D = D"
lemma has_field_derivative_inverse_strong_x: fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a" shows "\<lbrakk>DERIV f (g y) :> f'; f' \<noteq> 0; open S; continuous_on S f; g y \<in> S; f(g y) = y; \<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk> \<Longrightarrow> DERIV g y :> inverse (f')"
lemma nxtIN[intro]: fixes c::'id and t::"nat \<Rightarrow> cnf" and t'::"nat \<Rightarrow> 'cmp" and n::nat assumes "\<not>(\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)" and "eval c t t' (Suc n) \<gamma>" shows "eval c t t' n (\<circle>\<^sub>b(\<gamma>))"
lemma prod_lens_sublens_cong: "\<lbrakk> X\<^sub>1 \<subseteq>\<^sub>L X\<^sub>2; Y\<^sub>1 \<subseteq>\<^sub>L Y\<^sub>2 \<rbrakk> \<Longrightarrow> (X\<^sub>1 \<times>\<^sub>L Y\<^sub>1) \<subseteq>\<^sub>L (X\<^sub>2 \<times>\<^sub>L Y\<^sub>2)"
lemma Equiv_Exec_output_eqI: " \<lbrakk> equiv_states (localState1 c1) (localState2 c2); Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 \<rbrakk> \<Longrightarrow> last_message (map output_fun1 ( f_Exec_Comp_Stream trans_fun1 (input_fun1 m # \<NoMsg>\<^bsup>k1 - Suc 0\<^esup>) c1)) = last_message (map output_fun2 ( f_Exec_Comp_Stream trans_fun2 (input_fun2 m # \<NoMsg>\<^bsup>k2 - Suc 0\<^esup>) c2))"
lemma fset_finite_supp: fixes S::"('a::fs) fset" shows "finite (supp S)"
lemma Invariants_measurable_func: assumes "f \<in> measurable Invariants N" shows "f \<in> measurable M N"
lemma red_diff_rtrancl': assumes "(red F)\<^sup>\<^sup> (p - q) r" obtains p' q' where "(red F)\<^sup>\<^sup> p p'" and "(red F)\<^sup>\<^sup> q q'" and "r = p' - q'"
theorem lucas_lehmer_sufficient: assumes "prime p" "odd p" assumes "(2 ^ p - 1) dvd gen_lucas_lehmer_sequence 4 (p - 2)" shows "prime (2 ^ p - 1 :: nat)"
lemma take_from_set_correct: assumes "s \<noteq> {}" shows "take_from_set s \<le> SPEC (\<lambda> (x, s'). x \<in> s \<and> s' = s - {x})"
lemma hmac_trans_1_4_skr_extr_fake: "hmac X K \<in> parts (extr (bad s') (ik s') (chan s')) \<Longrightarrow> K \<notin> synth (analz (extr (bad s) (ik s) (chan s))) \<Longrightarrow> \<comment> \<open>necessary for the \<open>dy_fake_msg\<close> case\<close> s \<in> l2_inv2 \<Longrightarrow> \<comment> \<open>necessary for the \<open>skr\<close> case\<close> (s, s') \<in> l2_step1 Ra A B \<union> l2_step4 Rb A B Ni gnx \<union> l2_skr R KK \<union> l2_dy_fake_msg M \<union> l2_dy_fake_chan MM \<Longrightarrow> hmac X K \<in> parts (extr (bad s) (ik s) (chan s))"
lemma replacement_monotonic : shows "\<And> t s. ((subst v \<sigma>), (subst u \<sigma>)) \<in> trm_ord \<Longrightarrow> subterm t p u \<Longrightarrow> replace_subterm t p v s \<Longrightarrow> ((subst s \<sigma>), (subst t \<sigma>)) \<in> trm_ord"
lemma NS_staticSecret_parts_Spy: "\<lbrakk>m \<in> parts (knows Spy evs); m \<in> staticSecret A; evs \<in> ns_public\<rbrakk> \<Longrightarrow> A \<in> bad"
lemma valid_sops_stmt_invariant: assumes step: "\<theta>\<turnstile> (s,t) \<rightarrow>\<^sub>s ((s',t'),is)" shows "valid_sops_stmt t s \<Longrightarrow> valid_sops_stmt t' s'"
lemma (in poly_mod_prime_type) finite_field_factorization_modulo_ring: assumes g: "(g :: 'a mod_ring poly) = of_int_poly f" and sf: "square_free_m f" and fact: "finite_field_factorization g = (d,gs)" and c: "c = to_int_mod_ring d" and fs: "fs = map to_int_poly gs" shows "unique_factorization_m f (c, mset fs)"
lemma VPow_top: "A \<in>\<^sub>\<circ> VPow A"
lemma rel_frext_miracle [frame]: "a:[false]\<^sup>+ = false"
lemma RPDs_ok: "ok m mr \<Longrightarrow> ms \<in> RPDs mr \<Longrightarrow> ok m ms"
lemma sat_plan_to_dimacs_works: "valid_state_var sv \<Longrightarrow> dimacs_to_var (var_to_dimacs sv) = sv"
lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"
lemma mset_subset_eq_exists_conv: "(A::'a multiset) \<subseteq># B \<longleftrightarrow> (\<exists>C. B = A + C)"
lemma orthogonal_split: assumes "(embP Q \<and>* $ n) = (embP P \<and>* $ m)" shows "(Q = P \<and> n = m) \<or> Q = (\<lambda>s. False) \<and> P = (\<lambda>s. False)"
lemma pos_meas_self: assumes "pos_meas_set E" shows "0 \<le> \<mu> E"
lemma subprob_algebra_cong: "sets M = sets N \<Longrightarrow> subprob_algebra M = subprob_algebra N"
lemma vifintersection_vintersection: assumes "I \<noteq> 0" and "J \<noteq> 0" shows "(\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. f i) \<inter>\<^sub>\<circ> (\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>J. f i) = (\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>I \<union>\<^sub>\<circ> J. f i)"
lemma mor_unfold: "mor UNIV UNIV s1 s2 UNIV UNIV dtor1 dtor2 (unfold1 s1 s2) (unfold2 s1 s2)"
lemma check_mult_alt_def: \<open>check_mult A \<V> p q i r \<ge> do { b \<leftarrow> SPEC(\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars q \<subseteq> \<V> \<and> vars r \<subseteq> \<V>); if \<not>b then RETURN False else do { ASSERT (p \<in># dom_m A); let p = the (fmlookup A p); pq \<leftarrow> mult_poly_spec p q; p \<leftarrow> weak_equality pq r; RETURN p } }\<close>
lemma hypext_sin_e12 [simp]: "(h sin) (x * e12) = e12 * x"
lemma defect_less: assumes b: "\<And>t. t0 < t \<Longrightarrow> t \<le> t1 \<Longrightarrow> v' t - f t (v t) < w' t - f t (w t)" notes [continuous_intros] = vderiv_on_continuous_on[OF v', THEN continuous_on_subset] vderiv_on_continuous_on[OF w', THEN continuous_on_subset] shows "\<forall>t \<in> {t0 <.. t1}. v t < w t"
lemma (in group) trivial_group: "trivial_group G \<longleftrightarrow> (\<exists>a. carrier G = {a})"
lemma defines "prec \<equiv> ((\<lambda>f g. (pr_strict' f g, pr_weak' f g)))" and "prl \<equiv> (\<lambda>(f, n). n = 0 \<and> least f)" shows kbo_encoding_is_valid_wpo: "wpo_with_assms weight_S weight_NS prec prl full_status False (\<lambda>f. False)" and kbo_as_wpo: "bounded_arity n (funas_term t) \<Longrightarrow> kbo s t = wpo.wpo n weight_S weight_NS prec prl full_status (\<lambda>_. Lex) False (\<lambda>f. False) s t"
lemma ntadj_sub_diff_assign:"\<And>e \<sigma> x. (\<Union>y\<in>{y. Inl y \<in> SIGT e}. FVT (\<sigma> y)) \<subseteq> (\<Union>y\<in>{y. Inl (Inl y) \<in> SIGP (DiffAssign x e)}. FVT (\<sigma> y))"
lemma sortedByWeight_Cons_imp_forall_weight_ge: "sortedByWeight (t # ts) \<Longrightarrow> \<forall>u \<in> set ts. weight u \<ge> weight t"
lemma symmetric_mpoly_mult [intro]: "symmetric_mpoly A p \<Longrightarrow> symmetric_mpoly A q \<Longrightarrow> symmetric_mpoly A (p * q)"
lemma estimate'_bounds: "prob {\<omega>. of_rat \<delta> * real_of_rat (F 0 as) < \<bar>estimate' (sketch_rv' \<omega>) - of_rat (F 0 as)\<bar>} \<le> 1/3"
lemma (in comm_monoid) multlist_perm_cong: assumes prm: "as <~~> bs" and ascarr: "set as \<subseteq> carrier G" shows "foldr (\<otimes>) as \<one> = foldr (\<otimes>) bs \<one>"
lemma minus_one_sdiv_word_eq [simp]: \<open>- 1 sdiv w = - (1 sdiv w)\<close> for w :: \<open>'a::len word\<close>
lemma Lyndon_rec_all: assumes "Lyndon_rec (a # w) (\<^bold>\|w\<^bold>\|)" shows "n < \<^bold>\|a#w\<^bold>\| \<Longrightarrow> 0 < n \<Longrightarrow> Lyndon_rec (a#w) n"
lemma dickson_less_pD2: assumes "dickson_less_p d m p q" shows "q \<in> dgrad_p_set d m"
lemma qbs_morphism_Pair1: assumes "x \<in> qbs_space X" shows "Pair x \<in> Y \<rightarrow>\<^sub>Q X \<Otimes>\<^sub>Q Y"
lemma M_neg_imp_z_non_zero: assumes "v \<bullet> (M *v v) < 0" shows "v$3 \<noteq> 0"
lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
lemma hext_typeof_mono: "\<lbrakk> h \<unlhd> h'; typeof\<^bsub>h\<^esub> v = Some T \<rbrakk> \<Longrightarrow> typeof\<^bsub>h'\<^esub> v = Some T"
lemma compact_sums': fixes S :: "'a::real_normed_vector set" assumes "compact S" and "compact T" shows "compact (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
lemma has_disc_negated_alist_and': "has_disc_negated disc neg (alist_and' as) \<longleftrightarrow> (\<exists> a \<in> set as. has_disc_negated disc neg (negation_type_to_match_expr a))"
lemma map_pair_fst_helper : "map fst (map (\<lambda> (x1,x2) . ((x1,x2), f x1 x2)) xs) = xs"
lemma const_poly_nonzero_coeff : assumes "degree p = 0" "p \<noteq> 0" shows "coeff p 0 \<noteq> 0"
lemma restore_new_paras: "ffp \<ge> length gs \<Longrightarrow> \<lbrace>\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_exec i xs) gs @ 0 # xs @ anything\<rbrace> mv_boxes (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) 0 (length gs) \<lbrace>\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ 0 # xs @ anything\<rbrace>"
lemma set_pmf_of_set_lessThan_length [simp]: "xs \<noteq> [] \<Longrightarrow> set_pmf (pmf_of_set {..<length xs}) = {..<length xs}"
lemma less_eq_mctxt_MVarE1: assumes "MVar v \<le> D" obtains (MVar) "D = MVar v"
lemma neg_length_pos: assumes i0: "bv_to_int w < -1" shows "0 < length w"
lemma Program_code [code]: "Program = program \<circ> tabulate_program"
lemma tensor_extensionality3': fixes F G :: \<open>(('a::finite\<times>'b::finite)\<times>'c::finite) update \<Rightarrow> 'd::finite update\<close> assumes [simp]: \<open>register F\<close> \<open>register G\<close> assumes "\<And>f g h. F ((f \<otimes>\<^sub>u g) \<otimes>\<^sub>u h) = G ((f \<otimes>\<^sub>u g) \<otimes>\<^sub>u h)" shows "F = G"
lemma r_leap_total: "eval r_leap [t, i, x] \<down>"

lemma next_preserves_inv_implications_nonneg: assumes "next s" and "holds inv_implications_nonneg s" and "holds inv_imp_plus_work_nonneg s" shows "nxt (holds inv_implications_nonneg) s"

lemma card_Int_conv: " \<lbrakk> finite A; finite B \<rbrakk> \<Longrightarrow> card (A \<inter> B) = card A + card B - card (A \<union> B)"

lemma prv_imp_neg_imp_cnjL: assumes "\<phi> \<in> fmla" "\<phi>1 \<in> fmla" "\<phi>2 \<in> fmla" and "prv (imp \<phi> (neg \<phi>1))" shows "prv (imp \<phi> (neg (cnj \<phi>1 \<phi>2)))"

lemma sig_red_zeroE: assumes "sig_red_zero sing_reg F r" obtains s where "(sig_red sing_reg (\<preceq>) F)\<^sup>*\<^sup>* r s" and "rep_list s = 0"

lemma load_word_mem_10_low_equal: assumes a1: "low_equal s1 s2" shows "load_word_mem s1 address 10 = load_word_mem s2 address 10"

lemma selector_pushout: assumes "valid_selector Rs selector" "selector G'' = Some (R,f)" defines "G \<equiv> graph_of G''" assumes "graph G" defines "g \<equiv> extend (fst G'') R f" defines "G' \<equiv> LG (on_triple g `` edges (snd R) \<union> (snd G'')) {0..<max (fst G'') (nextMax (Range g))}" shows "pushout_step (t:: 'x itself) R G G'"

lemma path_connectedin_absolute [simp]: "path_connectedin (subtopology X S) S \<longleftrightarrow> path_connectedin X S"

lemma check_poly_eq: fixes p :: "('v :: linorder,'a :: poly_carrier)poly" assumes chk: "check_poly_eq p q" shows "p =p q"

lemma Arg2pi_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg2pi (z * of_real r) = Arg2pi z"

lemma states_closed: assumes "s \<in> states" assumes "(s, t) \<in> acc_on (- {error, ok})" shows "t \<in> states"

lemma gsfw_join_valid: "gsfw_valid f1 \<Longrightarrow> gsfw_valid f2 \<Longrightarrow> gsfw_valid (generalized_fw_join f1 f2)"

theorem fw_shortest_path_up_to: "D m i j k = fw m n k i j" if "cyc_free_subs n {0..k} m" "i \<le> n" "j \<le> n" "k \<le> n"

lemma sound_opr_alt: "sound_opr opr f = ((\<forall>s. s \<Turnstile>\<^sub>= (precondition opr) \<longrightarrow> (\<exists>s'. f s = (Some s') \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set(dels (effect opr)) \<and> s \<Turnstile>\<^sub>= atm \<longrightarrow> s' \<Turnstile>\<^sub>= atm) \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set (adds (effect opr)) \<and> s \<Turnstile>\<^sub>= Not atm \<longrightarrow> s' \<Turnstile>\<^sub>= Not atm) \<and> (\<forall>atm. atm \<in> set(adds (effect opr)) \<longrightarrow> s' \<Turnstile>\<^sub>= atm) \<and> (\<forall>fmla. fmla \<in> set (dels (effect opr)) \<and> fmla \<notin> set(adds (effect opr)) \<longrightarrow> s' \<Turnstile>\<^sub>= (Not fmla)) \<and> (\<forall>a b. s \<Turnstile>\<^sub>= Atom (Eq a b) \<longrightarrow> s' \<Turnstile>\<^sub>= Atom (Eq a b)) \<and> (\<forall>a b. s \<Turnstile>\<^sub>= Not (Atom (Eq a b)) \<longrightarrow> s' \<Turnstile>\<^sub>= Not (Atom (Eq a b))) )) \<and> (\<forall>fmla\<in>set(adds (effect opr)). is_predAtom fmla))"

lemma subtree_rank_ge_if_mdeg_le1: "\<lbrakk>is_subtree (Node r {|(t1,e1)|}) t; max_deg (Node r {|(t1,e1)|}) \<le> 1; v \<noteq> r; v \<in> dverts t; \<exists>y \<in> set v. \<not>(\<exists>x'\<in>set (Dtree.root t1). x' \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y) \<and> (\<exists>x\<in>set r. x \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y)\<rbrakk> \<Longrightarrow> rank (rev (Dtree.root t1)) \<le> rank (rev v)"

lemma subst_liftn: "i \<le> n + k \<and> k \<le> i \<Longrightarrow> (liftn (Suc n) s k)[t/i] = liftn n s k"

lemma coeff_mult_degree_sum: "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)"

lemma fixes s :: complex assumes s: "Re s > 1" shows abs_summable_Dirichlet_L: "summable (\<lambda>n. norm (\<chi> n * of_nat n powr -s))" and summable_Dirichlet_L: "summable (\<lambda>n. \<chi> n * of_nat n powr -s)" and sums_Dirichlet_L: "(\<lambda>n. \<chi> n * n powr -s) sums Dirichlet_L n \<chi> s" and Dirichlet_L_conv_eval_fds_weak: "Dirichlet_L n \<chi> s = eval_fds (fds \<chi>) s"

lemma mult_div_mono_left: fixes c::real assumes nnc: "0 \<le> c" and nzc: "c \<noteq> 0" and inv: "a \<le> inverse c * b" shows "c * a \<le> b"

lemma if_Red1_mthr_imp_if_Red1_mthr': assumes lok: "if_lock_oks1 (locks s) (thr s)" and elo: "ts_ok (init_fin_lift (\<lambda>t exexs h. el_loc_ok1 exexs)) (thr s) (shr s)" and Red: "Red1_mthr.if.redT uf P s tta s'" shows "Red1_mthr.if.redT (\<not> uf) P s tta s'"

lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"

lemma st_vec_nonneg[simp]: "st_vec x $ i \<ge> 0"

lemma change_respecting_doesnt_modify: assumes cr: "change_respecting (cms, mem) (cms', mem') X g" assumes eval: "(cms, mem) \<leadsto> (cms', mem')" assumes domf: "dom f = X" assumes x_in_dom: "x \<in> dom (g f)" assumes noread: "doesnt_read (fst cms) x" shows "mem x = mem' x"

lemma COND_terms_hf: assumes "hf_valid ainfo uinfo hf nxt" "terms_hf hf \<subseteq> analz ik" "no_oracle ainfo uinfo" shows "\<exists>hfs. hf \<in> set hfs \<and> (\<exists>uinfo' . (ainfo, hfs) \<in> (auth_seg2 uinfo'))"

lemma round_unique: "of_int y > x - 1/2 \<Longrightarrow> of_int y \<le> x + 1/2 \<Longrightarrow> round x = y"

lemma (in jozsa) jozsa_algo_result [simp]: shows "jozsa_algo = \<psi>\<^sub>3"

lemma qfa_enq_correct: "list (enq x (qfa l)) = (list (qfa l)) @ [x]"

lemma monad_state_prob_stateT [locale_witness]: "monad_state_prob return_state bind_state get_state put_state sample_state"

lemma not_diff_distrib: \<open>NOT (a - b) = NOT a + b\<close>

theorem SF1: assumes h1: "|~ P \<and> [N]_v \<longrightarrow> \<circle>P \<or> \<circle>Q" and h2: "|~ P \<and> \<langle>N \<and> A\<rangle>_v \<longrightarrow> \<circle>Q" and h3: "\<turnstile> \<box>P \<and> \<box>[N]_v \<and> \<box>F \<longrightarrow> \<diamond>Enabled \<langle>A\<rangle>_v" and h4: "|~ P \<and> Unchanged v \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>[N]_v \<and> SF(A)_v \<and> \<box>F \<longrightarrow> (P \<leadsto> Q)"

lemma emb_step_equiv': "emb_step t s \<longleftrightarrow> (\<exists>p. p \<noteq> [] \<and> emb_step_at' p t = s) \<and> t \<noteq> s"

lemma hcomplex_of_hypreal_pow: "\<And>x. hcomplex_of_hypreal (x ^ n) = hcomplex_of_hypreal x ^ n"

lemma compact_trapC: shows "compact trapC"

lemma nabla_sum_expand [simp]: "|x\<rangle> \<nabla> (x + y) + |y\<rangle> \<nabla> (x + y) = \<nabla> (x + y)"

lemma f_join_singleton_if: " xs \<Join>\<^sub>f {n} = (if n < length xs then [xs ! n] else [])"

lemma mem_card1_singleton: "\<lbrakk> u \<in> U; card U = 1 \<rbrakk> \<Longrightarrow> U = {u}"

lemma costBIT_1y: assumes "x\<noteq>y" "x : {x0,y0}" "y\<in>{x0,y0}" shows "E (type1 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s y \<bind> (\<lambda>(a, is'). return_pmf (real (t\<^sub>p (fst s) y a))))) = 3/4"

lemma effect_apply_eqvt' [eqvt]: "p \<bullet> \<langle>f\<rangle>P = \<langle>p \<bullet> f\<rangle>(p \<bullet> P)"

lemma HEndPhase0_blocksRead: assumes act: "HEndPhase0 s s' p" shows "\<exists>d. blocksRead s p d \<noteq> {}"

lemma mset_le_single_iff[iff]: "{#x#} \<le> {#y#} \<longleftrightarrow> x \<le> y" for x y :: "'a::order"

lemma consistent_imp_nonneg_constant: assumes "consistent x t T B p incr" assumes "t < T" shows "B \<ge> 0"

lemma p_inf_pp_pp [simp]: "-(--x \<sqinter> --y) = -(x \<sqinter> y)"

lemma iso_Euclidean_complements_lemma1: assumes S: "closedin (Euclidean_space m) S" and cmf: "continuous_map(subtopology (Euclidean_space m) S) (Euclidean_space n) f" obtains g where "continuous_map (Euclidean_space m) (Euclidean_space n) g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"

lemma fall2_imp_alw_index: assumes 0: "\<And> \<pi> \<pi>'. wfp AP' \<pi> \<and> wfp AP' \<pi>' \<longrightarrow> \<phi> [] [\<pi>,\<pi>'] = f (stateOf \<pi>) (stateOf \<pi>') \<and> \<psi> [] [\<pi>,\<pi>'] = g (stateOf \<pi>) (stateOf \<pi>')" shows "fall2 (\<lambda> \<pi>' \<pi> \<pi>l. imp (alw (\<phi> \<pi>l)) (alw (\<psi> \<pi>l)) (\<pi>l @ [\<pi>,\<pi>'])) [] \<longleftrightarrow> (\<forall> \<pi> \<pi>'. wfp AP' \<pi> \<and> wfp AP' \<pi>' \<and> stateOf \<pi> = s0 \<and> stateOf \<pi>' = s0 \<longrightarrow> (\<forall> i. (\<forall> j \<le> i. f (fst (\<pi> !! j)) (fst (\<pi>' !! j))) \<longrightarrow> g (fst (\<pi> !! i)) (fst (\<pi>' !! i))) )" (is "?L \<longleftrightarrow> ?R")

lemma fetch_wcode_adjust_tm[simp]: "fetch wcode_adjust_tm (Suc 0) Bk = (WO, 1)" "fetch wcode_adjust_tm (Suc 0) Oc = (R, 2)" "fetch wcode_adjust_tm (Suc (Suc 0)) Oc = (R, 3)" "fetch wcode_adjust_tm (Suc (Suc (Suc 0))) Oc = (R, 4)" "fetch wcode_adjust_tm (Suc (Suc (Suc 0))) Bk = (R, 3)" "fetch wcode_adjust_tm 4 Bk = (L, 8)" "fetch wcode_adjust_tm 4 Oc = (L, 5)" "fetch wcode_adjust_tm 5 Oc = (WB, 5)" "fetch wcode_adjust_tm 5 Bk = (L, 6)" "fetch wcode_adjust_tm 6 Oc = (R, 7)" "fetch wcode_adjust_tm 6 Bk = (L, 6)" "fetch wcode_adjust_tm 7 Bk = (WO, 2)" "fetch wcode_adjust_tm 8 Bk = (L, 9)" "fetch wcode_adjust_tm 8 Oc = (WB, 8)" "fetch wcode_adjust_tm 9 Oc = (L, 10)" "fetch wcode_adjust_tm 9 Bk = (L, 9)" "fetch wcode_adjust_tm 10 Bk = (L, 11)" "fetch wcode_adjust_tm 10 Oc = (L, 10)" "fetch wcode_adjust_tm 11 Oc = (L, 11)" "fetch wcode_adjust_tm 11 Bk = (R, 0)"

theorem Completeness: "I \<noteq> {} \<Longrightarrow> finite I \<Longrightarrow> Kripke {s :: ('s :: state). \<exists> i \<in> I. (i \<rightarrow>\<^sub>i* s)} (I :: ('s :: state)set) \<turnstile> EF s \<Longrightarrow> \<exists> (A :: ('s :: state) attree). \<turnstile> A \<and> attack A = (I,s)"

lemma inorder_baliR: "inorder(baliR l a r) = inorder l @ a # inorder r"

lemma diagonalize_is_idempotent: shows "D o D = D"

lemma has_field_derivative_inverse_strong_x: fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a" shows "\<lbrakk>DERIV f (g y) :> f'; f' \<noteq> 0; open S; continuous_on S f; g y \<in> S; f(g y) = y; \<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk> \<Longrightarrow> DERIV g y :> inverse (f')"

lemma nxtIN[intro]: fixes c::'id and t::"nat \<Rightarrow> cnf" and t'::"nat \<Rightarrow> 'cmp" and n::nat assumes "\<not>(\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)" and "eval c t t' (Suc n) \<gamma>" shows "eval c t t' n (\<circle>\<^sub>b(\<gamma>))"

lemma prod_lens_sublens_cong: "\<lbrakk> X\<^sub>1 \<subseteq>\<^sub>L X\<^sub>2; Y\<^sub>1 \<subseteq>\<^sub>L Y\<^sub>2 \<rbrakk> \<Longrightarrow> (X\<^sub>1 \<times>\<^sub>L Y\<^sub>1) \<subseteq>\<^sub>L (X\<^sub>2 \<times>\<^sub>L Y\<^sub>2)"

lemma Equiv_Exec_output_eqI: " \<lbrakk> equiv_states (localState1 c1) (localState2 c2); Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 \<rbrakk> \<Longrightarrow> last_message (map output_fun1 ( f_Exec_Comp_Stream trans_fun1 (input_fun1 m # \<NoMsg>\<^bsup>k1 - Suc 0\<^esup>) c1)) = last_message (map output_fun2 ( f_Exec_Comp_Stream trans_fun2 (input_fun2 m # \<NoMsg>\<^bsup>k2 - Suc 0\<^esup>) c2))"

lemma fset_finite_supp: fixes S::"('a::fs) fset" shows "finite (supp S)"

lemma Invariants_measurable_func: assumes "f \<in> measurable Invariants N" shows "f \<in> measurable M N"

lemma red_diff_rtrancl': assumes "(red F)\<^sup>*\<^sup>* (p - q) r" obtains p' q' where "(red F)\<^sup>*\<^sup>* p p'" and "(red F)\<^sup>*\<^sup>* q q'" and "r = p' - q'"

theorem lucas_lehmer_sufficient: assumes "prime p" "odd p" assumes "(2 ^ p - 1) dvd gen_lucas_lehmer_sequence 4 (p - 2)" shows "prime (2 ^ p - 1 :: nat)"

lemma take_from_set_correct: assumes "s \<noteq> {}" shows "take_from_set s \<le> SPEC (\<lambda> (x, s'). x \<in> s \<and> s' = s - {x})"

lemma hmac_trans_1_4_skr_extr_fake: "hmac X K \<in> parts (extr (bad s') (ik s') (chan s')) \<Longrightarrow> K \<notin> synth (analz (extr (bad s) (ik s) (chan s))) \<Longrightarrow> \<comment> \<open>necessary for the \<open>dy_fake_msg\<close> case\<close> s \<in> l2_inv2 \<Longrightarrow> \<comment> \<open>necessary for the \<open>skr\<close> case\<close> (s, s') \<in> l2_step1 Ra A B \<union> l2_step4 Rb A B Ni gnx \<union> l2_skr R KK \<union> l2_dy_fake_msg M \<union> l2_dy_fake_chan MM \<Longrightarrow> hmac X K \<in> parts (extr (bad s) (ik s) (chan s))"

lemma replacement_monotonic : shows "\<And> t s. ((subst v \<sigma>), (subst u \<sigma>)) \<in> trm_ord \<Longrightarrow> subterm t p u \<Longrightarrow> replace_subterm t p v s \<Longrightarrow> ((subst s \<sigma>), (subst t \<sigma>)) \<in> trm_ord"

lemma NS_staticSecret_parts_Spy: "\<lbrakk>m \<in> parts (knows Spy evs); m \<in> staticSecret A; evs \<in> ns_public\<rbrakk> \<Longrightarrow> A \<in> bad"

lemma valid_sops_stmt_invariant: assumes step: "\<theta>\<turnstile> (s,t) \<rightarrow>\<^sub>s ((s',t'),is)" shows "valid_sops_stmt t s \<Longrightarrow> valid_sops_stmt t' s'"

lemma (in poly_mod_prime_type) finite_field_factorization_modulo_ring: assumes g: "(g :: 'a mod_ring poly) = of_int_poly f" and sf: "square_free_m f" and fact: "finite_field_factorization g = (d,gs)" and c: "c = to_int_mod_ring d" and fs: "fs = map to_int_poly gs" shows "unique_factorization_m f (c, mset fs)"

lemma VPow_top: "A \<in>\<^sub>\<circ> VPow A"

lemma rel_frext_miracle [frame]: "a:[false]\<^sup>+ = false"

lemma RPDs_ok: "ok m mr \<Longrightarrow> ms \<in> RPDs mr \<Longrightarrow> ok m ms"

lemma sat_plan_to_dimacs_works: "valid_state_var sv \<Longrightarrow> dimacs_to_var (var_to_dimacs sv) = sv"

lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"

lemma mset_subset_eq_exists_conv: "(A::'a multiset) \<subseteq># B \<longleftrightarrow> (\<exists>C. B = A + C)"

lemma orthogonal_split: assumes "(embP Q \<and>* $ n) = (embP P \<and>* $ m)" shows "(Q = P \<and> n = m) \<or> Q = (\<lambda>s. False) \<and> P = (\<lambda>s. False)"

lemma pos_meas_self: assumes "pos_meas_set E" shows "0 \<le> \<mu> E"

lemma subprob_algebra_cong: "sets M = sets N \<Longrightarrow> subprob_algebra M = subprob_algebra N"

lemma vifintersection_vintersection: assumes "I \<noteq> 0" and "J \<noteq> 0" shows "(\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. f i) \<inter>\<^sub>\<circ> (\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>J. f i) = (\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>I \<union>\<^sub>\<circ> J. f i)"

lemma mor_unfold: "mor UNIV UNIV s1 s2 UNIV UNIV dtor1 dtor2 (unfold1 s1 s2) (unfold2 s1 s2)"

lemma check_mult_alt_def: \<open>check_mult A \<V> p q i r \<ge> do { b \<leftarrow> SPEC(\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars q \<subseteq> \<V> \<and> vars r \<subseteq> \<V>); if \<not>b then RETURN False else do { ASSERT (p \<in># dom_m A); let p = the (fmlookup A p); pq \<leftarrow> mult_poly_spec p q; p \<leftarrow> weak_equality pq r; RETURN p } }\<close>

lemma hypext_sin_e12 [simp]: "(*h* sin) (x * e12) = e12 * x"

lemma defect_less: assumes b: "\<And>t. t0 < t \<Longrightarrow> t \<le> t1 \<Longrightarrow> v' t - f t (v t) < w' t - f t (w t)" notes [continuous_intros] = vderiv_on_continuous_on[OF v', THEN continuous_on_subset] vderiv_on_continuous_on[OF w', THEN continuous_on_subset] shows "\<forall>t \<in> {t0 <.. t1}. v t < w t"

lemma (in group) trivial_group: "trivial_group G \<longleftrightarrow> (\<exists>a. carrier G = {a})"

lemma defines "prec \<equiv> ((\<lambda>f g. (pr_strict' f g, pr_weak' f g)))" and "prl \<equiv> (\<lambda>(f, n). n = 0 \<and> least f)" shows kbo_encoding_is_valid_wpo: "wpo_with_assms weight_S weight_NS prec prl full_status False (\<lambda>f. False)" and kbo_as_wpo: "bounded_arity n (funas_term t) \<Longrightarrow> kbo s t = wpo.wpo n weight_S weight_NS prec prl full_status (\<lambda>_. Lex) False (\<lambda>f. False) s t"

lemma ntadj_sub_diff_assign:"\<And>e \<sigma> x. (\<Union>y\<in>{y. Inl y \<in> SIGT e}. FVT (\<sigma> y)) \<subseteq> (\<Union>y\<in>{y. Inl (Inl y) \<in> SIGP (DiffAssign x e)}. FVT (\<sigma> y))"

lemma sortedByWeight_Cons_imp_forall_weight_ge: "sortedByWeight (t # ts) \<Longrightarrow> \<forall>u \<in> set ts. weight u \<ge> weight t"

lemma symmetric_mpoly_mult [intro]: "symmetric_mpoly A p \<Longrightarrow> symmetric_mpoly A q \<Longrightarrow> symmetric_mpoly A (p * q)"

lemma estimate'_bounds: "prob {\<omega>. of_rat \<delta> * real_of_rat (F 0 as) < \<bar>estimate' (sketch_rv' \<omega>) - of_rat (F 0 as)\<bar>} \<le> 1/3"

lemma (in comm_monoid) multlist_perm_cong: assumes prm: "as <~~> bs" and ascarr: "set as \<subseteq> carrier G" shows "foldr (\<otimes>) as \<one> = foldr (\<otimes>) bs \<one>"

lemma minus_one_sdiv_word_eq [simp]: \<open>- 1 sdiv w = - (1 sdiv w)\<close> for w :: \<open>'a::len word\<close>

lemma Lyndon_rec_all: assumes "Lyndon_rec (a # w) (\<^bold>|w\<^bold>|)" shows "n < \<^bold>|a#w\<^bold>| \<Longrightarrow> 0 < n \<Longrightarrow> Lyndon_rec (a#w) n"

lemma dickson_less_pD2: assumes "dickson_less_p d m p q" shows "q \<in> dgrad_p_set d m"

lemma qbs_morphism_Pair1: assumes "x \<in> qbs_space X" shows "Pair x \<in> Y \<rightarrow>\<^sub>Q X \<Otimes>\<^sub>Q Y"

lemma M_neg_imp_z_non_zero: assumes "v \<bullet> (M *v v) < 0" shows "v$3 \<noteq> 0"

lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"

lemma hext_typeof_mono: "\<lbrakk> h \<unlhd> h'; typeof\<^bsub>h\<^esub> v = Some T \<rbrakk> \<Longrightarrow> typeof\<^bsub>h'\<^esub> v = Some T"

lemma compact_sums': fixes S :: "'a::real_normed_vector set" assumes "compact S" and "compact T" shows "compact (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"

lemma has_disc_negated_alist_and': "has_disc_negated disc neg (alist_and' as) \<longleftrightarrow> (\<exists> a \<in> set as. has_disc_negated disc neg (negation_type_to_match_expr a))"

lemma map_pair_fst_helper : "map fst (map (\<lambda> (x1,x2) . ((x1,x2), f x1 x2)) xs) = xs"

lemma const_poly_nonzero_coeff : assumes "degree p = 0" "p \<noteq> 0" shows "coeff p 0 \<noteq> 0"

lemma restore_new_paras: "ffp \<ge> length gs \<Longrightarrow> \<lbrace>\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_exec i xs) gs @ 0 # xs @ anything\<rbrace> mv_boxes (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) 0 (length gs) \<lbrace>\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ 0 # xs @ anything\<rbrace>"

lemma set_pmf_of_set_lessThan_length [simp]: "xs \<noteq> [] \<Longrightarrow> set_pmf (pmf_of_set {..<length xs}) = {..<length xs}"

lemma less_eq_mctxt_MVarE1: assumes "MVar v \<le> D" obtains (MVar) "D = MVar v"

lemma neg_length_pos: assumes i0: "bv_to_int w < -1" shows "0 < length w"

lemma Program_code [code]: "Program = program \<circ> tabulate_program"

lemma tensor_extensionality3': fixes F G :: \<open>(('a::finite\<times>'b::finite)\<times>'c::finite) update \<Rightarrow> 'd::finite update\<close> assumes [simp]: \<open>register F\<close> \<open>register G\<close> assumes "\<And>f g h. F ((f \<otimes>\<^sub>u g) \<otimes>\<^sub>u h) = G ((f \<otimes>\<^sub>u g) \<otimes>\<^sub>u h)" shows "F = G"

lemma r_leap_total: "eval r_leap [t, i, x] \<down>"