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

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lemma ta_seq_consist_mrw_before: assumes sc: "ta_seq_consist P Map.empty (lmap snd E)" and new_actions_for_fun: "\<And>adal a a'. \<lbrakk> a \<in> new_actions_for P E adal; a' \<in> new_actions_for P E adal \<rbrakk> \<Longrightarrow> a = a'" and mrw: "P,E \<turnstile> r \<leadsto>mrw w" shows "w < r"
lemma reach_reach12: assumes "reach s" obtains "One.reach (fst s)" and "Two.reach (snd s)"
lemma perm_cases: assumes pi: "set pi \<subseteq> A \<times> A" shows "((pi::name prm) \<bullet> B) \<subseteq> A \<union> B"
lemma exec_return: "exec (IO.return a) world = world" and "eval (IO.return a) world = a"
lemma Ml_Rel_components: shows "Ml_Rel \<CC> a\<lparr>NTMap\<rparr> = (\<lambda>B\<in>\<^sub>\<circ>\<CC>\<lparr>Obj\<rparr>. vsnd_arrow (set {a}) B)" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTDom\<rparr> = cf_prod_2_Rel \<CC>\<^bsub>\<CC>,\<CC>\<^esub>(set {a},-)\<^sub>C\<^sub>F" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTCod\<rparr> = cf_id \<CC>" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTDGDom\<rparr> = \<CC>" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTDGCod\<rparr> = \<CC>"
lemma card_permutations_of_multiset_insert_aux: "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)"
lemma Mempty_parametric [transfer_rule]: "rel_mset A {#} {#}"
lemma fixes f :: "real \<Rightarrow> real" assumes M: "sets M = sets borel" assumes nonneg: "AE x in M. 0 \<le> f x" assumes borel: "f \<in> borel_measurable borel" assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)" assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) \<longlongrightarrow> x) at_top" shows has_bochner_integral_monotone_convergence_at_top: "has_bochner_integral M f x" and integrable_monotone_convergence_at_top: "integrable M f" and integral_monotone_convergence_at_top:"integral\<^sup>L M f = x"
lemma lmap_eq_LCons_conv: "lmap f xs = LCons y ys \<longleftrightarrow> (\<exists>x xs'. xs = LCons x xs' \<and> y = f x \<and> ys = lmap f xs')"
lemma starfun_inverse_real_of_nat_eq: "N \<in> HNatInfinite \<Longrightarrow> ( f (\<lambda>x::nat. inverse (real x))) N = inverse (hypreal_of_hypnat N)"
lemma max_less_imp_conj:"max x y < b \<Longrightarrow> x < (b::('a::linorder)) \<and> y < b"
lemma map_mpoly_Var: "f 1 = 1 \<Longrightarrow> map_mpoly (f :: 'b :: zero_neq_one \<Rightarrow> _) (Var i) = Var i"
lemma product_states[simp]: "c.states (product A B) w (p, q) = a.states A w p \|\| b.states B w q"
lemma [simp]: "safe (Exit g r # t) r' = safe t r'"
lemma size_0_same': "size w = 0 \<Longrightarrow> w = v" for v w :: "'a::len word"
lemma vec_index_vCons: "vCons a v $ n = (if n = 0 then a else v $ (n - 1))"
lemma cIsInvar_isRevNth_isPC: "cIsInvar isRevNth_isPC"
lemma of_complex_neg_numeral [simp]: "of_complex (- numeral w) = - numeral w"
lemma HEndPhase0_HInv4b_p_dblock: assumes act: "HEndPhase0 s s' p" and inv1: "Inv1 s" and inv2a: "Inv2a s" and inv2c: "Inv2c_inner s p" shows "bal(dblock s' p) < mbal(dblock s' p)"
lemma wwf_J_mdecl[simp]: "wwf_J_mdecl P C (M,Ts,T,pns,body) = (length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> fv body \<subseteq> {this} \<union> set pns)"
lemma plus_Box_strict1[simp]: "\<bottom> + (y :: 'a::{predomain, plus} Box) = \<bottom>"
lemma while_decompose_8: "(x \<squnion> y) \<star> z = (x \<squnion> y) \<star> (x \<star> (y \<star> z))"
lemma safe_rpd: "(\<forall>x \<in> atms r. safe Strict x \<longrightarrow> safe Lax x) \<Longrightarrow> safe_regex Past Strict r \<Longrightarrow> s \<in> rpd test i r \<Longrightarrow> safe_regex Past Strict s"
lemma encode_0_iff: "encode e n = 0 \<longleftrightarrow> n = 0"
lemma root_normalize1_eq2: "\<forall>x. xs \<noteq> {\|x\|} \<Longrightarrow> root (normalize1 (Node r xs)) = r"
lemma wf_mdecl_wwf_mdecl: "wf_J_mdecl P C Md \<Longrightarrow> wwf_J_mdecl P C Md"
lemma eps_free_AGTT_trancl_eps_free: "is_gtt_eps_free \<G> \<Longrightarrow> is_gtt_eps_free (AGTT_trancl_eps_free \<G>)"
lemma eval_wp_Embed: "wp (Embed t) = t"
lemma node_sos_never_newpkt [simp]: assumes "(s, a, s') \<in> node_sos S" shows "a \<noteq> i:newpkt(d, di)"
lemma fps_of_poly_0 [simp]: "fps_of_poly 0 = 0"
lemma all_shared_dropWhile_subset: "all_shared (dropWhile P sb) \<subseteq> all_shared sb"
lemma eq_comps_compare: assumes "sorted l" and "a#m = eq_comps l" and "i < a" and "a \<le> j" and "j < length l" shows "nth l i < nth l j"
lemma length_plus_28_elim2[elim]: "(a, b) \<in> set (shift (adjust0 fourtimes_compile_tm) (twice_tm_len + 13)) \<Longrightarrow> b \<le> (28 + (length twice_compile_tm + length fourtimes_compile_tm)) div 2"
lemma valid_refs_inv_load: "\<lbrakk> valid_refs_inv s; sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref r'; r \<in> mut_roots s \<rbrakk> \<Longrightarrow> valid_refs_inv (s(mutator m := s (mutator m)\<lparr>roots := mut_roots s \<union> Option.set_option r'\<rparr>))"
lemma is_Done_map_resumption [simp]: "is_Done (map_resumption f1 f2 r) \<longleftrightarrow> is_Done r"
lemma NSCauchy_NSBseq: "NSCauchy X \<Longrightarrow> NSBseq X"
lemma merge_comm_if_not_equiv: "\<forall>x \<in> set xs. \<forall>y \<in> set ys. compare cmp x y \<noteq> Equiv \<Longrightarrow> Sorting_Algorithms.merge cmp xs ys = Sorting_Algorithms.merge cmp ys xs"
lemma closed_proj_rel: "closed {(x::'a::euclidean_space nonzero, y::'a nonzero). proj_rel x y}"
lemma callee_lcl_VNam_simp [simp]: "callee_lcl C sig m (EName (VNam v)) = (table_of (lcls (mbody m))((pars m)[\<mapsto>](parTs sig))) v"
lemma P_LCons': "P = LCons v0 (LCons w0 (ltl (ltl P)))"
lemma set_fmap_inv1: "\<lbrakk> fst x \<in> dom F; snd x = the (F (fst x)) \<rbrakk> \<Longrightarrow> (F -- x)(fst x \<mapsto> snd x) = F"
lemma "(0::int) < 1"
lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
lemma (in loc1) is_path_f: "infinite (deriv s) ==> \<forall>n. f n \<in> deriv s & fst (f n) = n & (snd (f (Suc n))) : set (subs (snd (f n))) & infinite (deriv (snd (f n)))"
lemma path_image_join: assumes "pathfinish g1 = pathstart g2" shows "path_image(g1 +++ g2) = path_image g1 \<union> path_image g2"
lemma tmap_transfer [transfer_rule]: "((=) ===> (=) ===> pcr_tllist (=) (=) ===> pcr_tllist (=) (=)) (map_prod \<circ> lmap) tmap"
lemma face_cong_if_norm_eq: "\<lbrakk> rotate_min xs = rotate_min ys; xs \<noteq> []; ys \<noteq> [] \<rbrakk> \<Longrightarrow> xs \<cong> ys"
lemma labeled_saturation_lifting: "saturated NL \<Longrightarrow> no_labels.saturated (fst ` NL)"
lemma iter_space_notempty: "iter_space \<noteq> {}"
lemma DIGeq_valid:"valid DIGeqaxiom"
lemma pmf_bind_bernoulli: assumes "x \<in> {0..1}" shows "pmf (bernoulli_pmf x \<bind> f) y = x * pmf (f True) y + (1 - x) * pmf (f False) y"
lemma distance_attains_inf: fixes a :: "'a::heine_borel" assumes "closed s" and "s \<noteq> {}" obtains x where "x\<in>s" "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y"
lemma fpxs_X_power_conv_shift: "fpxs_X_power r = fpxs_shift (-r) 1"
lemma S'_univalent: "univalent S'"
lemma ipt_iprange_to_set_nonempty: "ipt_iprange_to_set ip = {} \<longleftrightarrow> (\<exists>ip1 ip2. ip = IpAddrRange ip1 ip2 \<and> ip1 > ip2)"
lemma powreal_le_cancel: "\<lbrakk> a pow\<^sub>\<real> r \<le> a pow\<^sub>\<real> s; a > 1 \<rbrakk> \<Longrightarrow> r \<le> s"
lemma "P x \<Longrightarrow> P (The P)"
lemma LIMSEQ_inverse_real_of_nat_add_minus: "(\<lambda>n. r + -inverse (real (Suc n))) \<longlonglongrightarrow> r"
lemma preserves_equation: "-y * x \<le> x * -y \<longleftrightarrow> -y * x = -y * x * -y"
lemma (in Corps) nsum_in_Vr:"\<lbrakk>valuation K v; \<forall>j \<le> n. f j \<in> carrier K; \<forall>j \<le> n. 0 \<le> (v (f j))\<rbrakk> \<Longrightarrow> (nsum K f n) \<in> carrier (Vr K v)"
lemma [simp]: \<open>bij assoc\<close>
lemma ring_iso_set_sym: assumes "ring R" and h: "h \<in> ring_iso R S" shows "(inv_into (carrier R) h) \<in> ring_iso S R"
lemma leq_mask_shift: "(x :: 'a :: len word) \<le> mask (low_bits + high_bits) \<Longrightarrow> (x >> low_bits) \<le> mask high_bits"
lemma analytic_on_compose_gen: "f analytic_on S \<Longrightarrow> g analytic_on T \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<in> T) \<Longrightarrow> g o f analytic_on S"
lemma pair_declclass_simp[simp]: "declclass (c,x) = declclass c"
lemma ins_sorted_inorder: "sorted_less (inorder t) \<Longrightarrow> (inorder_up\<^sub>i (ins k (x::('a::linorder)) t)) = ins_list x (inorder t)"
lemma traces_nil [simp, intro!]: "init E s \<Longrightarrow> [] \<in> traces E"
lemma(in padic_integers) p_intnatpow_prod: assumes "(n::int) \<ge> 0" shows "\<p>[^]n \<otimes> \<p>[^](m::nat) = \<p>[^](m + n)"
lemma "Fr_1b \<F> \<Longrightarrow> \<forall>A. Op(A) \<longrightarrow> DNI\<^sup>A \<^bold>\<not>\<^sup>I"
lemma Log_nat_power: assumes "0 < x" and "1 < b" and "b \<noteq> 1" shows " Log b (x ^ n) = real n * Log b x"
lemma stails_strict[simp]: "stails\<cdot>\<bottom> = \<bottom>"
lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
lemma Ipoly_snd_tmf_polys: "set_of (horner_eval (real_interval o centered o nth z) x (length z)) \<subseteq> set_of (Ipoly [x] (map_poly real_interval (snd (tmf_polys z))))"
lemma exists_split_characterisations: shows "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> (\<exists>x . linear_orderable_2 x)" and "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> (\<exists>x . linear_orderable_3 x)" and "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> (\<exists>x . linear_orderable_3a x)" and "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> transitively_orientable (-1)" and "(\<exists>x . linear_orderable_1 x) \<Longrightarrow> (\<exists>x . orientable_11 x)" and "(\<exists>x . orientable_11 x) \<longleftrightarrow> (\<exists>x . orientable_12 x)"
lemma degree_eq_0_mpoly_to_nested_polyI: "mpoly_to_nested_poly mp v = 0 \<Longrightarrow> MPoly_Type.degree mp v = 0"
lemma sumset_translate_eq_left: assumes "A \<subseteq> G" and "B \<subseteq> G" and "x \<in> G" shows " (sumset {x} A = sumset {x} B) \<longleftrightarrow> A = B"
lemma paths_of_length_1: "paths_of_length 1 e s = fimage (\<lambda>(d, t, id). [id]) (outgoing_transitions s e)"
lemma ccTTree_mono1: "S \<subseteq> S' \<Longrightarrow> ccTTree S G \<sqsubseteq> ccTTree S' G"
lemma set_empty_choose_code [code]: "(set_empty_choose :: 'a :: {ceq, ccompare} set) = (case CCOMPARE('a) of Some _ \<Rightarrow> RBT_set RBT_Set2.empty \| None \<Rightarrow> case CEQ('a) of None \<Rightarrow> Set_Monad [] \| Some _ \<Rightarrow> DList_set (DList_Set.empty))"
lemma minarc_bot_iff: "minarc x = bot \<longleftrightarrow> x = bot"
lemma addr_locs_compP [simp]: "addr_locs (compP f P) = addr_locs P"
lemma strand_sem_model_swap: assumes "\<forall>x \<in> fv\<^sub>s\<^sub>s\<^sub>t S. I x = J x" and "\<lbrakk>M; D; S\<rbrakk>\<^sub>s I" shows "\<lbrakk>M; D; S\<rbrakk>\<^sub>s J"
lemma assumes a: "Quotient R Abs Rep T" shows "symp R"
lemma lower_or_eq_monotonic: assumes "lower_or_eq t1 s1" assumes "lower_or_eq t2 s2" shows "lower_or_eq (Comb t1 t2) (Comb s1 s2)"
lemma mk_rtrancl_complete: assumes a: "a \<in> R^* `` set_of init" shows "\<exists> b. b \<in> mk_rtrancl init \<and> subsumes b a"
lemma prefs_from_table_swap: "x \<noteq> y \<Longrightarrow> prefs_from_table ((x,x')#(y,y')#xs) = prefs_from_table ((y,y')#(x,x')#xs)"
lemma (in Worder) Word_compareTr2:"\<lbrakk>Worder E; ord_equiv D E; \<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b))\<rbrakk> \<Longrightarrow> False"
lemma iter_widen_pfp: "iter_widen f x = Some p \<Longrightarrow> f p \<le> p"
lemma (in monoid) units_of_pow: fixes n :: nat shows "x \<in> Units G \<Longrightarrow> x [^]\<^bsub>units_of G\<^esub> n = x [^]\<^bsub>G\<^esub> n"
lemma invfun_mem:"\<lbrakk> f \<in> A \<rightarrow> B; inj_on f A; surj_to f A B; b \<in> B \<rbrakk> \<Longrightarrow> (invfun A B f) b \<in> A"
lemma wp_g_odei: "\<lceil>P\<rceil> \<le> \<lceil>I\<rceil> \<Longrightarrow> \<lceil>I\<rceil> \<le> wp (x\<acute>= f & G on U S @ t\<^sub>0) \<lceil>I\<rceil> \<Longrightarrow> \<lceil>\<lambda>s. I s \<and> G s\<rceil> \<le> \<lceil>Q\<rceil> \<Longrightarrow> \<lceil>P\<rceil> \<le> wp (x\<acute>= f & G on U S @ t\<^sub>0 DINV I) \<lceil>Q\<rceil>"
lemma dsum_bound: assumes p: "p division_of (cbox a b)" and "norm c \<le> e" shows "norm (sum (\<lambda>l. content l \<^sub>R c) p) \<le> e content(cbox a b)"
lemma prog_configs_append: "\<And>ys. prog_configs (xs@ys) = prog_configs xs \<union> prog_configs ys"
lemma agtt_grrstep_eps_trancl [simp]: "(eps (fst (agtt_grrstep \<R> \<F>)))\|\<^sup>+\| = eps (fst (agtt_grrstep \<R> \<F>))" "(eps (snd (agtt_grrstep \<R> \<F>))) = {\|\|}"
lemma row_rank_eq_col_rank: fixes A::"'a::{field}^'n::{mod_type}^'m::{mod_type}" shows "row_rank A = col_rank A"
lemma measure_restrict_space: assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>" shows "measure (restrict_space M \<Omega>) A = measure M A"
lemma proj_lnil [simp,intro]: "\<pi>\<^bsub>c\<^esub>([]\<^sub>l) = []\<^sub>l"
lemma cring_class: "cring cring_class_ops"
lemma word_rec_in: "f (word_rec z (\<lambda>_. f) n) = word_rec (f z) (\<lambda>_. f) n"
lemma has_field_derivative_stirling_integral_complex: fixes x :: complex assumes "x \<notin> \<real>\<^sub>\<le>\<^sub>0" "n > 0" shows "(stirling_integral n has_field_derivative deriv (stirling_integral n) x) (at x)"

lemma ta_seq_consist_mrw_before: assumes sc: "ta_seq_consist P Map.empty (lmap snd E)" and new_actions_for_fun: "\<And>adal a a'. \<lbrakk> a \<in> new_actions_for P E adal; a' \<in> new_actions_for P E adal \<rbrakk> \<Longrightarrow> a = a'" and mrw: "P,E \<turnstile> r \<leadsto>mrw w" shows "w < r"

lemma reach_reach12: assumes "reach s" obtains "One.reach (fst s)" and "Two.reach (snd s)"

lemma perm_cases: assumes pi: "set pi \<subseteq> A \<times> A" shows "((pi::name prm) \<bullet> B) \<subseteq> A \<union> B"

lemma exec_return: "exec (IO.return a) world = world" and "eval (IO.return a) world = a"

lemma Ml_Rel_components: shows "Ml_Rel \<CC> a\<lparr>NTMap\<rparr> = (\<lambda>B\<in>\<^sub>\<circ>\<CC>\<lparr>Obj\<rparr>. vsnd_arrow (set {a}) B)" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTDom\<rparr> = cf_prod_2_Rel \<CC>\<^bsub>\<CC>,\<CC>\<^esub>(set {a},-)\<^sub>C\<^sub>F" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTCod\<rparr> = cf_id \<CC>" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTDGDom\<rparr> = \<CC>" and [cat_cs_simps]: "Ml_Rel \<CC> a\<lparr>NTDGCod\<rparr> = \<CC>"

lemma card_permutations_of_multiset_insert_aux: "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)"

lemma Mempty_parametric [transfer_rule]: "rel_mset A {#} {#}"

lemma fixes f :: "real \<Rightarrow> real" assumes M: "sets M = sets borel" assumes nonneg: "AE x in M. 0 \<le> f x" assumes borel: "f \<in> borel_measurable borel" assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)" assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) \<longlongrightarrow> x) at_top" shows has_bochner_integral_monotone_convergence_at_top: "has_bochner_integral M f x" and integrable_monotone_convergence_at_top: "integrable M f" and integral_monotone_convergence_at_top:"integral\<^sup>L M f = x"

lemma lmap_eq_LCons_conv: "lmap f xs = LCons y ys \<longleftrightarrow> (\<exists>x xs'. xs = LCons x xs' \<and> y = f x \<and> ys = lmap f xs')"

lemma starfun_inverse_real_of_nat_eq: "N \<in> HNatInfinite \<Longrightarrow> ( *f* (\<lambda>x::nat. inverse (real x))) N = inverse (hypreal_of_hypnat N)"

lemma max_less_imp_conj:"max x y x < (b::('a::linorder)) \<and> y < b"

lemma map_mpoly_Var: "f 1 = 1 \<Longrightarrow> map_mpoly (f :: 'b :: zero_neq_one \<Rightarrow> _) (Var i) = Var i"

lemma product_states[simp]: "c.states (product A B) w (p, q) = a.states A w p || b.states B w q"

lemma [simp]: "safe (Exit g r # t) r' = safe t r'"

lemma size_0_same': "size w = 0 \<Longrightarrow> w = v" for v w :: "'a::len word"

lemma vec_index_vCons: "vCons a v $ n = (if n = 0 then a else v $ (n - 1))"

lemma cIsInvar_isRevNth_isPC: "cIsInvar isRevNth_isPC"

lemma of_complex_neg_numeral [simp]: "of_complex (- numeral w) = - numeral w"

lemma HEndPhase0_HInv4b_p_dblock: assumes act: "HEndPhase0 s s' p" and inv1: "Inv1 s" and inv2a: "Inv2a s" and inv2c: "Inv2c_inner s p" shows "bal(dblock s' p) < mbal(dblock s' p)"

lemma wwf_J_mdecl[simp]: "wwf_J_mdecl P C (M,Ts,T,pns,body) = (length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> fv body \<subseteq> {this} \<union> set pns)"

lemma plus_Box_strict1[simp]: "\<bottom> + (y :: 'a::{predomain, plus} Box) = \<bottom>"

lemma while_decompose_8: "(x \<squnion> y) \<star> z = (x \<squnion> y) \<star> (x \<star> (y \<star> z))"

lemma safe_rpd: "(\<forall>x \<in> atms r. safe Strict x \<longrightarrow> safe Lax x) \<Longrightarrow> safe_regex Past Strict r \<Longrightarrow> s \<in> rpd test i r \<Longrightarrow> safe_regex Past Strict s"

lemma encode_0_iff: "encode e n = 0 \<longleftrightarrow> n = 0"

lemma root_normalize1_eq2: "\<forall>x. xs \<noteq> {|x|} \<Longrightarrow> root (normalize1 (Node r xs)) = r"

lemma wf_mdecl_wwf_mdecl: "wf_J_mdecl P C Md \<Longrightarrow> wwf_J_mdecl P C Md"

lemma eps_free_AGTT_trancl_eps_free: "is_gtt_eps_free \<G> \<Longrightarrow> is_gtt_eps_free (AGTT_trancl_eps_free \<G>)"

lemma eval_wp_Embed: "wp (Embed t) = t"

lemma node_sos_never_newpkt [simp]: assumes "(s, a, s') \<in> node_sos S" shows "a \<noteq> i:newpkt(d, di)"

lemma fps_of_poly_0 [simp]: "fps_of_poly 0 = 0"

lemma all_shared_dropWhile_subset: "all_shared (dropWhile P sb) \<subseteq> all_shared sb"

lemma eq_comps_compare: assumes "sorted l" and "a#m = eq_comps l" and "i < a" and "a \<le> j" and "j < length l" shows "nth l i < nth l j"

lemma length_plus_28_elim2[elim]: "(a, b) \<in> set (shift (adjust0 fourtimes_compile_tm) (twice_tm_len + 13)) \<Longrightarrow> b \<le> (28 + (length twice_compile_tm + length fourtimes_compile_tm)) div 2"

lemma valid_refs_inv_load: "\<lbrakk> valid_refs_inv s; sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref r'; r \<in> mut_roots s \<rbrakk> \<Longrightarrow> valid_refs_inv (s(mutator m := s (mutator m)\<lparr>roots := mut_roots s \<union> Option.set_option r'\<rparr>))"

lemma is_Done_map_resumption [simp]: "is_Done (map_resumption f1 f2 r) \<longleftrightarrow> is_Done r"

lemma NSCauchy_NSBseq: "NSCauchy X \<Longrightarrow> NSBseq X"

lemma merge_comm_if_not_equiv: "\<forall>x \<in> set xs. \<forall>y \<in> set ys. compare cmp x y \<noteq> Equiv \<Longrightarrow> Sorting_Algorithms.merge cmp xs ys = Sorting_Algorithms.merge cmp ys xs"

lemma closed_proj_rel: "closed {(x::'a::euclidean_space nonzero, y::'a nonzero). proj_rel x y}"

lemma callee_lcl_VNam_simp [simp]: "callee_lcl C sig m (EName (VNam v)) = (table_of (lcls (mbody m))((pars m)[\<mapsto>](parTs sig))) v"

lemma P_LCons': "P = LCons v0 (LCons w0 (ltl (ltl P)))"

lemma set_fmap_inv1: "\<lbrakk> fst x \<in> dom F; snd x = the (F (fst x)) \<rbrakk> \<Longrightarrow> (F -- x)(fst x \<mapsto> snd x) = F"

lemma "(0::int) < 1"

lemma cmod_unit_one: "cmod (cos a + \ * sin a) = 1"

lemma (in loc1) is_path_f: "infinite (deriv s) ==> \<forall>n. f n \<in> deriv s & fst (f n) = n & (snd (f (Suc n))) : set (subs (snd (f n))) & infinite (deriv (snd (f n)))"

lemma path_image_join: assumes "pathfinish g1 = pathstart g2" shows "path_image(g1 +++ g2) = path_image g1 \<union> path_image g2"

lemma tmap_transfer [transfer_rule]: "((=) ===> (=) ===> pcr_tllist (=) (=) ===> pcr_tllist (=) (=)) (map_prod \<circ> lmap) tmap"

lemma face_cong_if_norm_eq: "\<lbrakk> rotate_min xs = rotate_min ys; xs \<noteq> []; ys \<noteq> [] \<rbrakk> \<Longrightarrow> xs \<cong> ys"

lemma labeled_saturation_lifting: "saturated NL \<Longrightarrow> no_labels.saturated (fst ` NL)"

lemma iter_space_notempty: "iter_space \<noteq> {}"

lemma DIGeq_valid:"valid DIGeqaxiom"

lemma pmf_bind_bernoulli: assumes "x \<in> {0..1}" shows "pmf (bernoulli_pmf x \<bind> f) y = x * pmf (f True) y + (1 - x) * pmf (f False) y"

lemma distance_attains_inf: fixes a :: "'a::heine_borel" assumes "closed s" and "s \<noteq> {}" obtains x where "x\<in>s" "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y"

lemma fpxs_X_power_conv_shift: "fpxs_X_power r = fpxs_shift (-r) 1"

lemma S'_univalent: "univalent S'"

lemma ipt_iprange_to_set_nonempty: "ipt_iprange_to_set ip = {} \<longleftrightarrow> (\<exists>ip1 ip2. ip = IpAddrRange ip1 ip2 \<and> ip1 > ip2)"

lemma powreal_le_cancel: "\<lbrakk> a pow\<^sub>\<real> r \<le> a pow\<^sub>\<real> s; a > 1 \<rbrakk> \<Longrightarrow> r \<le> s"

lemma "P x \<Longrightarrow> P (The P)"

lemma LIMSEQ_inverse_real_of_nat_add_minus: "(\<lambda>n. r + -inverse (real (Suc n))) \<longlonglongrightarrow> r"

lemma preserves_equation: "-y * x \<le> x * -y \<longleftrightarrow> -y * x = -y * x * -y"

lemma (in Corps) nsum_in_Vr:"\<lbrakk>valuation K v; \<forall>j \<le> n. f j \<in> carrier K; \<forall>j \<le> n. 0 \<le> (v (f j))\<rbrakk> \<Longrightarrow> (nsum K f n) \<in> carrier (Vr K v)"

lemma [simp]: \<open>bij assoc\<close>

lemma ring_iso_set_sym: assumes "ring R" and h: "h \<in> ring_iso R S" shows "(inv_into (carrier R) h) \<in> ring_iso S R"

lemma leq_mask_shift: "(x :: 'a :: len word) \<le> mask (low_bits + high_bits) \<Longrightarrow> (x >> low_bits) \<le> mask high_bits"

lemma analytic_on_compose_gen: "f analytic_on S \<Longrightarrow> g analytic_on T \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<in> T) \<Longrightarrow> g o f analytic_on S"

lemma pair_declclass_simp[simp]: "declclass (c,x) = declclass c"

lemma ins_sorted_inorder: "sorted_less (inorder t) \<Longrightarrow> (inorder_up\<^sub>i (ins k (x::('a::linorder)) t)) = ins_list x (inorder t)"

lemma traces_nil [simp, intro!]: "init E s \<Longrightarrow> [] \<in> traces E"

lemma(in padic_integers) p_intnatpow_prod: assumes "(n::int) \<ge> 0" shows "\[^]n \<otimes> \[^](m::nat) = \[^](m + n)"

lemma "Fr_1b \<F> \<Longrightarrow> \<forall>A. Op(A) \<longrightarrow> DNI\<^sup>A \<^bold>\<not>\<^sup>I"

lemma Log_nat_power: assumes "0 < x" and "1 < b" and "b \<noteq> 1" shows " Log b (x ^ n) = real n * Log b x"

lemma stails_strict[simp]: "stails\<cdot>\<bottom> = \<bottom>"

lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"

lemma Ipoly_snd_tmf_polys: "set_of (horner_eval (real_interval o centered o nth z) x (length z)) \<subseteq> set_of (Ipoly [x] (map_poly real_interval (snd (tmf_polys z))))"

lemma exists_split_characterisations: shows "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> (\<exists>x . linear_orderable_2 x)" and "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> (\<exists>x . linear_orderable_3 x)" and "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> (\<exists>x . linear_orderable_3a x)" and "(\<exists>x . linear_orderable_1 x) \<longleftrightarrow> transitively_orientable (-1)" and "(\<exists>x . linear_orderable_1 x) \<Longrightarrow> (\<exists>x . orientable_11 x)" and "(\<exists>x . orientable_11 x) \<longleftrightarrow> (\<exists>x . orientable_12 x)"

lemma degree_eq_0_mpoly_to_nested_polyI: "mpoly_to_nested_poly mp v = 0 \<Longrightarrow> MPoly_Type.degree mp v = 0"

lemma sumset_translate_eq_left: assumes "A \<subseteq> G" and "B \<subseteq> G" and "x \<in> G" shows " (sumset {x} A = sumset {x} B) \<longleftrightarrow> A = B"

lemma paths_of_length_1: "paths_of_length 1 e s = fimage (\<lambda>(d, t, id). [id]) (outgoing_transitions s e)"

lemma ccTTree_mono1: "S \<subseteq> S' \<Longrightarrow> ccTTree S G \<sqsubseteq> ccTTree S' G"

lemma set_empty_choose_code [code]: "(set_empty_choose :: 'a :: {ceq, ccompare} set) = (case CCOMPARE('a) of Some _ \<Rightarrow> RBT_set RBT_Set2.empty | None \<Rightarrow> case CEQ('a) of None \<Rightarrow> Set_Monad [] | Some _ \<Rightarrow> DList_set (DList_Set.empty))"

lemma minarc_bot_iff: "minarc x = bot \<longleftrightarrow> x = bot"

lemma addr_locs_compP [simp]: "addr_locs (compP f P) = addr_locs P"

lemma strand_sem_model_swap: assumes "\<forall>x \<in> fv\<^sub>s\<^sub>s\<^sub>t S. I x = J x" and "\<lbrakk>M; D; S\<rbrakk>\<^sub>s I" shows "\<lbrakk>M; D; S\<rbrakk>\<^sub>s J"

lemma assumes a: "Quotient R Abs Rep T" shows "symp R"

lemma lower_or_eq_monotonic: assumes "lower_or_eq t1 s1" assumes "lower_or_eq t2 s2" shows "lower_or_eq (Comb t1 t2) (Comb s1 s2)"

lemma mk_rtrancl_complete: assumes a: "a \<in> R^* `` set_of init" shows "\<exists> b. b \<in> mk_rtrancl init \<and> subsumes b a"

lemma prefs_from_table_swap: "x \<noteq> y \<Longrightarrow> prefs_from_table ((x,x')#(y,y')#xs) = prefs_from_table ((y,y')#(x,x')#xs)"

lemma (in Worder) Word_compareTr2:"\<lbrakk>Worder E; ord_equiv D E; \<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b))\<rbrakk> \<Longrightarrow> False"

lemma iter_widen_pfp: "iter_widen f x = Some p \<Longrightarrow> f p \<le> p"

lemma (in monoid) units_of_pow: fixes n :: nat shows "x \<in> Units G \<Longrightarrow> x [^]\<^bsub>units_of G\<^esub> n = x [^]\<^bsub>G\<^esub> n"

lemma invfun_mem:"\<lbrakk> f \<in> A \<rightarrow> B; inj_on f A; surj_to f A B; b \<in> B \<rbrakk> \<Longrightarrow> (invfun A B f) b \<in> A"

lemma wp_g_odei: "\<lceil>P\<rceil> \<le> \<lceil>I\<rceil> \<Longrightarrow> \<lceil>I\<rceil> \<le> wp (x\<acute>= f & G on U S @ t\<^sub>0) \<lceil>I\<rceil> \<Longrightarrow> \<lceil>\<lambda>s. I s \<and> G s\<rceil> \<le> \<lceil>Q\<rceil> \<Longrightarrow> \<lceil>P\<rceil> \<le> wp (x\<acute>= f & G on U S @ t\<^sub>0 DINV I) \<lceil>Q\<rceil>"

lemma dsum_bound: assumes p: "p division_of (cbox a b)" and "norm c \<le> e" shows "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"

lemma prog_configs_append: "\<And>ys. prog_configs (xs@ys) = prog_configs xs \<union> prog_configs ys"

lemma agtt_grrstep_eps_trancl [simp]: "(eps (fst (agtt_grrstep \<R> \<F>)))|\<^sup>+| = eps (fst (agtt_grrstep \<R> \<F>))" "(eps (snd (agtt_grrstep \<R> \<F>))) = {||}"

lemma row_rank_eq_col_rank: fixes A::"'a::{field}^'n::{mod_type}^'m::{mod_type}" shows "row_rank A = col_rank A"

lemma measure_restrict_space: assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>" shows "measure (restrict_space M \<Omega>) A = measure M A"

lemma proj_lnil [simp,intro]: "\<pi>\<^bsub>c\<^esub>([]\<^sub>l) = []\<^sub>l"

lemma cring_class: "cring cring_class_ops"

lemma word_rec_in: "f (word_rec z (\<lambda>_. f) n) = word_rec (f z) (\<lambda>_. f) n"

lemma has_field_derivative_stirling_integral_complex: fixes x :: complex assumes "x \<notin> \<real>\<^sub>\<le>\<^sub>0" "n > 0" shows "(stirling_integral n has_field_derivative deriv (stirling_integral n) x) (at x)"