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lemma transfer_FOREACH_nres[refine_transfer]:
assumes A: "set_iterator iterate s"
assumes R: "\<And>x \<sigma>. nres_of (fi x \<sigma>) \<le> f x \<sigma>"
shows "nres_of (dres_it iterate (\<lambda>_. True) fi \<sigma>) \<le> FOREACH s f \<sigma>" |
lemma sclose_subset_FV[rule_format]:
"FV ({n \<leftarrow> [s,p]} t) \<subseteq> FV t" |
lemma [iff]: "(P,E,h \<turnstile> e#es [:] T#Ts) = (P,E,h \<turnstile> e : T \<and> P,E,h \<turnstile> es [:] Ts)" |
lemma PO_l3_inv9 [iff]: "reach l3 \<subseteq> l3_inv9" |
lemma inext_less_mono_rev: "
\<lbrakk> inext a I < inext b I; a \<in> I; b \<in> I \<rbrakk> \<Longrightarrow> a < b" |
lemma rpr: "R \<subseteq> Id \<Longrightarrow> p2r (r2p R) = R" |
lemma arity_var\<^sub>h_if_\<delta>\<^sub>h_gt_0_E:
assumes \<delta>_gt_0: "\<delta>\<^sub>h > 0"
obtains n where "arity_var\<^sub>h f = of_nat n" |
lemma EmptyMap_ran [simp]:
"S \<noteq> {} \<Longrightarrow> ran (EmptyMap S) = {{}}" |
lemma inv_end5_exit_Bk_Oc_via_loop[elim]:
"\<lbrakk>0 < x; inv_end5_loop x (b, Oc # list); b \<noteq> []; hd b = Bk\<rbrakk> \<Longrightarrow>
inv_end5_exit x (tl b, Bk # Oc # list)" |
lemma OclExcluding_rep_set:
assumes S_def: "\<tau> \<Turnstile> \<delta> S"
shows "\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S->excluding\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. \<lfloor>\<lfloor>x\<rfloor>\<rfloor>) \<tau>)\<rceil>\<rceil> = \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S \<tau>)\<rceil>\<rceil> - {\<lfloor>\<lfloor>x\<rfloor>\<rfloor>}" |
lemma MOST_neq [simp]:
"MOST x. x \<noteq> a"
"MOST x. a \<noteq> x" |
lemma [code]: "prod_mset (mset xs) = fold times xs 1" |
lemma sum_with_closed:
assumes "g ` A \<subseteq> S"
shows "sum_with pls z g A \<in> S" |
lemma subgame_path_conforms_with_strategy:
assumes V': "V' \<subseteq> V" and P: "path_conforms_with_strategy p P \<sigma>" "lset P \<subseteq> V'"
shows "ParityGame.path_conforms_with_strategy (subgame V') p P \<sigma>" |
lemma board_exec_aux_leq_mem: "(i,j) \<in> board_exec_aux k M \<longleftrightarrow> 1 \<le> i \<and> i \<le> int k \<and> j \<in> M" |
lemma kernel_Proj[simp]: \<open>kernel (Proj S) = - S\<close> |
lemma assm8: "0 < arity f \<Longrightarrow> \<Gamma> (Fun f X) = TComp f (map \<Gamma> X)" |
theorem TBtheore54b_P:
assumes "eoutM P M E"
and "subcomponents PQ = {P,Q}"
and "correctCompositionOut PQ"
and "\<exists> ch. ((ch \<in> (out Q)) \<and> (exprChannel ch E) \<and>
(ch \<notin> (loc PQ)) \<and> (ch \<in> M) )"
shows "eoutM PQ M E" |
lemma right_segment_congruence:
assumes "{c, d} = {p, q}" and "a b \<congruent> p q"
shows "a b \<congruent> c d" |
lemma (in StdSet) set_i2_impl: "invar s \<Longrightarrow> invar (set_i2 l) \<and> \<alpha> (set_i2 l) = set_a2 l" |
lemma add_edge_maximally_connected:
assumes "maximally_connected H G"
assumes "subgraph H G"
assumes "(a, w, b) \<in> E"
shows "maximally_connected (add_edge a w b H) G" |
lemma OUTfromVCorrect1_data9: "OUTfromVCorrect1 data9" |
lemma iso_img_block_orig_exists: "x \<in># \<B>' \<Longrightarrow> \<exists> bl . bl \<in># \<B> \<and> x = \<pi> ` bl" |
lemma rec_inseparable_mono:
"rec_inseparable A B \<Longrightarrow> A \<subseteq> A' \<Longrightarrow> B \<subseteq> B' \<Longrightarrow> rec_inseparable A' B'" |
lemma set_interact [simp]:
shows "list.set (interact xs ys) = list.set (concat xs) \<union> list.set (concat ys)" |
lemma NS_subterm: assumes all: "\<And> f k. set (\<sigma> (f,k)) = {0 ..< k}"
shows "s \<unrhd> t \<Longrightarrow> (s,t) \<in> NS" |
lemma eccentricity_Max_alt:
assumes "v \<in> V"
assumes "V \<noteq> {v}"
shows "eccentricity v = Max ((\<lambda> u. shortest_path v u) ` (V - {v}))" |
lemma lang_eq_ext_Nil_fold_Deriv:
fixes K L A R
assumes
"\<And>w. in_language K w \<Longrightarrow> w \<in> lists A"
"\<And>w. in_language L w \<Longrightarrow> w \<in> lists B"
"\<And>a b. R a b \<Longrightarrow> a \<in> A \<longleftrightarrow> b \<in> B"
defines "\<BB> \<equiv> {(\<dd>s w1 K, \<dd>s w2 L) | w1 w2. w1 \<in> lists A \<and> w2 \<in> lists B \<and> list_all2 R w1 w2}"
shows "rel_language R K L \<longleftrightarrow> (\<forall>(K, L) \<in> \<BB>. \<oo> K \<longleftrightarrow> \<oo> L)" |
lemma unitary_unitary_gen [simp]:
assumes "unitary M"
shows "unitary_gen M" |
lemma establish_invarI[case_names init new_root finish cross_edge back_edge discover]:
\<comment> \<open>Establish a DFS invariant (explicit preconditions).\<close>
assumes init: "on_init param \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"
assumes new_root: "\<And>s s' v0.
\<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s;
stack s = []; v0 \<in> V0; v0 \<notin> dom (discovered s);
s' = new_root v0 s\<rbrakk>
\<Longrightarrow> on_new_root param v0 s' \<le>\<^sub>n
SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>)
\<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
assumes finish: "\<And>s s' u.
\<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s;
stack s \<noteq> []; u = hd (stack s);
pending s `` {u} = {};
s' = finish u s\<rbrakk>
\<Longrightarrow> on_finish param u s' \<le>\<^sub>n
SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>)
\<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
assumes cross_edge: "\<And>s s' u v.
\<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s;
stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s);
v \<in> dom (discovered s); v\<in>dom (finished s);
s' = cross_edge u v (s\<lparr>pending := pending s - {(u,v)}\<rparr>)\<rbrakk>
\<Longrightarrow> on_cross_edge param u v s' \<le>\<^sub>n
SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>)
\<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
assumes back_edge: "\<And>s s' u v.
\<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s;
stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s);
v \<in> dom (discovered s); v\<notin>dom (finished s);
s' = back_edge u v (s\<lparr>pending := pending s - {(u,v)}\<rparr>)\<rbrakk>
\<Longrightarrow> on_back_edge param u v s' \<le>\<^sub>n
SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>)
\<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
assumes discover: "\<And>s s' u v.
\<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s;
stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s);
v \<notin> dom (discovered s);
s' = discover u v (s\<lparr>pending := pending s - {(u,v)}\<rparr>)\<rbrakk>
\<Longrightarrow> on_discover param u v s' \<le>\<^sub>n
SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>)
\<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
shows "is_invar I" |
lemma Figure_6:
assumes "\<And>i. i \<le> k \<Longrightarrow> foldl1 f1 (map (h1 k i) js) = One"
and "i > k"
shows "i \<notin> set js" |
lemma "\<Gamma>\<turnstile>\<^sub>t \<lbrace>\<acute>M=0 \<and> \<acute>N=0\<rbrace>
WHILE (\<acute>M < i)
INV \<lbrace>\<acute>M \<le> i \<and> (\<acute>M \<noteq> 0 \<longrightarrow> \<acute>N = j) \<and> \<acute>N \<le> j\<rbrace>
VAR MEASURE (i - \<acute>M)
DO
\<acute>N :== 0;;
WHILE (\<acute>N < j)
FIX m.
INV \<lbrace>\<acute>M=m \<and> \<acute>N \<le> j\<rbrace>
VAR MEASURE (j - \<acute>N)
DO
\<acute>N :== \<acute>N + 1
OD;;
\<acute>M :== \<acute>M + 1
OD
\<lbrace>\<acute>M=i \<and> (\<acute>M\<noteq>0 \<longrightarrow> \<acute>N=j)\<rbrace>" |
lemma is_renamingD: "is_renaming \<sigma> \<Longrightarrow> (\<forall>A1 A2. A1 \<cdot>a \<sigma> = A2 \<cdot>a \<sigma> \<longleftrightarrow> A1 = A2)" |
lemma crsp_step_dec_b_e_pre:
assumes "ly = layout_of ap"
and inv_start: "inv_locate_b (as, lm) (n, la, ra) ires"
and dec_0: "abc_lm_v lm n = 0"
and fetch: "abc_fetch as ap = Some (Dec n e)"
shows "\<exists>stp lb rb.
steps (Suc (start_of ly as) + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e),
start_of ly as - Suc 0) stp = (start_of ly e, lb, rb) \<and>
dec_inv_1 ly n e (as, lm) (start_of ly e, lb, rb) ires" |
lemma the_wall_betw_adj_fundchamber:
"(f,g)\<in>fundfoldpairs \<Longrightarrow>
the_wall_betw C0 (Abs_induced_automorph f g `\<rightarrow> C0) = {f\<turnstile>\<C>,g\<turnstile>\<C>}" |
lemma map_transfer [transfer_rule]:
"rel_fun (=) (rel_fun (pcr_dnelist (=)) (pcr_dnelist (=))) (\<lambda>f xs. remdups (map f xs)) Applicative_DNEList.map" |
lemma lookup_zero_fun: "lookup 0 = 0" |
lemma (in fixed_carrier_mat) smult_mem:
assumes "A \<in> fc_mats"
shows "a \<cdot>\<^sub>m A \<in> fc_mats" |
lemma (in group) generate_sincl:
"A \<subseteq> generate G A" |
lemma "id (a \<and> b) \<Longrightarrow> id a" |
lemma pred_map_get:
assumes "pred_map P m" and "m x = Some y"
shows "P y" |
lemma horner_bounds:
fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
(lapprox_rat prec 1 k)
(- float_round_up prec (x * (ub n (F i) (G i k) x)))"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
(rapprox_rat prec 1 k)
(- float_round_down prec (x * (lb n (F i) (G i k) x)))"
shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j))"
(is "?lb")
and "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)"
(is "?ub") |
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)" |
lemma exec_final_notin_iff_execn: "\<Gamma>\<turnstile>\<langle>c,s\<rangle> \<Rightarrow>\<notin>T = (\<forall>n. \<Gamma>\<turnstile>\<langle>c,s\<rangle> =n\<Rightarrow>\<notin>T)" |
lemma DT_D: "P \<sqsubseteq>\<^sub>D\<^sub>T Q \<Longrightarrow> P \<sqsubseteq>\<^sub>D Q" |
lemma (in ring_hom_ring) img_is_subfield:
assumes "subfield K R" and "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"
shows "inj_on h K" and "subfield (h ` K) S" |
lemma HSigmaE [elim!]:
assumes "c \<^bold>\<in> HSigma A B"
obtains x y where "x \<^bold>\<in> A" "y \<^bold>\<in> B(x)" "c=\<langle>x,y\<rangle>" |
lemma invertible_Q_GS: "invertible\<^sub>L (Q_GS d)" for d |
lemma fbox_antitone_var: "x \<le> y \<Longrightarrow> |y] z \<le> |x] z" |
lemma overlap_imp_same:
assumes "u \<in> \<C>" "v \<in> \<C>"
and "p \<cdot> u \<bowtie> q \<cdot> v"
and "\<^bold>|p\<^bold>| < \<^bold>|q\<^bold>| + \<^bold>|v\<^bold>|" "\<^bold>|q\<^bold>| < \<^bold>|p\<^bold>| + \<^bold>|u\<^bold>|"
shows "u = v" |
lemma ccProd_mono1: "S' \<subseteq> S'' \<Longrightarrow> S' G\<times> S \<sqsubseteq> S'' G\<times> S" |
lemma bij_betw_partition_of:
"bij_betw (\<lambda>R. A // R) {R. equiv A R} {P. partition_on A P}" |
lemma trunc_bound_euclE:
obtains err where
"\<bar>err\<bar> \<le> snd (trunc_bound_eucl p x)"
"fst (trunc_bound_eucl p x) = x + err" |
lemma lesubstep_union:
"\<lbrakk> A\<^sub>1 {\<sqsubseteq>\<^bsub>r\<^esub>} B\<^sub>1; A\<^sub>2 {\<sqsubseteq>\<^bsub>r\<^esub>} B\<^sub>2 \<rbrakk> \<Longrightarrow> A\<^sub>1 \<union> A\<^sub>2 {\<sqsubseteq>\<^bsub>r\<^esub>} B\<^sub>1 \<union> B\<^sub>2" |
lemma is_pseudonatural_transformation:
shows "pseudonatural_transformation
V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F map\<^sub>0 map\<^sub>1" |
lemma iso_same_card: "G \<cong> H \<Longrightarrow> card (carrier G) = card (carrier H)" |
theorem regexp_ewp:
defines P_def: "P(t) \<equiv> \<exists> t'. t \<sim> ro(t) +\<^sub>r t' \<and> ro(t') = 0\<^sub>r"
shows "P t" |
lemma iT_Plus_neg_iprev: "
iprev (n - k) (I \<oplus>- k) = iprev n (I \<down>\<ge> k) - k" |
lemma bij_betw_subset_of:
assumes "finite A" "finite B"
shows "bij_betw (subset_of A) ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \<subseteq> B \<and> card X = card A}" |
lemma freshBinp_empBinp[simp]:
"freshBinp xs x empBinp" |
lemma sorted_wrt_less_sum_mono_lowerbound:
fixes f :: "nat \<Rightarrow> ('b::ordered_comm_monoid_add)"
assumes mono: "\<And>x y. x\<le>y \<Longrightarrow> f x \<le> f y"
shows "sorted_wrt (<) ns \<Longrightarrow>
(\<Sum>i\<in>{0..<length ns}. f i) \<le> (\<Sum>i\<leftarrow>ns. f i)" |
lemma term_beta_trans[trans]: "s \<leftrightarrow> t \<Longrightarrow> t \<rightarrow>\<^sub>\<beta> u \<Longrightarrow> s \<leftrightarrow> u" |
lemma lprefix_refl [intro, simp]: "xs \<sqsubseteq> xs" |
lemma prime_square_sum_nat_decomp_code [code]:
"prime_square_sum_nat_decomp p =
(if prime p \<and> (p = 2 \<or> [p = 1] (mod 4))
then the_elem (Set.filter (\<lambda>(x,y). x ^ 2 + y ^ 2 = p) (SIGMA x:{0..p}. {x..p}))
else (0, 0))" |
lemma lfinite_Sts: "lfinite Sts" |
lemma RDirProd_list_iso1:
"(\<lambda>(a, as). a # as) \<in> ring_iso (RDirProd R (RDirProd_list Rs)) (RDirProd_list (R # Rs))" |
lemma nn_transfer_operator_mono:
assumes "AE x in M. f x \<le> g x"
and [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "AE x in M. nn_transfer_operator f x \<le> nn_transfer_operator g x" |
lemma list_of_oalist_empty [simp, code abstract]: "list_of_oalist (empty ko) = ([], ko)" |
lemma aux6:
assumes
"a \<in># wait"
"start_subsumed passed wait"
"set_mset wait' = set_mset (wait - {#a#}) \<union> Collect (E a)"
shows "start_subsumed (insert a passed) wait'" |
lemma (in Order) ord_isom_le_forall:"\<lbrakk>Order E; ord_isom D E f\<rbrakk> \<Longrightarrow>
\<forall>a \<in> carrier D. \<forall> b \<in> carrier D. (a \<preceq> b) = ((f a) \<preceq>\<^bsub>E\<^esub> (f b))" |
lemma bind_mono_strong:
assumes "mono_state m"
assumes "\<And>x s s'. run_state m s = (x, s') \<Longrightarrow> mono_state (f x)"
shows "mono_state (bind m f)" |
lemma oconf_hext: "P,h \<turnstile> obj \<surd> \<Longrightarrow> h \<unlhd> h' \<Longrightarrow> P,h' \<turnstile> obj \<surd>" |
lemma nempty_sl_in_state_set:
fixes sl
assumes "sl \<noteq> []"
shows "sl \<in> state_set sl" |
lemma height_ge_one:
shows "1 \<le> (height e)" |
lemma bnt_flip_is_iso_ntcf'[cat_cs_intros]:
assumes "category \<alpha> \<AA>"
and "category \<alpha> \<BB>"
and "\<NN> : \<SS> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>i\<^sub>s\<^sub>o \<SS>' : \<AA> \<times>\<^sub>C \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"
and "\<FF> = bifunctor_flip \<AA> \<BB> \<SS>"
and "\<GG> = bifunctor_flip \<AA> \<BB> \<SS>'"
and "\<DD> = \<BB> \<times>\<^sub>C \<AA>"
shows "bnt_flip \<AA> \<BB> \<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>i\<^sub>s\<^sub>o \<GG> : \<DD> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" |
lemma Or\<^sub>n_empty[simp]:
"Or\<^sub>n {} = False_ltln" |
lemma sucs_encoding_inv: "sucs_decoding \<circ> sucs_encoding = id" |
lemma slice1_param_in_slice1:
"\<lbrakk>nx \<in> sum_SDG_slice1 n; n s-p:V\<rightarrow>\<^bsub>in\<^esub> n'\<rbrakk> \<Longrightarrow> nx \<in> sum_SDG_slice1 n'" |
lemma maxim_mono: "\<lbrakk>X \<subseteq> Y; finite Y; X \<noteq> {}; Y \<subseteq> Field r\<rbrakk> \<Longrightarrow> (maxim X, maxim Y) \<in> r" |
lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
\<comment> \<open>Example 3.\<close> |
lemma validFrom_comp: "validFrom s tr \<Longrightarrow> comp s tr" |
lemma somewhere_leq:
"v \<le> v' \<longleftrightarrow> (\<exists>v1 v2 v3 vl vr vu vd.
(v'=vl\<parallel>v1) \<and> (v1=v2\<parallel>vr) \<and> (v2=vd--v3) \<and> (v3=v--vu))" |
lemma take_apply_is_hom:
shows "take_apply (n + k) k \<in> ring_hom (SA k) (SA (n + k))" |
lemma cl_idem [simp]: "cl L (cl L r) = cl L r" |
lemma ereal_add_diff_cancel:
fixes a b :: ereal
shows "\<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> (a + b) - b = a" |
lemma join_ir_nth [simp]:
"i < length is \<Longrightarrow> join_ir is r (I i) = Some (is ! i)" |
lemma module_hom_neg: "module_hom s1 s2 f \<Longrightarrow> module_hom s1 s2 (\<lambda>x. - f x)" |
lemma chamber_card: "chamber C \<Longrightarrow> chamber D \<Longrightarrow> card C = card D" |
lemma mset_map_remdups_gen:
"mset (map f (remdups_gen f xs)) = mset (remdups_gen (\<lambda>x. x) (map f xs))" |
lemma compose_surj: "\<lbrakk>f:A \<rightarrow> B; surj_to f A B; g : B \<rightarrow> C; surj_to g B C \<rbrakk>
\<Longrightarrow> surj_to (compose A g f) A C " |
lemma filter_compliant_stateful_ACS_subseteq_input: "set (filter_compliant_stateful_ACS G M Es) \<subseteq> set Es" |
lemma upto_rec2: "i \<le> j \<Longrightarrow> [i..j] = [i..j - 1]@[j]" |
lemma pos_cons [simp]:
"xs \<noteq> [] \<longrightarrow> pos (x#xs) =
(if (x>0) then pos xs else False)"
(is "?P x xs" is "?A xs \<longrightarrow> ?S x xs") |
lemma c_IKKBZ_cf'_eq': "valid_tree t \<Longrightarrow> c_IKKBZ h cf' sf t = c_IKKBZ h cf sf t" |
lemma inj_map_total[simp]: "inj_map (Some o \<pi>) = inj \<pi>" |
lemma ordered_subdag: "\<lbrakk>ordered t var; not <= t\<rbrakk> \<Longrightarrow> ordered not var" for not |
lemma chain_mono: assumes "R' \<subseteq> R" "chain R' seq" shows "chain R seq" |
lemma os_distincts:
assumes "A B OS X Y"
shows "A \<noteq> B \<and> A \<noteq> X \<and> A \<noteq> Y \<and> B \<noteq> X \<and> B \<noteq> Y" |
lemma infsetsum_nat:
assumes "f abs_summable_on A"
shows "infsetsum f A = (\<Sum>n. if n \<in> A then f n else 0)" |
lemma "nats = (\<lambda>n. n \<in> {0, 1, 2, 3, 4})" |
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