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

Statement: stringlengths 7 24.3k
lemma init_InvInfoAtBetaUpright: "init_config c \<Longrightarrow> InvInfoAtBetaUpright c"
lemma Hintikka_Extend: assumes \<open>consistent H\<close> and \<open>maximal H\<close> and \<open>saturated H\<close> shows \<open>Hintikka H\<close>
lemma "is_cons_cluster C \<Longrightarrow> quorum C" \<comment> \<open>Every consensus cluster is a quorum.\<close>
lemma in_set_real_plus_interval[intro, simp]: "x + y \<in>\<^sub>r X + Y" if "x \<in>\<^sub>r X" "y \<in>\<^sub>r Y"
lemma b_def: "b = 2^(Suc c)"
lemma finite_enumerate_Ex: fixes S :: "'a::wellorder set" assumes S: "finite S" and s: "s \<in> S" shows "\<exists>n<card S. enumerate S n = s"
lemma ubs_ex_n_max:"\<lbrakk>A \<noteq> {}; A \<subseteq> {i. i \<le> (n::nat)}\<rbrakk> \<Longrightarrow> \<exists>!m. m\<in>A \<and> (\<forall>x\<in>A. x \<le> m)"
lemma eq_inc_same: "eq a v1 v2 \<Longrightarrow> eq (inc\<cdot>a) v1 v2"
lemma set_partition_on_insert_with_fixed_card_eq: assumes "finite A" assumes "a \<notin> A" shows "{P. partition_on (insert a A) P \<and> card P = Suc k} = (do { P <- {P. partition_on A P \<and> card P = Suc k}; p <- P; {insert (insert a p) (P - {p})} }) \<union> (do { P <- {P. partition_on A P \<and> card P = k}; {insert {a} P} })" (is "?S = ?T")
lemma(in UP_cring) poly_comp_finsum: assumes "\<And>i::nat. i \<le> n \<Longrightarrow> g i \<in> carrier P" assumes "q \<in> carrier P" assumes "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. g i)" shows "p of q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. (g i) of q)"
lemma preserves_cong: assumes "A.cong t u" shows "B.cong (F t) (F u)"
lemma sim_move10_CallParams: "sim_moves10 P t ta1 es1 es1' es h xs ta es' h' xs' \<Longrightarrow> sim_move10 P t ta1 (Val v\<bullet>M(es1)) (Val v\<bullet>M(es1')) (Val v\<bullet>M(es)) h xs ta (Val v\<bullet>M(es')) h' xs'"
lemma ta_subset_states: "ta_subset \<A> \<B> \<Longrightarrow> \<Q> \<A> \|\<subseteq>\| \<Q> \<B>"
lemma list_all2_approx: "list_all2 (approx_val G hp) s (map OK S) = list_all2 (conf G hp) s S"
lemma measurable_count_space_eq2_countable: fixes f :: "'a => 'c::countable" shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
lemma instance_of\<^sub>l_trans : assumes l\<^sub>1\<^sub>2: "instance_of\<^sub>l l\<^sub>1 l\<^sub>2" assumes l\<^sub>2\<^sub>3: "instance_of\<^sub>l l\<^sub>2 l\<^sub>3" shows "instance_of\<^sub>l l\<^sub>1 l\<^sub>3"
lemma homotopic_with_subset_left: "\<lbrakk>homotopic_with_canon P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with_canon P Z Y f g"
lemma comp_eqI: assumes "t \<lesssim> v" and "u = v \\ t" shows "t \<cdot> u = v"
lemma int_less_real_le: "n < m \<longleftrightarrow> real_of_int n + 1 \<le> real_of_int m"
lemma pure_false[simp]: "\<up>False = false"
lemma list_dtree_sub: "is_subtree x t \<Longrightarrow> list_dtree x"
lemma fixes a::real assumes "a \<le> b" shows ge_midpoint_1: "a \<le> midpoint a b" and le_midpoint_1: "midpoint a b \<le> b"
lemma gcd_fib_mod: "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" if "0 < m"
lemma convex_hull_3: "convex hull {a,b,c} = { u \<^sub>R a + v \<^sub>R b + w *\<^sub>R c \| u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e" for x :: "'a::real_normed_vector"
lemma DEG_eq [code]: "nat_term_order_eq (DEG to) (LEX::'a nat_term_order) dg ps = nat_term_order_eq LEX (DEG to) dg ps" "nat_term_order_eq (DEG to) (DRLEX::'a nat_term_order) dg ps = nat_term_order_eq DRLEX (DEG to) dg ps" "nat_term_order_eq (DEG to1) (DEG (to2::'a nat_term_order)) dg ps = nat_term_order_eq to1 to2 True ps" (is ?thesis3) "nat_term_order_eq (DEG to1) (POT (to2::'a nat_term_order)) dg ps = (if dg then nat_term_order_eq to1 (POT to2) dg ps else ((ps \<or> is_scalar TYPE('a::nat_term_compare)) \<and> nat_term_order_eq (DEG to1) to2 dg ps))" (is ?thesis4)
lemma finite_minusset: "finite A \<Longrightarrow> finite (minusset A)"
lemma Disj_Neg_1: "H \<turnstile> A OR B \<Longrightarrow> H \<turnstile> Neg B \<Longrightarrow> H \<turnstile> A"
lemma ts_process_traces_implies_d: "ts_process (traces P) = P \<Longrightarrow> deterministic P"
lemma \<alpha>_inner: "inner_prod \<alpha> \<alpha> = 1"
lemma uinter_empty_1 [simp]: "x \<inter>\<^sub>u {}\<^sub>u = {}\<^sub>u"
lemma conflictClauseIsDecreasedByExplain: fixes stateA::State and stateB::State assumes "appliedExplain stateA stateB" shows "getM stateA = getM stateB" and "getConflictFlag stateA = getConflictFlag stateB" and "(getC stateB, getC stateA) \<in> multLess (getM stateA)"
lemma upper_bound2: fixes b::nat and c::int assumes "b > 0" and "c < 2^b" and "c \<ge> 0" shows "c - (2^(b-1)) < 2^(b-1)"
lemma nth_root_nat_aux2: assumes "k > 0" shows "finite {m::nat. m ^ k \<le> n}" "{m::nat. m ^ k \<le> n} \<noteq> {}"
lemma stream_all2_equiv'_D: "stream_all2 (\<sim>) xs ys" if "stream_all2 A_B.equiv' xs ys"
lemma "lCoP1(\<^bold>\<rightarrow>) \<eta> \<Longrightarrow> CoP1 \<eta>"
lemma D_incr: "D A c A' \<Longrightarrow> A \<subseteq> A'"
lemma domo_join: "domo (S \<squnion> T) \<subseteq> domo S \<union> domo T"
lemma ctxt_ctxt_compose [simp]: "(C \<circ>\<^sub>c D)\<langle>t\<rangle> = C\<langle>D\<langle>t\<rangle>\<rangle>"
lemma WildPos: "A, \<Delta> \<turnstile> B \<Longrightarrow> (Wildcard A, \<Delta> \<turnstile> Wildcard B)"
theorem (in Congruence_Rule) Ang_split : assumes "Def (Ang (An h1 o1 k1))" "Def (Ang (An h2 o2 k2))" "Cong (Geos (Ang (An h1 o1 k1)) add Emp) (Geos (Ang (An h2 o2 k2)) add Emp)" "Ang_inside (An h1 o1 k1) l1" shows "\<exists>p. Ang_inside (An h2 o2 k2) p \<and> Cong (Geos (Ang (An h1 o1 l1)) add Emp) (Geos (Ang (An h2 o2 p)) add Emp) \<and> Cong (Geos (Ang (An k1 o1 l1)) add Emp) (Geos (Ang (An k2 o2 p)) add Emp)"
lemma (in CLF) lubH_least_fixf: "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
lemma (in CRR_market) risky_asset_martingale_only_if: assumes "N = bernoulli_stream q" and "0 < q" and "q < 1" and "martingale N G (discounted_value r (prices Mkt stk))" shows "q = (1 + r - d) / (u - d)"
lemma LI_acyclic: "acyclic LI_rel"
lemma (in PolynRg) Slide_pol_coeff:"\<lbrakk>pol_coeff S c; n < (fst c)\<rbrakk> \<Longrightarrow> pol_coeff S (((fst c) - Suc n), (\<lambda>x. (snd c) (Suc (n + x))))"
lemma "ipset_from_cidr (ipv4addr_of_dotdecimal (192,168,0,42)) 16 = {ipv4addr_of_dotdecimal (192,168,0,0) .. ipv4addr_of_dotdecimal (192,168,255,255)}"
lemma fls_regpart_deriv: "fls_regpart (fls_deriv f) = fps_deriv (fls_regpart f)"
lemma swap_preserves_wls3: assumes "good X" and "good Y" and "(X #[x1 \<and> x2]_xs) = (Y #[y1 \<and> y2]_ys)" shows "wls s X = wls s Y"
lemma discovered_not_stack_imp_finished: "x \<in> dom (discovered s) \<Longrightarrow> x \<notin> set (stack s) \<Longrightarrow> x \<in> dom (finished s)"
lemma reduce_not_value: assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_i \<lparr>s';vs';es'\<rparr>" shows "es \<noteq> $$* ves"
lemma omega_sum_less_eqlen_iff_cases [simp]: assumes "length ms = length ns" shows "omega_sum (m#ms) < omega_sum (n#ns) \<longleftrightarrow> m<n \<or> m=n \<and> omega_sum ms < omega_sum ns" (is "?lhs = ?rhs")
lemma totient_power: "m > 0 \<Longrightarrow> totient (n ^ m) = n ^ (m - 1) * totient n"
lemma dyn_accessible_static_field_Protected: assumes dyn_acc: "G \<turnstile> f in C dyn_accessible_from accC" and prot: "accmodi f = Protected" and field: "is_field f" and static_field: "is_static f" and outside: "pid (declclass f) \<noteq> pid accC" shows "G\<turnstile> accC \<preceq>\<^sub>C declclass f \<and> G\<turnstile>C \<preceq>\<^sub>C declclass f"
lemma le_minus_plus_same_hmset: "m \<le> m - n + n" for m n :: hmultiset
lemma H_on_ket_one_is_state: shows "state 1 (H * \|one\<rangle>)"
lemma iprod_left_diff_distrib: "\<langle>xs - ys, zs\<rangle> = \<langle>xs,zs\<rangle> - \<langle>ys,zs\<rangle>"
lemma chinese_remainder_coprime_unique: fixes a :: nat assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b \<noteq> 0" and ma: "coprime m a" and nb: "coprime n b" shows "\<exists>!x. coprime x (a * b) \<and> x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs"
lemma as_rel_def: "\<langle>R\<rangle>as_rel \<equiv> br as_raw_\<alpha> as_raw_invar O \<langle>R\<rangle>list_rel"
lemma f_last_message_hold_take: "xs \<down> n \<longmapsto>\<^sub>f k = xs \<longmapsto>\<^sub>f k \<down> n"
lemma DynProcModifyReturnNoAbr: assumes to_prove: "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (dynCall init p return' c) Q,A" assumes ret_nrm_modif: "\<forall>s t. t \<in> (Modif (init s)) \<longrightarrow> return' s t = return s t" assumes modif_clause: "\<forall>s \<in> P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),{}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (dynCall init p return c) Q,A"
lemma length_generate_PS: "length (generate_PS emBits 160) = (roundup emBits 8)*8 - sLen - 160 - 16"
lemma cartesian_product_binary_union_left: assumes "C \<subseteq> carrier (R\<^bsup>n\<^esup>)" assumes "D \<subseteq> carrier (R\<^bsup>n\<^esup>)" shows "cartesian_product (C \<union> D) A = (cartesian_product C A) \<union> (cartesian_product D A)"
lemma smult_mat_mult_mat_vec_assoc: fixes A :: "'a::comm_ring_1 mat" assumes A: "A \<in> carrier_mat n m" and w: "w \<in> carrier_vec m" shows "a \<cdot>\<^sub>m A \<^sub>v w = a \<cdot>\<^sub>v (A \<^sub>v w)"
lemma is_separator_simps : assumes "is_separator M q1 q2 A t1 t2" shows "single_input A" and "acyclic A" and "observable A" and "deadlock_state A t1" and "deadlock_state A t2" and "t1 \<in> reachable_states A" and "t2 \<in> reachable_states A" and "\<And> t . t \<in> reachable_states A \<Longrightarrow> t \<noteq> t1 \<Longrightarrow> t \<noteq> t2 \<Longrightarrow> \<not> deadlock_state A t" and "\<And> io x yq yt . io@[(x,yq)] \<in> LS M q1 \<Longrightarrow> io@[(x,yt)] \<in> L A \<Longrightarrow> (io@[(x,yq)] \<in> L A)" and "\<And> io x yq yt . io@[(x,yq)] \<in> LS M q2 \<Longrightarrow> io@[(x,yt)] \<in> L A \<Longrightarrow> (io@[(x,yq)] \<in> L A)" and "\<And> p . path A (initial A) p \<Longrightarrow> target (initial A) p = t1 \<Longrightarrow> p_io p \<in> LS M q1 - LS M q2" and "\<And> p . path A (initial A) p \<Longrightarrow> target (initial A) p = t2 \<Longrightarrow> p_io p \<in> LS M q2 - LS M q1" and "\<And> p . path A (initial A) p \<Longrightarrow> target (initial A) p \<noteq> t1 \<Longrightarrow> target (initial A) p \<noteq> t2 \<Longrightarrow> p_io p \<in> LS M q1 \<inter> LS M q2" and "q1 \<noteq> q2" and "t1 \<noteq> t2" and "(inputs A) \<subseteq> (inputs M)"
lemma is_iso_arr_is_iso_ntcf: assumes "\<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" and "\<MM> : \<GG> \<mapsto>\<^sub>C\<^sub>F \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" and "\<NN> \<bullet>\<^sub>N\<^sub>T\<^sub>C\<^sub>F \<MM> = ntcf_id \<GG>" and "\<MM> \<bullet>\<^sub>N\<^sub>T\<^sub>C\<^sub>F \<NN> = ntcf_id \<FF>" shows "\<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>i\<^sub>s\<^sub>o \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
lemma silent_moves_no_slice_edges: "\<lbrakk>S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s'); tl ms = targetnodes rs; length rs = length cs; \<forall>i<length cs. call_of_return_node (tl ms!i) (sourcenode (cs!i))\<rbrakk> \<Longrightarrow> slice_edges S cs as = [] \<and> (\<exists>rs'. tl ms' = targetnodes rs' \<and> length rs' = length (upd_cs cs as) \<and> (\<forall>i<length (upd_cs cs as). call_of_return_node (tl ms'!i) (sourcenode ((upd_cs cs as)!i))))"
lemma mask_range_subsetD: "\<lbrakk> p' \<in> mask_range p n; x' \<in> mask_range p' n'; n' \<le> n; is_aligned p n; is_aligned p' n' \<rbrakk> \<Longrightarrow> x' \<in> mask_range p n"
lemma semilat_opt [intro, simp]: "err_semilat L \<Longrightarrow> err_semilat (Opt.esl L)"
lemma (in Corps) Cauchy_down:"\<lbrakk>Corps K'; valuation K v; valuation K' v'; subfield K K'; \<forall>x\<in>carrier K. v x = v' x; \<forall>j. f j \<in> carrier K; Cauchy\<^bsub>K' v' \<^esub>f\<rbrakk> \<Longrightarrow> Cauchy\<^bsub>K v \<^esub>f"
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
lemma absCase_Abs_subst[simp]: assumes isInBar: "isInBar (xs,s)" and X: "wls s X" and f_compat: "compatAbsSubst xs s f" shows "absCase xs s f (Abs xs x X) = f x X"
lemma cr_vreal_transfer_domain_rule[transfer_domain_rule]: "Domainp cr_vreal = (\<lambda>x. x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>)"
lemma subst_bb_commute [simp]: "atom j \<sharp> A \<Longrightarrow> (subst_bb (subst_bb A i t ) j u ) = subst_bb A i (subst_bb t j u) "
lemma PO_m2_inv2_keys_for_init [iff]: "init m2 \<subseteq> m2_inv2_keys_for"
lemma next_Null[simp]: assumes "xs \<noteq> \<bottom>" shows "next\<cdot>xs\<cdot>Null = Null"
lemma tendsto_le_ccpo: fixes f g :: "'a \<Rightarrow> 'b::ccpo_topology" assumes F: "\<not> trivial_limit F" assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F" assumes ev: "eventually (\<lambda>x. g x \<le> f x) F" shows "y \<le> x"
lemma flush_reads_program: "\<And>\<O> \<S> \<R> . \<forall>r \<in> set sb. is_Read\<^sub>s\<^sub>b r \<or> is_Prog\<^sub>s\<^sub>b r \<or> is_Ghost\<^sub>s\<^sub>b r \<Longrightarrow> \<exists>\<O>' \<R>' \<S>'. (m,sb,\<O>,\<R>,\<S>) \<rightarrow>\<^sub>f\<^sup>* (m,[],\<O>',\<R>',\<S>')"
theorem IK_real_card: shows "card {f::real set \<Rightarrow> real set. IK f} = 7" (is "?lhs = ?rhs")
lemma remdups_fwd_acc_append[simp]: "remdups_fwd_acc Acc (xs@ys) = (remdups_fwd_acc Acc xs) @ (remdups_fwd_acc (Acc \<union> set xs) ys)"
lemma update_acyclic_3: assumes "acyclic (p - 1)" and "point y" and "point w" and "y \<le> p\<^sup>T\<^sup>\<star> * w" shows "acyclic ((p[w\<longmapsto>y]) - 1)"
lemma odlist_induct [case_names empty insert, cases type: odlist]: assumes empty: "\<And>dxs. dxs = empty \<Longrightarrow> P dxs" assumes insrt: "\<And>dxs x xs. \<lbrakk> dxs = fromList (x # xs); distinct (x # xs); sorted (x # xs); P (fromList xs) \<rbrakk> \<Longrightarrow> P dxs" shows "P dxs"
lemma decE_decE [simp]: "\<down>\<^sub>e n k (\<down>\<^sub>e n' (k + n) \<Gamma>) = \<down>\<^sub>e (n + n') k \<Gamma>"
lemma (in idom_char_0) linear_pow_mul_degree: "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
lemma COND_path_prefix_extr: "prefix (AHIS (hfs_valid_prefix ainfo uinfo l nxt)) (extr_from_hd l)"
lemma "z \<cdot> x \<le> y \<cdot> z \<Longrightarrow> z \<cdot> x\<^sup>\<omega> \<le> y\<^sup>\<omega> \<cdot> z"
lemma minus_minus_float[simp]: "- (-f) = f" for f::"('e, 'f)float"
lemma IdNatTransNatTrans: "Functor F \<Longrightarrow> NatTrans (IdNatTrans F)"
lemma chinese_remainder_poly: fixes A :: "'a set" and m :: "'a \<Rightarrow> 'b::{field_gcd} poly" and u :: "'a \<Rightarrow> 'b poly" assumes fin: "finite A" and cop: "\<forall>i\<in>A. (\<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))" shows "\<exists>x. (\<forall>i\<in>A. [x = u i] (mod m i))"
lemma list_of_lappend: assumes "lfinite xs" "lfinite ys" shows "list_of (lappend xs ys) = list_of xs @ list_of ys"
lemma continuous_on_compact_bound: assumes "compact A" "continuous_on A f" obtains B where "B \<ge> 0" "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> B"
lemma Im_interval_times: "Im_interval (A * B) = Re_interval A * Im_interval B + Im_interval A * Re_interval B"
lemma insert_before_removes_child: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "ptr \<noteq> ptr'" assumes "h \<turnstile> insert_before ptr node child \<rightarrow>\<^sub>h h'" assumes "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r node # children" shows "h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
lemma unification': assumes "finite ts" assumes "unifier\<^sub>t\<^sub>s \<sigma> ts" shows "\<exists>\<theta>. mgu\<^sub>t\<^sub>s \<theta> ts"
lemma lexext_singleton: "lexext gt [y] [x] \<longleftrightarrow> gt y x"
lemma proposition3_4_inv_step: assumes t: "trans r" and i: "irrefl r" and k:"(r\|\<sigma>\| -s r \<down>l [\<beta>], {#\<alpha>#}) \<in> mul_eq r" (is "(?M,_) \<in> _") shows "\<exists> \<sigma>1 \<sigma>2 \<sigma>3. ((\<sigma> = \<sigma>1@\<sigma>2@\<sigma>3) \<and> LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3)"
lemma powsum_ext: "x \<le> x\<^bsub>0\<^esub>\<^bsup>Suc n\<^esup>"
lemma finite_subset[trans]: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
lemma "\<sim>ECQ \<^bold>\<not>\<^sup>C \<and> Fr_1 \<F> \<and> Fr_2 \<F> \<and> Fr_4 \<F> \<and> lCoPw(\<^bold>\<rightarrow>) \<^bold>\<not>\<^sup>C"
lemma fimage_eq_to_f: assumes "f1 \|`\| S1 = f2 \|`\| S2" obtains f where "\<And> x. x \|\<in>\| S2 \<Longrightarrow> f x \|\<in>\| S1 \<and> f1 (f x) = f2 x"

lemma init_InvInfoAtBetaUpright: "init_config c \<Longrightarrow> InvInfoAtBetaUpright c"

lemma Hintikka_Extend: assumes \<open>consistent H\<close> and \<open>maximal H\<close> and \<open>saturated H\<close> shows \<open>Hintikka H\<close>

lemma "is_cons_cluster C \<Longrightarrow> quorum C" \<comment> \<open>Every consensus cluster is a quorum.\<close>

lemma in_set_real_plus_interval[intro, simp]: "x + y \<in>\<^sub>r X + Y" if "x \<in>\<^sub>r X" "y \<in>\<^sub>r Y"

lemma b_def: "b = 2^(Suc c)"

lemma finite_enumerate_Ex: fixes S :: "'a::wellorder set" assumes S: "finite S" and s: "s \<in> S" shows "\<exists>n<card S. enumerate S n = s"

lemma ubs_ex_n_max:"\<lbrakk>A \<noteq> {}; A \<subseteq> {i. i \<le> (n::nat)}\<rbrakk> \<Longrightarrow> \<exists>!m. m\<in>A \<and> (\<forall>x\<in>A. x \<le> m)"

lemma eq_inc_same: "eq a v1 v2 \<Longrightarrow> eq (inc\<cdot>a) v1 v2"

lemma set_partition_on_insert_with_fixed_card_eq: assumes "finite A" assumes "a \<notin> A" shows "{P. partition_on (insert a A) P \<and> card P = Suc k} = (do { P <- {P. partition_on A P \<and> card P = Suc k}; p <- P; {insert (insert a p) (P - {p})} }) \<union> (do { P <- {P. partition_on A P \<and> card P = k}; {insert {a} P} })" (is "?S = ?T")

lemma(in UP_cring) poly_comp_finsum: assumes "\<And>i::nat. i \<le> n \<Longrightarrow> g i \<in> carrier P" assumes "q \<in> carrier P" assumes "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. g i)" shows "p of q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. (g i) of q)"

lemma preserves_cong: assumes "A.cong t u" shows "B.cong (F t) (F u)"

lemma sim_move10_CallParams: "sim_moves10 P t ta1 es1 es1' es h xs ta es' h' xs' \<Longrightarrow> sim_move10 P t ta1 (Val v\<bullet>M(es1)) (Val v\<bullet>M(es1')) (Val v\<bullet>M(es)) h xs ta (Val v\<bullet>M(es')) h' xs'"

lemma ta_subset_states: "ta_subset \<A> \<B> \<Longrightarrow> \<Q> \<A> |\<subseteq>| \<Q> \<B>"

lemma list_all2_approx: "list_all2 (approx_val G hp) s (map OK S) = list_all2 (conf G hp) s S"

lemma measurable_count_space_eq2_countable: fixes f :: "'a => 'c::countable" shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"

lemma instance_of\<^sub>l_trans : assumes l\<^sub>1\<^sub>2: "instance_of\<^sub>l l\<^sub>1 l\<^sub>2" assumes l\<^sub>2\<^sub>3: "instance_of\<^sub>l l\<^sub>2 l\<^sub>3" shows "instance_of\<^sub>l l\<^sub>1 l\<^sub>3"

lemma homotopic_with_subset_left: "\<lbrakk>homotopic_with_canon P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with_canon P Z Y f g"

lemma comp_eqI: assumes "t \<lesssim> v" and "u = v \\ t" shows "t \<cdot> u = v"

lemma int_less_real_le: "n < m \<longleftrightarrow> real_of_int n + 1 \<le> real_of_int m"

lemma pure_false[simp]: "\<up>False = false"

lemma list_dtree_sub: "is_subtree x t \<Longrightarrow> list_dtree x"

lemma fixes a::real assumes "a \<le> b" shows ge_midpoint_1: "a \<le> midpoint a b" and le_midpoint_1: "midpoint a b \<le> b"

lemma gcd_fib_mod: "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" if "0 < m"

lemma convex_hull_3: "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"

lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e" for x :: "'a::real_normed_vector"

lemma DEG_eq [code]: "nat_term_order_eq (DEG to) (LEX::'a nat_term_order) dg ps = nat_term_order_eq LEX (DEG to) dg ps" "nat_term_order_eq (DEG to) (DRLEX::'a nat_term_order) dg ps = nat_term_order_eq DRLEX (DEG to) dg ps" "nat_term_order_eq (DEG to1) (DEG (to2::'a nat_term_order)) dg ps = nat_term_order_eq to1 to2 True ps" (is ?thesis3) "nat_term_order_eq (DEG to1) (POT (to2::'a nat_term_order)) dg ps = (if dg then nat_term_order_eq to1 (POT to2) dg ps else ((ps \<or> is_scalar TYPE('a::nat_term_compare)) \<and> nat_term_order_eq (DEG to1) to2 dg ps))" (is ?thesis4)

lemma finite_minusset: "finite A \<Longrightarrow> finite (minusset A)"

lemma Disj_Neg_1: "H \<turnstile> A OR B \<Longrightarrow> H \<turnstile> Neg B \<Longrightarrow> H \<turnstile> A"

lemma ts_process_traces_implies_d: "ts_process (traces P) = P \<Longrightarrow> deterministic P"

lemma \<alpha>_inner: "inner_prod \<alpha> \<alpha> = 1"

lemma uinter_empty_1 [simp]: "x \<inter>\<^sub>u {}\<^sub>u = {}\<^sub>u"

lemma conflictClauseIsDecreasedByExplain: fixes stateA::State and stateB::State assumes "appliedExplain stateA stateB" shows "getM stateA = getM stateB" and "getConflictFlag stateA = getConflictFlag stateB" and "(getC stateB, getC stateA) \<in> multLess (getM stateA)"

lemma upper_bound2: fixes b::nat and c::int assumes "b > 0" and "c < 2^b" and "c \<ge> 0" shows "c - (2^(b-1)) < 2^(b-1)"

lemma nth_root_nat_aux2: assumes "k > 0" shows "finite {m::nat. m ^ k \<le> n}" "{m::nat. m ^ k \<le> n} \<noteq> {}"

lemma stream_all2_equiv'_D: "stream_all2 (\<sim>) xs ys" if "stream_all2 A_B.equiv' xs ys"

lemma "lCoP1(\<^bold>\<rightarrow>) \<eta> \<Longrightarrow> CoP1 \<eta>"

lemma D_incr: "D A c A' \<Longrightarrow> A \<subseteq> A'"

lemma domo_join: "domo (S \<squnion> T) \<subseteq> domo S \<union> domo T"

lemma ctxt_ctxt_compose [simp]: "(C \<circ>\<^sub>c D)\<langle>t\<rangle> = C\<langle>D\<langle>t\<rangle>\<rangle>"

lemma WildPos: "A, \<Delta> \<turnstile> B \<Longrightarrow> (Wildcard A, \<Delta> \<turnstile> Wildcard B)"

theorem (in Congruence_Rule) Ang_split : assumes "Def (Ang (An h1 o1 k1))" "Def (Ang (An h2 o2 k2))" "Cong (Geos (Ang (An h1 o1 k1)) add Emp) (Geos (Ang (An h2 o2 k2)) add Emp)" "Ang_inside (An h1 o1 k1) l1" shows "\<exists>p. Ang_inside (An h2 o2 k2) p \<and> Cong (Geos (Ang (An h1 o1 l1)) add Emp) (Geos (Ang (An h2 o2 p)) add Emp) \<and> Cong (Geos (Ang (An k1 o1 l1)) add Emp) (Geos (Ang (An k2 o2 p)) add Emp)"

lemma (in CLF) lubH_least_fixf: "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"

lemma (in CRR_market) risky_asset_martingale_only_if: assumes "N = bernoulli_stream q" and "0 < q" and "q < 1" and "martingale N G (discounted_value r (prices Mkt stk))" shows "q = (1 + r - d) / (u - d)"

lemma LI_acyclic: "acyclic LI_rel"

lemma (in PolynRg) Slide_pol_coeff:"\<lbrakk>pol_coeff S c; n < (fst c)\<rbrakk> \<Longrightarrow> pol_coeff S (((fst c) - Suc n), (\<lambda>x. (snd c) (Suc (n + x))))"

lemma "ipset_from_cidr (ipv4addr_of_dotdecimal (192,168,0,42)) 16 = {ipv4addr_of_dotdecimal (192,168,0,0) .. ipv4addr_of_dotdecimal (192,168,255,255)}"

lemma fls_regpart_deriv: "fls_regpart (fls_deriv f) = fps_deriv (fls_regpart f)"

lemma swap_preserves_wls3: assumes "good X" and "good Y" and "(X #[x1 \<and> x2]_xs) = (Y #[y1 \<and> y2]_ys)" shows "wls s X = wls s Y"

lemma discovered_not_stack_imp_finished: "x \<in> dom (discovered s) \<Longrightarrow> x \<notin> set (stack s) \<Longrightarrow> x \<in> dom (finished s)"

lemma reduce_not_value: assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_i \<lparr>s';vs';es'\<rparr>" shows "es \<noteq> $$* ves"

lemma omega_sum_less_eqlen_iff_cases [simp]: assumes "length ms = length ns" shows "omega_sum (m#ms) < omega_sum (n#ns) \<longleftrightarrow> m<n \<or> m=n \<and> omega_sum ms < omega_sum ns" (is "?lhs = ?rhs")

lemma totient_power: "m > 0 \<Longrightarrow> totient (n ^ m) = n ^ (m - 1) * totient n"

lemma dyn_accessible_static_field_Protected: assumes dyn_acc: "G \<turnstile> f in C dyn_accessible_from accC" and prot: "accmodi f = Protected" and field: "is_field f" and static_field: "is_static f" and outside: "pid (declclass f) \<noteq> pid accC" shows "G\<turnstile> accC \<preceq>\<^sub>C declclass f \<and> G\<turnstile>C \<preceq>\<^sub>C declclass f"

lemma le_minus_plus_same_hmset: "m \<le> m - n + n" for m n :: hmultiset

lemma H_on_ket_one_is_state: shows "state 1 (H * |one\<rangle>)"

lemma iprod_left_diff_distrib: "\<langle>xs - ys, zs\<rangle> = \<langle>xs,zs\<rangle> - \<langle>ys,zs\<rangle>"

lemma chinese_remainder_coprime_unique: fixes a :: nat assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b \<noteq> 0" and ma: "coprime m a" and nb: "coprime n b" shows "\<exists>!x. coprime x (a * b) \<and> x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"

lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs"

lemma as_rel_def: "\<langle>R\<rangle>as_rel \<equiv> br as_raw_\<alpha> as_raw_invar O \<langle>R\<rangle>list_rel"

lemma f_last_message_hold_take: "xs \<down> n \<longmapsto>\<^sub>f k = xs \<longmapsto>\<^sub>f k \<down> n"

lemma DynProcModifyReturnNoAbr: assumes to_prove: "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (dynCall init p return' c) Q,A" assumes ret_nrm_modif: "\<forall>s t. t \<in> (Modif (init s)) \<longrightarrow> return' s t = return s t" assumes modif_clause: "\<forall>s \<in> P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),{}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (dynCall init p return c) Q,A"

lemma length_generate_PS: "length (generate_PS emBits 160) = (roundup emBits 8)*8 - sLen - 160 - 16"

lemma cartesian_product_binary_union_left: assumes "C \<subseteq> carrier (R\<^bsup>n\<^esup>)" assumes "D \<subseteq> carrier (R\<^bsup>n\<^esup>)" shows "cartesian_product (C \<union> D) A = (cartesian_product C A) \<union> (cartesian_product D A)"

lemma smult_mat_mult_mat_vec_assoc: fixes A :: "'a::comm_ring_1 mat" assumes A: "A \<in> carrier_mat n m" and w: "w \<in> carrier_vec m" shows "a \<cdot>\<^sub>m A *\<^sub>v w = a \<cdot>\<^sub>v (A *\<^sub>v w)"

lemma is_separator_simps : assumes "is_separator M q1 q2 A t1 t2" shows "single_input A" and "acyclic A" and "observable A" and "deadlock_state A t1" and "deadlock_state A t2" and "t1 \<in> reachable_states A" and "t2 \<in> reachable_states A" and "\<And> t . t \<in> reachable_states A \<Longrightarrow> t \<noteq> t1 \<Longrightarrow> t \<noteq> t2 \<Longrightarrow> \<not> deadlock_state A t" and "\<And> io x yq yt . io@[(x,yq)] \<in> LS M q1 \<Longrightarrow> io@[(x,yt)] \<in> L A \<Longrightarrow> (io@[(x,yq)] \<in> L A)" and "\<And> io x yq yt . io@[(x,yq)] \<in> LS M q2 \<Longrightarrow> io@[(x,yt)] \<in> L A \<Longrightarrow> (io@[(x,yq)] \<in> L A)" and "\<And> p . path A (initial A) p \<Longrightarrow> target (initial A) p = t1 \<Longrightarrow> p_io p \<in> LS M q1 - LS M q2" and "\<And> p . path A (initial A) p \<Longrightarrow> target (initial A) p = t2 \<Longrightarrow> p_io p \<in> LS M q2 - LS M q1" and "\<And> p . path A (initial A) p \<Longrightarrow> target (initial A) p \<noteq> t1 \<Longrightarrow> target (initial A) p \<noteq> t2 \<Longrightarrow> p_io p \<in> LS M q1 \<inter> LS M q2" and "q1 \<noteq> q2" and "t1 \<noteq> t2" and "(inputs A) \<subseteq> (inputs M)"

lemma is_iso_arr_is_iso_ntcf: assumes "\<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" and "\<MM> : \<GG> \<mapsto>\<^sub>C\<^sub>F \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" and "\<NN> \<bullet>\<^sub>N\<^sub>T\<^sub>C\<^sub>F \<MM> = ntcf_id \<GG>" and "\<MM> \<bullet>\<^sub>N\<^sub>T\<^sub>C\<^sub>F \<NN> = ntcf_id \<FF>" shows "\<NN> : \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>i\<^sub>s\<^sub>o \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"

lemma silent_moves_no_slice_edges: "\<lbrakk>S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s'); tl ms = targetnodes rs; length rs = length cs; \<forall>i<length cs. call_of_return_node (tl ms!i) (sourcenode (cs!i))\<rbrakk> \<Longrightarrow> slice_edges S cs as = [] \<and> (\<exists>rs'. tl ms' = targetnodes rs' \<and> length rs' = length (upd_cs cs as) \<and> (\<forall>i<length (upd_cs cs as). call_of_return_node (tl ms'!i) (sourcenode ((upd_cs cs as)!i))))"

lemma mask_range_subsetD: "\<lbrakk> p' \<in> mask_range p n; x' \<in> mask_range p' n'; n' \<le> n; is_aligned p n; is_aligned p' n' \<rbrakk> \<Longrightarrow> x' \<in> mask_range p n"

lemma semilat_opt [intro, simp]: "err_semilat L \<Longrightarrow> err_semilat (Opt.esl L)"

lemma (in Corps) Cauchy_down:"\<lbrakk>Corps K'; valuation K v; valuation K' v'; subfield K K'; \<forall>x\<in>carrier K. v x = v' x; \<forall>j. f j \<in> carrier K; Cauchy\<^bsub>K' v' \<^esub>f\<rbrakk> \<Longrightarrow> Cauchy\<^bsub>K v \<^esub>f"

lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"

lemma absCase_Abs_subst[simp]: assumes isInBar: "isInBar (xs,s)" and X: "wls s X" and f_compat: "compatAbsSubst xs s f" shows "absCase xs s f (Abs xs x X) = f x X"

lemma cr_vreal_transfer_domain_rule[transfer_domain_rule]: "Domainp cr_vreal = (\<lambda>x. x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>)"

lemma subst_bb_commute [simp]: "atom j \<sharp> A \<Longrightarrow> (subst_bb (subst_bb A i t ) j u ) = subst_bb A i (subst_bb t j u) "

lemma PO_m2_inv2_keys_for_init [iff]: "init m2 \<subseteq> m2_inv2_keys_for"

lemma next_Null[simp]: assumes "xs \<noteq> \<bottom>" shows "next\<cdot>xs\<cdot>Null = Null"

lemma tendsto_le_ccpo: fixes f g :: "'a \<Rightarrow> 'b::ccpo_topology" assumes F: "\<not> trivial_limit F" assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F" assumes ev: "eventually (\<lambda>x. g x \<le> f x) F" shows "y \<le> x"

lemma flush_reads_program: "\<And>\<O> \<S> \<R> . \<forall>r \<in> set sb. is_Read\<^sub>s\<^sub>b r \<or> is_Prog\<^sub>s\<^sub>b r \<or> is_Ghost\<^sub>s\<^sub>b r \<Longrightarrow> \<exists>\<O>' \<R>' \<S>'. (m,sb,\<O>,\<R>,\<S>) \<rightarrow>\<^sub>f\<^sup>* (m,[],\<O>',\<R>',\<S>')"

theorem IK_real_card: shows "card {f::real set \<Rightarrow> real set. IK f} = 7" (is "?lhs = ?rhs")

lemma remdups_fwd_acc_append[simp]: "remdups_fwd_acc Acc (xs@ys) = (remdups_fwd_acc Acc xs) @ (remdups_fwd_acc (Acc \<union> set xs) ys)"

lemma update_acyclic_3: assumes "acyclic (p - 1)" and "point y" and "point w" and "y \<le> p\<^sup>T\<^sup>\<star> * w" shows "acyclic ((p[w\<longmapsto>y]) - 1)"

lemma odlist_induct [case_names empty insert, cases type: odlist]: assumes empty: "\<And>dxs. dxs = empty \<Longrightarrow> P dxs" assumes insrt: "\<And>dxs x xs. \<lbrakk> dxs = fromList (x # xs); distinct (x # xs); sorted (x # xs); P (fromList xs) \<rbrakk> \<Longrightarrow> P dxs" shows "P dxs"

lemma decE_decE [simp]: "\<down>\<^sub>e n k (\<down>\<^sub>e n' (k + n) \<Gamma>) = \<down>\<^sub>e (n + n') k \<Gamma>"

lemma (in idom_char_0) linear_pow_mul_degree: "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"

lemma COND_path_prefix_extr: "prefix (AHIS (hfs_valid_prefix ainfo uinfo l nxt)) (extr_from_hd l)"

lemma "z \<cdot> x \<le> y \<cdot> z \<Longrightarrow> z \<cdot> x\<^sup>\<omega> \<le> y\<^sup>\<omega> \<cdot> z"

lemma minus_minus_float[simp]: "- (-f) = f" for f::"('e, 'f)float"

lemma IdNatTransNatTrans: "Functor F \<Longrightarrow> NatTrans (IdNatTrans F)"

lemma chinese_remainder_poly: fixes A :: "'a set" and m :: "'a \<Rightarrow> 'b::{field_gcd} poly" and u :: "'a \<Rightarrow> 'b poly" assumes fin: "finite A" and cop: "\<forall>i\<in>A. (\<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))" shows "\<exists>x. (\<forall>i\<in>A. [x = u i] (mod m i))"

lemma list_of_lappend: assumes "lfinite xs" "lfinite ys" shows "list_of (lappend xs ys) = list_of xs @ list_of ys"

lemma continuous_on_compact_bound: assumes "compact A" "continuous_on A f" obtains B where "B \<ge> 0" "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> B"

lemma Im_interval_times: "Im_interval (A * B) = Re_interval A * Im_interval B + Im_interval A * Re_interval B"

lemma insert_before_removes_child: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "ptr \<noteq> ptr'" assumes "h \<turnstile> insert_before ptr node child \<rightarrow>\<^sub>h h'" assumes "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r node # children" shows "h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"

lemma unification': assumes "finite ts" assumes "unifier\<^sub>t\<^sub>s \<sigma> ts" shows "\<exists>\<theta>. mgu\<^sub>t\<^sub>s \<theta> ts"

lemma lexext_singleton: "lexext gt [y] [x] \<longleftrightarrow> gt y x"

lemma proposition3_4_inv_step: assumes t: "trans r" and i: "irrefl r" and k:"(r|\<sigma>| -s r \<down>l [\<beta>], {#\<alpha>#}) \<in> mul_eq r" (is "(?M,_) \<in> _") shows "\<exists> \<sigma>1 \<sigma>2 \<sigma>3. ((\<sigma> = \<sigma>1@\<sigma>2@\<sigma>3) \<and> LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3)"

lemma powsum_ext: "x \<le> x\<^bsub>0\<^esub>\<^bsup>Suc n\<^esup>"

lemma finite_subset[trans]: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"

lemma "\<sim>ECQ \<^bold>\<not>\<^sup>C \<and> Fr_1 \<F> \<and> Fr_2 \<F> \<and> Fr_4 \<F> \<and> lCoPw(\<^bold>\<rightarrow>) \<^bold>\<not>\<^sup>C"

lemma fimage_eq_to_f: assumes "f1 |`| S1 = f2 |`| S2" obtains f where "\<And> x. x |\<in>| S2 \<Longrightarrow> f x |\<in>| S1 \<and> f1 (f x) = f2 x"