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

Statement: stringlengths 7 24.3k
lemma subterms_singleton: assumes "(\<exists>v. t = Var v) \<or> (\<exists>f. t = Fun f [])" shows "subterms t = {t}"
lemma insert_prefix_trie_same: "insert ps (prefix_trie ps (Nd b lr)) = prefix_trie ps (Nd True lr)"
lemma Ch_XH_largest_Xd: assumes "x \<in> Ch h XH_largest" shows "Xd x \<in> ds"
lemma start_end_implies_terminating: assumes "has_start_points x" and "has_end_points x" shows "terminating x"
lemma smc_Set_composable_arrs_dg_Set: "composable_arrs (dg_Set \<alpha>) = composable_arrs (smc_Set \<alpha>)"
lemma \<J>_induct[consumes 1, case_names Induct]: assumes "x \<in> \<J> k u" assumes induct: "\<And> x k u . (\<And> x' k' u'. x' \<in> \<J> k' u' \<Longrightarrow> indexlt k' u' k u \<Longrightarrow> P x' k' u') \<Longrightarrow> x \<in> \<J> k u \<Longrightarrow> P x k u" shows "P x k u"
lemma ide_char': shows "ide a \<longleftrightarrow> arr a \<and> (dom a = a \<or> cod a = a)"
lemma member_paperIDAsStr_iff[simp]: "str \<in> paperIDAsStr ` paperIDs \<longleftrightarrow> PaperID str \<in> paperIDs"
lemma sdrop_snth: "sdrop n s !! m = s !! (n + m)"
lemma find_policy_QR_error_bound: assumes "eps > 0" "2 * QR_disc * dist v (\<L>\<^sub>b_split v) < eps * (1-QR_disc)" assumes am: "\<And>s. is_arg_max (\<lambda>d. L_split d (\<L>\<^sub>b_split v) s) (\<lambda>d. d \<in> D\<^sub>D) d" shows "dist (\<nu>\<^sub>b (mk_stationary_det d)) \<nu>\<^sub>b_opt < eps"
lemma OF_spm3_noa_none: assumes no: "no_overlaps \<gamma> ft" shows "OF_priority_match \<gamma> ft p = NoAction \<Longrightarrow> \<forall>e \<in> set ft. \<not>\<gamma> (ofe_fields e) p"
lemma ahm_invar_auxE: assumes "ahm_invar_aux bhc n a" obtains "\<forall>h. h < array_length a \<longrightarrow> bucket_ok bhc (array_length a) h (array_get a h) \<and> list_map_invar (array_get a h)" and "n = array_foldl (\<lambda>_ n kvs. n + length kvs) 0 a" and "array_length a > 1"
lemma ex_conga_ts: assumes "\<not> Col A B C" and "\<not> Col A' B' P" shows "\<exists> C'. A B C CongA A' B' C' \<and> A' B' TS C' P"
lemma distinctPermSimps[simp]: fixes p :: "name prm" and a :: name and b :: name shows "distinctPerm([]::name prm)" and "(distinctPerm((a, b)#p)) = (distinctPerm p \<and> a \<noteq> b \<and> a \<sharp> p \<and> b \<sharp> p)"
lemma in_mktop: "(x,y) \<in> mktop L z \<longleftrightarrow> x\<noteq>z \<and> (if y=z then x\<noteq>y else (x,y) \<in> L)"
lemma used_Gets_rev: "used (evs @ [Gets B X]) = used evs"
lemma [sepref_fr_rules]: " hn_refine (hn_ctxt (is_graph n R) G Gi * hn_ctxt (node_assn n) v vi) (succi Gi vi) (hn_ctxt (is_graph n R) G Gi * hn_ctxt (node_assn n) v vi) (pure (\<langle>R \<times>\<^sub>r node_rel n\<rangle>list_set_rel)) (RETURN$(succ$G$v))"
lemma xor_identity2[simp]: "xor {\|\|} xs = xs"
lemma final_simulation2: "\<lbrakk> s1 \<approx> s2; s2 -\<tau>2-tls2\<rightarrow>* s2'; final2 s2' \<rbrakk> \<Longrightarrow> \<exists>s1' tls1. s1 -\<tau>1-tls1\<rightarrow>* s1' \<and> s1' \<approx> s2' \<and> final1 s1' \<and> tls1 [\<sim>] tls2"
lemma Gen_Shleg_n_0 [simp]: "Gen_Shleg n 0 = fact n"
lemma wf_measures[simp]: "wf (measures fs)"
lemma Skip_is_action: "(R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not>wait A' \<and> more A = more A')) \<in> {p. is_CSP_process p}"
lemma (in Ring) n_prod_idealTr: "(\<forall>k \<le> n. ideal R (J k)) \<longrightarrow> ideal R (ideal_n_prod R n J)"
lemma lm148: assumes "card (Pow X) = 1 \<or> card (Pow X) = 2" shows "trivial X"
lemma indicator_image: "inj f \<Longrightarrow> indicator (f ` X) (f x) = (indicator X x::_::zero_neq_one)"
lemma unique_Poincare: defines "\<Sigma> \<equiv> {(1::real, 2.25::real) .. (2, 2.25)}" shows "\<exists>!x. x \<in> {(1.41::real, 2.25::real) .. (1.42, 2.25)} \<and> vdp.poincare_map \<Sigma> x = x"
lemma gauss_int_norm_eq_0_iff [simp]: "gauss_int_norm z = 0 \<longleftrightarrow> z = 0"
lemma impl_of_empty [code abstract]: "impl_of empty = empty_trie"
lemma div_const_unit_poly: "is_unit c \<Longrightarrow> p div [:c:] = smult (1 div c) p"
lemma "rec_BitList nil bit0 bit1 (Bit1 xs) = bit1 xs (rec_BitList nil bit0 bit1 xs)"
lemma sum_to_zero: "(a :: 'a :: ring) + b = 0 \<Longrightarrow> a = (- b)"
lemma is_subprob_density_imp_has_density: "\<lbrakk>is_subprob_density N f; M = density N f\<rbrakk> \<Longrightarrow> has_density M N f"
lemma normal_drop [simp]: "normal (dropWhile (\<lambda>n. n=0) ns)"
lemma below_meet2[simp]: fixes x y :: "'a :: Finite_Meet_cpo" assumes "z \<sqsubseteq> x" shows "(x \<sqinter> z) = z"
lemma hfset_hinsert: "hfset (b \<triangleleft> a) = insert a (hfset b)"
lemma emp_to_emp[simp]: "f \<epsilon> = \<epsilon>"
lemma HNatInfinite_inverse_Infinitesimal [simp]: assumes "n \<in> HNatInfinite" shows "inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
lemma liftWT_return: "liftWT\<cdot>(return\<cdot>x) = unitWT\<cdot>x"
lemma "\<lfloor>D\<rfloor>"
lemma lfinite_lmap[simp]: "lfinite (lmap f xs) = lfinite xs"
lemma alluopairs_ex: "\<forall>x y. P x y = P y x \<Longrightarrow> (\<exists>x \<in> set xs. \<exists>y \<in> set xs. P x y) = (\<exists>(x, y) \<in> set (alluopairs xs). P x y)"
theorem chain_union_closed: assumes \<open>finite_char C\<close> and \<open>is_chain f\<close> and \<open>\<forall>n. f n \<in> C\<close> shows \<open>(\<Union>n. f n) \<in> C\<close>
lemma Leaf_mirror[simp]: "\<langle>\<rangle> = mirror t \<longleftrightarrow> t = \<langle>\<rangle>"
lemma lp_monom_code[code]: "linear_poly_map (lp_monom c x) = (if c = 0 then fmempty else fmupd x c fmempty)"
lemma extr_mono_chan [dest]: "G \<subseteq> H \<Longrightarrow> extr bad IK G \<subseteq> extr bad IK H"
lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"
lemma whn_eq_\<omega>m1: "hypreal_of_hypnat whn = \<omega> - 1"
lemma lowner_le_antisym: assumes A: "A \<in> carrier_mat n n" and B: "B \<in> carrier_mat n n" and L1: "A \<le>\<^sub>L B" and L2: "B \<le>\<^sub>L A" shows "A = B"
lemma subtensor0: assumes "ds \<noteq> []" and "i<hd ds" shows "subtensor (tensor0 ds) i = tensor0 (tl ds)"
lemma parts_preserves_unaffected: assumes \<open>\<not> affects r p\<close> \<open>z' \<in> set (parts A r p)\<close> shows \<open>p \<in> set z'\<close>
lemma gauss_jordan_complete: "m \<le> n \<Longrightarrow> usolution A m n x \<Longrightarrow> \<exists>B. gauss_jordan A m = Some B"
lemma simple_resolution_sound: assumes C\<^sub>1sat: "eval\<^sub>c F G C\<^sub>1" assumes C\<^sub>2sat: "eval\<^sub>c F G C\<^sub>2" assumes l\<^sub>1inc\<^sub>1: "l\<^sub>1 \<in> C\<^sub>1" assumes l\<^sub>2inc\<^sub>2: "l\<^sub>2 \<in> C\<^sub>2" assumes comp: "l\<^sub>1\<^sup>c = l\<^sub>2" shows "eval\<^sub>c F G ((C\<^sub>1 - {l\<^sub>1}) \<union> (C\<^sub>2 - {l\<^sub>2}))"
lemma convert_extTA_eq_conv: "convert_extTA f ta = ta' \<longleftrightarrow> \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>w\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>i\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub> \<and> length \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> = length \<lbrace>ta'\<rbrace>\<^bsub>t\<^esub> \<and> (\<forall>n < length \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>. convert_new_thread_action f (\<lbrace>ta\<rbrace>\<^bsub>t\<^esub> ! n) = \<lbrace>ta'\<rbrace>\<^bsub>t\<^esub> ! n)"
lemma test_comp_anti_iff: "test p \<Longrightarrow> test q \<Longrightarrow> p \<le> q \<longleftrightarrow> !q \<le> !p"
lemma termMOD_igOpIPresIGWls: "igOpIPresIGWls termMOD"
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
lemma reduce_element_mod_D_preserves_dimensions: shows [simp]: "dim_row (reduce_element_mod_D A a j D m) = dim_row A" and [simp]: "dim_col (reduce_element_mod_D A a j D m) = dim_col A" and [simp]: "dim_row (reduce_element_mod_D_abs A a j D m) = dim_row A" and [simp]: "dim_col (reduce_element_mod_D_abs A a j D m) = dim_col A"
lemma Count: "\<turnstile> Count \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace>"
lemma of_nat_diff [simp]: "n \<le> m \<Longrightarrow> \<guillemotleft>m - n\<guillemotright>\<^sub>\<nat> = \<guillemotleft>m\<guillemotright>\<^sub>\<nat> \<ominus> \<guillemotleft>n\<guillemotright>\<^sub>\<nat>"
lemma in_set_left: "y \<in> set_of l \<Longrightarrow> y \<in> set_of (Node l x False r)"
lemma eefm_clean_mem: assumes "si' \<le> si" and "eq_except_for_mem (\<sigma> with updates) (\<sigma> with updates') a si v b" shows "eq_except_for_mem (\<sigma> with ((\<lbrakk>a,si'\<rbrakk> :=\<^sub>m v')#updates)) (\<sigma> with updates') a si v b"
lemma args_are_strictly_lower: assumes "is_compound t" shows "(lhs t,t) \<in> trm_ord \<and> (rhs t,t) \<in> trm_ord"
lemma obj_ty_widenD: "G\<turnstile>obj_ty obj\<preceq>RefT t \<Longrightarrow> (\<exists>C. tag obj = CInst C) \<or> (\<exists>T k. tag obj = Arr T k)"
lemma res_uminus: assumes "k > 0" assumes "f \<in> carrier Z\<^sub>p" assumes "c \<in> carrier (Zp_res_ring k)" assumes "c = \<ominus>\<^bsub>Zp_res_ring k\<^esub> (f k)" shows "c = ((\<ominus>\<^bsub>Z\<^sub>p\<^esub> f) k)"
lemma \<pi>_is_natural_transformation: shows "natural_transformation CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>"
lemma distinct_permutations_of_list_impl_aux: "distinct (permutations_of_list_impl_aux acc xs)"
lemma lipschitz_on_cong[cong]: "C-lipschitz_on U g \<longleftrightarrow> D-lipschitz_on V f" if "C = D" "U = V" "\<And>x. x \<in> V \<Longrightarrow> g x = f x"
lemma heaps_eq_DagI1: "\<lbrakk>Dag p l r t; \<forall>x\<in>set_of t. l x = l' x \<and> r x = r' x\<rbrakk> \<Longrightarrow> Dag p l' r' t"
lemma vrestriction_vempty[simp]: "0 \<restriction>\<^sub>\<circ> A = 0"
lemma sup_loc_some [rule_format]: "\<forall>y n. (G \<turnstile> b <=l y) \<longrightarrow> n < length y \<longrightarrow> y!n = OK t \<longrightarrow> (\<exists>t. b!n = OK t \<and> (G \<turnstile> (b!n) <=o (y!n)))"
lemma drinks_rejects_future: "\<not> recognises_execution drinks 2 d ((l, i)#t)"
lemma (in is_ntcf) ntcf_of_ntcf_arrow_is_ntcf': assumes "\<NN>' = ntcf_arrow \<NN>" and "\<AA>' = \<AA>" and "\<BB>' = \<BB>" shows "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>' : \<FF> \<mapsto>\<^sub>C\<^sub>F \<GG> : \<AA>' \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>'"
lemma Nonce_secrecy_lemma: "P \<longrightarrow> (X \<in> analz (G \<union> H)) \<longrightarrow> (X \<in> analz H) \<Longrightarrow> P \<longrightarrow> (X \<in> analz (G \<union> H)) = (X \<in> analz H)"
lemma fps_right_inverse: fixes f :: "'a::ring_1 fps" assumes f0: "f$0 * y = 1" shows "f * fps_right_inverse f y = 1"
lemma New_correct: "\<lbrakk> wf_prog wt G; method (G,C) sig = Some (C,rT,maxs,maxl,ins,et); ins!pc = New X; wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) et pc; Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ; G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd>; fst (exec_instr (ins!pc) G hp stk loc C sig pc frs) = None \<rbrakk> \<Longrightarrow> G,phi \<turnstile>JVM state'\<surd>"
lemma add_row_to_multiple_index_unchanged [simp]: "i < dim_row A \<Longrightarrow> j < dim_col A \<Longrightarrow> i \<notin> set ks \<Longrightarrow> add_row_to_multiple a ks l A $$ (i,j) = A $$(i,j)"
lemma staticSecretA_notin_parts_initStateB: "m \<in> staticSecret A \<Longrightarrow> m \<in> parts(initState B) = (A=B)"
theorem compliant_stateful_ACS_no_side_effects: "\<forall> E \<subseteq> backflows (flows_state \<T>). \<forall> F \<in> get_offending_flows(get_ACS M) \<lparr> nodes = hosts \<T>, edges = flows_fix \<T> \<union> E \<rparr>. F \<subseteq> E"
lemma card1_eE: "finite S \<Longrightarrow> \<exists>y. y \<in> S \<Longrightarrow> 1 \<le> card S"
lemma distinct_mtf[simp]: "distinct (mtf x xs) = distinct xs"
lemma perm_ty[simp]: fixes T::"ty" and pi::"name prm" shows "pi\<bullet>T = T"
lemma insert_body_stop_iteration: assumes "fst e > fst x" shows "insert_body (x#xs) e = e#x#xs"
lemma le_defined: fixes x :: Integer shows "le\<cdot>x\<cdot>y = TT \<Longrightarrow> (x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>)" "le\<cdot>x\<cdot>y = FF \<Longrightarrow> (x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>)"
lemma size_simp6: "s \<in> set ss \<Longrightarrow> s \<unrhd> t \<Longrightarrow> size t \<le> size s \<Longrightarrow> size t \<le> Suc (size_list size ss)"
lemma DERIV_inverse_function: fixes f g :: "real \<Rightarrow> real" assumes der: "DERIV f (g x) :> D" and neq: "D \<noteq> 0" and x: "a < x" "x < b" and inj: "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> f (g y) = y" and cont: "isCont g x" shows "DERIV g x :> inverse D"
lemma mono2mono_enat_minus1 [THEN lfp.mono2mono, cont_intro, simp]: shows monotone_enat_minus1: "monotone (\<le>) (\<le>) (\<lambda>n. n - m :: enat)"
lemma wlp_Seq: assumes ent_a: "P \<tturnstile> wlp a Q" and ent_b: "Q \<tturnstile> wlp b R" and wa: "well_def a" and wb: "well_def b" and u_Q: "unitary Q" and u_R: "unitary R" shows "P \<tturnstile> wlp (a ;; b) R"
lemma generalise_all: assumes i: "PROP Pure.prop (\<And>s. PROP Pure.prop (PROP P s) \<Longrightarrow> PROP Pure.prop (PROP Q s))" shows "PROP Pure.prop ((PROP Pure.prop (\<And>s. PROP P s)) \<Longrightarrow> (PROP Pure.prop (\<And>s. PROP Q s)))"
lemma at_ds6: fixes a :: "'x" and b :: "'x" and c :: "'x" assumes at: "at TYPE('x)" and a: "distinct [a,b,c]" shows "[(a,c),(a,b)] \<triangleq> [(b,c),(a,c)]"
lemma ring_transfer[transfer_rule]: assumes[transfer_rule]: "bi_unique A" "right_total A" shows "( (A ===> A ===> A) ===> A ===> (A ===> A ===> A) ===> (A ===> A) ===> (A ===> A ===> A) ===> (=) ) (ring_ow (Collect (Domainp A))) class.ring"
lemma reach_snoc: "reach tr p bs q \<Longrightarrow> reach tr p (bs @ [b]) (tr q b)"
lemma vinsert_iff[simp]: "x \<in>\<^sub>\<circ> vinsert y A \<longleftrightarrow> x = y \<or> x \<in>\<^sub>\<circ> A"
lemma float_eq_refl [simp]: "a \<doteq> a \<longleftrightarrow> \<not> is_nan a"
lemma subst_typ'_AList_clearjunk: "subst_typ' insts t = subst_typ' (AList.clearjunk insts) t"
lemma VARusefulSYSTEM_holds: "VARusefulSYSTEM"
lemma hom_f: "homogeneous f"
lemma fold_invalid_means_one_invalid: "gval (fold gAnd G (Bc True)) s = invalid \<Longrightarrow> \<exists>g \<in> set G. gval g s = invalid"
lemma other_net_tree_ips_par_right: assumes "other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) \<sigma> \<sigma>'" and "\<And>\<xi>. U \<xi> \<xi>" shows "other U (net_tree_ips p\<^sub>2) \<sigma> \<sigma>'"
lemma is_parser_exactly [intro]: "is_parser (exactly xs)"
lemma infinite_int_iff_infinite_nat_abs: "infinite S \<longleftrightarrow> infinite ((nat \<circ> abs) ` S)" for S :: "int set"

lemma subterms_singleton: assumes "(\<exists>v. t = Var v) \<or> (\<exists>f. t = Fun f [])" shows "subterms t = {t}"

lemma insert_prefix_trie_same: "insert ps (prefix_trie ps (Nd b lr)) = prefix_trie ps (Nd True lr)"

lemma Ch_XH_largest_Xd: assumes "x \<in> Ch h XH_largest" shows "Xd x \<in> ds"

lemma start_end_implies_terminating: assumes "has_start_points x" and "has_end_points x" shows "terminating x"

lemma smc_Set_composable_arrs_dg_Set: "composable_arrs (dg_Set \<alpha>) = composable_arrs (smc_Set \<alpha>)"

lemma \<J>_induct[consumes 1, case_names Induct]: assumes "x \<in> \<J> k u" assumes induct: "\<And> x k u . (\<And> x' k' u'. x' \<in> \<J> k' u' \<Longrightarrow> indexlt k' u' k u \<Longrightarrow> P x' k' u') \<Longrightarrow> x \<in> \<J> k u \<Longrightarrow> P x k u" shows "P x k u"

lemma ide_char': shows "ide a \<longleftrightarrow> arr a \<and> (dom a = a \<or> cod a = a)"

lemma member_paperIDAsStr_iff[simp]: "str \<in> paperIDAsStr ` paperIDs \<longleftrightarrow> PaperID str \<in> paperIDs"

lemma sdrop_snth: "sdrop n s !! m = s !! (n + m)"

lemma find_policy_QR_error_bound: assumes "eps > 0" "2 * QR_disc * dist v (\<L>\<^sub>b_split v) < eps * (1-QR_disc)" assumes am: "\<And>s. is_arg_max (\<lambda>d. L_split d (\<L>\<^sub>b_split v) s) (\<lambda>d. d \<in> D\<^sub>D) d" shows "dist (\<nu>\<^sub>b (mk_stationary_det d)) \<nu>\<^sub>b_opt < eps"

lemma OF_spm3_noa_none: assumes no: "no_overlaps \<gamma> ft" shows "OF_priority_match \<gamma> ft p = NoAction \<Longrightarrow> \<forall>e \<in> set ft. \<not>\<gamma> (ofe_fields e) p"

lemma ahm_invar_auxE: assumes "ahm_invar_aux bhc n a" obtains "\<forall>h. h < array_length a \<longrightarrow> bucket_ok bhc (array_length a) h (array_get a h) \<and> list_map_invar (array_get a h)" and "n = array_foldl (\<lambda>_ n kvs. n + length kvs) 0 a" and "array_length a > 1"

lemma ex_conga_ts: assumes "\<not> Col A B C" and "\<not> Col A' B' P" shows "\<exists> C'. A B C CongA A' B' C' \<and> A' B' TS C' P"

lemma distinctPermSimps[simp]: fixes p :: "name prm" and a :: name and b :: name shows "distinctPerm([]::name prm)" and "(distinctPerm((a, b)#p)) = (distinctPerm p \<and> a \<noteq> b \<and> a \<sharp> p \<and> b \<sharp> p)"

lemma in_mktop: "(x,y) \<in> mktop L z \<longleftrightarrow> x\<noteq>z \<and> (if y=z then x\<noteq>y else (x,y) \<in> L)"

lemma used_Gets_rev: "used (evs @ [Gets B X]) = used evs"

lemma [sepref_fr_rules]: " hn_refine (hn_ctxt (is_graph n R) G Gi * hn_ctxt (node_assn n) v vi) (succi Gi vi) (hn_ctxt (is_graph n R) G Gi * hn_ctxt (node_assn n) v vi) (pure (\<langle>R \<times>\<^sub>r node_rel n\<rangle>list_set_rel)) (RETURN$(succ$G$v))"

lemma xor_identity2[simp]: "xor {||} xs = xs"

lemma final_simulation2: "\<lbrakk> s1 \<approx> s2; s2 -\<tau>2-tls2\<rightarrow>* s2'; final2 s2' \<rbrakk> \<Longrightarrow> \<exists>s1' tls1. s1 -\<tau>1-tls1\<rightarrow>* s1' \<and> s1' \<approx> s2' \<and> final1 s1' \<and> tls1 [\<sim>] tls2"

lemma Gen_Shleg_n_0 [simp]: "Gen_Shleg n 0 = fact n"

lemma wf_measures[simp]: "wf (measures fs)"

lemma Skip_is_action: "(R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not>wait A' \<and> more A = more A')) \<in> {p. is_CSP_process p}"

lemma (in Ring) n_prod_idealTr: "(\<forall>k \<le> n. ideal R (J k)) \<longrightarrow> ideal R (ideal_n_prod R n J)"

lemma lm148: assumes "card (Pow X) = 1 \<or> card (Pow X) = 2" shows "trivial X"

lemma indicator_image: "inj f \<Longrightarrow> indicator (f ` X) (f x) = (indicator X x::_::zero_neq_one)"

lemma unique_Poincare: defines "\<Sigma> \<equiv> {(1::real, 2.25::real) .. (2, 2.25)}" shows "\<exists>!x. x \<in> {(1.41::real, 2.25::real) .. (1.42, 2.25)} \<and> vdp.poincare_map \<Sigma> x = x"

lemma gauss_int_norm_eq_0_iff [simp]: "gauss_int_norm z = 0 \<longleftrightarrow> z = 0"

lemma impl_of_empty [code abstract]: "impl_of empty = empty_trie"

lemma div_const_unit_poly: "is_unit c \<Longrightarrow> p div [:c:] = smult (1 div c) p"

lemma "rec_BitList nil bit0 bit1 (Bit1 xs) = bit1 xs (rec_BitList nil bit0 bit1 xs)"

lemma sum_to_zero: "(a :: 'a :: ring) + b = 0 \<Longrightarrow> a = (- b)"

lemma is_subprob_density_imp_has_density: "\<lbrakk>is_subprob_density N f; M = density N f\<rbrakk> \<Longrightarrow> has_density M N f"

lemma normal_drop [simp]: "normal (dropWhile (\<lambda>n. n=0) ns)"

lemma below_meet2[simp]: fixes x y :: "'a :: Finite_Meet_cpo" assumes "z \<sqsubseteq> x" shows "(x \<sqinter> z) = z"

lemma hfset_hinsert: "hfset (b \<triangleleft> a) = insert a (hfset b)"

lemma emp_to_emp[simp]: "f \<epsilon> = \<epsilon>"

lemma HNatInfinite_inverse_Infinitesimal [simp]: assumes "n \<in> HNatInfinite" shows "inverse (hypreal_of_hypnat n) \<in> Infinitesimal"

lemma liftWT_return: "liftWT\<cdot>(return\<cdot>x) = unitWT\<cdot>x"

lemma "\<lfloor>D\<rfloor>"

lemma lfinite_lmap[simp]: "lfinite (lmap f xs) = lfinite xs"

lemma alluopairs_ex: "\<forall>x y. P x y = P y x \<Longrightarrow> (\<exists>x \<in> set xs. \<exists>y \<in> set xs. P x y) = (\<exists>(x, y) \<in> set (alluopairs xs). P x y)"

theorem chain_union_closed: assumes \<open>finite_char C\<close> and \<open>is_chain f\<close> and \<open>\<forall>n. f n \<in> C\<close> shows \<open>(\<Union>n. f n) \<in> C\<close>

lemma Leaf_mirror[simp]: "\<langle>\<rangle> = mirror t \<longleftrightarrow> t = \<langle>\<rangle>"

lemma lp_monom_code[code]: "linear_poly_map (lp_monom c x) = (if c = 0 then fmempty else fmupd x c fmempty)"

lemma extr_mono_chan [dest]: "G \<subseteq> H \<Longrightarrow> extr bad IK G \<subseteq> extr bad IK H"

lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"

lemma whn_eq_\<omega>m1: "hypreal_of_hypnat whn = \<omega> - 1"

lemma lowner_le_antisym: assumes A: "A \<in> carrier_mat n n" and B: "B \<in> carrier_mat n n" and L1: "A \<le>\<^sub>L B" and L2: "B \<le>\<^sub>L A" shows "A = B"

lemma subtensor0: assumes "ds \<noteq> []" and "i<hd ds" shows "subtensor (tensor0 ds) i = tensor0 (tl ds)"

lemma parts_preserves_unaffected: assumes \<open>\<not> affects r p\<close> \<open>z' \<in> set (parts A r p)\<close> shows \<open>p \<in> set z'\<close>

lemma gauss_jordan_complete: "m \<le> n \<Longrightarrow> usolution A m n x \<Longrightarrow> \<exists>B. gauss_jordan A m = Some B"

lemma simple_resolution_sound: assumes C\<^sub>1sat: "eval\<^sub>c F G C\<^sub>1" assumes C\<^sub>2sat: "eval\<^sub>c F G C\<^sub>2" assumes l\<^sub>1inc\<^sub>1: "l\<^sub>1 \<in> C\<^sub>1" assumes l\<^sub>2inc\<^sub>2: "l\<^sub>2 \<in> C\<^sub>2" assumes comp: "l\<^sub>1\<^sup>c = l\<^sub>2" shows "eval\<^sub>c F G ((C\<^sub>1 - {l\<^sub>1}) \<union> (C\<^sub>2 - {l\<^sub>2}))"

lemma convert_extTA_eq_conv: "convert_extTA f ta = ta' \<longleftrightarrow> \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>w\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> \<and> \<lbrace>ta\<rbrace>\<^bsub>i\<^esub> = \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub> \<and> length \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> = length \<lbrace>ta'\<rbrace>\<^bsub>t\<^esub> \<and> (\<forall>n < length \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>. convert_new_thread_action f (\<lbrace>ta\<rbrace>\<^bsub>t\<^esub> ! n) = \<lbrace>ta'\<rbrace>\<^bsub>t\<^esub> ! n)"

lemma test_comp_anti_iff: "test p \<Longrightarrow> test q \<Longrightarrow> p \<le> q \<longleftrightarrow> !q \<le> !p"

lemma termMOD_igOpIPresIGWls: "igOpIPresIGWls termMOD"

lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"

lemma reduce_element_mod_D_preserves_dimensions: shows [simp]: "dim_row (reduce_element_mod_D A a j D m) = dim_row A" and [simp]: "dim_col (reduce_element_mod_D A a j D m) = dim_col A" and [simp]: "dim_row (reduce_element_mod_D_abs A a j D m) = dim_row A" and [simp]: "dim_col (reduce_element_mod_D_abs A a j D m) = dim_col A"

lemma Count: "\<turnstile> Count \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace>"

lemma of_nat_diff [simp]: "n \<le> m \<Longrightarrow> \<guillemotleft>m - n\<guillemotright>\<^sub>\<nat> = \<guillemotleft>m\<guillemotright>\<^sub>\<nat> \<ominus> \<guillemotleft>n\<guillemotright>\<^sub>\<nat>"

lemma in_set_left: "y \<in> set_of l \<Longrightarrow> y \<in> set_of (Node l x False r)"

lemma eefm_clean_mem: assumes "si' \<le> si" and "eq_except_for_mem (\<sigma> with updates) (\<sigma> with updates') a si v b" shows "eq_except_for_mem (\<sigma> with ((\<lbrakk>a,si'\<rbrakk> :=\<^sub>m v')#updates)) (\<sigma> with updates') a si v b"

lemma args_are_strictly_lower: assumes "is_compound t" shows "(lhs t,t) \<in> trm_ord \<and> (rhs t,t) \<in> trm_ord"

lemma obj_ty_widenD: "G\<turnstile>obj_ty obj\<preceq>RefT t \<Longrightarrow> (\<exists>C. tag obj = CInst C) \<or> (\<exists>T k. tag obj = Arr T k)"

lemma res_uminus: assumes "k > 0" assumes "f \<in> carrier Z\<^sub>p" assumes "c \<in> carrier (Zp_res_ring k)" assumes "c = \<ominus>\<^bsub>Zp_res_ring k\<^esub> (f k)" shows "c = ((\<ominus>\<^bsub>Z\<^sub>p\<^esub> f) k)"

lemma \<pi>_is_natural_transformation: shows "natural_transformation CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>"

lemma distinct_permutations_of_list_impl_aux: "distinct (permutations_of_list_impl_aux acc xs)"

lemma lipschitz_on_cong[cong]: "C-lipschitz_on U g \<longleftrightarrow> D-lipschitz_on V f" if "C = D" "U = V" "\<And>x. x \<in> V \<Longrightarrow> g x = f x"

lemma heaps_eq_DagI1: "\<lbrakk>Dag p l r t; \<forall>x\<in>set_of t. l x = l' x \<and> r x = r' x\<rbrakk> \<Longrightarrow> Dag p l' r' t"

lemma vrestriction_vempty[simp]: "0 \<restriction>\<^sub>\<circ> A = 0"

lemma sup_loc_some [rule_format]: "\<forall>y n. (G \<turnstile> b <=l y) \<longrightarrow> n < length y \<longrightarrow> y!n = OK t \<longrightarrow> (\<exists>t. b!n = OK t \<and> (G \<turnstile> (b!n) <=o (y!n)))"

lemma drinks_rejects_future: "\<not> recognises_execution drinks 2 d ((l, i)#t)"

lemma (in is_ntcf) ntcf_of_ntcf_arrow_is_ntcf': assumes "\<NN>' = ntcf_arrow \<NN>" and "\<AA>' = \<AA>" and "\<BB>' = \<BB>" shows "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>' : \<FF> \<mapsto>\<^sub>C\<^sub>F \<GG> : \<AA>' \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>'"

lemma Nonce_secrecy_lemma: "P \<longrightarrow> (X \<in> analz (G \<union> H)) \<longrightarrow> (X \<in> analz H) \<Longrightarrow> P \<longrightarrow> (X \<in> analz (G \<union> H)) = (X \<in> analz H)"

lemma fps_right_inverse: fixes f :: "'a::ring_1 fps" assumes f0: "f$0 * y = 1" shows "f * fps_right_inverse f y = 1"

lemma New_correct: "\<lbrakk> wf_prog wt G; method (G,C) sig = Some (C,rT,maxs,maxl,ins,et); ins!pc = New X; wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) et pc; Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ; G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd>; fst (exec_instr (ins!pc) G hp stk loc C sig pc frs) = None \<rbrakk> \<Longrightarrow> G,phi \<turnstile>JVM state'\<surd>"

lemma add_row_to_multiple_index_unchanged [simp]: "i < dim_row A \<Longrightarrow> j < dim_col A \<Longrightarrow> i \<notin> set ks \<Longrightarrow> add_row_to_multiple a ks l A $$ (i,j) = A $$(i,j)"

lemma staticSecretA_notin_parts_initStateB: "m \<in> staticSecret A \<Longrightarrow> m \<in> parts(initState B) = (A=B)"

theorem compliant_stateful_ACS_no_side_effects: "\<forall> E \<subseteq> backflows (flows_state \<T>). \<forall> F \<in> get_offending_flows(get_ACS M) \<lparr> nodes = hosts \<T>, edges = flows_fix \<T> \<union> E \<rparr>. F \<subseteq> E"

lemma card1_eE: "finite S \<Longrightarrow> \<exists>y. y \<in> S \<Longrightarrow> 1 \<le> card S"

lemma distinct_mtf[simp]: "distinct (mtf x xs) = distinct xs"

lemma perm_ty[simp]: fixes T::"ty" and pi::"name prm" shows "pi\<bullet>T = T"

lemma insert_body_stop_iteration: assumes "fst e > fst x" shows "insert_body (x#xs) e = e#x#xs"

lemma le_defined: fixes x :: Integer shows "le\<cdot>x\<cdot>y = TT \<Longrightarrow> (x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>)" "le\<cdot>x\<cdot>y = FF \<Longrightarrow> (x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>)"

lemma size_simp6: "s \<in> set ss \<Longrightarrow> s \<unrhd> t \<Longrightarrow> size t \<le> size s \<Longrightarrow> size t \<le> Suc (size_list size ss)"

lemma DERIV_inverse_function: fixes f g :: "real \<Rightarrow> real" assumes der: "DERIV f (g x) :> D" and neq: "D \<noteq> 0" and x: "a < x" "x < b" and inj: "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> f (g y) = y" and cont: "isCont g x" shows "DERIV g x :> inverse D"

lemma mono2mono_enat_minus1 [THEN lfp.mono2mono, cont_intro, simp]: shows monotone_enat_minus1: "monotone (\<le>) (\<le>) (\<lambda>n. n - m :: enat)"

lemma wlp_Seq: assumes ent_a: "P \<tturnstile> wlp a Q" and ent_b: "Q \<tturnstile> wlp b R" and wa: "well_def a" and wb: "well_def b" and u_Q: "unitary Q" and u_R: "unitary R" shows "P \<tturnstile> wlp (a ;; b) R"

lemma generalise_all: assumes i: "PROP Pure.prop (\<And>s. PROP Pure.prop (PROP P s) \<Longrightarrow> PROP Pure.prop (PROP Q s))" shows "PROP Pure.prop ((PROP Pure.prop (\<And>s. PROP P s)) \<Longrightarrow> (PROP Pure.prop (\<And>s. PROP Q s)))"

lemma at_ds6: fixes a :: "'x" and b :: "'x" and c :: "'x" assumes at: "at TYPE('x)" and a: "distinct [a,b,c]" shows "[(a,c),(a,b)] \<triangleq> [(b,c),(a,c)]"

lemma ring_transfer[transfer_rule]: assumes[transfer_rule]: "bi_unique A" "right_total A" shows "( (A ===> A ===> A) ===> A ===> (A ===> A ===> A) ===> (A ===> A) ===> (A ===> A ===> A) ===> (=) ) (ring_ow (Collect (Domainp A))) class.ring"

lemma reach_snoc: "reach tr p bs q \<Longrightarrow> reach tr p (bs @ [b]) (tr q b)"

lemma vinsert_iff[simp]: "x \<in>\<^sub>\<circ> vinsert y A \<longleftrightarrow> x = y \<or> x \<in>\<^sub>\<circ> A"

lemma float_eq_refl [simp]: "a \<doteq> a \<longleftrightarrow> \<not> is_nan a"

lemma subst_typ'_AList_clearjunk: "subst_typ' insts t = subst_typ' (AList.clearjunk insts) t"

lemma VARusefulSYSTEM_holds: "VARusefulSYSTEM"

lemma hom_f: "homogeneous f"

lemma fold_invalid_means_one_invalid: "gval (fold gAnd G (Bc True)) s = invalid \<Longrightarrow> \<exists>g \<in> set G. gval g s = invalid"

lemma other_net_tree_ips_par_right: assumes "other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) \<sigma> \<sigma>'" and "\<And>\<xi>. U \<xi> \<xi>" shows "other U (net_tree_ips p\<^sub>2) \<sigma> \<sigma>'"

lemma is_parser_exactly [intro]: "is_parser (exactly xs)"

lemma infinite_int_iff_infinite_nat_abs: "infinite S \<longleftrightarrow> infinite ((nat \<circ> abs) ` S)" for S :: "int set"