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

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lemma "\<CC> \<C> \<Longrightarrow> \<forall>A. Fr(\<F> A)"
lemma E_inf_Skip: "step.E_inf (Skip, s) (r f) = f s"
lemma norm_ge_zero [simp]: "0 \<le> norm x"
lemma LI_preproc_sem_eq: "\<lbrakk>M; S\<rbrakk>\<^sub>c \<I> \<longleftrightarrow> \<lbrakk>M; LI_preproc S\<rbrakk>\<^sub>c \<I>" (is "?A \<longleftrightarrow> ?B")
lemma minimize_wrt_eq: assumes "distinct xs" and "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> P x y \<longleftrightarrow> Q x y \<or> x = y" shows "minimize_wrt P xs = minimize_wrt Q xs"
lemma divideC_field_splits_simps_1 [field_split_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed ) "a = b /\<^sub>C c \<longleftrightarrow> (if c = 0 then a = 0 else c \<^sub>C a = b)" "b /\<^sub>C c = a \<longleftrightarrow> (if c = 0 then a = 0 else b = c \<^sub>C a)" "a + b /\<^sub>C c = (if c = 0 then a else (c \<^sub>C a + b) /\<^sub>C c)" "a /\<^sub>C c + b = (if c = 0 then b else (a + c \<^sub>C b) /\<^sub>C c)" "a - b /\<^sub>C c = (if c = 0 then a else (c \<^sub>C a - b) /\<^sub>C c)" "a /\<^sub>C c - b = (if c = 0 then - b else (a - c \<^sub>C b) /\<^sub>C c)" "- (a /\<^sub>C c) + b = (if c = 0 then b else (- a + c \<^sub>C b) /\<^sub>C c)" "- (a /\<^sub>C c) - b = (if c = 0 then - b else (- a - c *\<^sub>C b) /\<^sub>C c)" for a b :: "'a :: complex_vector"
lemma "(P :: nat \<Rightarrow> bool) (The P)"
lemma compact_AR: fixes S :: "'a::euclidean_space set" shows "compact S \<and> AR S \<longleftrightarrow> compact S \<and> S retract_of UNIV"
lemma unrest_uop [unrest]: "x \<sharp> e \<Longrightarrow> x \<sharp> uop f e"
lemma valid_regions_distinct: "valid_region X k I r \<Longrightarrow> valid_region X k I' r' \<Longrightarrow> v \<in> region X I r \<Longrightarrow> v \<in> region X I' r' \<Longrightarrow> region X I r = region X I' r'"
lemma CompDiag_Diag_Ide [simp]: assumes "Diag t" and "Ide a" and "Dom t = Cod a" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> a = t"
lemma path_valid_edges:"n -as\<rightarrow>* n' \<Longrightarrow> \<forall>a \<in> set as. valid_edge a"
lemma map_entries_parametric: "((A ===> B) ===> (A ===> C ===> rel_option D) ===> (B ===> rel_option C) ===> A ===> rel_option D) (\<lambda>f g m x. case (m \<circ> f) x of None \<Rightarrow> None \| Some y \<Rightarrow> g x y) (\<lambda>f g m x. case (m \<circ> f) x of None \<Rightarrow> None \| Some y \<Rightarrow> g x y)"
lemma downsetI[intro]: assumes "\<And> xs. xs \<in> xss \<Longrightarrow> xs \<noteq> [] \<Longrightarrow> butlast xs \<in> xss" shows "downset xss"
lemma Figure_4: assumes "foldl1 f1 (map (h1 k i) js) = One" and "js \<noteq> []" shows "i \<in> set js"
lemma map_of_subst: "map_of (\<Gamma>[x::h=y]) k = map_option (\<lambda> e . e[x::=y]) (map_of \<Gamma> k)"
lemma tfun_pre_expression: "x \<in> While_program \<Longrightarrow> p \<in> Pre_expression \<Longrightarrow> q \<in> Pre_expression \<Longrightarrow> r \<in> Pre_expression \<Longrightarrow> tfun p x q r \<in> Pre_expression"
lemma i_hcomplex_of_hypreal [simp]: "\<And>r. iii * hcomplex_of_hypreal r = HComplex 0 r"
lemma bisim_inv: "bisim_inv"
lemma (in anon_papp) is_pref_profile_replace: assumes "is_pref_profile A" and "X \<in># A" and "Y \<noteq> {}" and "Y \<subseteq> parties" shows "is_pref_profile (A - {#X#} + {#Y#})"
lemma fr_Send2: assumes h1:"\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)" and h2:"DisjointSchedules n nC" and h3:"IdenticCycleLength n nC" and h4:"t mod cycleLength (nC k) mem schedule (nC k)" and h5:"k < n" shows "nSend k t = nReturn k t"
lemma seq_tm_next: assumes a_ht: "steps0 (1, tap) A n = (0, tap')" and a_composable: "composable_tm (A, 0)" obtains n' where "steps0 (1, tap) (A \|+\| B) n' = (Suc (length A div 2), tap')"
lemma Var_injD: "Var x = Var y \<Longrightarrow> x \<in> var \<Longrightarrow> y \<in> var \<Longrightarrow> x = y"
lemma generaliseConj: assumes i1: "PROP Pure.prop (PROP Pure.prop (Trueprop P) \<Longrightarrow> PROP Pure.prop (Trueprop Q))" assumes i2: "PROP Pure.prop (PROP Pure.prop (Trueprop P') \<Longrightarrow> PROP Pure.prop (Trueprop Q'))" shows "PROP Pure.prop (PROP Pure.prop (Trueprop (P \<and> P')) \<Longrightarrow> (PROP Pure.prop (Trueprop (Q \<and> Q'))))"
lemma eff_children: assumes \<open>\<not> branchDone z\<close> \<open>eff r (A, z) ss\<close> shows \<open>\<forall>z' \<in> set (children (remdups (A @ subtermFms z)) r z). \<exists>B. (B, z') \|\<in>\| ss\<close>
lemma blinfun_scaleR_transfer[transfer_rule]: "rel_fun (pcr_blinfun (=) (=)) (rel_fun (=) (pcr_blinfun (=) (=))) (\<lambda>a b c. a c *\<^sub>R b) blinfun_scaleR"
lemma fold_Option_bind_eq_Some_start_not_None: "fold (\<lambda>new option . Option.bind option (f new)) list start = Some res \<Longrightarrow> start \<noteq> None"
lemma rem_effectless_works_5_i: shows "subseq (rem_effectless_act as) as"
lemma Quotient_lmap_Abs_Rep: "Quotient3 R Abs Rep \<Longrightarrow> lmap Abs (lmap Rep a) = a"
lemma nth_is_and_neq_0: "bit (x::'a::len word) n = (x AND 2 ^ n \<noteq> 0)"
lemma apps_append[simp]: "apps s (ss @ ts) = apps (apps s ss) ts"
lemma preregister_mult_left: \<open>preregister (\<lambda>a. z \<circ>\<^sub>m a)\<close>
lemma valid_slide: assumes "0 < fst w" "fst w \<le> snd w" "valid t" "l t \<le> fst w" "r t \<le> snd w" "snd w \<le> length as" shows "valid (slide t w) \<and> l (slide t w) = fst w \<and> r (slide t w) = snd w"
lemma CompDiag_Ide_Diag [simp]: assumes "Diag t" and "Ide a" and "Dom a = Cod t" shows "a \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
lemma scale_flow: assumes f: "flow \<Delta> f" and c: "c \<le> 1" shows "flow \<Delta> (\<lambda>e. c * f e)"
lemma H_eq: "P \<subseteq> Id \<Longrightarrow> Q \<subseteq> Id \<Longrightarrow> rel_kat.H P X Q = rel_antidomain_kleene_algebra.H P X Q"
lemma Interleaves_all_nil_1 [rule_format]: "xs \<cong> {xs, [], P} \<longrightarrow> (\<forall>n < length xs. P (xs ! n) (drop (Suc n) xs))"
lemma relax_outgoing''_refine: assumes "set l = {(d,v). w (u,v) = enat d}" shows "relax_outgoing'' l du V Q = relax_outgoing' u du V Q"
lemma ipassmt_sanity_nowildcards_match_iface: "ipassmt_sanity_nowildcards ipassmt \<Longrightarrow> ipassmt (Iface ifce2) = None \<Longrightarrow> ipassmt ifce = Some a \<Longrightarrow> \<not> match_iface ifce ifce2"
lemma (in PolynRg) add_cf_len:"\<lbrakk>pol_coeff S c; pol_coeff S d\<rbrakk> \<Longrightarrow> fst (add_cf S c d) = (max (fst c) (fst d))"
lemma poly_prime_factorization_exists_content_1: fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly" assumes "p \<noteq> 0" "content p = 1" shows "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
lemma successive_stepI: "successive P xs \<Longrightarrow> \<not> P x \<Longrightarrow> successive P (x # xs)"
lemma rb_Nil_sp : assumes "RedBlack prb" shows "subpath (red prb) rv1 [] rv2 (subs prb) = (rv1 \<in> red_vertices prb \<and> (rv1 = rv2 \<or> (rv1,rv2) \<in> (subs prb)))"
lemma composite_of_arr_ide: assumes "ide b" shows "composite_of t b t \<longleftrightarrow> t \\ t \<frown> b"
lemma gfp_dual: "(\<partial>::'a::complete_lattice_with_dual \<Rightarrow> 'a) \<circ> gfp = lfp \<circ> \<partial>\<^sub>F"
lemma single_step: "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow> \<langle>x4,x5\<rangle> \<Longrightarrow> x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow>* \<langle>x4,x5\<rangle>"
lemma latin_rect_iff: "m\<le>n \<and> partial_latin_square s n \<and> card s = n*m \<and> (\<forall>e\<in>s. e Row < m) \<longleftrightarrow> latin_rect s m n"
lemma pairs_substI'[intro]: "subst_domain \<theta> \<inter> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = {} \<Longrightarrow> F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta> = F"
theorem list_dtree_comb: "list_dtree (combine x y t)"
lemma gen_eq_gen_ML: "gen A t = gen_ML A t"
lemma exec_move_Cond2: assumes pc: "pc < length (compE2 e1)" shows "exec_move ci P t (if (e) e1 else e2) h (stk, loc, (Suc (length (compE2 e) + pc)), xcp) ta h' (stk', loc', (Suc (length (compE2 e) + pc')), xcp') = exec_move ci P t e1 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" (is "?lhs = ?rhs")
lemma main_step_aux2: fixes qs:: "rat poly list" assumes lenh: "length (fst(factorize_polys qs)) > 0" shows "set (find_consistent_signs qs) = consistent_sign_vectors qs UNIV"
lemma ntsmcf_vcomp_NTMap_vsv[dg_shared_cs_intros, smc_cs_intros]: "vsv ((\<MM> \<bullet>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTMap\<rparr>)"
lemma spmf_geometric_nonpos: "p \<le> 0 \<Longrightarrow> geometric_spmf p = return_pmf None"
lemma open_map_id: "open_map X X id"
lemma \<Phi>2_X\<Phi>: \<open>\<Phi>2 a = X\<Phi> (id_cblinfun \<otimes>\<^sub>o (id_cblinfun \<otimes>\<^sub>o a))\<close>
lemma OUTfromVCorrect1_data15: "OUTfromVCorrect1 data15"
lemma strictly_subsumes_subsumes_list[code_unfold]: "strictly_subsumes (mset Ls) (mset Ks) = (subsumes_list Ls Ks Map.empty \<and> \<not> subsumes_list Ks Ls Map.empty)"
lemma PROB_DOM_PROJ_DIFF: fixes P vs shows "prob_dom (prob_proj PROB (prob_dom PROB - vs)) = (prob_dom PROB) - vs"
lemma fun_typ_eq_body_unique: fixes v::v and x1::x and x2::x and s1'::s and s2'::s assumes "(AF_fun_typ x1 b1 c1 \<tau>1' s1') = (AF_fun_typ x2 b2 c2 \<tau>2' s2')" shows "s1'[x1::=v]\<^sub>s\<^sub>v = s2'[x2::=v]\<^sub>s\<^sub>v"
lemma seg_start_indep[simp]: "GS.seg_start (S',B,I',P') = seg_start"
lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al"
lemma continuous_openin_preimage_eq: "continuous_on S f \<longleftrightarrow> (\<forall>T. open T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
lemma jmm_JVM_heap_conf: "JVM_heap_conf addr2thread_id thread_id2addr jmm_empty jmm_allocate (jmm_typeof_addr P) jmm_heap_write jmm_hconf P"
lemma wt_instrs_ext: "\<lbrakk> \<turnstile> is\<^sub>1,xt\<^sub>1 [::] \<tau>s\<^sub>1@\<tau>s\<^sub>2; \<turnstile> is\<^sub>2,xt\<^sub>2 [::] \<tau>s\<^sub>2; size \<tau>s\<^sub>1 = size is\<^sub>1 \<rbrakk> \<Longrightarrow> \<turnstile> is\<^sub>1@is\<^sub>2, xt\<^sub>1 @ shift (size is\<^sub>1) xt\<^sub>2 [::] \<tau>s\<^sub>1@\<tau>s\<^sub>2"
lemma ereal_of_enat_simps[simp]: "ereal_of_enat (enat n) = ereal n" "ereal_of_enat \<infinity> = \<infinity>"
lemma query_prog[named_ss vcg_bb]: "(\<pi>(k\<mapsto>v)) k' = (if k'=k then Some v else \<pi> k')" for \<pi> :: program
lemma infdist_Un_min: assumes "A \<noteq> {}" "B \<noteq> {}" shows "infdist x (A \<union> B) = min (infdist x A) (infdist x B)"
lemma unzip_3_tailrec [code]: "unzip_3 l = unzip_3_tailrec l"
lemma gu_condition_imply_secure_2 [rule_format]: assumes RUC: "ref_union_closed P" and VP: "view_partition P D R" and WFC: "weakly_future_consistent P I D R" and WSC: "weakly_step_consistent P D R" and LR: "locally_respects P I D R" and Y: "xs @ [y] \<in> traces P" shows "(xs @ zs, Z) \<in> failures P \<longrightarrow> (xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> failures P"
lemma lead_monom_numeral [simp]: "lead_monom (numeral n) = 0"
lemma basis_reduction_mod_add_row: assumes Linv: "LLL_invariant_mod_weak fs mfs dmu p first b" and res: "basis_reduction_mod_add_row p mfs dmu i j = (mfs', dmu')" and i: "i < m" and j: "j < i" and igtz: "i \<noteq> 0" shows "(\<exists>fs'. LLL_invariant_mod_weak fs' mfs' dmu' p first b \<and> LLL_measure i fs' = LLL_measure i fs \<and> (\<mu>_small_row i fs (Suc j) \<longrightarrow> \<mu>_small_row i fs' j) \<and> \<bar>\<mu> fs' i j\<bar> \<le> 1 / 2 \<and> (\<forall>i' j'. i' < i \<longrightarrow> j' \<le> i' \<longrightarrow> \<mu> fs' i' j' = \<mu> fs i' j') \<and> (LLL_invariant_mod fs mfs dmu p first b i \<longrightarrow> LLL_invariant_mod fs' mfs' dmu' p first b i) \<and> (\<forall>ii \<le> m. d fs' ii = d fs ii))"
lemma [autoref_rules]: "(bs_inter,(\<inter>))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
lemma substT_atrm[simp]: assumes "r \<in> atrm" and "x \<in> var" and "t \<in> atrm" shows "substT r t x \<in> atrm"
lemma dec_interp_enc_Inr: "\<lbrakk>dec_interp n FO (enc (w, I)) ! i = Inr P'; I ! i = Inr P; i \<notin> FO; i < n; length I = n; \<forall>p \<in> P. p < length w\<rbrakk> \<Longrightarrow> P = P'"
lemma gt_comm: assumes "A B Gt C D" shows "B A Gt D C"
lemma lsl_Inf_closed: "Inf_closed_set (Fix (\<nu>::'a::unital_quantale \<Rightarrow> 'a))"
lemma lift_Pareto_SD_iff: assumes "p \<in> lotteries_on alts" "q \<in> lotteries_on alts" shows "p \<preceq>[Pareto(SD \<circ> R')] q \<longleftrightarrow> p \<preceq>[Pareto(SD \<circ> R)] q"
lemma connected_ball [iff]: fixes x :: "'a::real_normed_vector" shows "connected (ball x e)"
lemma representation_neg: "independent basis \<Longrightarrow> v \<in> span basis \<Longrightarrow> representation basis (- v) = (\<lambda>b. - representation basis v b)"
lemma [code abstract]: "integer_of_nat (m mod n) = of_nat m mod of_nat n"
lemma uniformly_continuous_on_cmul_left[continuous_intros]: fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra" assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
theorem weak_decoupling: assumes leadsTo: "F \<in> A leadsTo B" and stable: "F\<squnion>G \<in> stable B" and dec: "decoupled F (F\<squnion>G)" shows "F\<squnion>G \<in> A leadsTo B"
lemma meyer_1: "x\<^sup>\<star> = (1 + x) \<cdot> (x \<cdot> x)\<^sup>\<star>"
lemma conj_assoc:"(((P::'\<alpha> upred) \<and> Q) \<and> S) = (P \<and> (Q \<and> S))"
lemma spectrum_non_empty: assumes A: "(A :: complex mat) \<in> carrier_mat n n" and n: "n > 0" shows "spectrum A \<noteq> {}"
lemma continuous_map_cases_alt: assumes f: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. P x})) Y f" and g: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. ~P x})) Y g" and fg: "\<And>x. x \<in> X frontier_of {x \<in> topspace X. P x} \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
lemma fA_rel_inv[intro]: notes fun_upd_apply[simp] shows "\<lbrace> LSTP fA_rel_inv \<rbrace> mutator m"
lemma heap_copies_known_addrs_WriteMem: assumes "heap_copies a a' als h obs h'" and "obs ! n = WriteMem ad al (Addr a'')" "n < length obs" shows "a'' \<in> new_obs_addrs (take n obs)"
lemma statImpEnt: fixes \<Psi> :: 'b and \<Psi>' :: 'b and \<Phi> :: 'c assumes "\<Psi> \<hookrightarrow> \<Psi>'" and "\<Psi> \<turnstile> \<Phi>" shows "\<Psi>' \<turnstile> \<Phi>"
lemma ex_x_axis_poincare_distance_positive: assumes "d \<ge> 0" shows "\<exists> z. is_real z \<and> Re z \<ge> 0 \<and> Re z < 1 \<and> of_complex z \<in> unit_disc \<and> of_complex z \<in> circline_set x_axis \<and> poincare_distance 0\<^sub>h (of_complex z) = d" (is "\<exists> z. is_real z \<and> Re z \<ge> 0 \<and> Re z < 1 \<and> ?P z")
lemma enn2real_positive_iff: "0 < enn2real x \<longleftrightarrow> (0 < x \<and> x < top)"
lemma (in encoding) indRelTEQ_impl_TRel_is_weak_reduction_coupled_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes couSim: "weak_reduction_coupled_simulation (indRelTEQ TRel) (STCal Source Target)" shows "weak_reduction_coupled_simulation (TRel\<^sup>*) Target"
lemma eqExcPID2_imp2: assumes "eqExcPID2 s s1" and "pid \<noteq> PID \<or> PID \<noteq> pid" shows "getReviewersReviews s cid pid = getReviewersReviews s1 cid pid"
lemma jvm_make_test_prog_sees_Test_main: assumes P: "P \<in> jvm_progs" and t: "t \<in> jvm_tests" shows "\<exists>m. jvm_make_test_prog P t \<turnstile> Test sees main, Static : []\<rightarrow>Void = m in Test"
lemma i_set_Div_not_closed: "Suc 0 < k \<Longrightarrow> \<exists>I\<in>i_set. I \<oslash> k \<notin> i_set"
lemma MGT_implies_complete: "{} \<turnstile>\<^sub>t MGT\<^sub>t c \<Longrightarrow> {} \<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> {} \<turnstile>\<^sub>t {P}c{Q::state assn}"
lemma cnt_distinct_intro: "\<forall> x \<in> set xs. cnt x xs \<le> 1 \<Longrightarrow> distinct xs"
lemma ls_delete_distinct: "distinct (map fst l) \<Longrightarrow> distinct (map fst (fst (ls_delete k l)))"
lemma n_n_L: "n(n(x) * L) = n(x)"

lemma "\<CC> \<C> \<Longrightarrow> \<forall>A. Fr(\<F> A)"

lemma E_inf_Skip: "step.E_inf (Skip, s) (r f) = f s"

lemma norm_ge_zero [simp]: "0 \<le> norm x"

lemma LI_preproc_sem_eq: "\<lbrakk>M; S\<rbrakk>\<^sub>c \<I> \<longleftrightarrow> \<lbrakk>M; LI_preproc S\<rbrakk>\<^sub>c \<I>" (is "?A \<longleftrightarrow> ?B")

lemma minimize_wrt_eq: assumes "distinct xs" and "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> P x y \<longleftrightarrow> Q x y \<or> x = y" shows "minimize_wrt P xs = minimize_wrt Q xs"

lemma divideC_field_splits_simps_1 [field_split_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *) "a = b /\<^sub>C c \<longleftrightarrow> (if c = 0 then a = 0 else c *\<^sub>C a = b)" "b /\<^sub>C c = a \<longleftrightarrow> (if c = 0 then a = 0 else b = c *\<^sub>C a)" "a + b /\<^sub>C c = (if c = 0 then a else (c *\<^sub>C a + b) /\<^sub>C c)" "a /\<^sub>C c + b = (if c = 0 then b else (a + c *\<^sub>C b) /\<^sub>C c)" "a - b /\<^sub>C c = (if c = 0 then a else (c *\<^sub>C a - b) /\<^sub>C c)" "a /\<^sub>C c - b = (if c = 0 then - b else (a - c *\<^sub>C b) /\<^sub>C c)" "- (a /\<^sub>C c) + b = (if c = 0 then b else (- a + c *\<^sub>C b) /\<^sub>C c)" "- (a /\<^sub>C c) - b = (if c = 0 then - b else (- a - c *\<^sub>C b) /\<^sub>C c)" for a b :: "'a :: complex_vector"

lemma "(P :: nat \<Rightarrow> bool) (The P)"

lemma compact_AR: fixes S :: "'a::euclidean_space set" shows "compact S \<and> AR S \<longleftrightarrow> compact S \<and> S retract_of UNIV"

lemma unrest_uop [unrest]: "x \<sharp> e \<Longrightarrow> x \<sharp> uop f e"

lemma valid_regions_distinct: "valid_region X k I r \<Longrightarrow> valid_region X k I' r' \<Longrightarrow> v \<in> region X I r \<Longrightarrow> v \<in> region X I' r' \<Longrightarrow> region X I r = region X I' r'"

lemma CompDiag_Diag_Ide [simp]: assumes "Diag t" and "Ide a" and "Dom t = Cod a" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> a = t"

lemma path_valid_edges:"n -as\<rightarrow>* n' \<Longrightarrow> \<forall>a \<in> set as. valid_edge a"

lemma map_entries_parametric: "((A ===> B) ===> (A ===> C ===> rel_option D) ===> (B ===> rel_option C) ===> A ===> rel_option D) (\<lambda>f g m x. case (m \<circ> f) x of None \<Rightarrow> None | Some y \<Rightarrow> g x y) (\<lambda>f g m x. case (m \<circ> f) x of None \<Rightarrow> None | Some y \<Rightarrow> g x y)"

lemma downsetI[intro]: assumes "\<And> xs. xs \<in> xss \<Longrightarrow> xs \<noteq> [] \<Longrightarrow> butlast xs \<in> xss" shows "downset xss"

lemma Figure_4: assumes "foldl1 f1 (map (h1 k i) js) = One" and "js \<noteq> []" shows "i \<in> set js"

lemma map_of_subst: "map_of (\<Gamma>[x::h=y]) k = map_option (\<lambda> e . e[x::=y]) (map_of \<Gamma> k)"

lemma tfun_pre_expression: "x \<in> While_program \<Longrightarrow> p \<in> Pre_expression \<Longrightarrow> q \<in> Pre_expression \<Longrightarrow> r \<in> Pre_expression \<Longrightarrow> tfun p x q r \<in> Pre_expression"

lemma i_hcomplex_of_hypreal [simp]: "\<And>r. iii * hcomplex_of_hypreal r = HComplex 0 r"

lemma bisim_inv: "bisim_inv"

lemma (in anon_papp) is_pref_profile_replace: assumes "is_pref_profile A" and "X \<in># A" and "Y \<noteq> {}" and "Y \<subseteq> parties" shows "is_pref_profile (A - {#X#} + {#Y#})"

lemma fr_Send2: assumes h1:"\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)" and h2:"DisjointSchedules n nC" and h3:"IdenticCycleLength n nC" and h4:"t mod cycleLength (nC k) mem schedule (nC k)" and h5:"k < n" shows "nSend k t = nReturn k t"

lemma seq_tm_next: assumes a_ht: "steps0 (1, tap) A n = (0, tap')" and a_composable: "composable_tm (A, 0)" obtains n' where "steps0 (1, tap) (A |+| B) n' = (Suc (length A div 2), tap')"

lemma Var_injD: "Var x = Var y \<Longrightarrow> x \<in> var \<Longrightarrow> y \<in> var \<Longrightarrow> x = y"

lemma generaliseConj: assumes i1: "PROP Pure.prop (PROP Pure.prop (Trueprop P) \<Longrightarrow> PROP Pure.prop (Trueprop Q))" assumes i2: "PROP Pure.prop (PROP Pure.prop (Trueprop P') \<Longrightarrow> PROP Pure.prop (Trueprop Q'))" shows "PROP Pure.prop (PROP Pure.prop (Trueprop (P \<and> P')) \<Longrightarrow> (PROP Pure.prop (Trueprop (Q \<and> Q'))))"

lemma eff_children: assumes \<open>\<not> branchDone z\<close> \<open>eff r (A, z) ss\<close> shows \<open>\<forall>z' \<in> set (children (remdups (A @ subtermFms z)) r z). \<exists>B. (B, z') |\<in>| ss\<close>

lemma blinfun_scaleR_transfer[transfer_rule]: "rel_fun (pcr_blinfun (=) (=)) (rel_fun (=) (pcr_blinfun (=) (=))) (\<lambda>a b c. a c *\<^sub>R b) blinfun_scaleR"

lemma fold_Option_bind_eq_Some_start_not_None: "fold (\<lambda>new option . Option.bind option (f new)) list start = Some res \<Longrightarrow> start \<noteq> None"

lemma rem_effectless_works_5_i: shows "subseq (rem_effectless_act as) as"

lemma Quotient_lmap_Abs_Rep: "Quotient3 R Abs Rep \<Longrightarrow> lmap Abs (lmap Rep a) = a"

lemma nth_is_and_neq_0: "bit (x::'a::len word) n = (x AND 2 ^ n \<noteq> 0)"

lemma apps_append[simp]: "apps s (ss @ ts) = apps (apps s ss) ts"

lemma preregister_mult_left: \<open>preregister (\<lambda>a. z \<circ>\<^sub>m a)\<close>

lemma valid_slide: assumes "0 < fst w" "fst w \<le> snd w" "valid t" "l t \<le> fst w" "r t \<le> snd w" "snd w \<le> length as" shows "valid (slide t w) \<and> l (slide t w) = fst w \<and> r (slide t w) = snd w"

lemma CompDiag_Ide_Diag [simp]: assumes "Diag t" and "Ide a" and "Dom a = Cod t" shows "a \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"

lemma scale_flow: assumes f: "flow \<Delta> f" and c: "c \<le> 1" shows "flow \<Delta> (\<lambda>e. c * f e)"

lemma H_eq: "P \<subseteq> Id \<Longrightarrow> Q \<subseteq> Id \<Longrightarrow> rel_kat.H P X Q = rel_antidomain_kleene_algebra.H P X Q"

lemma Interleaves_all_nil_1 [rule_format]: "xs \<cong> {xs, [], P} \<longrightarrow> (\<forall>n < length xs. P (xs ! n) (drop (Suc n) xs))"

lemma relax_outgoing''_refine: assumes "set l = {(d,v). w (u,v) = enat d}" shows "relax_outgoing'' l du V Q = relax_outgoing' u du V Q"

lemma ipassmt_sanity_nowildcards_match_iface: "ipassmt_sanity_nowildcards ipassmt \<Longrightarrow> ipassmt (Iface ifce2) = None \<Longrightarrow> ipassmt ifce = Some a \<Longrightarrow> \<not> match_iface ifce ifce2"

lemma (in PolynRg) add_cf_len:"\<lbrakk>pol_coeff S c; pol_coeff S d\<rbrakk> \<Longrightarrow> fst (add_cf S c d) = (max (fst c) (fst d))"

lemma poly_prime_factorization_exists_content_1: fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly" assumes "p \<noteq> 0" "content p = 1" shows "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"

lemma successive_stepI: "successive P xs \<Longrightarrow> \<not> P x \<Longrightarrow> successive P (x # xs)"

lemma rb_Nil_sp : assumes "RedBlack prb" shows "subpath (red prb) rv1 [] rv2 (subs prb) = (rv1 \<in> red_vertices prb \<and> (rv1 = rv2 \<or> (rv1,rv2) \<in> (subs prb)))"

lemma composite_of_arr_ide: assumes "ide b" shows "composite_of t b t \<longleftrightarrow> t \\ t \<frown> b"

lemma gfp_dual: "(\<partial>::'a::complete_lattice_with_dual \<Rightarrow> 'a) \<circ> gfp = lfp \<circ> \<partial>\<^sub>F"

lemma single_step: "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow> \<langle>x4,x5\<rangle> \<Longrightarrow> x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow>* \<langle>x4,x5\<rangle>"

lemma latin_rect_iff: "m\<le>n \<and> partial_latin_square s n \<and> card s = n*m \<and> (\<forall>e\<in>s. e Row < m) \<longleftrightarrow> latin_rect s m n"

lemma pairs_substI'[intro]: "subst_domain \<theta> \<inter> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = {} \<Longrightarrow> F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta> = F"

theorem list_dtree_comb: "list_dtree (combine x y t)"

lemma gen_eq_gen_ML: "gen A t = gen_ML A t"

lemma exec_move_Cond2: assumes pc: "pc < length (compE2 e1)" shows "exec_move ci P t (if (e) e1 else e2) h (stk, loc, (Suc (length (compE2 e) + pc)), xcp) ta h' (stk', loc', (Suc (length (compE2 e) + pc')), xcp') = exec_move ci P t e1 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" (is "?lhs = ?rhs")

lemma main_step_aux2: fixes qs:: "rat poly list" assumes lenh: "length (fst(factorize_polys qs)) > 0" shows "set (find_consistent_signs qs) = consistent_sign_vectors qs UNIV"

lemma ntsmcf_vcomp_NTMap_vsv[dg_shared_cs_intros, smc_cs_intros]: "vsv ((\<MM> \<bullet>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTMap\<rparr>)"

lemma spmf_geometric_nonpos: "p \<le> 0 \<Longrightarrow> geometric_spmf p = return_pmf None"

lemma open_map_id: "open_map X X id"

lemma \<Phi>2_X\<Phi>: \<open>\<Phi>2 a = X\<Phi> (id_cblinfun \<otimes>\<^sub>o (id_cblinfun \<otimes>\<^sub>o a))\<close>

lemma OUTfromVCorrect1_data15: "OUTfromVCorrect1 data15"

lemma strictly_subsumes_subsumes_list[code_unfold]: "strictly_subsumes (mset Ls) (mset Ks) = (subsumes_list Ls Ks Map.empty \<and> \<not> subsumes_list Ks Ls Map.empty)"

lemma PROB_DOM_PROJ_DIFF: fixes P vs shows "prob_dom (prob_proj PROB (prob_dom PROB - vs)) = (prob_dom PROB) - vs"

lemma fun_typ_eq_body_unique: fixes v::v and x1::x and x2::x and s1'::s and s2'::s assumes "(AF_fun_typ x1 b1 c1 \<tau>1' s1') = (AF_fun_typ x2 b2 c2 \<tau>2' s2')" shows "s1'[x1::=v]\<^sub>s\<^sub>v = s2'[x2::=v]\<^sub>s\<^sub>v"

lemma seg_start_indep[simp]: "GS.seg_start (S',B,I',P') = seg_start"

lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al"

lemma continuous_openin_preimage_eq: "continuous_on S f \<longleftrightarrow> (\<forall>T. open T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"

lemma jmm_JVM_heap_conf: "JVM_heap_conf addr2thread_id thread_id2addr jmm_empty jmm_allocate (jmm_typeof_addr P) jmm_heap_write jmm_hconf P"

lemma wt_instrs_ext: "\<lbrakk> \<turnstile> is\<^sub>1,xt\<^sub>1 [::] \<tau>s\<^sub>1@\<tau>s\<^sub>2; \<turnstile> is\<^sub>2,xt\<^sub>2 [::] \<tau>s\<^sub>2; size \<tau>s\<^sub>1 = size is\<^sub>1 \<rbrakk> \<Longrightarrow> \<turnstile> is\<^sub>1@is\<^sub>2, xt\<^sub>1 @ shift (size is\<^sub>1) xt\<^sub>2 [::] \<tau>s\<^sub>1@\<tau>s\<^sub>2"

lemma ereal_of_enat_simps[simp]: "ereal_of_enat (enat n) = ereal n" "ereal_of_enat \<infinity> = \<infinity>"

lemma query_prog[named_ss vcg_bb]: "(\<pi>(k\<mapsto>v)) k' = (if k'=k then Some v else \<pi> k')" for \<pi> :: program

lemma infdist_Un_min: assumes "A \<noteq> {}" "B \<noteq> {}" shows "infdist x (A \<union> B) = min (infdist x A) (infdist x B)"

lemma unzip_3_tailrec [code]: "unzip_3 l = unzip_3_tailrec l"

lemma gu_condition_imply_secure_2 [rule_format]: assumes RUC: "ref_union_closed P" and VP: "view_partition P D R" and WFC: "weakly_future_consistent P I D R" and WSC: "weakly_step_consistent P D R" and LR: "locally_respects P I D R" and Y: "xs @ [y] \<in> traces P" shows "(xs @ zs, Z) \<in> failures P \<longrightarrow> (xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> failures P"

lemma lead_monom_numeral [simp]: "lead_monom (numeral n) = 0"

lemma basis_reduction_mod_add_row: assumes Linv: "LLL_invariant_mod_weak fs mfs dmu p first b" and res: "basis_reduction_mod_add_row p mfs dmu i j = (mfs', dmu')" and i: "i < m" and j: "j < i" and igtz: "i \<noteq> 0" shows "(\<exists>fs'. LLL_invariant_mod_weak fs' mfs' dmu' p first b \<and> LLL_measure i fs' = LLL_measure i fs \<and> (\<mu>_small_row i fs (Suc j) \<longrightarrow> \<mu>_small_row i fs' j) \<and> \<bar>\<mu> fs' i j\<bar> \<le> 1 / 2 \<and> (\<forall>i' j'. i' < i \<longrightarrow> j' \<le> i' \<longrightarrow> \<mu> fs' i' j' = \<mu> fs i' j') \<and> (LLL_invariant_mod fs mfs dmu p first b i \<longrightarrow> LLL_invariant_mod fs' mfs' dmu' p first b i) \<and> (\<forall>ii \<le> m. d fs' ii = d fs ii))"

lemma [autoref_rules]: "(bs_inter,(\<inter>))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"

lemma substT_atrm[simp]: assumes "r \<in> atrm" and "x \<in> var" and "t \<in> atrm" shows "substT r t x \<in> atrm"

lemma dec_interp_enc_Inr: "\<lbrakk>dec_interp n FO (enc (w, I)) ! i = Inr P'; I ! i = Inr P; i \<notin> FO; i < n; length I = n; \<forall>p \<in> P. p < length w\<rbrakk> \<Longrightarrow> P = P'"

lemma gt_comm: assumes "A B Gt C D" shows "B A Gt D C"

lemma lsl_Inf_closed: "Inf_closed_set (Fix (\<nu>::'a::unital_quantale \<Rightarrow> 'a))"

lemma lift_Pareto_SD_iff: assumes "p \<in> lotteries_on alts" "q \<in> lotteries_on alts" shows "p \<preceq>[Pareto(SD \<circ> R')] q \<longleftrightarrow> p \<preceq>[Pareto(SD \<circ> R)] q"

lemma connected_ball [iff]: fixes x :: "'a::real_normed_vector" shows "connected (ball x e)"

lemma representation_neg: "independent basis \<Longrightarrow> v \<in> span basis \<Longrightarrow> representation basis (- v) = (\<lambda>b. - representation basis v b)"

lemma [code abstract]: "integer_of_nat (m mod n) = of_nat m mod of_nat n"

lemma uniformly_continuous_on_cmul_left[continuous_intros]: fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra" assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (\<lambda>x. c * f x)"

theorem weak_decoupling: assumes leadsTo: "F \<in> A leadsTo B" and stable: "F\<squnion>G \<in> stable B" and dec: "decoupled F (F\<squnion>G)" shows "F\<squnion>G \<in> A leadsTo B"

lemma meyer_1: "x\<^sup>\<star> = (1 + x) \<cdot> (x \<cdot> x)\<^sup>\<star>"

lemma conj_assoc:"(((P::'\<alpha> upred) \<and> Q) \<and> S) = (P \<and> (Q \<and> S))"

lemma spectrum_non_empty: assumes A: "(A :: complex mat) \<in> carrier_mat n n" and n: "n > 0" shows "spectrum A \<noteq> {}"

lemma continuous_map_cases_alt: assumes f: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. P x})) Y f" and g: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. ~P x})) Y g" and fg: "\<And>x. x \<in> X frontier_of {x \<in> topspace X. P x} \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"

lemma fA_rel_inv[intro]: notes fun_upd_apply[simp] shows "\<lbrace> LSTP fA_rel_inv \<rbrace> mutator m"

lemma heap_copies_known_addrs_WriteMem: assumes "heap_copies a a' als h obs h'" and "obs ! n = WriteMem ad al (Addr a'')" "n < length obs" shows "a'' \<in> new_obs_addrs (take n obs)"

lemma statImpEnt: fixes \<Psi> :: 'b and \<Psi>' :: 'b and \<Phi> :: 'c assumes "\<Psi> \<hookrightarrow> \<Psi>'" and "\<Psi> \<turnstile> \<Phi>" shows "\<Psi>' \<turnstile> \<Phi>"

lemma ex_x_axis_poincare_distance_positive: assumes "d \<ge> 0" shows "\<exists> z. is_real z \<and> Re z \<ge> 0 \<and> Re z < 1 \<and> of_complex z \<in> unit_disc \<and> of_complex z \<in> circline_set x_axis \<and> poincare_distance 0\<^sub>h (of_complex z) = d" (is "\<exists> z. is_real z \<and> Re z \<ge> 0 \<and> Re z < 1 \<and> ?P z")

lemma enn2real_positive_iff: "0 < enn2real x \<longleftrightarrow> (0 < x \<and> x < top)"

lemma (in encoding) indRelTEQ_impl_TRel_is_weak_reduction_coupled_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes couSim: "weak_reduction_coupled_simulation (indRelTEQ TRel) (STCal Source Target)" shows "weak_reduction_coupled_simulation (TRel\<^sup>*) Target"

lemma eqExcPID2_imp2: assumes "eqExcPID2 s s1" and "pid \<noteq> PID \<or> PID \<noteq> pid" shows "getReviewersReviews s cid pid = getReviewersReviews s1 cid pid"

lemma jvm_make_test_prog_sees_Test_main: assumes P: "P \<in> jvm_progs" and t: "t \<in> jvm_tests" shows "\<exists>m. jvm_make_test_prog P t \<turnstile> Test sees main, Static : []\<rightarrow>Void = m in Test"

lemma i_set_Div_not_closed: "Suc 0 < k \<Longrightarrow> \<exists>I\<in>i_set. I \<oslash> k \<notin> i_set"

lemma MGT_implies_complete: "{} \<turnstile>\<^sub>t MGT\<^sub>t c \<Longrightarrow> {} \<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> {} \<turnstile>\<^sub>t {P}c{Q::state assn}"

lemma cnt_distinct_intro: "\<forall> x \<in> set xs. cnt x xs \<le> 1 \<Longrightarrow> distinct xs"

lemma ls_delete_distinct: "distinct (map fst l) \<Longrightarrow> distinct (map fst (fst (ls_delete k l)))"

lemma n_n_L: "n(n(x) * L) = n(x)"