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

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lemma ex1_poincare_line_general: assumes "u \<noteq> v" "u \<noteq> inversion v" shows "\<exists>! l. is_poincare_line l \<and> u \<in> circline_set l \<and> v \<in> circline_set l"
lemma RS2Set[rule_format]: "set (removeShadowRules2 p) \<subseteq> set p"
lemma eLet_rule': assumes "\<And>x. x=v \<Longrightarrow> f x \<le> ESPEC \<Phi> \<Psi>" shows "Let v (\<lambda>x. f x) \<le> ESPEC \<Phi> \<Psi>"
lemma vfsequence_iff: "vfsequence xs \<longleftrightarrow> vsv xs \<and> \<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"
lemma set_in_finite_cone: assumes Vs: "Vs \<subseteq> carrier_vec n" and fin: "finite Vs" shows "Vs \<subseteq> finite_cone Vs"
lemma supt_list_sound [simp]: "set (supt_list t) = {s. t \<rhd> s}"
lemma (in Ring) submodule_over_zeroring:"\<lbrakk>zeroring R; R module M; submodule R M N\<rbrakk> \<Longrightarrow> N = {\<zero>\<^bsub>M\<^esub>}"
lemma (in Corps) n_value_x_1:"\<lbrakk>valuation K v; 0 \<le> n; x \<in> (vp K v) \<^bsup>(Vr K v) n\<^esup>\<rbrakk> \<Longrightarrow> n \<le> (n_val K v x)"
lemma sub_add: assumes "g \<in> carrier P" assumes "f \<in> carrier P" assumes "h \<in>carrier P" shows "((f \<oplus>\<^bsub>P\<^esub> h) of g) = ((f of g) \<oplus>\<^bsub>P\<^esub> (h of g))"
lemma cf_smcf_cf_id[slicing_commute]: "smcf_id (cat_smc \<CC>) = cf_smcf (cf_id \<CC>)"
lemma support_postList: "support (postList xs) \<subseteq> lesvars xs"
lemma equivP_pick_preserves: assumes "equivP P \<phi> " and "(P///\<phi>) X" shows "P (pick X)"
lemma pmdl_struct: assumes "struct_spec sel ap ab compl" and "compl_pmdl compl" and "is_Groebner_basis (fst ` set gs)" and "ps \<noteq> []" and "set ps \<subseteq> (set bs) \<times> (set gs \<union> set bs)" and "unique_idx (gs @ bs) (snd data)" and "sps = sel gs bs ps (snd data)" and "aux = compl gs bs (ps -- sps) sps (snd data)" and "(hs, data') = add_indices aux (snd data)" shows "pmdl (fst ` set (gs @ ab gs bs hs data')) = pmdl (fst ` set (gs @ bs))"
lemma next'_next: assumes "v \<noteq> \<bottom>" assumes "vs \<noteq> \<bottom>" shows "next'\<cdot>v\<cdot>(tree_map'\<cdot>csnd\<cdot>t) = tree_map'\<cdot>csnd\<cdot>(next\<cdot>(v :# vs)\<cdot>t)"
lemma (in encoding) indRelRPO_iff_exists_source_target_relation: fixes Pred :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) \<Rightarrow> bool" shows "(\<forall>(P, Q) \<in> indRelRPO. Pred (P, Q)) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q) \<in> Rel. Pred (P, Q)) \<and> preorder Rel)"
lemma poly_map_pullback_char: assumes "is_poly_tuple n fs" assumes "length fs = m" assumes "is_poly_tuple k gs" assumes "length gs = n" shows "(pullback (R\<^bsup>k\<^esup>) (poly_map k gs) (poly_map n fs)) = poly_map k (map (poly_compose n k gs) fs)"
lemma tfin_enat_code[code]: "(tfin :: enat set) = Collect_set (\<lambda>x. x \<noteq> \<infinity>)"
lemma append_queue_rep: "linearize (append_queue a q) = linearize q @ [a]"
lemma sum_Basis_sum_nth_Basis_list: "(\<Sum>i\<in>Basis. f i) = (\<Sum>i<DIM('a::executable_euclidean_space). f ((Basis_list::'a list) ! i))"
lemma border_short_dec: assumes border: "x \<le>b w" and short: "\<^bold>\|x\<^bold>\| + \<^bold>\|x\<^bold>\| \<le> \<^bold>\|w\<^bold>\|" shows "x \<cdot> x\<inverse>\<^sup>>(w\<^sup><\<inverse>x) \<cdot> x = w"
theorem Szemeredi_Regularity_Lemma: assumes "\<epsilon> > 0" obtains M where "\<And>G. card (uverts G) > 0 \<Longrightarrow> \<exists>P. regular_partition \<epsilon> G P \<and> card P \<le> M"
lemma [code]: "IArray.exists p as \<longleftrightarrow> exists_upto p (length' as) as"
lemma ttree_join_transfer[transfer_rule]: "rel_fun (pcr_ttree (=)) (rel_fun (pcr_ttree (=)) (pcr_ttree (=))) (\<union>) (\<squnion>)"
lemma wf_sees_method_fun: "\<lbrakk>P \<turnstile> C has least M = mthd via Cs; P \<turnstile> C has least M = mthd' via Cs'; wf_prog wf_md P\<rbrakk> \<Longrightarrow> mthd = mthd' \<and> Cs = Cs'"
lemma lns_distI [intro]: assumes "\<And>x e. x \<in> B \<Longrightarrow> e > 0 \<Longrightarrow> (\<exists>y\<in>B. (dist y x) \<le> e \<and> y \<succ>[P] x)" shows "local_nonsatiation B P"
lemma path2_transfer [transfer_rule]: assumes [transfer_rule]: "right_total A" and [transfer_rule]: "(A ===> (=)) \<alpha>e \<alpha>e2" and [transfer_rule]: "(A ===> (=)) \<alpha>n \<alpha>n2" and [transfer_rule]: "(A ===> (=)) invar invar2" and [transfer_rule]: "(A ===> (=)) inEdges' inEdges2" shows "(A ===> (=)) path2 (graph_path_base.path2 \<alpha>n2 invar2 inEdges2)"
lemma \<L>\<^sub>b_split_tendsto_opt: "(\<lambda>n. (\<L>\<^sub>b_split ^^ n) v) \<longlonglongrightarrow> \<nu>\<^sub>b_opt"
lemma(in UP_cring) UP_subring_taylor_appr: assumes "subring S R" assumes "g \<in> carrier (UP (R \<lparr> carrier := S \<rparr>))" assumes "a \<in> S" assumes "b \<in> S" shows "\<exists>c \<in> S. to_fun g a= to_fun g b \<oplus> (deriv g b)\<otimes> (a \<ominus> b) \<oplus> (c \<otimes> (a \<ominus> b)[^](2::nat))"
lemma pop_list [simp]: "invar common \<Longrightarrow> 0 < size common \<Longrightarrow> pop common = (x, common') \<Longrightarrow> x # list common' = list common"
lemma PiE_defaut_undefined_eq: "PiE_dflt I undefined M = PiE I M"
lemma find_first_distinct_ofsm_table_is_first : assumes "q1 \<in> FSM.states M" and "q2 \<in> FSM.states M" and "ofsm_table_fix M (\<lambda>q . states M) 0 q1 \<noteq> ofsm_table_fix M (\<lambda>q . states M) 0 q2" shows "ofsm_table M (\<lambda>q . states M) (find_first_distinct_ofsm_table M q1 q2) q1 \<noteq> ofsm_table M (\<lambda>q . states M) (find_first_distinct_ofsm_table M q1 q2) q2" and "k' < (find_first_distinct_ofsm_table M q1 q2) \<Longrightarrow> ofsm_table M (\<lambda>q . states M) k' q1 = ofsm_table M (\<lambda>q . states M) k' q2"
lemma collect_subfmlas_app: "\<exists>phis'. collect_subfmlas r phis = phis @ phis'"
lemma costBIT_4y: assumes "x\<noteq>y" "x : {x0,y0}" "y\<in>{x0,y0}" shows "E (type4 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s y \<bind> (\<lambda>(a, is'). return_pmf (real (t\<^sub>p (fst s) y a))))) = 0.5"
lemma holomorphic_on_Un [holomorphic_intros]: assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B" shows "f holomorphic_on (A \<union> B)"
lemma finite_cycles: "finite (cycles)"
lemma [simp,code_unfold]: "\<upsilon> \<two> = true"
lemma k_adju_var: "\<exists>F. \<forall>x.\<forall>f::'a::order \<Rightarrow> 'b::complete_lattice. (F x \<le> f) = (x \<le> (\<lambda>k. k y) f)"
lemma cf_list_entries: assumes "i \<le> deg R p" shows "(cf_list R p)!i = p i"
lemma unfold2' : assumes context_ok: "cp E" and args_def_or_valid: "(\<tau> \<Turnstile> \<delta> self) \<and> (f_\<upsilon> a1 \<tau>)" and pre_satisfied: "\<tau> \<Turnstile> PRE self a1" and postsplit_satisfied: "\<tau> \<Turnstile> POST' self a1" (* split constraint holds on post-state *) and post_decomposable : "\<And> res. (POST self a1 res) = ((POST' self a1) and (res \<triangleq> (BODY self a1)))" shows "(\<tau> \<Turnstile> E(f self a1)) = (\<tau> \<Turnstile> E(BODY self a1))"
lemma id_WHILEIT[id_rules]: "PR_CONST (WHILEIT I) ::\<^sub>i TYPE(('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a nres) \<Rightarrow> 'a \<Rightarrow> 'a nres)"
lemma equiv_up_to_refl [simp, intro!]: "P \<Turnstile> c \<sim> c"
lemma r_child_lower_bound: "l \<le> p \<Longrightarrow> p < r_child l p"
lemma eventually_in_cballs: assumes "d > 0" "c > 0" shows "eventually (\<lambda>e. cball t0 (c * e) \<times> (cball x0 e) \<subseteq> cball (t0, x0) d) (at_right 0)"
lemma LinLs: "L0 i j : Ls & L1 i j : Ls & L2 i j : Ls & L3 i j : Ls"
lemma wf_tuple_Suc_fvi_SomeI: "0 \<in> MFOTL.fvi b \<phi> \<Longrightarrow> wf_tuple n (MFOTL.fvi (Suc b) \<phi>) v \<Longrightarrow> wf_tuple (Suc n) (MFOTL.fvi b \<phi>) (Some x # v)"
lemma separated_by_wall_ex_foldpair: assumes "H\<in>walls" "separated_by H C D" shows "\<exists>(f,g)\<in>foldpairs. H = {f\<turnstile>\<C>,g\<turnstile>\<C>} \<and> C\<in>f\<turnstile>\<C> \<and> D\<in>g\<turnstile>\<C>"
lemma R_loop_mono: "X \<le> X' \<Longrightarrow> LOOP X INV I \<subseteq> LOOP X' INV I"
lemma eps_nfa'_step_eps_closure_cong: "step_eps_closure bs q q' \<Longrightarrow> q \<in> nfa'.Q \<Longrightarrow> (q' \<in> nfa'.Q \<and> nfa'.step_eps_closure bs q q') \<or> (nfa'.step_eps_closure bs q qf' \<and> step_eps_closure bs qf' q')"
lemma sees_method_mono2: "\<lbrakk> P \<turnstile> C' \<preceq>\<^sup>* C; wf_prog wf_md P; P \<turnstile> C sees M:Ts\<rightarrow>T = m in D; P \<turnstile> C' sees M:Ts'\<rightarrow>T' = m' in D' \<rbrakk> \<Longrightarrow> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
lemma true_dsij_zero:"(P \<or> true) = true"
lemma comparator_lex_comp_aux: "comparator (lex_comp_aux::'a::nat_term comparator)"
lemma closed_funI: assumes "\<And>x. g x \<in> carrier R" shows "closed_fun R g"
lemma REFL_IMP_3_CONJ_1: fixes R P x y assumes "((\<lambda>x y. R x y \<and> P x y)\<^sup>+\<^sup>+ x y)" shows "R\<^sup>+\<^sup>+ x y"
lemma (in \<Z>) ord_of_nat_in_Vset[simp]: "a\<^sub>\<nat> \<in>\<^sub>\<circ> Vset \<alpha>"
lemma "x\<^sup>\<omega> \<cdot> y = x\<^sup>\<omega>"
lemma shiftr_uint8_code [code]: "drop_bit n x = (if n < 8 then uint8_shiftr x (integer_of_nat n) else 0)"
lemma prj1_simps [simp]: assumes "cospan f g" shows "arr \<p>\<^sub>1[f, g]" and "dom \<p>\<^sub>1[f, g] = f \<down>\<down> g" and "cod \<p>\<^sub>1[f, g] = dom f"
lemma real_poly_uminus: assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (-p)) \<subseteq> \<real>"
lemma finite_tvs[simp]: "finite (tvs t)"
lemma min_element: fixes k :: nat assumes "\<exists> (m::nat). P m" shows "\<exists> mm. P mm \<and> (\<forall> m'. m' < mm \<longrightarrow> \<not> P m')"
lemma Snoc_step1_SnocD: "step1 r (ys @ [y]) (xs @ [x]) \<Longrightarrow> (step1 r ys xs \<and> y = x \<or> ys = xs \<and> r y x)"
lemma "((a::nat) - (b - (c - (d - e)))) = (a - (b - (c - (d - e))))"
lemma lift_Postdomination: assumes wf:"CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" and pd:"Postdomination sourcenode targetnode kind valid_edge Entry Exit" and inner:"CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "Postdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"
lemma lowner_lub_add: assumes "matrix_seq d f" "matrix_seq d g" "\<forall> n. trace (f n + g n) \<le> 1" shows "matrix_seq.lowner_lub (\<lambda>n. f n + g n) = matrix_seq.lowner_lub f + matrix_seq.lowner_lub g"
lemma merge_mdeg_le_1: "max_deg (merge t1) \<le> 1"
lemma Fls_E [intro!]: "insert Fls H \<turnstile> A"
lemma (in bounded_linear) has_derivative: "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
lemma interpretation_grounds_all: "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<Longrightarrow> (\<And>v. fv (\<theta> v) = {})"
lemma det3_nonneg_scaleR_segment2: assumes "det3 x y z \<ge> 0" assumes "a > 0" shows "det3 x ((1 - a) \<^sub>R x + a \<^sub>R y) z \<ge> 0"
lemma lossless_spmf_inline1: assumes lossless: "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (the_gpv (callee s x))" shows "lossless_spmf (inline1 callee gpv s)"
lemma sub2_closed [simp]: "sub2 d l (r,A) = (r',A') \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> A' \<in> carrier_mat m n"
lemma ord_term_canc_left: assumes "t \<oplus> v \<preceq>\<^sub>t s \<oplus> v" shows "t \<preceq> s"
lemma itrev: "itrev xs ys = rev xs @ ys"
lemma space_x0[simp]: "x0 \<in> space (prob_algebra Ms) \<Longrightarrow> space x0 = space Ms"
lemma wt_bij_finite_tatom_subst_exists: assumes "finite (S::'var set)" "finite (T::('fun,'var) terms)" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>a. \<Gamma> (Var x) = TAtom a" shows "\<exists>\<sigma>::('fun,'var) subst. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> ((\<lambda>c. Fun c []) ` \<C>\<^sub>p\<^sub>u\<^sub>b) - T \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
lemma nsubst_ode: fixes I::"('sf, 'sc, 'sz) interp" fixes \<nu>::"'sz state" fixes \<nu>'::"'sz state" assumes good_interp:"is_interp I" shows "osafe ODE \<Longrightarrow> OadmitFO \<sigma> ODE U \<Longrightarrow> (\<And>i. dsafe (\<sigma> i)) \<Longrightarrow> ODE_sem I (OsubstFO ODE \<sigma>) (fst \<nu>)= ODE_sem (adjointFO I \<sigma> \<nu>) ODE (fst \<nu>)"
lemma has_type_le_env [rule_format (no_asm)]: "A \<turnstile> e::t \<Longrightarrow> \<forall>B. A \<le> B \<longrightarrow> B \<turnstile> e::t"
lemma snd_msum_aform[simp]: "snd (msum_aform n f g) = msum_pdevs n (snd f) (snd g)"
lemma d0_omega_mult: "d(x\<^sup>\<omega> * y * bot) = d(x\<^sup>\<omega> * bot)"
lemma unitary11_gen_iff: shows "unitary11_gen M \<longleftrightarrow> (\<exists> k a b. k \<noteq> 0 \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a))" (is "?lhs = ?rhs")
lemma affine_independent_span_eq: fixes S :: "'a::euclidean_space set" assumes "\<not> affine_dependent S" "card S = Suc (DIM ('a))" shows "affine hull S = UNIV"
lemma af_subsequence_W_GF_advice: assumes "i \<le> n" assumes "suffix n w \<Turnstile>\<^sub>n ((af \<psi> (w [i \<rightarrow> n]))[X]\<^sub>\<nu>)" assumes "\<And>j. j < i \<Longrightarrow> suffix n w \<Turnstile>\<^sub>n ((af \<phi> (w [j \<rightarrow> n]))[X]\<^sub>\<nu>)" shows "suffix (Suc n) w \<Turnstile>\<^sub>n (af (\<phi> W\<^sub>n \<psi>) (prefix (Suc n) w))[X]\<^sub>\<nu>"
lemma L6_Boiler: assumes h1:"SteamBoiler x s y" and h2:"ts x" and h3:"hd (x j) = Zero" shows "(hd (s j)) - (10::nat) \<le> hd (s (Suc j))"
lemma smooth_on_sqrt: "k-smooth_on S (\<lambda>x. sqrt (f x))" if "k-smooth_on S f" "0 \<notin> f ` S" "open S"
lemma Span_eq_combine_set: assumes "set Us \<subseteq> carrier R" shows "Span K Us = { combine Ks Us \| Ks. set Ks \<subseteq> K }"
lemma DERIV_mirror: "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x)) x :> - y)" for f :: "real \<Rightarrow> real" and x y :: real
lemma npos_len: "\<^bold>\|u\<^bold>\| \<le> 0 \<Longrightarrow> u = \<epsilon>"
lemma degree_shleg_poly [simp]: "degree shleg_poly = n"
lemma after_summary_union: "after_summary (M + N) S = after_summary M S + after_summary N S"
lemma foreach_impl_correct_presentation: fixes Qi Vi \<pi>i defines "Q \<equiv> Q_\<alpha> Qi" and "\<pi> \<equiv> M_lookup \<pi>i" assumes A: "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi',\<pi>i')" assumes I: "prim_invar_impl Qi \<pi>i" shows "Q_invar Qi' \<and> M_invar \<pi>i' \<and> Q_\<alpha> Qi' = Qinter Q \<pi> u \<and> M_lookup \<pi>i' = \<pi>' Q \<pi> u"
lemma sum_in_Rats [intro]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<rat>) \<Longrightarrow> sum f A \<in> \<rat>"
lemma LI_preproc_wf\<^sub>s\<^sub>t: assumes "wf\<^sub>s\<^sub>t X S" shows "wf\<^sub>s\<^sub>t X (LI_preproc S)"
lemma CLS_eqI: assumes "B.ide f" shows "\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>g\<rbrakk>\<rbrakk> \<longleftrightarrow> f \<cong>\<^sub>B g"
lemma is_scc_ex: "\<exists>scc. is_scc E scc \<and> v \<in> scc"
lemma prom_neq_reqm [iff]: "prom B' ofr A' r' I' (cons M L) J C \<noteq> reqm A r n I B"
lemma obj_MkObj: assumes "A \<in> Obj" shows "H.obj (MkObj A)"
lemma zero_in_succ [simp,intro]: "Ord i \<Longrightarrow> 0 \<^bold>\<in> succ i"
lemma ik\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_subset_rcv: assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t#S \<in> \<S>" "\<S>' = update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t#S)" "\<A>' = \<A>@[Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)]" shows "(\<Union>(ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>')) \<union> (ik\<^sub>e\<^sub>s\<^sub>t \<A>') \<subseteq> (\<Union>(ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>)) \<union> (ik\<^sub>e\<^sub>s\<^sub>t \<A>)" (is ?A) "(\<Union>(assignment_rhs\<^sub>s\<^sub>t ` \<S>')) \<union> (assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>') \<subseteq> (\<Union>(assignment_rhs\<^sub>s\<^sub>t ` \<S>)) \<union> (assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>)" (is ?B)
lemma rel_FGco_neg_distr_cond_eq: fixes tytok :: "('l1 \<times> 'l1' \<times> 'l1'' \<times> 'f1 \<times> 'f2) itself" shows "rel_FGco_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) tytok"
lemma rel_witness_option: shows set_rel_witness_option: "\<lbrakk> rel_option A x y; (a, b) \<in> set_option (rel_witness_option (x, y)) \<rbrakk> \<Longrightarrow> A a b" and map1_rel_witness_option: "rel_option A x y \<Longrightarrow> map_option fst (rel_witness_option (x, y)) = x" and map2_rel_witness_option: "rel_option A x y \<Longrightarrow> map_option snd (rel_witness_option (x, y)) = y"

lemma ex1_poincare_line_general: assumes "u \<noteq> v" "u \<noteq> inversion v" shows "\<exists>! l. is_poincare_line l \<and> u \<in> circline_set l \<and> v \<in> circline_set l"

lemma RS2Set[rule_format]: "set (removeShadowRules2 p) \<subseteq> set p"

lemma eLet_rule': assumes "\<And>x. x=v \<Longrightarrow> f x \<le> ESPEC \<Phi> \<Psi>" shows "Let v (\<lambda>x. f x) \<le> ESPEC \<Phi> \<Psi>"

lemma vfsequence_iff: "vfsequence xs \<longleftrightarrow> vsv xs \<and> \<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"

lemma set_in_finite_cone: assumes Vs: "Vs \<subseteq> carrier_vec n" and fin: "finite Vs" shows "Vs \<subseteq> finite_cone Vs"

lemma supt_list_sound [simp]: "set (supt_list t) = {s. t \<rhd> s}"

lemma (in Ring) submodule_over_zeroring:"\<lbrakk>zeroring R; R module M; submodule R M N\<rbrakk> \<Longrightarrow> N = {\<zero>\<^bsub>M\<^esub>}"

lemma (in Corps) n_value_x_1:"\<lbrakk>valuation K v; 0 \<le> n; x \<in> (vp K v) \<^bsup>(Vr K v) n\<^esup>\<rbrakk> \<Longrightarrow> n \<le> (n_val K v x)"

lemma sub_add: assumes "g \<in> carrier P" assumes "f \<in> carrier P" assumes "h \<in>carrier P" shows "((f \<oplus>\<^bsub>P\<^esub> h) of g) = ((f of g) \<oplus>\<^bsub>P\<^esub> (h of g))"

lemma cf_smcf_cf_id[slicing_commute]: "smcf_id (cat_smc \<CC>) = cf_smcf (cf_id \<CC>)"

lemma support_postList: "support (postList xs) \<subseteq> lesvars xs"

lemma equivP_pick_preserves: assumes "equivP P \<phi> " and "(P///\<phi>) X" shows "P (pick X)"

lemma pmdl_struct: assumes "struct_spec sel ap ab compl" and "compl_pmdl compl" and "is_Groebner_basis (fst ` set gs)" and "ps \<noteq> []" and "set ps \<subseteq> (set bs) \<times> (set gs \<union> set bs)" and "unique_idx (gs @ bs) (snd data)" and "sps = sel gs bs ps (snd data)" and "aux = compl gs bs (ps -- sps) sps (snd data)" and "(hs, data') = add_indices aux (snd data)" shows "pmdl (fst ` set (gs @ ab gs bs hs data')) = pmdl (fst ` set (gs @ bs))"

lemma next'_next: assumes "v \<noteq> \<bottom>" assumes "vs \<noteq> \<bottom>" shows "next'\<cdot>v\<cdot>(tree_map'\<cdot>csnd\<cdot>t) = tree_map'\<cdot>csnd\<cdot>(next\<cdot>(v :# vs)\<cdot>t)"

lemma (in encoding) indRelRPO_iff_exists_source_target_relation: fixes Pred :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) \<Rightarrow> bool" shows "(\<forall>(P, Q) \<in> indRelRPO. Pred (P, Q)) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q) \<in> Rel. Pred (P, Q)) \<and> preorder Rel)"

lemma poly_map_pullback_char: assumes "is_poly_tuple n fs" assumes "length fs = m" assumes "is_poly_tuple k gs" assumes "length gs = n" shows "(pullback (R\<^bsup>k\<^esup>) (poly_map k gs) (poly_map n fs)) = poly_map k (map (poly_compose n k gs) fs)"

lemma tfin_enat_code[code]: "(tfin :: enat set) = Collect_set (\<lambda>x. x \<noteq> \<infinity>)"

lemma append_queue_rep: "linearize (append_queue a q) = linearize q @ [a]"

lemma sum_Basis_sum_nth_Basis_list: "(\<Sum>i\<in>Basis. f i) = (\<Sum>i<DIM('a::executable_euclidean_space). f ((Basis_list::'a list) ! i))"

theorem Szemeredi_Regularity_Lemma: assumes "\<epsilon> > 0" obtains M where "\<And>G. card (uverts G) > 0 \<Longrightarrow> \<exists>P. regular_partition \<epsilon> G P \<and> card P \<le> M"

lemma [code]: "IArray.exists p as \<longleftrightarrow> exists_upto p (length' as) as"

lemma ttree_join_transfer[transfer_rule]: "rel_fun (pcr_ttree (=)) (rel_fun (pcr_ttree (=)) (pcr_ttree (=))) (\<union>) (\<squnion>)"

lemma wf_sees_method_fun: "\<lbrakk>P \<turnstile> C has least M = mthd via Cs; P \<turnstile> C has least M = mthd' via Cs'; wf_prog wf_md P\<rbrakk> \<Longrightarrow> mthd = mthd' \<and> Cs = Cs'"

lemma lns_distI [intro]: assumes "\<And>x e. x \<in> B \<Longrightarrow> e > 0 \<Longrightarrow> (\<exists>y\<in>B. (dist y x) \<le> e \<and> y \<succ>[P] x)" shows "local_nonsatiation B P"

lemma path2_transfer [transfer_rule]: assumes [transfer_rule]: "right_total A" and [transfer_rule]: "(A ===> (=)) \<alpha>e \<alpha>e2" and [transfer_rule]: "(A ===> (=)) \<alpha>n \<alpha>n2" and [transfer_rule]: "(A ===> (=)) invar invar2" and [transfer_rule]: "(A ===> (=)) inEdges' inEdges2" shows "(A ===> (=)) path2 (graph_path_base.path2 \<alpha>n2 invar2 inEdges2)"

lemma \<L>\<^sub>b_split_tendsto_opt: "(\<lambda>n. (\<L>\<^sub>b_split ^^ n) v) \<longlonglongrightarrow> \<nu>\<^sub>b_opt"

lemma(in UP_cring) UP_subring_taylor_appr: assumes "subring S R" assumes "g \<in> carrier (UP (R \<lparr> carrier := S \<rparr>))" assumes "a \<in> S" assumes "b \<in> S" shows "\<exists>c \<in> S. to_fun g a= to_fun g b \<oplus> (deriv g b)\<otimes> (a \<ominus> b) \<oplus> (c \<otimes> (a \<ominus> b)[^](2::nat))"

lemma pop_list [simp]: "invar common \<Longrightarrow> 0 < size common \<Longrightarrow> pop common = (x, common') \<Longrightarrow> x # list common' = list common"

lemma PiE_defaut_undefined_eq: "PiE_dflt I undefined M = PiE I M"

lemma find_first_distinct_ofsm_table_is_first : assumes "q1 \<in> FSM.states M" and "q2 \<in> FSM.states M" and "ofsm_table_fix M (\<lambda>q . states M) 0 q1 \<noteq> ofsm_table_fix M (\<lambda>q . states M) 0 q2" shows "ofsm_table M (\<lambda>q . states M) (find_first_distinct_ofsm_table M q1 q2) q1 \<noteq> ofsm_table M (\<lambda>q . states M) (find_first_distinct_ofsm_table M q1 q2) q2" and "k' < (find_first_distinct_ofsm_table M q1 q2) \<Longrightarrow> ofsm_table M (\<lambda>q . states M) k' q1 = ofsm_table M (\<lambda>q . states M) k' q2"

lemma collect_subfmlas_app: "\<exists>phis'. collect_subfmlas r phis = phis @ phis'"

lemma costBIT_4y: assumes "x\<noteq>y" "x : {x0,y0}" "y\<in>{x0,y0}" shows "E (type4 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s y \<bind> (\<lambda>(a, is'). return_pmf (real (t\<^sub>p (fst s) y a))))) = 0.5"

lemma holomorphic_on_Un [holomorphic_intros]: assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B" shows "f holomorphic_on (A \<union> B)"

lemma finite_cycles: "finite (cycles)"

lemma [simp,code_unfold]: "\<upsilon> \<two> = true"

lemma k_adju_var: "\<exists>F. \<forall>x.\<forall>f::'a::order \<Rightarrow> 'b::complete_lattice. (F x \<le> f) = (x \<le> (\<lambda>k. k y) f)"

lemma cf_list_entries: assumes "i \<le> deg R p" shows "(cf_list R p)!i = p i"

lemma unfold2' : assumes context_ok: "cp E" and args_def_or_valid: "(\<tau> \<Turnstile> \<delta> self) \<and> (f_\<upsilon> a1 \<tau>)" and pre_satisfied: "\<tau> \<Turnstile> PRE self a1" and postsplit_satisfied: "\<tau> \<Turnstile> POST' self a1" (* split constraint holds on post-state *) and post_decomposable : "\<And> res. (POST self a1 res) = ((POST' self a1) and (res \<triangleq> (BODY self a1)))" shows "(\<tau> \<Turnstile> E(f self a1)) = (\<tau> \<Turnstile> E(BODY self a1))"

lemma id_WHILEIT[id_rules]: "PR_CONST (WHILEIT I) ::\<^sub>i TYPE(('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a nres) \<Rightarrow> 'a \<Rightarrow> 'a nres)"

lemma equiv_up_to_refl [simp, intro!]: "P \<Turnstile> c \<sim> c"

lemma r_child_lower_bound: "l \<le> p \<Longrightarrow> p < r_child l p"

lemma eventually_in_cballs: assumes "d > 0" "c > 0" shows "eventually (\<lambda>e. cball t0 (c * e) \<times> (cball x0 e) \<subseteq> cball (t0, x0) d) (at_right 0)"

lemma LinLs: "L0 i j : Ls & L1 i j : Ls & L2 i j : Ls & L3 i j : Ls"

lemma wf_tuple_Suc_fvi_SomeI: "0 \<in> MFOTL.fvi b \<phi> \<Longrightarrow> wf_tuple n (MFOTL.fvi (Suc b) \<phi>) v \<Longrightarrow> wf_tuple (Suc n) (MFOTL.fvi b \<phi>) (Some x # v)"

lemma separated_by_wall_ex_foldpair: assumes "H\<in>walls" "separated_by H C D" shows "\<exists>(f,g)\<in>foldpairs. H = {f\<turnstile>\<C>,g\<turnstile>\<C>} \<and> C\<in>f\<turnstile>\<C> \<and> D\<in>g\<turnstile>\<C>"

lemma R_loop_mono: "X \<le> X' \<Longrightarrow> LOOP X INV I \<subseteq> LOOP X' INV I"

lemma eps_nfa'_step_eps_closure_cong: "step_eps_closure bs q q' \<Longrightarrow> q \<in> nfa'.Q \<Longrightarrow> (q' \<in> nfa'.Q \<and> nfa'.step_eps_closure bs q q') \<or> (nfa'.step_eps_closure bs q qf' \<and> step_eps_closure bs qf' q')"

lemma sees_method_mono2: "\<lbrakk> P \<turnstile> C' \<preceq>\<^sup>* C; wf_prog wf_md P; P \<turnstile> C sees M:Ts\<rightarrow>T = m in D; P \<turnstile> C' sees M:Ts'\<rightarrow>T' = m' in D' \<rbrakk> \<Longrightarrow> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"

lemma true_dsij_zero:"(P \<or> true) = true"

lemma comparator_lex_comp_aux: "comparator (lex_comp_aux::'a::nat_term comparator)"

lemma closed_funI: assumes "\<And>x. g x \<in> carrier R" shows "closed_fun R g"

lemma REFL_IMP_3_CONJ_1: fixes R P x y assumes "((\<lambda>x y. R x y \<and> P x y)\<^sup>+\<^sup>+ x y)" shows "R\<^sup>+\<^sup>+ x y"

lemma (in \<Z>) ord_of_nat_in_Vset[simp]: "a\<^sub>\<nat> \<in>\<^sub>\<circ> Vset \<alpha>"

lemma "x\<^sup>\<omega> \<cdot> y = x\<^sup>\<omega>"

lemma shiftr_uint8_code [code]: "drop_bit n x = (if n < 8 then uint8_shiftr x (integer_of_nat n) else 0)"

lemma prj1_simps [simp]: assumes "cospan f g" shows "arr \\<^sub>1[f, g]" and "dom \\<^sub>1[f, g] = f \<down>\<down> g" and "cod \\<^sub>1[f, g] = dom f"

lemma real_poly_uminus: assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (-p)) \<subseteq> \<real>"

lemma finite_tvs[simp]: "finite (tvs t)"

lemma min_element: fixes k :: nat assumes "\<exists> (m::nat). P m" shows "\<exists> mm. P mm \<and> (\<forall> m'. m' < mm \<longrightarrow> \<not> P m')"

lemma Snoc_step1_SnocD: "step1 r (ys @ [y]) (xs @ [x]) \<Longrightarrow> (step1 r ys xs \<and> y = x \<or> ys = xs \<and> r y x)"

lemma "((a::nat) - (b - (c - (d - e)))) = (a - (b - (c - (d - e))))"

lemma lift_Postdomination: assumes wf:"CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" and pd:"Postdomination sourcenode targetnode kind valid_edge Entry Exit" and inner:"CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "Postdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"

lemma lowner_lub_add: assumes "matrix_seq d f" "matrix_seq d g" "\<forall> n. trace (f n + g n) \<le> 1" shows "matrix_seq.lowner_lub (\<lambda>n. f n + g n) = matrix_seq.lowner_lub f + matrix_seq.lowner_lub g"

lemma merge_mdeg_le_1: "max_deg (merge t1) \<le> 1"

lemma Fls_E [intro!]: "insert Fls H \<turnstile> A"

lemma (in bounded_linear) has_derivative: "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"

lemma interpretation_grounds_all: "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<Longrightarrow> (\<And>v. fv (\<theta> v) = {})"

lemma det3_nonneg_scaleR_segment2: assumes "det3 x y z \<ge> 0" assumes "a > 0" shows "det3 x ((1 - a) *\<^sub>R x + a *\<^sub>R y) z \<ge> 0"

lemma lossless_spmf_inline1: assumes lossless: "\<And>s x. x \<in> outs_\ \ \<Longrightarrow> lossless_spmf (the_gpv (callee s x))" shows "lossless_spmf (inline1 callee gpv s)"

lemma sub2_closed [simp]: "sub2 d l (r,A) = (r',A') \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> A' \<in> carrier_mat m n"

lemma ord_term_canc_left: assumes "t \<oplus> v \<preceq>\<^sub>t s \<oplus> v" shows "t \<preceq> s"

lemma itrev: "itrev xs ys = rev xs @ ys"

lemma space_x0[simp]: "x0 \<in> space (prob_algebra Ms) \<Longrightarrow> space x0 = space Ms"

lemma wt_bij_finite_tatom_subst_exists: assumes "finite (S::'var set)" "finite (T::('fun,'var) terms)" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>a. \<Gamma> (Var x) = TAtom a" shows "\<exists>\<sigma>::('fun,'var) subst. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> ((\<lambda>c. Fun c []) ` \<C>\<^sub>p\<^sub>u\<^sub>b) - T \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"

lemma nsubst_ode: fixes I::"('sf, 'sc, 'sz) interp" fixes \<nu>::"'sz state" fixes \<nu>'::"'sz state" assumes good_interp:"is_interp I" shows "osafe ODE \<Longrightarrow> OadmitFO \<sigma> ODE U \<Longrightarrow> (\<And>i. dsafe (\<sigma> i)) \<Longrightarrow> ODE_sem I (OsubstFO ODE \<sigma>) (fst \<nu>)= ODE_sem (adjointFO I \<sigma> \<nu>) ODE (fst \<nu>)"

lemma has_type_le_env [rule_format (no_asm)]: "A \<turnstile> e::t \<Longrightarrow> \<forall>B. A \<le> B \<longrightarrow> B \<turnstile> e::t"

lemma snd_msum_aform[simp]: "snd (msum_aform n f g) = msum_pdevs n (snd f) (snd g)"

lemma d0_omega_mult: "d(x\<^sup>\<omega> * y * bot) = d(x\<^sup>\<omega> * bot)"

lemma unitary11_gen_iff: shows "unitary11_gen M \<longleftrightarrow> (\<exists> k a b. k \<noteq> 0 \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a))" (is "?lhs = ?rhs")

lemma affine_independent_span_eq: fixes S :: "'a::euclidean_space set" assumes "\<not> affine_dependent S" "card S = Suc (DIM ('a))" shows "affine hull S = UNIV"

lemma af_subsequence_W_GF_advice: assumes "i \<le> n" assumes "suffix n w \<Turnstile>\<^sub>n ((af \<psi> (w [i \<rightarrow> n]))[X]\<^sub>\<nu>)" assumes "\<And>j. j suffix n w \<Turnstile>\<^sub>n ((af \<phi> (w [j \<rightarrow> n]))[X]\<^sub>\<nu>)" shows "suffix (Suc n) w \<Turnstile>\<^sub>n (af (\<phi> W\<^sub>n \<psi>) (prefix (Suc n) w))[X]\<^sub>\<nu>"

lemma L6_Boiler: assumes h1:"SteamBoiler x s y" and h2:"ts x" and h3:"hd (x j) = Zero" shows "(hd (s j)) - (10::nat) \<le> hd (s (Suc j))"

lemma smooth_on_sqrt: "k-smooth_on S (\<lambda>x. sqrt (f x))" if "k-smooth_on S f" "0 \<notin> f ` S" "open S"

lemma Span_eq_combine_set: assumes "set Us \<subseteq> carrier R" shows "Span K Us = { combine Ks Us | Ks. set Ks \<subseteq> K }"

lemma DERIV_mirror: "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x)) x :> - y)" for f :: "real \<Rightarrow> real" and x y :: real

lemma npos_len: "\<^bold>|u\<^bold>| \<le> 0 \<Longrightarrow> u = \<epsilon>"

lemma degree_shleg_poly [simp]: "degree shleg_poly = n"

lemma after_summary_union: "after_summary (M + N) S = after_summary M S + after_summary N S"

lemma foreach_impl_correct_presentation: fixes Qi Vi \<pi>i defines "Q \<equiv> Q_\<alpha> Qi" and "\<pi> \<equiv> M_lookup \<pi>i" assumes A: "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi',\<pi>i')" assumes I: "prim_invar_impl Qi \<pi>i" shows "Q_invar Qi' \<and> M_invar \<pi>i' \<and> Q_\<alpha> Qi' = Qinter Q \<pi> u \<and> M_lookup \<pi>i' = \<pi>' Q \<pi> u"

lemma sum_in_Rats [intro]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<rat>) \<Longrightarrow> sum f A \<in> \<rat>"

lemma LI_preproc_wf\<^sub>s\<^sub>t: assumes "wf\<^sub>s\<^sub>t X S" shows "wf\<^sub>s\<^sub>t X (LI_preproc S)"

lemma CLS_eqI: assumes "B.ide f" shows "\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>g\<rbrakk>\<rbrakk> \<longleftrightarrow> f \<cong>\<^sub>B g"

lemma is_scc_ex: "\<exists>scc. is_scc E scc \<and> v \<in> scc"

lemma prom_neq_reqm [iff]: "prom B' ofr A' r' I' (cons M L) J C \<noteq> reqm A r n I B"

lemma obj_MkObj: assumes "A \<in> Obj" shows "H.obj (MkObj A)"

lemma zero_in_succ [simp,intro]: "Ord i \<Longrightarrow> 0 \<^bold>\<in> succ i"

lemma ik\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_subset_rcv: assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t#S \<in> \<S>" "\<S>' = update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t#S)" "\<A>' = \<A>@[Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)]" shows "(\<Union>(ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>')) \<union> (ik\<^sub>e\<^sub>s\<^sub>t \<A>') \<subseteq> (\<Union>(ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>)) \<union> (ik\<^sub>e\<^sub>s\<^sub>t \<A>)" (is ?A) "(\<Union>(assignment_rhs\<^sub>s\<^sub>t ` \<S>')) \<union> (assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>') \<subseteq> (\<Union>(assignment_rhs\<^sub>s\<^sub>t ` \<S>)) \<union> (assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>)" (is ?B)

lemma rel_FGco_neg_distr_cond_eq: fixes tytok :: "('l1 \<times> 'l1' \<times> 'l1'' \<times> 'f1 \<times> 'f2) itself" shows "rel_FGco_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) tytok"

lemma rel_witness_option: shows set_rel_witness_option: "\<lbrakk> rel_option A x y; (a, b) \<in> set_option (rel_witness_option (x, y)) \<rbrakk> \<Longrightarrow> A a b" and map1_rel_witness_option: "rel_option A x y \<Longrightarrow> map_option fst (rel_witness_option (x, y)) = x" and map2_rel_witness_option: "rel_option A x y \<Longrightarrow> map_option snd (rel_witness_option (x, y)) = y"