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

Statement: stringlengths 7 24.3k
lemma binop_bijE[elim]: assumes "binop_bij A f" obtains "nop_bij A (2\<^sub>\<nat>) f"
lemma lm1: "pr_gr (Suc x) = pr_gr x \<or> c_tl (pr_gr (Suc x)) = pr_gr x"
lemma enum_less: "a \<in> s \<Longrightarrow> i < n \<Longrightarrow> enum i < a \<longleftrightarrow> enum (Suc i) \<le> a"
lemma isCont_real_sqrt: "isCont sqrt x"
lemma disjoint_projection: "X \<inter> Y = {} \<Longrightarrow> (l \<upharpoonleft> X) \<upharpoonleft> Y = []"
lemma preSetpreList_eq: "preList xs C l s = preSet (set xs) C l s"
lemma nnegD[dest]: "nneg P \<Longrightarrow> 0 \<le> P x"
lemma finite_conj : assumes "finite E" assumes "\<forall> e \<in> E. finite (vars e)" shows "finite (vars (conjunct E))"
lemma differentiable_cinner [simp]: "f differentiable (at x within s) \<Longrightarrow> g differentiable at x within s \<Longrightarrow> (\<lambda>x. cinner (f x) (g x)) differentiable at x within s"
lemma check_instr'_ins_jump_ok: "check_instr' ins P h stk loc C M pc frs \<Longrightarrow> ins_jump_ok ins pc"
lemma infsetsum_altdef': "A \<subseteq> B \<Longrightarrow> infsetsum f A = set_lebesgue_integral (count_space B) A f"
lemma norm_notMT: "DenyAll \<in> set (policy2list p) \<Longrightarrow> normalizePr p \<noteq> []"
lemma approx_floatarith_Elem: assumes "approx_floatarith p ra VS = Some X" assumes e: "e \<in> UNIV \<rightarrow> {-1 .. 1}" assumes "vs \<in> aforms_err e VS" shows "interpret_floatarith ra vs \<in> aform_err e X"
lemma some_ord0rel: "(x, SOME y. (x,y) \<in> ord0rel) \<in> ord0rel"
lemma dg_prod_Obj_cong: assumes "g \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>I. \<AA> i)\<lparr>Obj\<rparr>" and "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>I. \<AA> i)\<lparr>Obj\<rparr>" and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> g\<lparr>i\<rparr> = f\<lparr>i\<rparr>" shows "g = f"
lemma "x - (0::int) = x"
lemma (in gcd_condition_monoid) division_weak_lower_semilattice [simp]: "weak_lower_semilattice (division_rel G)"
lemma G_eq: "G cf = (if proper (fst cf) then {cont_eff cf i \| i. i < brn (fst cf) \<and> 0 < wt (fst cf) (snd cf) i } else {cf})"
lemma open_map_imp_subset_topspace: "open_map X1 X2 f \<Longrightarrow> f ` (topspace X1) \<subseteq> topspace X2"
lemma (in ell_field) pdouble_correct: "a \<in> carrier R \<Longrightarrow> in_carrierp p \<Longrightarrow> make_affine (pdouble a p) = add a (make_affine p) (make_affine p)"
lemma optim_instrs_ConsD[dest]: assumes "optim_instrs F ret pc \<Sigma>i (x # xs) = Some (\<Sigma>o, ys)" shows "\<exists>y ys' \<Sigma>. ys = y # ys' \<and> optim_instr F ret pc x \<Sigma>i = Some (y, \<Sigma>) \<and> optim_instrs F ret (Suc pc) \<Sigma> xs = Some (\<Sigma>o, ys')"
lemma index_lfp: "lfp f i : L i"
lemma tau_seq_precongl: "\<tau> x \<le> \<tau> y \<Longrightarrow> \<tau> (z \<cdot> x) \<le> \<tau> (z \<cdot> y)"
lemma mult_ge_zero[intro]: "(a :: 'a :: ordered_semiring_1) \<ge> 0 \<Longrightarrow> b \<ge> 0 \<Longrightarrow> a * b \<ge> 0"
lemma awalk_to_path_no_neg_cyc_cost: assumes p_props:"awalk u p v" assumes no_neg_cyc: "\<not> (\<exists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0)" shows "awalk_cost f (awalk_to_apath p) \<le> awalk_cost f p"
lemma create_graph_finite: "create_graph \<phi> \<le> SPEC finite"
lemma thm_relation_negation_1_1[PLM]: "[\<lparr>F\<^sup>-, x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F, x\<^sup>P\<rparr> in v]"
lemma hcomplex_numeral_hcnj [simp]: "hcnj (numeral v :: hcomplex) = numeral v"
lemma simplex_extremal_le_exists: fixes S :: "'a::real_inner set" shows "finite S \<Longrightarrow> x \<in> convex hull S \<Longrightarrow> y \<in> convex hull S \<Longrightarrow> \<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"
lemma lists_accI: "Wellfounded.accp (step1 r) xs ==> listsp (Wellfounded.accp r) xs"
lemma llist_corec_transfer [transfer_rule]: "((A ===> (=)) ===> (A ===> B) ===> (A ===> (=)) ===> (A ===> llist_all2 B) ===> (A ===> A) ===> A ===> llist_all2 B) corec_llist corec_llist"
lemma resumption_ord_antisym: "\<lbrakk> resumption_ord r r'; resumption_ord r' r \<rbrakk> \<Longrightarrow> r = r'"
lemma permutes_insert: "{p. p permutes (insert a S)} = (\<lambda>(b, p). transpose a b \<circ> p) ` {(b, p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
lemma JVM_states_unfold: "states S maxs maxr == err(opt((\<Union>{list n (types S) \|n. n <= maxs}) \<times> list maxr (err(types S))))"
lemma dotprod_le_drop: assumes "length a = length b" and "k \<le> length a" shows "drop k a \<bullet> drop k b \<le> a \<bullet> b"
lemma (in Group) smallest_sg_gen:"\<lbrakk>A \<subseteq> carrier G; G \<guillemotright> H; A \<subseteq> H\<rbrakk> \<Longrightarrow> sg_gen G A \<subseteq> H"
lemma emp_hcomp: "\<epsilon> \<otimes> P = P"
lemma liset_suffix: assumes "i \<in> liset A u" "u \<le> v" shows "i \<in> liset A v"
lemma term_ok'_contained_tvars_typ_ok_sig: "term_ok' \<Sigma> t \<Longrightarrow> (idn, S) \<in> tvs t \<Longrightarrow> typ_ok_sig \<Sigma> (Tv idn S)"
lemma NSDeMoivre_ext: "\<And>a n. (hcis a) pow n = hcis (hypreal_of_hypnat n * a)"
lemma conf_trpl_ex: "\<exists> p q r. conf m (bl2wc (<lm>)) stp = trpl p q r"
lemma untilE: assumes "(P \<U> Q) \<sigma>" obtains i where "Q (\<sigma> \|\<^sub>s i)" and "\<forall>k<i. P (\<sigma> \|\<^sub>s k)"
lemma (in pell) find_nth_solution_correct: "find_nth_solution D n = nth_solution n"
lemma consistent_sign_vectors_consistent_sign_vectors_r: shows"consistent_sign_vectors_r (cast_rat_list qs) S = consistent_sign_vectors qs S"
lemma ex_countable_subset_image: "(\<exists>T. countable T \<and> T \<subseteq> f ` S \<and> P T) \<longleftrightarrow> (\<exists>T. countable T \<and> T \<subseteq> S \<and> P (f ` T))"
lemma test_valid: "valid test_axiom"
lemma exec_assign_global : assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> ( _ \<leftarrow> assign_global upd rhs; M)) = ( \<sigma> \<Turnstile> M)"
lemma objI_trg: assumes "arr a" and "trg a = a" shows "obj a"
lemma rb_aux_inv_invariant: assumes "rb_aux_inv (fst args)" shows "rb_aux_inv (fst (rb_aux args))"
theorem lucas_theorem: fixes n k d::nat assumes "n < p ^ (Suc d)" assumes "k < p ^ (Suc d)" assumes "prime p" shows "(n choose k) mod p = (\<Prod>i\<le>d. ((nth_digit_general n i p) choose (nth_digit_general k i p))) mod p"
lemma of_real_euler_mascheroni [simp]: "of_real euler_mascheroni = euler_mascheroni"
lemma disc_rfr_proc_positive: assumes "-1 < r" shows "\<And>n w . 0 < disc_rfr_proc r n w"
lemma vint_one_closed: "1\<^sub>\<int> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
lemma mtx_times_scaleR_commute: "A * (c \<^sub>R B) = c \<^sub>R (A * B)" for A::"('n::finite) sq_mtx"
lemma size_fset_of_list: "size (fset_of_list l) = length (remdups l)"
lemma size_list_delete: "size_list f (AList.delete a al) \<le> size_list f al"
lemma \<Delta>_factor: "\<Delta> (\<lambda>k. c * k) k = c"
lemma disjoint_dverts_if_wf_aux: assumes "wf_dverts (Node r xs)" and "(t1,e1) \<in> fset xs" and "(t2,e2) \<in> fset xs" and "(t1,e1) \<noteq> (t2,e2)" shows "dverts t1 \<inter> dverts t2 = {}"
lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
lemma ipoly_poly_inverse[simp]: assumes "coeff p 0 \<noteq> 0" shows "ipoly (poly_inverse p) (x :: 'a :: field_char_0) = 0 \<longleftrightarrow> ipoly p (inverse x) = 0"
lemma im_diff: "T (g - g') = T g - T g'"
lemma outside_list_coeff0: assumes "i \<ge> dim_vec xs" shows "coeff (vec_to_lpoly xs) i = 0"
lemma toplevel_summands_nullable: "nullable s = (\<exists>r\<in>toplevel_summands s. nullable r)"
lemma small_ordinal_arrs[simp]: "small {[a, b]\<^sub>\<circ> \| a b. a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> A \<and> a \<le> b}"
lemma nat_mult_not_infty[simp]: assumes "c \<noteq> \<infinity>" shows "(eint n) * c \<noteq> \<infinity>"
lemma fold_fold_prod_conv: "fold (\<lambda>i. fold (f i) l1) l2 s = fold (\<lambda>(i,j). f i j) (List.product l2 l1) s"
lemma (in rpo_with_assms) rpo_vars_term: "rpo_s s t \<or> rpo_ns s t \<Longrightarrow> vars_term s \<supseteq> vars_term t"
lemma starlike_imp_Borsukian: fixes S :: "'a::real_normed_vector set" shows "starlike S \<Longrightarrow> Borsukian S"
lemma Qp_cong_set_is_univ_semialgebraic: assumes "a \<in> carrier Z\<^sub>p" shows "is_univ_semialgebraic (Qp_cong_set \<alpha> a)"
lemma pnet_init_net_ips_net_tree_ips: assumes "s \<in> init (pnet np p)" shows "net_ips s = net_tree_ips p"
lemma integral_cos [simp]: fixes a::real assumes "a \<le> b" shows "integral {a..b} cos = sin b - sin a"
lemma "nths' n m xs = take (m - n) (drop n xs)"
lemma DynCastThrow': "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow>' \<langle>throw e',s\<^sub>1\<rangle> \<Longrightarrow> P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow>' \<langle>throw e',s\<^sub>1\<rangle>"
lemma bottom_pow: "order.bottom (Pow A) = {}"
lemma random_member_None[simp]: "random_member ss = None = (ss = {\|\|})"
lemma ket_01_dim: shows "dim_vec ket_01 = 4"
lemma OclExcluding_commute[simp,code_unfold]: "((S :: ('\<AA>, 'a::null) Set)->excluding\<^sub>S\<^sub>e\<^sub>t(i)->excluding\<^sub>S\<^sub>e\<^sub>t(j) = (S->excluding\<^sub>S\<^sub>e\<^sub>t(j)->excluding\<^sub>S\<^sub>e\<^sub>t(i)))"
lemma kill_short_order_of_graph: assumes "finite (uverts G)" shows "card (uverts G) - card (short_cycles G k) \<le> card (uverts (kill_short G k))"
lemma tl_eq2[rule_format]: "tl sl = [] \<longrightarrow> sl ! (0) = sl ! (length sl - (1))"
lemma proots_line_smods: assumes "poly p st \<noteq>0" "poly p tt \<noteq> 0" "st\<noteq>tt" shows "proots_line p st tt = (let pc = pcompose p [:st, tt - st:]; pR = map_poly Re pc; pI = map_poly Im pc; g = gcd pR pI in nat (changes_itv_smods_ext 0 1 g (pderiv g)))" (is "_=?R")
lemma n_omega_L_star: "x\<^sup>\<star> \<squnion> n(x\<^sup>\<omega>) * L = x\<^sup>\<star> \<squnion> x * n(x\<^sup>\<omega>) * L"
lemma filter_diff_mset': "filter_mset P (X - Y) = filter_mset P X - Y"
lemma dres_Sup_cases: obtains "S\<subseteq>{dSUCCEED}" and "Sup S = dSUCCEED" \| "dFAIL\<in>S" and "Sup S = dFAIL" \| a b where "a\<noteq>b" "dRETURN a\<in>S" "dRETURN b\<in>S" "dFAIL\<notin>S" "Sup S = dFAIL" \| a where "S \<subseteq> {dSUCCEED, dRETURN a}" "dRETURN a\<in>S" "Sup S = dRETURN a"
lemma lifted_less_eq_anti_sym [trans]: "lifted_less_eq x y \<Longrightarrow> lifted_less_eq y x \<Longrightarrow> x = y"
lemma subs_head_redex: shows "Arr t \<Longrightarrow> head_redex t \<sqsubseteq> t"
lemma coerce_errorT_abs [simp]: "coerce\<cdot>(errorT_abs\<cdot>x) = errorT_abs\<cdot>(coerce\<cdot>x)"
lemma getMinTree_cons: "prio (getMinTree (y # x # xs)) \<le> prio (getMinTree (x # xs))"
lemma proj2_abs_abs_mult: assumes "proj2_abs v = proj2_abs w" and "w \<noteq> 0" shows "\<exists> c. v = c *\<^sub>R w"
lemma decisive_coalitionD[dest]: "decisive scf A Is C x y \<Longrightarrow> C \<subseteq> Is"
lemma mv_box_correct': "\<lbrakk>length lm > max m n; m \<noteq> n\<rbrakk> \<Longrightarrow> \<exists> stp. abc_steps_l (0::nat, lm) (mv_box m n) stp = (3, (lm[n := (lm ! m + lm ! n)])[m := 0::nat])"
lemma execl_all_Bc [simplified, intro]: "\<lbrakk>if v = f then [JMP i] else [] \<Turnstile> cfs\<box>; 0 \<le> i\<rbrakk> \<Longrightarrow> bpred (Bc v, f, i) (cfs ! 0) (cfs ! (length cfs - 1))"
lemma eval_formula_conv: "eval_formula \<F> Rs \<alpha> f = eval \<alpha> undefined (for_eval_rel \<F> Rs) (form_of_formula f)"
lemma fds_const_of_real [simp]: "fds_const (of_real c) = of_real c"
lemma fls_subdegree_integral_ge: "fls_integral f \<noteq> 0 \<Longrightarrow> fls_subdegree (fls_integral f) \<ge> fls_subdegree f + 1"
lemma val_in_type_universe[simp]: "v \<in> type_universe (val_type v)"
lemma l9_19_3: assumes "Coplanar A B C P" and "Col X Y P" shows "A B C OSP X Y \<longleftrightarrow> (P Out X Y \<and> \<not> Coplanar A B C X)"
lemma (in itrace_top) topology [iff]: "topology T"
lemma ccomp_exec_n: "ccomp c \<turnstile> (0,s,stk) \<rightarrow>^n (size(ccomp c),t,stk') \<Longrightarrow> (c,s) \<Rightarrow> t \<and> stk'=stk"
lemma ack_3: "ack (Suc (Suc (Suc 0))) j = 2 ^ (j+3) - 3"
lemma param_list_bex[param]: "\<lbrakk>(P,P')\<in>(Ra\<rightarrow>Id); (l,l')\<in>\<langle>Ra\<rangle> list_rel\<rbrakk> \<Longrightarrow> (\<exists>x\<in>set l. P x, \<exists>x\<in>set l'. P' x) \<in> Id"

lemma binop_bijE[elim]: assumes "binop_bij A f" obtains "nop_bij A (2\<^sub>\<nat>) f"

lemma lm1: "pr_gr (Suc x) = pr_gr x \<or> c_tl (pr_gr (Suc x)) = pr_gr x"

lemma enum_less: "a \<in> s \<Longrightarrow> i < n \<Longrightarrow> enum i < a \<longleftrightarrow> enum (Suc i) \<le> a"

lemma isCont_real_sqrt: "isCont sqrt x"

lemma disjoint_projection: "X \<inter> Y = {} \<Longrightarrow> (l \<upharpoonleft> X) \<upharpoonleft> Y = []"

lemma preSetpreList_eq: "preList xs C l s = preSet (set xs) C l s"

lemma nnegD[dest]: "nneg P \<Longrightarrow> 0 \<le> P x"

lemma finite_conj : assumes "finite E" assumes "\<forall> e \<in> E. finite (vars e)" shows "finite (vars (conjunct E))"

lemma differentiable_cinner [simp]: "f differentiable (at x within s) \<Longrightarrow> g differentiable at x within s \<Longrightarrow> (\<lambda>x. cinner (f x) (g x)) differentiable at x within s"

lemma check_instr'_ins_jump_ok: "check_instr' ins P h stk loc C M pc frs \<Longrightarrow> ins_jump_ok ins pc"

lemma infsetsum_altdef': "A \<subseteq> B \<Longrightarrow> infsetsum f A = set_lebesgue_integral (count_space B) A f"

lemma norm_notMT: "DenyAll \<in> set (policy2list p) \<Longrightarrow> normalizePr p \<noteq> []"

lemma approx_floatarith_Elem: assumes "approx_floatarith p ra VS = Some X" assumes e: "e \<in> UNIV \<rightarrow> {-1 .. 1}" assumes "vs \<in> aforms_err e VS" shows "interpret_floatarith ra vs \<in> aform_err e X"

lemma some_ord0rel: "(x, SOME y. (x,y) \<in> ord0rel) \<in> ord0rel"

lemma dg_prod_Obj_cong: assumes "g \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>I. \<AA> i)\<lparr>Obj\<rparr>" and "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>I. \<AA> i)\<lparr>Obj\<rparr>" and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> g\<lparr>i\<rparr> = f\<lparr>i\<rparr>" shows "g = f"

lemma "x - (0::int) = x"

lemma (in gcd_condition_monoid) division_weak_lower_semilattice [simp]: "weak_lower_semilattice (division_rel G)"

lemma G_eq: "G cf = (if proper (fst cf) then {cont_eff cf i | i. i < brn (fst cf) \<and> 0 < wt (fst cf) (snd cf) i } else {cf})"

lemma open_map_imp_subset_topspace: "open_map X1 X2 f \<Longrightarrow> f ` (topspace X1) \<subseteq> topspace X2"

lemma (in ell_field) pdouble_correct: "a \<in> carrier R \<Longrightarrow> in_carrierp p \<Longrightarrow> make_affine (pdouble a p) = add a (make_affine p) (make_affine p)"

lemma optim_instrs_ConsD[dest]: assumes "optim_instrs F ret pc \<Sigma>i (x # xs) = Some (\<Sigma>o, ys)" shows "\<exists>y ys' \<Sigma>. ys = y # ys' \<and> optim_instr F ret pc x \<Sigma>i = Some (y, \<Sigma>) \<and> optim_instrs F ret (Suc pc) \<Sigma> xs = Some (\<Sigma>o, ys')"

lemma index_lfp: "lfp f i : L i"

lemma tau_seq_precongl: "\<tau> x \<le> \<tau> y \<Longrightarrow> \<tau> (z \<cdot> x) \<le> \<tau> (z \<cdot> y)"

lemma mult_ge_zero[intro]: "(a :: 'a :: ordered_semiring_1) \<ge> 0 \<Longrightarrow> b \<ge> 0 \<Longrightarrow> a * b \<ge> 0"

lemma awalk_to_path_no_neg_cyc_cost: assumes p_props:"awalk u p v" assumes no_neg_cyc: "\<not> (\<exists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0)" shows "awalk_cost f (awalk_to_apath p) \<le> awalk_cost f p"

lemma create_graph_finite: "create_graph \<phi> \<le> SPEC finite"

lemma thm_relation_negation_1_1[PLM]: "[\<lparr>F\<^sup>-, x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F, x\<^sup>P\<rparr> in v]"

lemma hcomplex_numeral_hcnj [simp]: "hcnj (numeral v :: hcomplex) = numeral v"

lemma simplex_extremal_le_exists: fixes S :: "'a::real_inner set" shows "finite S \<Longrightarrow> x \<in> convex hull S \<Longrightarrow> y \<in> convex hull S \<Longrightarrow> \<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"

lemma lists_accI: "Wellfounded.accp (step1 r) xs ==> listsp (Wellfounded.accp r) xs"

lemma llist_corec_transfer [transfer_rule]: "((A ===> (=)) ===> (A ===> B) ===> (A ===> (=)) ===> (A ===> llist_all2 B) ===> (A ===> A) ===> A ===> llist_all2 B) corec_llist corec_llist"

lemma resumption_ord_antisym: "\<lbrakk> resumption_ord r r'; resumption_ord r' r \<rbrakk> \<Longrightarrow> r = r'"

lemma permutes_insert: "{p. p permutes (insert a S)} = (\<lambda>(b, p). transpose a b \<circ> p) ` {(b, p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"

lemma JVM_states_unfold: "states S maxs maxr == err(opt((\<Union>{list n (types S) |n. n <= maxs}) \<times> list maxr (err(types S))))"

lemma dotprod_le_drop: assumes "length a = length b" and "k \<le> length a" shows "drop k a \<bullet> drop k b \<le> a \<bullet> b"

lemma (in Group) smallest_sg_gen:"\<lbrakk>A \<subseteq> carrier G; G \<guillemotright> H; A \<subseteq> H\<rbrakk> \<Longrightarrow> sg_gen G A \<subseteq> H"

lemma emp_hcomp: "\<epsilon> \<otimes> P = P"

lemma liset_suffix: assumes "i \<in> liset A u" "u \<le> v" shows "i \<in> liset A v"

lemma term_ok'_contained_tvars_typ_ok_sig: "term_ok' \<Sigma> t \<Longrightarrow> (idn, S) \<in> tvs t \<Longrightarrow> typ_ok_sig \<Sigma> (Tv idn S)"

lemma NSDeMoivre_ext: "\<And>a n. (hcis a) pow n = hcis (hypreal_of_hypnat n * a)"

lemma conf_trpl_ex: "\<exists> p q r. conf m (bl2wc (<lm>)) stp = trpl p q r"

lemma untilE: assumes "(P \<U> Q) \<sigma>" obtains i where "Q (\<sigma> |\<^sub>s i)" and "\<forall>k<i. P (\<sigma> |\<^sub>s k)"

lemma (in pell) find_nth_solution_correct: "find_nth_solution D n = nth_solution n"

lemma consistent_sign_vectors_consistent_sign_vectors_r: shows"consistent_sign_vectors_r (cast_rat_list qs) S = consistent_sign_vectors qs S"

lemma ex_countable_subset_image: "(\<exists>T. countable T \<and> T \<subseteq> f ` S \<and> P T) \<longleftrightarrow> (\<exists>T. countable T \<and> T \<subseteq> S \<and> P (f ` T))"

lemma test_valid: "valid test_axiom"

lemma exec_assign_global : assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> ( _ \<leftarrow> assign_global upd rhs; M)) = ( \<sigma> \<Turnstile> M)"

lemma objI_trg: assumes "arr a" and "trg a = a" shows "obj a"

lemma rb_aux_inv_invariant: assumes "rb_aux_inv (fst args)" shows "rb_aux_inv (fst (rb_aux args))"

theorem lucas_theorem: fixes n k d::nat assumes "n < p ^ (Suc d)" assumes "k < p ^ (Suc d)" assumes "prime p" shows "(n choose k) mod p = (\<Prod>i\<le>d. ((nth_digit_general n i p) choose (nth_digit_general k i p))) mod p"

lemma of_real_euler_mascheroni [simp]: "of_real euler_mascheroni = euler_mascheroni"

lemma disc_rfr_proc_positive: assumes "-1 < r" shows "\<And>n w . 0 < disc_rfr_proc r n w"

lemma vint_one_closed: "1\<^sub>\<int> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"

lemma mtx_times_scaleR_commute: "A * (c *\<^sub>R B) = c *\<^sub>R (A * B)" for A::"('n::finite) sq_mtx"

lemma size_fset_of_list: "size (fset_of_list l) = length (remdups l)"

lemma size_list_delete: "size_list f (AList.delete a al) \<le> size_list f al"

lemma \<Delta>_factor: "\<Delta> (\<lambda>k. c * k) k = c"

lemma disjoint_dverts_if_wf_aux: assumes "wf_dverts (Node r xs)" and "(t1,e1) \<in> fset xs" and "(t2,e2) \<in> fset xs" and "(t1,e1) \<noteq> (t2,e2)" shows "dverts t1 \<inter> dverts t2 = {}"

lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"

lemma ipoly_poly_inverse[simp]: assumes "coeff p 0 \<noteq> 0" shows "ipoly (poly_inverse p) (x :: 'a :: field_char_0) = 0 \<longleftrightarrow> ipoly p (inverse x) = 0"

lemma im_diff: "T (g - g') = T g - T g'"

lemma outside_list_coeff0: assumes "i \<ge> dim_vec xs" shows "coeff (vec_to_lpoly xs) i = 0"

lemma toplevel_summands_nullable: "nullable s = (\<exists>r\<in>toplevel_summands s. nullable r)"

lemma small_ordinal_arrs[simp]: "small {[a, b]\<^sub>\<circ> | a b. a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> A \<and> a \<le> b}"

lemma nat_mult_not_infty[simp]: assumes "c \<noteq> \<infinity>" shows "(eint n) * c \<noteq> \<infinity>"

lemma fold_fold_prod_conv: "fold (\<lambda>i. fold (f i) l1) l2 s = fold (\<lambda>(i,j). f i j) (List.product l2 l1) s"

lemma (in rpo_with_assms) rpo_vars_term: "rpo_s s t \<or> rpo_ns s t \<Longrightarrow> vars_term s \<supseteq> vars_term t"

lemma starlike_imp_Borsukian: fixes S :: "'a::real_normed_vector set" shows "starlike S \<Longrightarrow> Borsukian S"

lemma Qp_cong_set_is_univ_semialgebraic: assumes "a \<in> carrier Z\<^sub>p" shows "is_univ_semialgebraic (Qp_cong_set \<alpha> a)"

lemma pnet_init_net_ips_net_tree_ips: assumes "s \<in> init (pnet np p)" shows "net_ips s = net_tree_ips p"

lemma integral_cos [simp]: fixes a::real assumes "a \<le> b" shows "integral {a..b} cos = sin b - sin a"

lemma "nths' n m xs = take (m - n) (drop n xs)"

lemma DynCastThrow': "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow>' \<langle>throw e',s\<^sub>1\<rangle> \<Longrightarrow> P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow>' \<langle>throw e',s\<^sub>1\<rangle>"

lemma bottom_pow: "order.bottom (Pow A) = {}"

lemma random_member_None[simp]: "random_member ss = None = (ss = {||})"

lemma ket_01_dim: shows "dim_vec ket_01 = 4"

lemma OclExcluding_commute[simp,code_unfold]: "((S :: ('\<AA>, 'a::null) Set)->excluding\<^sub>S\<^sub>e\<^sub>t(i)->excluding\<^sub>S\<^sub>e\<^sub>t(j) = (S->excluding\<^sub>S\<^sub>e\<^sub>t(j)->excluding\<^sub>S\<^sub>e\<^sub>t(i)))"

lemma kill_short_order_of_graph: assumes "finite (uverts G)" shows "card (uverts G) - card (short_cycles G k) \<le> card (uverts (kill_short G k))"

lemma tl_eq2[rule_format]: "tl sl = [] \<longrightarrow> sl ! (0) = sl ! (length sl - (1))"

lemma proots_line_smods: assumes "poly p st \<noteq>0" "poly p tt \<noteq> 0" "st\<noteq>tt" shows "proots_line p st tt = (let pc = pcompose p [:st, tt - st:]; pR = map_poly Re pc; pI = map_poly Im pc; g = gcd pR pI in nat (changes_itv_smods_ext 0 1 g (pderiv g)))" (is "_=?R")

lemma n_omega_L_star: "x\<^sup>\<star> \<squnion> n(x\<^sup>\<omega>) * L = x\<^sup>\<star> \<squnion> x * n(x\<^sup>\<omega>) * L"

lemma filter_diff_mset': "filter_mset P (X - Y) = filter_mset P X - Y"

lemma dres_Sup_cases: obtains "S\<subseteq>{dSUCCEED}" and "Sup S = dSUCCEED" | "dFAIL\<in>S" and "Sup S = dFAIL" | a b where "a\<noteq>b" "dRETURN a\<in>S" "dRETURN b\<in>S" "dFAIL\<notin>S" "Sup S = dFAIL" | a where "S \<subseteq> {dSUCCEED, dRETURN a}" "dRETURN a\<in>S" "Sup S = dRETURN a"

lemma lifted_less_eq_anti_sym [trans]: "lifted_less_eq x y \<Longrightarrow> lifted_less_eq y x \<Longrightarrow> x = y"

lemma subs_head_redex: shows "Arr t \<Longrightarrow> head_redex t \<sqsubseteq> t"

lemma coerce_errorT_abs [simp]: "coerce\<cdot>(errorT_abs\<cdot>x) = errorT_abs\<cdot>(coerce\<cdot>x)"

lemma getMinTree_cons: "prio (getMinTree (y # x # xs)) \<le> prio (getMinTree (x # xs))"

lemma proj2_abs_abs_mult: assumes "proj2_abs v = proj2_abs w" and "w \<noteq> 0" shows "\<exists> c. v = c *\<^sub>R w"

lemma decisive_coalitionD[dest]: "decisive scf A Is C x y \<Longrightarrow> C \<subseteq> Is"

lemma mv_box_correct': "\<lbrakk>length lm > max m n; m \<noteq> n\<rbrakk> \<Longrightarrow> \<exists> stp. abc_steps_l (0::nat, lm) (mv_box m n) stp = (3, (lm[n := (lm ! m + lm ! n)])[m := 0::nat])"

lemma execl_all_Bc [simplified, intro]: "\<lbrakk>if v = f then [JMP i] else [] \<Turnstile> cfs\<box>; 0 \<le> i\<rbrakk> \<Longrightarrow> bpred (Bc v, f, i) (cfs ! 0) (cfs ! (length cfs - 1))"

lemma eval_formula_conv: "eval_formula \<F> Rs \<alpha> f = eval \<alpha> undefined (for_eval_rel \<F> Rs) (form_of_formula f)"

lemma fds_const_of_real [simp]: "fds_const (of_real c) = of_real c"

lemma fls_subdegree_integral_ge: "fls_integral f \<noteq> 0 \<Longrightarrow> fls_subdegree (fls_integral f) \<ge> fls_subdegree f + 1"

lemma val_in_type_universe[simp]: "v \<in> type_universe (val_type v)"

lemma l9_19_3: assumes "Coplanar A B C P" and "Col X Y P" shows "A B C OSP X Y \<longleftrightarrow> (P Out X Y \<and> \<not> Coplanar A B C X)"

lemma (in itrace_top) topology [iff]: "topology T"

lemma ccomp_exec_n: "ccomp c \<turnstile> (0,s,stk) \<rightarrow>^n (size(ccomp c),t,stk') \<Longrightarrow> (c,s) \<Rightarrow> t \<and> stk'=stk"

lemma ack_3: "ack (Suc (Suc (Suc 0))) j = 2 ^ (j+3) - 3"

lemma param_list_bex[param]: "\<lbrakk>(P,P')\<in>(Ra\<rightarrow>Id); (l,l')\<in>\<langle>Ra\<rangle> list_rel\<rbrakk> \<Longrightarrow> (\<exists>x\<in>set l. P x, \<exists>x\<in>set l'. P' x) \<in> Id"