Statement:
stringlengths
7
24.3k
lemma "let x = y in [x, x] = [y, y]"
lemma "Ex (\<lambda>x. f x y \<noteq> f x y) = False"
lemma (in topology) ex_dense_closure_interE: assumes ssub: "S \<subseteq> carrier" and H: "\<And>D C. \<lbrakk> D \<subseteq> carrier; C \<subseteq> carrier; D dense; C closed; S = D \<inter> C \<rbrakk> \<Longrightarrow> R" shows "R"
lemma update_twice: "apply_updates [(r, a), (r, b)] s regs = regs (r $:= aval b s)"
lemma T_nonTickFree_imp_decomp: "\<lbrakk>t \<in> T P; \<not> tickFree t\<rbrakk> \<Longrightarrow> \<exists>s. t = s @ [tick]"
lemma eval_ajoin: fixes \<phi> :: "('a :: infinite, 'b) fo_fmla" assumes wf: "fo_wf \<phi> I t\<phi>" "fo_wf \<psi> I t\<psi>" shows "fo_wf (Conj \<phi> (Neg \<psi>)) I (eval_ajoin (fv_fo_fmla_list \<phi>) t\<phi> (fv_fo_fmla_list \<psi>) t\<psi>)"
lemma (in cpx_sq_mat) lhv_lsum_scal_integrable: assumes "lhv M A B R XA XB" and "a\<in> spectrum A" shows "integrable M (\<lambda>x. \<Sum>b\<in>spectrum B. c * XA a x * (f b) * XB b x)"
lemma nested_case_bind: "(case p of (a,b) \<Rightarrow> bind (case a of (a1,a2) \<Rightarrow> m a b a1 a2) (f a b)) = (case p of ((a1,a2),b) \<Rightarrow> bind (m (a1,a2) b a1 a2) (f (a1,a2) b))" "(case p of (a,b) \<Rightarrow> bind (case b of (b1,b2) \<Rightarrow> m a b b1 b2) (f a b)) = (case p of (a,b1,b2) \<Rightarrow> bind (m a (b1,b2) b1 b2) (f a (b1,b2)))"
lemma integrable_bernoulli_pmf [intro]: fixes f :: "bool \<Rightarrow> 'a :: {banach, second_countable_topology}" shows "integrable (bernoulli_pmf p) f"
lemma bisimParNil: fixes P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> \<zero> \<sim> P"
lemma equivalence_relation_on_states_ran : assumes "equivalence_relation_on_states M f" and "q \<in> states M" shows "f q \<subseteq> states M"
lemma wens_Id [simp]: "wens F Id B = B"
lemma filter_distinct_at4: "distinct xs \<Longrightarrow> xs = (as @ u # bs) \<Longrightarrow> [v\<leftarrow>xs. v = u \<or> v \<in> set us] = u # us \<Longrightarrow> set zs \<inter> set us \<subseteq> {u} \<union> set as \<Longrightarrow> [v \<leftarrow> zs@bs. v \<in> set us] = us"
lemma ri_runs2sigs_upd_init_none [simp]: "\<lbrakk> Na \<notin> dom runz \<rbrakk> \<Longrightarrow> ri_runs2sigs (runz(Na \<mapsto> (Init, [A, B], []))) = ri_runs2sigs runz"
lemma size_fset_alt: "size_fset (size_prod snd (\<lambda>_. 0)) (map_prod (\<lambda>t. (t, size t)) (\<lambda>x. x) |`| xs) = (\<Sum>(x,y)\<in> fset xs. size x + 2)"
lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
lemma unstream_splice_trans_Right_only: "unstream (splice_trans g h) (Right_only sh) = unstream h sh"
theorem list_dtree_iff_wf_list: "wf_list_arcs xs \<and> (\<forall>v \<in> fst ` set xs. set r \<inter> set v = {}) \<and> r \<noteq> [] \<and> wf_list_lverts xs \<longleftrightarrow> list_dtree (dtree_from_list r xs)"
lemma step_nonneg: "i < k \<Longrightarrow> x \<ge> x\<^sub>1 \<Longrightarrow> bs ! i * x + (hs ! i) x \<ge> 0"
lemma stlI: "strict_lower_triangular_mat X \<Longrightarrow> lower_triangular_mat X"
lemma f: "current \<Gamma> f"
lemma surj_on_subset_right: "\<lbrakk> surj_on f A B; B' \<subseteq> B \<rbrakk> \<Longrightarrow> surj_on f A B'"
lemma r_amalgamate: "eval r_amalgamate [i, j] \<down>= amalgamate i j"
lemma ex1_poincare_line: assumes "u \<noteq> v" "u \<in> unit_disc" "v \<in> unit_disc" shows "\<exists>! l. is_poincare_line l \<and> u \<in> circline_set l \<and> v \<in> circline_set l"
lemma map2_elemE1: assumes "length xs = length ys" "x \<in> set xs" obtains y where "y \<in> set ys" "f x y \<in> set (map2 f xs ys)"
lemma monotone_directed_complete: assumes comp: "directed_complete A r" assumes fI: "f ` I \<subseteq> A" and dir: "directed I ri" and I0: "I \<noteq> {}" and mono: "monotone_on I ri r f" shows "\<exists>s. extreme_bound A r (f ` I) s"
lemma cong_power_nat_code [code_unfold]: "[b ^ e = (x ::nat)] (mod m) \<longleftrightarrow> mod_exp b e m = x mod m"
lemma outer_loop_rel_wf: assumes "finite V" shows "wf (outer_loop_rel src)"
lemma a_closure' [simp]: "d (ad x) = ad x"
lemma trace_positive_eq: fixes A :: "complex mat" assumes pos: "positive A" shows "trace (A * adjoint A) \<le> (trace A)\<^sup>2"
lemma ldropn_0 [simp]: "ldropn 0 xs = xs"
lemma sword32_integer_eq: "sword32_of_integer (integer_of_sword32 w) = w"
lemma bin_cat_foldl_lem: "foldl (\<lambda>u k. concat_bit n k u) x xs = concat_bit (size xs * n) (foldl (\<lambda>u k. concat_bit n k u) y xs) x"
lemma "rec_eval (Pr v va vb) [] = undefined"
lemma point_index_distrib: "(B1 + B2) index ps = B1 index ps + B2 index ps"
lemma classical_operator_Some[simp]: "classical_operator (Some::'a\<Rightarrow>_) = id_cblinfun"
lemma kauff_mat_swingneg: "kauff_mat (r_under_braid) = kauff_mat (l_under_braid)"
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
lemma isometry_on_inverse: assumes "isometry_on X f" shows "isometry_on (f`X) (inv_into X f)" "\<And>x. x \<in> X \<Longrightarrow> (inv_into X f) (f x) = x" "\<And>y. y \<in> f`X \<Longrightarrow> f (inv_into X f y) = y" "bij_betw f X (f`X)"
lemma trivial_limit_at_left_real [simp]: "\<not> trivial_limit (at_left x)" for x :: "'a::{no_bot,dense_order,linorder_topology}"
lemma "SML3 \<Longrightarrow> nNor \<^bold>\<not>"
lemma correctCompositionKS_exprChannel_s_P: assumes "subcomponents PQ = {P,Q}" and "correctCompositionKS PQ" and "sKS secret \<notin> LocalSecrets PQ" and "ch \<in> ins P" and "exprChannel ch (sE secret)" and "sKS secret \<notin> specKeysSecrets PQ" and "correctCompositionIn PQ" shows "ch \<in> ins PQ \<and> exprChannel ch (sE secret)"
theorem steps_approx_CS: "steps s c n \<le> CS c"
lemma intersection_number_empty_iff: assumes "finite b1" shows "b1 \<inter> b2 = {} \<longleftrightarrow> b1 |\<inter>| b2 = 0"
lemma exec_callUndefined: "\<lbrakk>\<Gamma> p=None\<rbrakk> \<Longrightarrow> \<Gamma>\<turnstile>\<langle>call init p return c,Normal s\<rangle> \<Rightarrow> Stuck"
lemma finite_carrier_vec[simp]: "finite (carrier_vec n:: 'b::finite vec set)"
lemma wt_subst_fv_termtype_subterm: assumes "x \<in> fv (\<theta> y)" and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" and "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> y)" shows "\<Gamma> (Var x) \<sqsubseteq> \<Gamma> (Var y)"
lemma robdd_invar_idsI: assumes "\<And>b1 b2. \<lbrakk>b1 \<in> (subrobdds_set bs); b2 \<in> (subrobdds_set bs)\<rbrakk> \<Longrightarrow> (robdd_\<alpha> b1 = robdd_\<alpha> b2) \<longleftrightarrow> (robdd_get_id b1 = robdd_get_id b2)" shows "robdd_invar_ids bs"
lemma fofu_II_III: "\<not> (\<exists> p. isAugmentingPath p) \<Longrightarrow> \<exists>k'. NCut c s t k' \<and> val = NCut.cap c k'"
lemma P_set_mult_closed: assumes "n \<noteq> 0" assumes "a \<in> P_set n" assumes "b \<in> P_set n" shows "a \<otimes> b \<in> P_set n"
lemma diagonal_imp_submatrix_element_not0: assumes dA: "diagonal_mat A" and A_carrier: "A \<in> carrier_mat n m" and Ik: "card I = k" and Jk: "card J = k" and I: "I \<subseteq> {0..<n}" and J: "J \<subseteq> {0..<m}" and b: "b < k" and ex_not0: "\<exists>i. i<k \<and> submatrix A I J $$ (i, b) \<noteq> 0" shows "\<exists>!i. i<k \<and> submatrix A I J $$ (i, b) \<noteq> 0"
lemma fixes h :: "('a::sep_algebra) * ('b::sep_algebra)" shows "((%(a,b). P a \<and> b = 0) ** (%(a,b). a = 0 \<and> Q b)) = (%(a,b). P a \<and> Q b)"
lemma var_not_app: shows "Var x \<noteq> App A B"
lemma one_zero_contra[dest,consumes 2]: "1 \<le> x \<Longrightarrow> (x :: QDelta) \<le> 0 \<Longrightarrow> False"
lemma index_drop: "\<And>x i. \<lbrakk> x \<in> set xs; index xs x < i \<rbrakk> \<Longrightarrow> x \<notin> set(drop i xs)"
lemma (in PolynRg) coeff_max_zeroTr:"pol_coeff S c \<Longrightarrow> \<forall>j. j \<le> (fst c) \<and> (c_max S c) < j \<longrightarrow> (snd c) j = \<zero>\<^bsub>S\<^esub>"
lemma scene_equiv_refl [simp]: "idem_scene a \<Longrightarrow> s \<approx>\<^sub>S s on a"
lemma many_letters [code, code_unfold]: "many_letters = many is_letter"
lemma add_independentS_ConsI : assumes "add_independentS As" "\<And>x a. \<lbrakk> x\<in>(\<Sum>X\<leftarrow>As. X); a \<in> A; a+x = 0 \<rbrakk> \<Longrightarrow> a = 0" shows "add_independentS (A#As)"
lemma avl_delete: "avl t \<Longrightarrow> avl (delete x t) \<and> height t = height (delete x t) + (if decr t (delete x t) then 1 else 0)"
lemma times_lepoll_mono: assumes "A \<lesssim> C" "B \<lesssim> D" shows "A \<times> B \<lesssim> C \<times> D"
lemma empty_set_less_eq_aux [simp]: "{} \<sqsubseteq>' A \<longleftrightarrow> finite A"
lemma (in rga) insert_commute_assms: assumes "{Deliver (i, Insert e n), Deliver (i', Insert e' n')} \<subseteq> set (history j)" and "hb.concurrent (i, Insert e n) (i', Insert e' n')" shows "n = None \<or> n \<noteq> Some (fst e')"
lemma rel_fundefs_next_instr1: assumes rel_F1_F2: "rel_fundefs F1 F2" and next_instr1: "next_instr F1 f l pc = Some instr1" shows "\<exists>instr2. next_instr F2 f l pc = Some instr2 \<and> norm_eq instr1 instr2"
lemma del_UnEdge_frame[intro]: "x\<in>edges g \<Longrightarrow> x\<noteq>(v,e,v') \<Longrightarrow>x\<noteq>(v',e,v) \<Longrightarrow> x\<in>edges (del_unEdge v e v' g)"
lemma list_concat_non_elem : "x \<notin> set xs \<Longrightarrow> x \<notin> set ys \<Longrightarrow> x \<notin> set (xs@ys)"
lemma resid_Arr_Ide: shows "\<lbrakk>Ide a; Coinitial t a\<rbrakk> \<Longrightarrow> t \\ a = t"
lemma join_dual [simp]: "p \<squnion>\<^bsub>inv_gorder L\<^esub> q = p \<sqinter>\<^bsub>L\<^esub> q"
lemma (in Module) mHom_ex_zero:"R module N \<Longrightarrow> mzeromap M N \<in> mHom R M N"
lemma turing_computable_partial_imp_turing_computable_partial': "turing_computable_partial f \<longrightarrow> turing_computable_partial' f"
lemma Jcc_pieces_LAss: assumes [simp]: "P \<equiv> compP\<^sub>2 P\<^sub>1" and "Jcc_pieces P\<^sub>1 E C M h\<^sub>0 vs ls\<^sub>0 pc ics frs sh\<^sub>0 I h\<^sub>1 ls\<^sub>1 sh\<^sub>1 v xa (i:=e) = (True, frs\<^sub>0, (xp',h',(v#vs',ls',C\<^sub>0,M',pc',ics')#frs',sh'), err)" shows "Jcc_pieces P\<^sub>1 E C M h\<^sub>0 vs ls\<^sub>0 pc ics frs sh\<^sub>0 I h\<^sub>1 ls\<^sub>1 sh\<^sub>1 v' xa e = (True, frs\<^sub>0, (xp',h',(v'#vs',ls',C\<^sub>0,M',pc' - 2,ics')#frs',sh'), (\<exists>pc\<^sub>1. pc \<le> pc\<^sub>1 \<and> pc\<^sub>1 < pc + size(compE\<^sub>2 e) \<and> \<not> caught P pc\<^sub>1 h\<^sub>1 xa (compxE\<^sub>2 e pc (size vs)) \<and> (\<exists>vs'. P \<turnstile> (None,h\<^sub>0,frs\<^sub>0,sh\<^sub>0) -jvm\<rightarrow> handle P C M xa h\<^sub>1 (vs'@vs) ls\<^sub>1 pc\<^sub>1 ics frs sh\<^sub>1)))"
lemma paths_for_input'_set : assumes "q \<in> states M" shows "paths_for_input' (h_from M) xs q prev = {prev@p | p . path M q p \<and> map fst (p_io p) = xs}"
lemma mapping_linorder_matrix: fixes f :: "('a::finite,'b::linorder_stone_relation_algebra_expansion) square" shows "matrix_stone_relation_algebra.mapping f \<longleftrightarrow> (\<forall>i . \<exists>j . f (i,j) = top \<and> (\<forall>k . j \<noteq> k \<longrightarrow> f (i,k) = bot))"
lemma qbs_closed1I: assumes "\<And>\<alpha> f. \<alpha> \<in> Mx \<Longrightarrow> f \<in> real_borel \<rightarrow>\<^sub>M real_borel \<Longrightarrow> \<alpha> \<circ> f \<in> Mx" shows "qbs_closed1 Mx"
lemma planning_dimacs_complete_code': "\<lbrakk>ast_problem.well_formed prob; (\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op); (\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> is_standard_operator op); ast_problem.valid_plan prob \<pi>s; length \<pi>s \<le> h\<rbrakk> \<Longrightarrow> let cnf_formula = (SASP_to_DIMACS' h prob) in \<exists>dimacs_M. dimacs_model dimacs_M cnf_formula"
lemma cl10_d_var3: "d (x \<sqinter> nc) = 1\<^sub>\<sigma> \<sqinter> x \<cdot> U"
lemma infsuff_inf: "x \<in> infsuff A s \<Longrightarrow> x \<in> A\<^sup>\<omega>"
lemma methd: "[| ws_prog G; (C,S,fs,mdecls) \<in> set G; (sig,rT,code) \<in> set mdecls |] ==> method (G,C) sig = Some(C,rT,code) \<and> is_class G C"
lemma fract_content_eq_0_iff [simp]: "fract_content p = 0 \<longleftrightarrow> p = 0"
lemma Inf_if_Inf_from: "\<iota> \<in> Inf_from N \<Longrightarrow> \<iota> \<in> Inf"
lemma sequence_number_increases': "paodv i \<TTurnstile>\<^sub>A (\<lambda>((\<xi>, _), _, (\<xi>', _)). sn \<xi> \<le> sn \<xi>')"
lemma twice_field_differentiable_at_sin [simp, intro]: "sin twice_field_differentiable_at x"
lemma dec_interp_enc_Inl: "\<lbrakk>dec_interp n FO (enc (w, I)) ! i = Inl p'; I ! i = Inl p; i \<in> FO; i < n; length I = n; p < length w; wf_interp w I\<rbrakk> \<Longrightarrow> p = p'"
lemma pow2_scaleR_mtx_cnst_acc: "(t *\<^sub>R K)\<^sup>2 = mtx ( [0,0,t\<^sup>2] # [0,0,0] # [0,0,0] # [])"
lemma last_prog_same_append: "\<And>xs p\<^sub>s\<^sub>b. last_prog p\<^sub>s\<^sub>b (sb@xs) = p\<^sub>s\<^sub>b \<Longrightarrow> last_prog p\<^sub>s\<^sub>b xs = p\<^sub>s\<^sub>b"
lemma uminus_interval_tendsto: fixes x :: "'a :: topological_group_add" assumes "X \<longlonglongrightarrow>\<^sub>i x" shows "(\<lambda>i. - X i) \<longlonglongrightarrow>\<^sub>i -x"
lemma map_some_list [simp]: "map the (map Some L) = L"
lemma REVERSE[import_const REVERSE : rev]: "rev [] = ([] :: 'A list) \<and> rev ((x::'A) # l) = rev l @ [x]"
lemma ReZ_of_int [simp]: "ReZ (of_int n) = n" and ImZ_of_int [simp]: "ImZ (of_int n) = 0"
lemma dord_pm_trans: assumes "ord s t \<Longrightarrow> ord t u \<Longrightarrow> ord s u" and "dord_pm ord s t" and "dord_pm ord t u" shows "dord_pm ord s u"
lemma gcsort_count_input: "count_inv (count (mset xs)) (0, [length xs], xs)"
lemma "\<turnstile> \<lbrace>\<acute>M = \<acute>N\<rbrace> \<acute>M := \<acute>M + 1 \<lbrace>\<acute>M \<noteq> \<acute>N\<rbrace>"
lemma mdeg_img_eq: assumes "\<forall>(t,e) \<in> fset xs. max_deg (fst (f (t,e))) = max_deg t" and "fcard (f |`| xs) = fcard xs" shows "max_deg (Node r (f |`| xs)) = max_deg (Node r xs)"
lemma L_left_zero_star: "L * x\<^sup>\<star> = L"
lemma inf_concat_surj: assumes j: "j < n i" shows "\<exists>k. inf_concat n k = (i, j)"
lemma unitary_diagD: assumes "unitary_diag A B U" shows "similar_mat_wit A B U (Complex_Matrix.adjoint U)" "diagonal_mat B" "unitary U"
lemma mult_right_inj: "0 < n \<Longrightarrow> inj (\<lambda>x. x * (n::nat))"
lemma gtrs_CE_aux: assumes step:"(s, t) \<in> srsteps_with_root_step \<F> (\<R>\<^sup>\<leftrightarrow>)" and gtrs: "ground_sys \<R>" "ground_sys \<S>" and ce: "CE (gsrstep \<F> \<R>) (gsrstep \<F> \<S>)" shows "(s, t) \<in> (srstep \<F> \<S>)\<^sup>\<leftrightarrow>\<^sup>*"
lemma pr_key_analz: "(evs, S, A, U) \<in> protocol \<Longrightarrow> (Key K \<in> analz (A \<union> spies evs)) = (Key K \<in> A)"
lemma "!!a::real. a + b + c \<le> i + j + k \<and> a \<le> b \<and> b \<le> c \<and> i \<le> j \<and> j \<le> k --> a + a + a \<le> k + k + k"