Statement:
stringlengths 7
24.3k
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lemma strand_sem_c_imp_ineqs_neq:
assumes "\<lbrakk>M; S\<rbrakk>\<^sub>c \<I>" "Inequality X [(t,t')] \<in> set S"
shows "t \<noteq> t' \<and> (\<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>)
\<longrightarrow> t \<cdot> \<delta> \<noteq> t' \<cdot> \<delta> \<and> t \<cdot> \<delta> \<cdot> \<I> \<noteq> t' \<cdot> \<delta> \<cdot> \<I>)"
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lemma continuous_finite_range_constant:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
assumes "connected S"
and "continuous_on S f"
and "finite (f ` S)"
shows "f constant_on S"
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theorem p50: "\<forall>x. F(a,x) \<or> (\<forall>y. F(x,y)) \<Longrightarrow> \<exists>x. \<forall>y. F(x,y)"
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lemma destrmi_someD: "destrmi' e bdd = Some x \<Longrightarrow> bdd_sane bdd \<and> bdd_node_valid bdd e"
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lemma induced_automorphism_halfmorphism_fimage_to_fopp:
"ChamberComplexMorphism (f\<turnstile>X) folding_f.opp_half_apartment \<s>"
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lemma fmember_implies_member: "e |\<in>| f \<Longrightarrow> e \<in> fset f"
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lemma SIod_isom_Iod:"\<lbrakk>Order D; T \<subseteq> carrier D \<rbrakk> \<Longrightarrow>
ord_isom (SIod D T) (Iod D T) (\<lambda>x\<in>T. x)"
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lemma dom_lookup_keys: "dom (DAList.lookup al) = keys al"
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lemma dense_lattice_char_1:
"(\<forall>x y . x \<sqinter> y = bot \<longrightarrow> x = bot \<or> y = bot) \<longleftrightarrow> (\<forall>x . x \<noteq> bot \<longrightarrow> dense x)"
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lemma set_disconnected_nodes_get_disconnected_nodes_l_set_disconnected_nodes_get_disconnected_nodes
[instances]:
"l_set_disconnected_nodes_get_disconnected_nodes ShadowRootClass.type_wf get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
|
lemma num_gholes_sup_gmctxt_args:
assumes "(C, D) \<in> comp_gmctxt"
shows "num_gholes C = length (sup_gmctxt_args C D)"
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lemma ta_der_monos:
"ta_der \<A> t |\<subseteq>| ta_der A t" "ta_der \<B> t |\<subseteq>| ta_der A t"
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theorem compiler_correctness_exec: "
\<lbrakk> G \<turnstile> Norm (hp, loc) -st-> Norm (hp', loc');
wf_java_prog G;
class_sig_defined G C S;
wtpd_stmt (env_of_jmb G C S) st;
(None,hp,loc) ::\<preceq> (env_of_jmb G C S) \<rbrakk> \<Longrightarrow>
{(TranslComp.comp G), C, S} \<turnstile>
{hp, os, (locvars_locals G C S loc)}
>- (compStmt (gmb G C S) st) \<rightarrow>
{hp', os, (locvars_locals G C S loc')}"
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lemma nonzero_lt_gtD: "(n::_::linorder) \<noteq> 0 \<Longrightarrow> 0 < n \<or> n < 0"
|
lemma freshInputAction:
fixes P :: pi
and a :: name
and b :: name
and P' :: pi
and c :: name
assumes "P \<longmapsto> a<b> \<prec> P'"
and "c \<sharp> P"
shows "c \<noteq> a"
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lemma interval_lowerbound[simp]:
"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
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lemma lrev_body2_strict[simp]: "lrev_body2\<cdot>r\<cdot>\<bottom> = \<bottom>"
|
theorem open_seq_step_invariant [intro]:
assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"
and "initiali i (init OA) (init A)"
and spo: "trans OA = oseqp_sos \<Gamma> i"
and sp: "trans A = seqp_sos \<Gamma>"
shows "OA \<Turnstile>\<^sub>A (act I, other ANY {i} \<rightarrow>) (seqll i P)"
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lemma thm_noncont_e_e_5[PLM]:
"[\<^bold>\<exists> (F::\<Pi>\<^sub>1) G . F \<^bold>\<noteq> G \<^bold>& NonContingent F \<^bold>& NonContingent G in v]"
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lemma zig_zig:
fixes s u r r1' r2' T a b
defines "t == Node s a (Node u b r)" and "t' == Node (Node s a u) b r1'"
assumes "size r1' \<le> size r"
"T_part p r + \<Phi> r1' + \<Phi> r2' - \<Phi> r \<le> 2 * \<phi> r + 1"
shows "T_part p r + 1 + \<Phi> t' + \<Phi> r2' - \<Phi> t \<le> 2 * \<phi> t + 1"
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lemma timpl_closure_set_timpls_trancl_eq:
"timpl_closure_set M (c\<^sup>+) = timpl_closure_set M c"
|
lemma length_lesss_less [intro]:
assumes "x \<in> set xs"
shows "length (lesss R x xs) < length xs"
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lemma AxB_AxPB: \<open>AxB = AxPB o lift\<close>
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lemma paths_cross_once:
assumes path_Q: "Q \<in> \<P>"
and path_R: "R \<in> \<P>"
and Q_neq_R: "Q \<noteq> R"
and QR_nonempty: "Q\<inter>R \<noteq> {}"
shows "\<exists>!a\<in>\<E>. Q\<inter>R = {a}"
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lemma set_insort_key_rel[simp]: \<open>set (insort_key_rel R x xs) = insert x (set xs)\<close>
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lemma currentD_weight_in: "current \<Gamma> h \<Longrightarrow> h (x, y) \<le> weight \<Gamma> y"
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lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
|
lemma Rep_hierauto_select:
"(HAInitValue HA, SAs HA, HAEvents HA, CompFun HA): hierauto"
|
lemma lcoeff_nonzero2: assumes p_in_R: "p \<in> carrier P" and p_not_zero: "p \<noteq> \<zero>\<^bsub>P\<^esub>" shows "lcoeff p \<noteq> \<zero>"
|
lemma regular_closed_star:
"regular x \<Longrightarrow> regular (x\<^sup>\<star>)"
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lemma C14_aux: "m \<le> n \<Longrightarrow> x\<^bsup>m\<^esup> \<cdot> (x\<^bsup>n\<^esup>)\<^sup>\<star> = (x\<^bsup>n\<^esup>)\<^sup>\<star> \<cdot> x\<^bsup>m\<^esup>"
|
lemma exp_similiar_sq_mtx_diag:
assumes "A \<sim> sq_mtx_diag f"
shows "exp A \<sim> exp (sq_mtx_diag f)"
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lemma op_dghm_dghm_0: "op_dghm (dghm_0 \<CC>) = dghm_0 (op_dg \<CC>)"
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lemma(in UP_cring) to_fun_finsum:
assumes "finite (Y::'d set)"
assumes "f \<in> UNIV \<rightarrow> carrier (UP R)"
assumes "t \<in> carrier R"
shows "to_fun (finsum (UP R) f Y) t = finsum R (\<lambda>i. (to_fun (f i) t)) Y"
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lemma nonconst_poly_irreducible_iff:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly"
assumes "degree p \<noteq> 0"
shows "irreducible p \<longleftrightarrow> irreducible (fract_poly p) \<and> content p = 1"
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lemma is_nextElem_between_empty': "between vs a b = [] \<Longrightarrow> distinct vs \<Longrightarrow> a \<in> set vs \<Longrightarrow> b \<in> set vs \<Longrightarrow>
a \<noteq> b \<Longrightarrow> is_nextElem vs a b"
|
lemma read_dcache_mod_state: "read_data_cache s addr = read_data_cache
(s\<lparr>cpu_reg := new_cpu_reg,
user_reg := new_user_reg,
dwrite := new_dwrite,
state_var := new_state_var,
traps := new_traps,
undef := new_undef\<rparr>) addr"
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lemma (in Corps) val_1_nonzero:"\<lbrakk>valuation K v; x \<in> carrier K; v x = 1\<rbrakk> \<Longrightarrow>
x \<noteq> \<zero>"
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lemma (in Module) l_span_closed1:"\<lbrakk>H \<subseteq> carrier M;
s \<in> {j. j \<le> (n::nat)} \<rightarrow> carrier R;
f \<in> {j. j \<le> n} \<rightarrow> linear_span R M (carrier R) H \<rbrakk> \<Longrightarrow>
\<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s (f j)) n \<in> linear_span R M (carrier R) H"
|
lemma compact_convex_collinear_segment:
fixes S :: "'a::euclidean_space set"
assumes "S \<noteq> {}" "compact S" "convex S" "collinear S"
obtains a b where "S = closed_segment a b"
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lemma coclop_lsl_lsu: "coclop (\<nu> \<circ> \<nu>\<^sup>\<natural>)"
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lemma ordinal_arrsD[dest]:
assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> ordinal_arrs A"
shows "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> A" and "a \<le> b"
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lemma norm_beta_complete_multi:
assumes "A \<turnstile> \<langle>l,[D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l',Z\<rangle>" "global_clock_numbering A v n" "valid_abstraction A X k"
and "valid_dbm D"
obtains D' where "A \<turnstile> \<langle>l,D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l',D'\<rangle>" "[D']\<^bsub>v,n\<^esub> = Z" "valid_dbm D'"
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lemma (in module) finite_span:
assumes fin: "finite S" and inC: "S\<subseteq>carrier M"
shows "span S = {lincomb a S | a. a\<in> (S\<rightarrow>carrier R)}"
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lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
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lemma lemma1:
"FP Rprg \<subseteq> fixedpoint"
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lemma [simp]: "safe t r \<Longrightarrow> safe (Enter g r c # t) r"
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lemma eq_adm: "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda> x. eq a (f x) (g x))"
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lemma aezfun_setspan_proj_sum_list :
"aezfun_setspan_proj A (\<Sum>x\<leftarrow>xs. f x) = (\<Sum>x\<leftarrow>xs. aezfun_setspan_proj A (f x))"
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lemma acc_nodes_le2: "acc (fst (snd nodes_semi))"
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lemma rhs_altdef: "rhs = 2 ^ T * (\<Prod>i=1..n. 2 ^ i - 1)"
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lemma lexp_lex: "lexp order xs ys \<longleftrightarrow> (xs, ys) \<in> lex {(x, y). order x y}"
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lemma finite_jumpFs_linear_pos:
assumes "c>0"
shows "finite_jumpFs (f o (\<lambda>x. c * x + b)) lb ub \<longleftrightarrow> finite_jumpFs f (c * lb +b) (c * ub + b)"
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lemma infsetsum_set_nn_integral_reals:
assumes "f abs_summable_on UNIV" "\<And>n. f n \<ge> 0"
shows "infsetsum f UNIV = set_nn_integral lborel {0::real..} (\<lambda>x. f (nat (floor x)))"
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lemma assumes "us = \<epsilon> \<or> last us \<noteq> w" and "\<epsilon> \<notin> set us"
shows emp_not_in_glue: "\<epsilon> \<notin> set (glue w us)"
and glued_not_in_glue: "w \<notin> set (glue w us)"
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lemma nth_partition: "i < length xs \<Longrightarrow>
(\<And>i'. i' < length (filter P xs) \<Longrightarrow> Q (filter P xs ! i')) \<Longrightarrow>
(\<And>i'. i' < length (filter (Not \<circ> P) xs) \<Longrightarrow> Q (filter (Not \<circ> P) xs ! i')) \<Longrightarrow> Q (xs ! i)"
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lemma exp_cf_cat_is_tiny_functor:
assumes "\<Z> \<beta>" and "\<alpha> \<in>\<^sub>\<circ> \<beta>" and "category \<alpha> \<AA>" and "\<KK> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"
shows "exp_cf_cat \<alpha> \<KK> \<AA> : cat_FUNCT \<alpha> \<AA> \<BB> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<beta>\<^esub> cat_FUNCT \<alpha> \<AA> \<CC>"
|
lemma bop_convr [simp]: "(bop f u v)\<^sup>- = bop f (u\<^sup>-) (v\<^sup>-)"
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lemma neighbors_ss_eq_neighborhoodY: "v \<in> Y \<Longrightarrow> neighborhood v = neighbors_ss v X"
|
lemma cong_gcd_eq_poly:
"gcd a m = gcd b m" if "[(a::'a mod_ring poly) = b] (mod m)"
|
lemma iMODb_inext_diff_const: "
\<lbrakk> t \<in> [r, mod m, c]; t < r + m * c \<rbrakk> \<Longrightarrow> inext t [r, mod m, c] - t = m"
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lemma min_ed_min_eds: "min_ed xs ys = cost(min_eds xs ys)"
|
lemma order_prod:
assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
shows "order x (\<Prod>y\<in>A. f y) = (\<Sum>y\<in>A. order x (f y))"
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lemma DerivProp_Aux: "(X,c,A):Deriv \<Longrightarrow> DProp A"
|
lemma Sigma_Suc1 [simp]:
"{0..< Suc n} \<times> B = ({n} \<times> B) \<union> ({0..<n} \<times> B)"
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lemma worecSL_isLim:
assumes a: "adm_woL L" and i: "isLim i"
shows "worecSL S L i = L (worecSL S L) i"
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lemma uniformly_convergent_eval_fds_aux':
assumes conv: "fds_converges f (s0 :: 'a)"
assumes B: "compact B" "\<And>z. z \<in> B \<Longrightarrow> z \<bullet> 1 > s0 \<bullet> 1"
shows "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
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lemma i0_iless_eSuc [simp]: "0 < eSuc n"
|
lemma has_sum_finite[simp]:
assumes "finite F"
shows "has_sum f F (sum f F)"
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lemma bits_minus_1_mod_2_eq [simp]:
\<open>(- 1) mod 2 = 1\<close>
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lemma diamond_lower_bound_right:
"|x>(d(y) * d(z)) \<le> |x>d(y)"
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lemma ptrm_prv_exi_imp_all:
assumes \<sigma>: "\<sigma> \<in> ptrm n" and [simp]: "\<phi> \<in> fmla"
shows "prv (imp (exi out (cnj \<sigma> \<phi>)) (all out (imp \<sigma> \<phi>)))"
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lemma (in is_dghm) dghm_is_tm_dghm_if_HomDom_finite_digraph:
assumes "finite_digraph \<alpha> \<AA>"
shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
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lemma mult_bounds_enclose_zero1:
"min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \<le> 0"
"0 \<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))"
if "la \<le> 0" "0 \<le> ua"
for la lb ua ub:: "'a::linordered_idom"
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lemma ldeep_s_eq_list_sel_aux'_split:
"y \<in> set xs \<Longrightarrow> \<exists>as bs. as @ y # bs = xs \<and> ldeep_s sel xs y = list_sel_aux' sel bs y"
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lemma minimal_unsat_state_core_translation: assumes
unsat: "minimal_unsat_state_core (s :: ('i,'a::lrv)state)" and
tabl: "\<forall>(v :: 'a valuation). v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t t" and
index: "index_valid as s" and
imp: "as \<Turnstile>\<^sub>i \<B>\<I> s" and
I: "I = the (\<U>\<^sub>c s)"
shows "minimal_unsat_core_tabl_atoms (set I) t as"
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lemma i_set_mult_mod_gr0_div_not_in_i_set:"
\<lbrakk> 0 < k; 0 < d; 0 < k mod d \<rbrakk> \<Longrightarrow> \<exists>I\<in>i_set_mult k. I \<oslash> d \<notin> i_set"
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lemma (in Universe) UniversePair: "\<lbrakk>Elem u U ; Elem v U\<rbrakk> \<Longrightarrow> Elem (Opair u v) U"
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lemma asprod_pos_pos:"0 \<le> x \<Longrightarrow> 0 \<le> int n *\<^sub>a x"
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lemma mopup_a_nth:
"\<lbrakk>q < n; x < 4\<rbrakk> \<Longrightarrow> mopup_a n ! (4 * q + x) =
mopup_a (Suc q) ! ((4 * q) + x)"
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lemma Expr2ZFDepthSuc: "ZFDepth (Expr2ZF e) = Suc n \<Longrightarrow> ZFType (Expr2ZF e) = 1"
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lemma tm_append_second_fetch0_eq:
assumes
even: "length A mod 2 = 0"
and off: "off = length A div 2"
and fetch: "fetch B s b = (ac, 0)"
and notfinal: "s \<noteq> 0"
shows "fetch (A @ shift B off) (s + off) b = (ac, 0)"
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theorem Gromov_hyperbolic_invariant_under_quasi_isometry:
assumes "quasi_isometric (UNIV::('a::geodesic_space) set) (UNIV::('b::Gromov_hyperbolic_space_geodesic) set)"
shows "\<exists>delta. Gromov_hyperbolic_subset delta (UNIV::'a set)"
|
lemma [code]: "iam_update k v a = upd_oo
(do {
l\<leftarrow>Array.len a;
let newsz = max (k+1) (2 * l + 3);
a\<leftarrow>array_grow a newsz None;
Array.upd k (Some v) a
})
k (Some v) a"
|
theorem prim_list_impl_correct_presentation:
shows "case prim_list_impl l r of
None \<Rightarrow> \<not>G.valid_wgraph_repr l \<comment> \<open>Invalid input\<close>
| Some \<pi>i \<Rightarrow> let
g=G.\<alpha>g (G.from_list l);
w=G.\<alpha>w (G.from_list l);
rg=component_of g r;
t=P.\<alpha>_MST r \<pi>i
in
G.valid_wgraph_repr l \<comment> \<open>Valid input\<close>
\<and> P.invar_MST \<pi>i \<comment> \<open>Output satisfies invariants\<close>
\<and> is_MST w rg t \<comment> \<open>and represents MST\<close>"
|
lemma mismatchId:
fixes a :: name
and b :: name
and P :: pi
assumes "a \<noteq> b"
shows "[a\<noteq>b]P \<sim>\<^sub>e P"
|
lemma mat_O_dim:
"mat_O \<in> carrier_mat N N"
|
theorem p6: "p \<or> \<not>p"
|
lemma a_comp_simp [simp]: "(ad x \<oplus> ad y) \<cdot> d x = ad y \<cdot> d x"
|
lemma utility_less_iff:
"x \<in> carrier \<Longrightarrow> y \<in> carrier \<Longrightarrow> u x < u y \<longleftrightarrow> x \<prec>[le] y"
|
lemma to_Zp_int_inc:
"to_Zp ([(a::int)]\<cdot>\<one>) = ([a]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one> \<^bsub>Z\<^sub>p\<^esub>)"
|
lemma conv_backward_finite_path:
"backward_finite_path x \<longleftrightarrow> forward_finite_path (x\<^sup>T)"
|
lemma FD2: "Fr_2(\<F>\<^sub>D \<D>)"
|
lemma spmf_spmf_of_pmf [simp]: "spmf (spmf_of_pmf p) x = pmf p x"
|
lemma lit_eq_negate_conv[simp]:
"Lit p v = negate l \<longleftrightarrow> l = Lit (\<not>p) v"
"negate l = Lit p v \<longleftrightarrow> l = Lit (\<not>p) v"
|
lemma closedin_Hausdorff_sing_eq:
"Hausdorff_space X \<Longrightarrow> closedin X {x} \<longleftrightarrow> x \<in> topspace X"
|
lemma partn_lst_VWF_nontriv:
assumes "partn_lst_VWF \<beta> \<alpha> n" "l = length \<alpha>" "Ord \<beta>" "l > 0"
obtains i where "i < l" "\<alpha>!i \<le> \<beta>"
|
lemma HaddP_Mem_contra:
assumes "H \<turnstile> HaddP x y z" "H \<turnstile> z IN x" "H \<turnstile> OrdP x"
shows "H \<turnstile> A"
|
lemma lead_coeff_rebase_poly[simp]:
"lead_coeff (#(p::'a mod_ring poly) :: 'b mod_ring poly) = @lead_coeff p"
|
lemma close_sub:"sublist (close \<Gamma> \<phi>) \<Gamma>"
|
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