Statement:
stringlengths 7
24.3k
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lemma aless_le_not_le: "((w::ant) < z) = (w \<le> z \<and> \<not> z \<le> w)"
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lemma ParCR_MtransC[simp]:
assumes "(c2, s) \<rightarrow>*c (c2', s')"
shows "(Par c1 c2, s) \<rightarrow>*c (Par c1 c2', s')"
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lemma lsumr_rsuml_left_gpv: "map_gpv' id lsumr rsuml (left_gpv gpv) = left_gpv (left_gpv gpv)"
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lemma is_leaf_refined[code] :
fixes m :: "('a :: ccompare, 'a prefix_tree) mapping_rbt"
shows "Prefix_Tree.is_leaf (MPT (RBT_Mapping m))
= (case ID CCOMPARE('a) of
None \<Rightarrow> Code.abort (STR ''is_leaf_MPT_RBT_Mapping: ccompare = None'') (\<lambda>_. Prefix_Tree.is_leaf (MPT (RBT_Mapping m)))
| Some _ \<Rightarrow> RBT_Mapping2.is_empty m)"
(is "?PT1 = ?PT2")
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lemma sH_g_orbital: "\<^bold>{P\<^bold>} (x\<acute>= f & G on U S @ t\<^sub>0) \<^bold>{Q\<^bold>} =
(\<forall>s. P s \<longrightarrow> (\<forall>X\<in>ivp_sols f U S t\<^sub>0 s. \<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (X \<tau>)) \<longrightarrow> Q (X t)))"
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lemma "\<forall>x::'a T2. P x"
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lemma normal_and[simp]:
"normal \<phi>\<^sub>1 \<Longrightarrow> normal \<phi>\<^sub>2 \<Longrightarrow> normal (and \<phi>\<^sub>1 \<phi>\<^sub>2)"
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lemma (in group) int_pow_neg_int: "x \<in> carrier G \<Longrightarrow> x [^] -(int n) = inv (x [^] n)"
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lemma Resid_along_normal_reflects_Cong\<^sub>0:
assumes "t \\ u \<approx>\<^sub>0 t' \\ u" and "u \<in> \<NN>"
shows "t \<approx>\<^sub>0 t'"
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lemma wcc_equivalence:
"equivalence (wcc x)"
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lemma s_ns_mul_ext:
assumes "(A, B) \<in> s_mul_ext ns s"
shows "(A, B) \<in> ns_mul_ext ns s"
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lemma pre_subdivFace_face_distinct: "pre_subdivFace_face f v vol \<Longrightarrow> distinct (removeNones vol)"
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lemma has_bochner_integral_mult_left[intro]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
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lemma inf_glb:"is_glb (s1 \<sqinter> s2) s1 s2"
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lemma is_ord_mkt_bal_r: "is_ord(MKT n l r h) \<Longrightarrow> is_ord (mkt_bal_r n l r)"
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lemma three_intervals_lemma:
fixes a b c::real
assumes a: "a \<in> A - B"
and b: "b \<in> B - A"
and c: "c \<in> A \<inter> B"
and iA: "is_interval A" and iB: "is_interval B"
and aI: "a \<in> I"
and bI: "b \<in> I"
and iI: "is_interval I"
shows "c \<in> I"
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lemma exists_cong_per:
"\<exists> C. Per A B C \<and> Cong B C X Y"
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lemma mono_D11:
"P \<sqsubseteq> Q \<Longrightarrow> D (Q `;` S) \<subseteq> D (P `;` S)"
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lemma periodic_set_tan_linear:
assumes "a\<noteq>0" "c\<noteq>0"
shows "periodic_set (roots (\<lambda>x. a*tan (x/c) + b)) (c*pi)"
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lemma (in Group) s_top_unit_closed:"\<lbrakk>G \<guillemotright> H; G \<guillemotright> K\<rbrakk> \<Longrightarrow> \<one> \<in> H \<diamondop>\<^bsub>G\<^esub> K"
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lemma strict_prefix_map_inj:
"\<lbrakk> inj_on f (set xs \<union> set ys); strict_prefix (map f xs) (map f ys) \<rbrakk> \<Longrightarrow>
strict_prefix xs ys"
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lemma first_node_in_verts_if_valid: "valid_tree t \<Longrightarrow> first_node t \<in> verts G"
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lemma cut_le_less_conv: "I \<down>\<le> t = ({t} \<inter> I) \<union> (I \<down>< t)"
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lemma OF_match_linear_append: "OF_match_linear \<gamma> (a @ b) p = (case OF_match_linear \<gamma> a p of NoAction \<Rightarrow> OF_match_linear \<gamma> b p | x \<Rightarrow> x)"
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lemma mat_mult_plus_distrib_right:
assumes wf1: "mat nr nc m1"
and wf2: "mat nc ncc m2"
and wf3: "mat nc ncc m3"
shows "mat_mult nr m1 (mat_plus m2 m3) = mat_plus (mat_mult nr m1 m2) (mat_mult nr m1 m3)" (is "mat_mult nr m1 ?m23 = mat_plus ?m12 ?m13")
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lemma cond_spmf_fst_map_Pair [simp]: "cond_spmf_fst (map_spmf (Pair x) p) x = mk_lossless p"
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lemma mset_le_distrib[consumes 1, case_names dist]: "\<lbrakk>(X::'a multiset)\<subseteq>#A+B; !!Xa Xb. \<lbrakk>X=Xa+Xb; Xa\<subseteq>#A; Xb\<subseteq>#B\<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
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lemma AAss_\<tau>red0t_xt3:
"\<tau>red0t extTA P t h (e, xs) (e', xs') \<Longrightarrow> \<tau>red0t extTA P t h (Val a\<lfloor>Val i\<rceil> := e, xs) (Val a\<lfloor>Val i\<rceil> := e', xs')"
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lemma enforce_spmf_alt_def:
"enforce_spmf P p = bind_spmf p (\<lambda>a. bind_spmf (assert_spmf (P a)) (\<lambda>_ :: unit. return_spmf a))"
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lemma disjoint_set_forest_update_square:
assumes "disjoint_set_forest x"
and "vector y"
and "regular y"
shows "disjoint_set_forest (x[y\<longmapsto>(x * x)\<^sup>T])"
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lemma (in Module) sc_un:" m \<in> carrier M \<Longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s m = m"
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lemma fails_assert'[fails_simp]: "fails (assert' P) h \<longleftrightarrow> \<not>P"
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lemma set_nebset_to_set_nebset: "A \<noteq> {} \<Longrightarrow> |A| <o natLeq +c |UNIV :: 'k set| \<Longrightarrow>
set_nebset (the_inv set_nebset A :: 'a set!['k]) = A"
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lemma polysub_normh:
fixes p::"'a::ring_1 poly"
shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
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lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"
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lemma eq_diff_eq': "x = y - z \<longleftrightarrow> y = x + z"
for x y z :: real
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lemma prefix_replicate_zero_coeff: "coeff p = coeff ((replicate n \<zero>) @ p)"
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lemma ipurge_tr_rev_aux_nil_1 [rule_format]:
"ipurge_tr_rev_aux I D U xs = [] \<longrightarrow> (\<forall>u \<in> U. \<not> (\<exists>v \<in> D ` set xs. (v, u) \<in> I))"
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lemma [simp]: "ispath (f(a := q)) p (Ref a) \<Longrightarrow> ispath (f(a := q)) p q"
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lemma isnpolyh_zero_iff:
assumes nq: "isnpolyh p n0"
and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0, power})"
shows "p = 0\<^sub>p"
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lemma vrrestriction_vinsert_nin[simp]:
assumes "b \<notin>\<^sub>\<circ> A"
shows "(vinsert \<langle>a, b\<rangle> r) \<restriction>\<^sup>r\<^sub>\<circ> A = r \<restriction>\<^sup>r\<^sub>\<circ> A"
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lemma (in cf_sspan) cf_sspan_\<gg>''[cat_ss_cs_intros]:
assumes "g = \<gg>" and "a = \<aa>"
shows "g : \<oo> \<mapsto>\<^bsub>\<CC>\<^esub> a"
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lemma pow_equals:
assumes "PoW t n = PoW t n'"
and "n'\<ge>n"
and "n''\<ge>n"
and "n''\<le>n'"
shows "PoW t n = PoW t n''"
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lemma intersect_segment_xline':
assumes "intersect_segment_xline' prec (p0, p1) x = Some (m, M)"
shows "closed_segment p0 p1 \<inter> {p. fst p = x} \<subseteq> {(x, m) .. (x, M)}"
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lemma D_not_diag_le': "set xs \<subseteq> {0..k} \<Longrightarrow> i \<notin> set xs \<Longrightarrow> j \<notin> set xs \<Longrightarrow> distinct xs
\<Longrightarrow> D m i j k \<le> len m i j xs"
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lemma urel32_inj: "urel32 x y \<Longrightarrow> urel32 x z \<Longrightarrow> y = z"
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lemma upd_vars_fm [simp]: \<open>max_list (vars_fm p) < n \<Longrightarrow> \<lbrakk>E(n := x), F, G\<rbrakk> p = \<lbrakk>E, F, G\<rbrakk> p\<close>
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lemma (in map_size) size_autoref[autoref_rules]:
"PREFER_id Rk \<Longrightarrow> (size,op_map_size)\<in>\<langle>Rk,Rv\<rangle>rel\<rightarrow>Id"
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lemma stakeWhile_K_True [simp]: "stakeWhile (\<lambda>_. True) xs = llist_of_stream xs"
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lemma Aizerman_on_Chernoff_on_path_independent_on:
assumes "f_range_on A f"
shows "Aizerman_on A f \<and> Chernoff_on A f \<longleftrightarrow> path_independent_on A f"
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lemma Qp_funs_Units_memI:
assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
assumes "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
shows "f \<in> (Units (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
"inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). inv\<^bsub>Q\<^sub>p\<^esub> (f x))"
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lemma init_state_present:
assumes
"execute (init_state k1 k2) heap = Some ((k_ref1, k_ref2, m_ref1, m_ref2), heap')"
shows
"Ref.present heap' k_ref1" "Ref.present heap' k_ref2"
"Ref.present heap' m_ref1" "Ref.present heap' m_ref2"
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lemma
"(\<lambda>s. 7/2) = wp (bind v at (\<lambda>s. Puniform {1..6} v) in red := (\<lambda>_. v)) red"
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lemma mset_trees_insert[simp]: "mset_trees (insert x t) = {#x#} + mset_trees t"
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lemma Bex_True: "RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t True = True"
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lemma(in monoid) int_pow_inv:
fixes n::int
assumes "a \<in> Units G"
shows "a[^] -n = inv a[^] n"
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lemma llist_all2_lnullD: "llist_all2 P xs ys \<Longrightarrow> lnull xs \<longleftrightarrow> lnull ys"
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lemma fls_subdegree_greaterI:
assumes "f \<noteq> 0" "\<And>k. k \<le> n \<Longrightarrow> f $$ k = 0"
shows "n < fls_subdegree f"
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lemma SetMem_strict[simp]: "SetMem\<cdot>x\<cdot>\<bottom> = \<bottom>"
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lemma (in deutsch) deutsch_algo_result [simp]:
shows "deutsch_algo = \<psi>\<^sub>3"
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lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p"
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lemma homologous_rel_set_eq_relboundary:
"homologous_rel_set p X S c = singular_relboundary_set p X S \<longleftrightarrow> singular_relboundary p X S c"
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lemma dyadics_in_open_unit_interval:
"{0<..<1} \<inter> (\<Union>k m. {real m / 2^k}) = (\<Union>k. \<Union>m \<in> {0<..<2^k}. {real m / 2^k})"
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lemma future_reach_And: "future_reach (And \<phi> \<psi>) = max (future_reach \<phi>) (future_reach \<psi>)"
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lemma mk_next_pow_semantics [simp]:
"w \<Turnstile>\<^sub>n mk_next_pow i x \<longleftrightarrow> suffix i w \<Turnstile>\<^sub>n x"
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lemma L_10_5_\<tau>_NTMap_app[cat_Kan_cs_simps]:
assumes "bf = [0, b, f]\<^sub>\<circ>" and "bf \<in>\<^sub>\<circ> c \<down>\<^sub>C\<^sub>F \<KK>\<lparr>Obj\<rparr>"
shows "L_10_5_\<tau> \<TT> \<KK> c \<upsilon> a\<lparr>NTMap\<rparr>\<lparr>bf\<rparr> = \<upsilon>\<lparr>NTMap\<rparr>\<lparr>b\<rparr>\<lparr>ArrVal\<rparr>\<lparr>f\<rparr>"
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lemma inf_greatest:
assumes "A \<subseteq> carrier L" "z \<in> carrier L"
"(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x)"
shows "z \<sqsubseteq> \<Sqinter>A"
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lemma globally_all_iff:"\<Turnstile> (\<^bold>G(\<^bold>\<forall>c. \<phi>)) \<^bold>\<leftrightarrow> (\<^bold>\<forall>c.( \<^bold>G \<phi>))"
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lemma squarefree_decompose: "\<Prod>(squarefree_part n) * square_part n ^ 2 = n"
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lemma a_kernel_def':
"a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
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theorem noninterference:
"\<lbrakk> (c,s) \<Rightarrow> s'; (c,t) \<Rightarrow> t'; 0 \<turnstile> c; s = t (\<le> l) \<rbrakk>
\<Longrightarrow> s' = t' (\<le> l)"
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lemma image_vector_to_cblinfun[simp]: "vector_to_cblinfun x *\<^sub>S \<top> = ccspan {x}"
\<comment> \<open>Not that the general case \<^term>\<open>vector_to_cblinfun x *\<^sub>S S\<close> can be handled by using
that \<open>S = \<top>\<close> or \<open>S = \<bottom>\<close> by @{thm [source] one_dim_ccsubspace_all_or_nothing}\<close>
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lemma path_connected_punctured_convex:
assumes "convex S" and aff: "aff_dim S \<noteq> 1"
shows "path_connected(S - {a})"
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lemma "neg\<cdot>(neq\<cdot>a\<cdot>b) = eq\<cdot>a\<cdot>b"
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lemma rtos_hom2:"\<lbrakk>(0::nat) < r; (0::nat) < s; l \<le> (r * s - Suc 0)\<rbrakk> \<Longrightarrow>
rtos r s l \<le> (r * s - Suc 0)"
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lemma rt_fresh_asD1 [dest]:
assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
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lemma unitary11_iff:
shows "unitary11 M \<longleftrightarrow>
(\<exists> a b k. (cmod a)\<^sup>2 > (cmod b)\<^sup>2 \<and>
(cmod k)\<^sup>2 = 1 / ((cmod a)\<^sup>2 - (cmod b)\<^sup>2) \<and>
M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a))" (is "?lhs = ?rhs")
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lemma tensor_op_cbilinear: \<open>cbilinear (tensor_op :: 'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::finite ell2
\<Rightarrow> 'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2 \<Rightarrow> _)\<close>
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lemma(in UP_domain) sub_monom:
assumes "a \<in> carrier R"
assumes "a \<noteq>\<zero>"
assumes "f = monom P a n"
assumes "g \<in> carrier P"
shows "degree (f of g) = n*(degree g)"
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lemma fdemodalisation2: "|x\<rangle> y \<le> d z \<longleftrightarrow> x \<cdot> d y \<le> d z \<cdot> x"
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lemma guard_match_symmetry: "(guardMatch t1 t2) = (guardMatch t2 t1)"
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lemma Rep_less_n: "Rep x < int CARD ('a)"
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lemma fun_eq_on_subset: "fun_eq_on f g A \<Longrightarrow> B\<subseteq>A \<Longrightarrow> fun_eq_on f g B"
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lemma measurement2_leq_one_mat:
assumes dP: "P \<in> carrier_mat d d" and dQ: "Q \<in> carrier_mat d d"
and leP: "P \<le>\<^sub>L 1\<^sub>m d" and leQ: "Q \<le>\<^sub>L 1\<^sub>m d" and m: "measurement d 2 M"
shows "(adjoint (M 0) * P * (M 0) + adjoint (M 1) * Q * (M 1)) \<le>\<^sub>L 1\<^sub>m d"
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theorem skeleton_spec: "skeleton \<le> SPEC (\<lambda>D. outer_invar {} D)"
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lemma norm_prod_le:
"norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))"
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lemma arr_mkarr [simp]:
assumes "Arr t"
shows "arr (mkarr t)"
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lemma inj_Pair[simp]:
"inj_on (\<lambda>x. (x,c x)) S"
"inj_on (\<lambda>x. (c x,x)) S"
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lemma no_change_eval_lt:
fixes x y:: "real"
assumes "x < y"
assumes "\<not>(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
shows "((aEvalUni (LessUni (a,b,c)) x = aEvalUni (LessUni (a,b,c)) y))"
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lemma has_hom_functor:
shows "hom_functor C S.comp S.setp (\<lambda>_. S.UP)"
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lemma combine_simps[simp]:
"alphabet\<^sub>2 (complement A) = alphabet\<^sub>1 A"
"initial\<^sub>2 (complement A) = initial\<^sub>1 A"
"transition\<^sub>2 (complement A) = transition\<^sub>1 A"
"condition\<^sub>2 (complement A) = condition (condition\<^sub>1 A)"
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lemma \<eta>_in_hom:
assumes "D.arr \<nu>"
shows [intro]: "\<guillemotleft>\<eta> \<nu> : src\<^sub>D \<nu> \<rightarrow>\<^sub>D trg\<^sub>D \<nu>\<guillemotright>"
and [intro]: "\<guillemotleft>\<eta> \<nu> : D.dom \<nu> \<Rightarrow>\<^sub>D F (G (D.cod \<nu>))\<guillemotright>"
and "D.ide \<nu> \<Longrightarrow> D.iso (\<eta> \<nu>)"
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lemma lead_coeff_i_def': "lead_coeff_i ops x = (coeff_i ops) x (degree_i x)"
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lemma subst_bvs1_step:
assumes "\<not> loose_bvar t lev"
shows "subst_bvs1 t lev (args@[u]) = subst_bv1 (subst_bvs1 t lev args) lev u"
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lemma Try_\<tau>ExectI1:
"\<tau>Exec_movet ci P t E h s s' \<Longrightarrow> \<tau>Exec_movet ci P t (try E catch(C' V) e) h s s'"
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lemma prime_elem_linear_field_poly:
"(b :: 'a :: field) \<noteq> 0 \<Longrightarrow> prime_elem [:a,b:]"
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lemma distrib_if_bind: "do { x \<leftarrow> if b then (c::_ Heap) else d; f x } = (if b then do {x \<leftarrow> c; f x} else do { x\<leftarrow>d; f x })"
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lemma double_orthogonal_complement_increasing[simp]:
shows "M \<subseteq> orthogonal_complement (orthogonal_complement M)"
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lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"
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lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
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