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

Statement: stringlengths 7 24.3k
lemma aless_le_not_le: "((w::ant) < z) = (w \<le> z \<and> \<not> z \<le> w)"
lemma ParCR_MtransC[simp]: assumes "(c2, s) \<rightarrow>c (c2', s')" shows "(Par c1 c2, s) \<rightarrow>c (Par c1 c2', s')"
lemma lsumr_rsuml_left_gpv: "map_gpv' id lsumr rsuml (left_gpv gpv) = left_gpv (left_gpv gpv)"
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")
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)))"
lemma "\<forall>x::'a T2. P x"
lemma normal_and[simp]: "normal \<phi>\<^sub>1 \<Longrightarrow> normal \<phi>\<^sub>2 \<Longrightarrow> normal (and \<phi>\<^sub>1 \<phi>\<^sub>2)"
lemma (in group) int_pow_neg_int: "x \<in> carrier G \<Longrightarrow> x [^] -(int n) = inv (x [^] n)"
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'"
lemma wcc_equivalence: "equivalence (wcc x)"
lemma s_ns_mul_ext: assumes "(A, B) \<in> s_mul_ext ns s" shows "(A, B) \<in> ns_mul_ext ns s"
lemma pre_subdivFace_face_distinct: "pre_subdivFace_face f v vol \<Longrightarrow> distinct (removeNones vol)"
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)"
lemma inf_glb:"is_glb (s1 \<sqinter> s2) s1 s2"
lemma is_ord_mkt_bal_r: "is_ord(MKT n l r h) \<Longrightarrow> is_ord (mkt_bal_r n l r)"
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"
lemma exists_cong_per: "\<exists> C. Per A B C \<and> Cong B C X Y"
lemma mono_D11: "P \<sqsubseteq> Q \<Longrightarrow> D (Q `;` S) \<subseteq> D (P `;` S)"
lemma periodic_set_tan_linear: assumes "a\<noteq>0" "c\<noteq>0" shows "periodic_set (roots (\<lambda>x. atan (x/c) + b)) (cpi)"
lemma (in Group) s_top_unit_closed:"\<lbrakk>G \<guillemotright> H; G \<guillemotright> K\<rbrakk> \<Longrightarrow> \<one> \<in> H \<diamondop>\<^bsub>G\<^esub> K"
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"
lemma first_node_in_verts_if_valid: "valid_tree t \<Longrightarrow> first_node t \<in> verts G"
lemma cut_le_less_conv: "I \<down>\<le> t = ({t} \<inter> I) \<union> (I \<down>< t)"
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)"
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")
lemma cond_spmf_fst_map_Pair [simp]: "cond_spmf_fst (map_spmf (Pair x) p) x = mk_lossless p"
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"
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')"
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))"
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])"
lemma (in Module) sc_un:" m \<in> carrier M \<Longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s m = m"
lemma fails_assert'[fails_simp]: "fails (assert' P) h \<longleftrightarrow> \<not>P"
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"
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)"
lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"
lemma eq_diff_eq': "x = y - z \<longleftrightarrow> y = x + z" for x y z :: real
lemma prefix_replicate_zero_coeff: "coeff p = coeff ((replicate n \<zero>) @ p)"
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))"
lemma [simp]: "ispath (f(a := q)) p (Ref a) \<Longrightarrow> ispath (f(a := q)) p q"
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"
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"
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"
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''"
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)}"
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"
lemma urel32_inj: "urel32 x y \<Longrightarrow> urel32 x z \<Longrightarrow> y = z"
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>
lemma (in map_size) size_autoref[autoref_rules]: "PREFER_id Rk \<Longrightarrow> (size,op_map_size)\<in>\<langle>Rk,Rv\<rangle>rel\<rightarrow>Id"
lemma stakeWhile_K_True [simp]: "stakeWhile (\<lambda>_. True) xs = llist_of_stream xs"
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"
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))"
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"
lemma "(\<lambda>s. 7/2) = wp (bind v at (\<lambda>s. Puniform {1..6} v) in red := (\<lambda>_. v)) red"
lemma mset_trees_insert[simp]: "mset_trees (insert x t) = {#x#} + mset_trees t"
lemma Bex_True: "RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t True = True"
lemma(in monoid) int_pow_inv: fixes n::int assumes "a \<in> Units G" shows "a[^] -n = inv a[^] n"
lemma llist_all2_lnullD: "llist_all2 P xs ys \<Longrightarrow> lnull xs \<longleftrightarrow> lnull ys"
lemma fls_subdegree_greaterI: assumes "f \<noteq> 0" "\<And>k. k \<le> n \<Longrightarrow> f $$ k = 0" shows "n < fls_subdegree f"
lemma SetMem_strict[simp]: "SetMem\<cdot>x\<cdot>\<bottom> = \<bottom>"
lemma (in deutsch) deutsch_algo_result [simp]: shows "deutsch_algo = \<psi>\<^sub>3"
lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p"
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"
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})"
lemma future_reach_And: "future_reach (And \<phi> \<psi>) = max (future_reach \<phi>) (future_reach \<psi>)"
lemma mk_next_pow_semantics [simp]: "w \<Turnstile>\<^sub>n mk_next_pow i x \<longleftrightarrow> suffix i w \<Turnstile>\<^sub>n x"
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>"
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"
lemma globally_all_iff:"\<Turnstile> (\<^bold>G(\<^bold>\<forall>c. \<phi>)) \<^bold>\<leftrightarrow> (\<^bold>\<forall>c.( \<^bold>G \<phi>))"
lemma squarefree_decompose: "\<Prod>(squarefree_part n) * square_part n ^ 2 = n"
lemma a_kernel_def': "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
theorem noninterference: "\<lbrakk> (c,s) \<Rightarrow> s'; (c,t) \<Rightarrow> t'; 0 \<turnstile> c; s = t (\<le> l) \<rbrakk> \<Longrightarrow> s' = t' (\<le> l)"
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>
lemma path_connected_punctured_convex: assumes "convex S" and aff: "aff_dim S \<noteq> 1" shows "path_connected(S - {a})"
lemma "neg\<cdot>(neq\<cdot>a\<cdot>b) = eq\<cdot>a\<cdot>b"
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)"
lemma rt_fresh_asD1 [dest]: assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
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")
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>
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)"
lemma fdemodalisation2: "\|x\<rangle> y \<le> d z \<longleftrightarrow> x \<cdot> d y \<le> d z \<cdot> x"
lemma guard_match_symmetry: "(guardMatch t1 t2) = (guardMatch t2 t1)"
lemma Rep_less_n: "Rep x < int CARD ('a)"
lemma fun_eq_on_subset: "fun_eq_on f g A \<Longrightarrow> B\<subseteq>A \<Longrightarrow> fun_eq_on f g B"
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"
theorem skeleton_spec: "skeleton \<le> SPEC (\<lambda>D. outer_invar {} D)"
lemma norm_prod_le: "norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))"
lemma arr_mkarr [simp]: assumes "Arr t" shows "arr (mkarr t)"
lemma inj_Pair[simp]: "inj_on (\<lambda>x. (x,c x)) S" "inj_on (\<lambda>x. (c x,x)) S"
lemma no_change_eval_lt: fixes x y:: "real" assumes "x < y" assumes "\<not>(\<exists>w. x \<le> w \<and> w \<le> y \<and> aw^2 + bw + c = 0)" shows "((aEvalUni (LessUni (a,b,c)) x = aEvalUni (LessUni (a,b,c)) y))"
lemma has_hom_functor: shows "hom_functor C S.comp S.setp (\<lambda>_. S.UP)"
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)"
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>)"
lemma lead_coeff_i_def': "lead_coeff_i ops x = (coeff_i ops) x (degree_i x)"
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"
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'"
lemma prime_elem_linear_field_poly: "(b :: 'a :: field) \<noteq> 0 \<Longrightarrow> prime_elem [:a,b:]"
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 })"
lemma double_orthogonal_complement_increasing[simp]: shows "M \<subseteq> orthogonal_complement (orthogonal_complement M)"
lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"
lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"

lemma aless_le_not_le: "((w::ant) < z) = (w \<le> z \<and> \<not> z \<le> w)"

lemma ParCR_MtransC[simp]: assumes "(c2, s) \<rightarrow>*c (c2', s')" shows "(Par c1 c2, s) \<rightarrow>*c (Par c1 c2', s')"

lemma lsumr_rsuml_left_gpv: "map_gpv' id lsumr rsuml (left_gpv gpv) = left_gpv (left_gpv gpv)"

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")

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)))"

lemma "\<forall>x::'a T2. P x"

lemma normal_and[simp]: "normal \<phi>\<^sub>1 \<Longrightarrow> normal \<phi>\<^sub>2 \<Longrightarrow> normal (and \<phi>\<^sub>1 \<phi>\<^sub>2)"

lemma (in group) int_pow_neg_int: "x \<in> carrier G \<Longrightarrow> x [^] -(int n) = inv (x [^] n)"

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'"

lemma wcc_equivalence: "equivalence (wcc x)"

lemma s_ns_mul_ext: assumes "(A, B) \<in> s_mul_ext ns s" shows "(A, B) \<in> ns_mul_ext ns s"

lemma pre_subdivFace_face_distinct: "pre_subdivFace_face f v vol \<Longrightarrow> distinct (removeNones vol)"

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)"

lemma inf_glb:"is_glb (s1 \<sqinter> s2) s1 s2"

lemma is_ord_mkt_bal_r: "is_ord(MKT n l r h) \<Longrightarrow> is_ord (mkt_bal_r n l r)"

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"

lemma exists_cong_per: "\<exists> C. Per A B C \<and> Cong B C X Y"

lemma mono_D11: "P \<sqsubseteq> Q \<Longrightarrow> D (Q `;` S) \<subseteq> D (P `;` S)"

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)"

lemma (in Group) s_top_unit_closed:"\<lbrakk>G \<guillemotright> H; G \<guillemotright> K\<rbrakk> \<Longrightarrow> \<one> \<in> H \<diamondop>\<^bsub>G\<^esub> K"

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"

lemma first_node_in_verts_if_valid: "valid_tree t \<Longrightarrow> first_node t \<in> verts G"

lemma cut_le_less_conv: "I \<down>\<le> t = ({t} \<inter> I) \<union> (I \<down>< t)"

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)"

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")

lemma cond_spmf_fst_map_Pair [simp]: "cond_spmf_fst (map_spmf (Pair x) p) x = mk_lossless p"

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"

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')"

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))"

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])"

lemma (in Module) sc_un:" m \<in> carrier M \<Longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s m = m"

lemma fails_assert'[fails_simp]: "fails (assert' P) h \<longleftrightarrow> \<not>P"

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"

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)"

lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"

lemma eq_diff_eq': "x = y - z \<longleftrightarrow> y = x + z" for x y z :: real

lemma prefix_replicate_zero_coeff: "coeff p = coeff ((replicate n \<zero>) @ p)"

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))"

lemma [simp]: "ispath (f(a := q)) p (Ref a) \<Longrightarrow> ispath (f(a := q)) p q"

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"

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"

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"

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''"

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)}"

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"

lemma urel32_inj: "urel32 x y \<Longrightarrow> urel32 x z \<Longrightarrow> y = z"

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>

lemma (in map_size) size_autoref[autoref_rules]: "PREFER_id Rk \<Longrightarrow> (size,op_map_size)\<in>\<langle>Rk,Rv\<rangle>rel\<rightarrow>Id"

lemma stakeWhile_K_True [simp]: "stakeWhile (\<lambda>_. True) xs = llist_of_stream xs"

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"

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))"

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"

lemma "(\<lambda>s. 7/2) = wp (bind v at (\<lambda>s. Puniform {1..6} v) in red := (\<lambda>_. v)) red"

lemma mset_trees_insert[simp]: "mset_trees (insert x t) = {#x#} + mset_trees t"

lemma Bex_True: "RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t True = True"

lemma(in monoid) int_pow_inv: fixes n::int assumes "a \<in> Units G" shows "a[^] -n = inv a[^] n"

lemma llist_all2_lnullD: "llist_all2 P xs ys \<Longrightarrow> lnull xs \<longleftrightarrow> lnull ys"

lemma fls_subdegree_greaterI: assumes "f \<noteq> 0" "\<And>k. k \<le> n \<Longrightarrow> f $$ k = 0" shows "n < fls_subdegree f"

lemma SetMem_strict[simp]: "SetMem\<cdot>x\<cdot>\<bottom> = \<bottom>"

lemma (in deutsch) deutsch_algo_result [simp]: shows "deutsch_algo = \<psi>\<^sub>3"

lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p"

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"

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})"

lemma future_reach_And: "future_reach (And \<phi> \<psi>) = max (future_reach \<phi>) (future_reach \<psi>)"

lemma mk_next_pow_semantics [simp]: "w \<Turnstile>\<^sub>n mk_next_pow i x \<longleftrightarrow> suffix i w \<Turnstile>\<^sub>n x"

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>"

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"

lemma globally_all_iff:"\<Turnstile> (\<^bold>G(\<^bold>\<forall>c. \<phi>)) \<^bold>\<leftrightarrow> (\<^bold>\<forall>c.( \<^bold>G \<phi>))"

lemma squarefree_decompose: "\<Prod>(squarefree_part n) * square_part n ^ 2 = n"

lemma a_kernel_def': "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"

theorem noninterference: "\<lbrakk> (c,s) \<Rightarrow> s'; (c,t) \<Rightarrow> t'; 0 \<turnstile> c; s = t (\<le> l) \<rbrakk> \<Longrightarrow> s' = t' (\<le> l)"

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>

lemma path_connected_punctured_convex: assumes "convex S" and aff: "aff_dim S \<noteq> 1" shows "path_connected(S - {a})"

lemma "neg\<cdot>(neq\<cdot>a\<cdot>b) = eq\<cdot>a\<cdot>b"

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)"

lemma rt_fresh_asD1 [dest]: assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"

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")

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>

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)"

lemma fdemodalisation2: "|x\<rangle> y \<le> d z \<longleftrightarrow> x \<cdot> d y \<le> d z \<cdot> x"

lemma guard_match_symmetry: "(guardMatch t1 t2) = (guardMatch t2 t1)"

lemma Rep_less_n: "Rep x < int CARD ('a)"

lemma fun_eq_on_subset: "fun_eq_on f g A \<Longrightarrow> B\<subseteq>A \<Longrightarrow> fun_eq_on f g B"

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"

theorem skeleton_spec: "skeleton \<le> SPEC (\<lambda>D. outer_invar {} D)"

lemma norm_prod_le: "norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))"

lemma arr_mkarr [simp]: assumes "Arr t" shows "arr (mkarr t)"

lemma inj_Pair[simp]: "inj_on (\<lambda>x. (x,c x)) S" "inj_on (\<lambda>x. (c x,x)) S"

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))"

lemma has_hom_functor: shows "hom_functor C S.comp S.setp (\<lambda>_. S.UP)"

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)"

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>)"

lemma lead_coeff_i_def': "lead_coeff_i ops x = (coeff_i ops) x (degree_i x)"

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"

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'"

lemma prime_elem_linear_field_poly: "(b :: 'a :: field) \<noteq> 0 \<Longrightarrow> prime_elem [:a,b:]"

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 })"

lemma double_orthogonal_complement_increasing[simp]: shows "M \<subseteq> orthogonal_complement (orthogonal_complement M)"

lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"

lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"