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

Statement: stringlengths 7 24.3k
lemma ifbd_inj_iff: "Sup_pres \<phi> \<Longrightarrow> Sup_pres \<psi> \<Longrightarrow> (bd\<^sup>-\<^sub>\<F> \<phi> = bd\<^sup>-\<^sub>\<F> \<psi>) = (\<phi> = \<psi>)"
lemma measurable_count_space_insert[measurable (raw)]: "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
lemma inverse_of_nat_tendsto_zero: "(\<lambda>x. inverse (of_nat x :: 'a :: real_normed_div_algebra)) \<longlonglongrightarrow> 0"
lemma map_entry_parametric [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B) (\<lambda>k f m. (case m k of None \<Rightarrow> m \| Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m \| Some v \<Rightarrow> m (k \<mapsto> (f v))))"
lemma safe_only_owner_enter_normal_aux[simp]: "\<lbrakk> s : reach; safe s r; (k',roomk s r) \<in> cards s g \<rbrakk> \<Longrightarrow> owns s r = Some g"
lemma min_Suc_gt[simp]: "a<b \<Longrightarrow> min (Suc a) b = Suc a" "a<b \<Longrightarrow> min b (Suc a) = Suc a"
lemma H_loop: "\<^bold>{P\<^bold>} X \<^bold>{P\<^bold>} \<Longrightarrow> \<^bold>{P\<^bold>} (LOOP X INV I) \<^bold>{P\<^bold>}"
lemma (in deutsch) adjoint_of_deutsch_transform: shows "(U\<^sub>f)\<^sup>\<dagger> = U\<^sub>f"
lemma atd_lem: "take n xs = t \<Longrightarrow> drop n xs = d \<Longrightarrow> xs = t @ d"
lemma field_differentiable_compose: "f field_differentiable at z \<Longrightarrow> g field_differentiable at (f z) \<Longrightarrow> (g o f) field_differentiable at z"
lemma two: "(2::nat) = Suc 1"
lemma reduce_below_abs_0_case_m1: assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n" and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))" and Aaj: "A $$ (a,0) \<noteq> 0" and mn: "m\<ge>n" assumes "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x" and "m\<noteq>a" and "D>0" shows "reduce_below_abs a (xs @ [m]) D A $$ (m,0) = 0"
lemma "(t \<in> B h \<longrightarrow> ins x t \<in> Bp h) \<and> (t \<in> U h \<longrightarrow> ins x t \<in> T h)"
lemma "Oc\<up>(0+1) = [Oc]"
lemma (in uprio_pop) autoref_prio_pop_min[autoref_rules]: "\<And>Re Ra. \<lbrakk>PREFER_id Re; PREFER_id Ra \<rbrakk> \<Longrightarrow> (\<lambda>s. RETURN (pop s),prio_pop_min)\<in>\<langle>Re,Ra\<rangle>rel\<rightarrow>\<langle>\<langle>Re,\<langle>Ra,\<langle>Re,Ra\<rangle>rel\<rangle>prod_rel\<rangle>prod_rel\<rangle>nres_rel"
lemma bin_split_minus1 [simp]: "bin_split n (- 1) = (- 1, (take_bit :: nat \<Rightarrow> int \<Rightarrow> int) n (- 1))"
lemma prv_eqv_prv: assumes "\<phi> \<in> fmla" and "\<chi> \<in> fmla" assumes "prv \<phi>" and "prv (eqv \<phi> \<chi>)" shows "prv \<chi>"
lemma circuit_in_elim: assumes "circuit_in \<E> C\<^sub>1" "circuit_in \<E> C\<^sub>2" "C\<^sub>1 \<noteq> C\<^sub>2" "x \<in> C\<^sub>1 \<inter> C\<^sub>2" shows "\<exists>C\<^sub>3 \<subseteq> (C\<^sub>1 \<union> C\<^sub>2) - {x}. circuit_in \<E> C\<^sub>3"
lemma post_meas0_index_0_alice: assumes "state 1 \<phi>" and "\<alpha> = \<phi> $$ (0,0)" and "\<beta> = \<phi> $$ (1,0)" shows "post_meas0 3 (alice \<phi>) 0 = mat_of_cols_list 8 [[\<alpha>/sqrt(2), \<beta>/sqrt(2), \<beta>/sqrt(2), \<alpha>/sqrt(2), 0, 0, 0, 0]]"
lemma in_singleton:" m \<^bold>\<in> Abs_nat_int{n} \<longrightarrow> m = n"
lemma infinite_times_eqpoll_self: assumes "infinite A" shows "A \<times> A \<approx> A"
lemma progress_le: "progress \<sigma> Map.empty \<phi> j \<le> j"
lemma ordst_trans: assumes As1: "ordst X Y" and As2: "ordst Y Z" shows "ordst X Z"
lemma wlp_skip_eq: "wlp SKIP Q s = Q s"
lemma strong_del_point_incidence_wf: "incidence_system (del_point p) (str_del_point_blocks p)"
lemma SplitGuards_sound: assumes valid_c1: "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,A" assumes valid_c2: "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c\<^sub>2 UNIV,UNIV" assumes c: "(c\<^sub>1 \<inter>\<^sub>g c\<^sub>2) = Some c" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
lemma ImplLeadsto_simple: "\<And>F G. \<turnstile> F \<longrightarrow> G \<Longrightarrow> \<turnstile> F \<leadsto> G"
lemma vsubset_vdiff: assumes "A \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> B"
lemma qp_D3_post: "is_quantum_predicate (adjoint exM0 * P' * exM0 + adjoint exM1 * Q * exM1)"
lemma lsl_top [simp]: "\<nu> \<top> = (\<top>::'a::unital_quantale)"
lemma fixbound_in: "Q \<in> fixbound \<Q> x \<Longrightarrow> Q \<in> \<Q>"
lemma tree_edge_disc: "(v,w) \<in> tree_edges s \<Longrightarrow> \<delta> s v < \<delta> s w"
lemma elementsButlastTrailIsButlastElementsTrail [simp]: shows "elements (butlast M) = butlast (elements M)"
lemma (in field) npepow_correct: "in_carrier xs \<Longrightarrow> peval xs (npepow e n) = peval xs (PExpr2 (PPow e n))"
lemma abupd_def2 [simp]: "abupd f (x,s) = (f x,s)"
lemma dom_empty [simp]: "dom empty = {}"
lemma "when" [simp]: "P \<Longrightarrow> (a when P) = a" "\<not> P \<Longrightarrow> (a when P) = 0"
lemma rbt_lookup_from_in_tree: assumes "rbt_sorted t1" "rbt_sorted t2" and "\<And>v. (k, v) \<in> set (entries t1) \<longleftrightarrow> (k, v) \<in> set (entries t2)" shows "rbt_lookup t1 k = rbt_lookup t2 k"
lemma fun_eq_on_cong: "fun_eq_on f h A \<Longrightarrow> fun_eq_on g h A \<Longrightarrow> fun_eq_on f g A"
lemma class_add_widens: "\<lbrakk> P \<turnstile> Ts [\<le>] Ts'; \<not> is_class P C \<rbrakk> \<Longrightarrow> (class_add P (C, cdec)) \<turnstile> Ts [\<le>] Ts'"
lemma alphas_abs_eqvt: shows "(bs, x) \<approx>abs_set (cs, y) \<Longrightarrow> (p \<bullet> bs, p \<bullet> x) \<approx>abs_set (p \<bullet> cs, p \<bullet> y)" and "(bs, x) \<approx>abs_res (cs, y) \<Longrightarrow> (p \<bullet> bs, p \<bullet> x) \<approx>abs_res (p \<bullet> cs, p \<bullet> y)" and "(ds, x) \<approx>abs_lst (es, y) \<Longrightarrow> (p \<bullet> ds, p \<bullet> x) \<approx>abs_lst (p \<bullet> es, p \<bullet> y)"
lemma scalingI[intro]: "\<lbrakk> \<And>P c x. \<lbrakk> sound P; 0 \<le> c \<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x \<rbrakk> \<Longrightarrow> scaling t"
lemma nonzero_fstI[intro, simp]: "fst x \<noteq> 0 \<Longrightarrow> x \<noteq> 0" and nonzero_sndI[intro, simp]: "snd x \<noteq> 0 \<Longrightarrow> x \<noteq> 0"
lemma cpx_length_of_vec_of_list [simp]: "\<parallel>vec_of_list l\<parallel> = sqrt(\<Sum>i<length l. (cmod (l ! i))\<^sup>2)"
lemma OclInt1_non_null [simp,code_unfold]: "(\<one> \<doteq> null) = false"
lemma ND0aux3[rule_format]: "AllowPortFromTo x y p \<in> set b \<Longrightarrow> x \<in> set (net_list_aux b)"
lemma semialg_intersect: assumes "A \<in> semialg_sets n" assumes "B \<in> semialg_sets n" shows "(A \<inter> B) \<in> semialg_sets n "
lemma box_import_shunting: "-p * -q \<le> \|x](-r) \<longleftrightarrow> -q \<le> \|-p * x](-r)"
lemma big_step_CS: "(c,s) \<Rightarrow> t \<Longrightarrow> t \<in> post(CS c)"
lemma nn_set_integral_cong: assumes "AE x in M. f x = g x" shows "(\<integral>\<^sup>+x \<in> A. f x \<partial>M) = (\<integral>\<^sup>+x \<in> A. g x \<partial>M)"
lemma rule_\<alpha>_type[simp]: "(N, \<alpha>) \<in> \<RR> \<Longrightarrow> is_sentence \<alpha>"
lemma weakPsiCongE: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" and \<Psi>' :: 'b assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> P \<approx> Q" and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q" and "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
lemma liftE_def2: "liftE f = (\<lambda>s. ((\<lambda>(v,s'). (Inr v, s')) (fst (f s)), snd (f s)))"
lemma HMA_V_diff [transfer_rule]: "(HMA_V ===> HMA_V ===> HMA_V) (-) (-)"
lemma dvd_associated2: fixes a::"'a::idom" assumes ab: "a dvd b" and ba: "b dvd a" and a: "a\<noteq>0" shows "\<exists>u. u dvd 1 \<and> a = u*b"
lemma has_bochner_integral_imp_has_integral': "has_bochner_integral lborel (\<lambda>x. indicator S x *\<^sub>R f x) I \<Longrightarrow> (f has_integral (I :: 'b :: euclidean_space)) S"
lemma value_class_inhabitants: "(\<forall>x. typeof x = CClassT typeId \<longrightarrow> P x) = (\<forall> a. P (objV typeId a))" (is "(\<forall> x. ?A x) = ?B")
lemma Assign_is_state_update_before: "ASSIGN x e = state_update_before (\<lambda> (s, s') . s' = (update x (\<lambda>_. (e s))) s) Skip"
lemma expand_prop_all: assumes "expand_assm_incoming n_ns \<and> expand_name_ident (snd n_ns)" (is "?Q n_ns") shows "expand n_ns \<le> SPEC (expand_rslt_all \<xi> n_ns)" (is "_ \<le> SPEC (?P n_ns)")
lemma subst_simps [simp]: "subst x t x = t" "subst x (Var x) = Var"
lemma enforce_option_bot [simp]: "enforce_option \<bottom> = (\<lambda>_. None)"
lemma path_vertices_shift: assumes "path v v' pth" shows "map fst (map fst pth)@[v'] = v#map fst (map snd pth)"
lemma extends_subst_cong_lit: "extends_subst \<sigma> \<tau> \<Longrightarrow> vars_lit L \<subseteq> dom \<sigma> \<Longrightarrow> L \<cdot>lit subst_of_map Var \<sigma> = L \<cdot>lit subst_of_map Var \<tau>"
lemma signed_drop_bit_word_numeral [simp]: \<open>signed_drop_bit (numeral n) (numeral k) = (word_of_int (drop_bit (numeral n) (signed_take_bit (LENGTH('a) - 1) (numeral k))) :: 'a::len word)\<close>
lemma l2norm_lmult: "f square_integrable S \<Longrightarrow> l2norm S (\<lambda>x. c * f x) = \<bar>c\<bar> * l2norm S f"
lemma open_chart_last_domain: "open (Collect chart_last_domainP)"
lemma mono_ndet_ref: " \<lbrakk>P \<sqsubseteq> P'; S \<sqsubseteq> S'\<rbrakk> \<Longrightarrow> (P \<sqinter> S) \<sqsubseteq> (P' \<sqinter> S')"
lemma parts_insert_subset_Un: "X \<in> G \<Longrightarrow> parts(insert X H) \<subseteq> parts G \<union> parts H"
lemma evaluate_iff_sym: "evaluate True env st e r \<longleftrightarrow> (eval env e st = r)" "evaluate_list True env st es r' \<longleftrightarrow> (eval_list env es st = r')" "evaluate_match True env st v pes v' r \<longleftrightarrow> (eval_match env v pes v' st = r)"
lemma insert_if: "finite A \<Longrightarrow> F g (insert x A) = (if x \<in> A then F g A else g x \<^bold>* F g A)"
lemma valid_UV_lists_alt: assumes "P = (\<lambda>x. (\<exists>as bs cs. as@U@bs@V@cs = x) \<and> set x = xs \<and> distinct x)" shows "{x. (\<exists>as bs cs. as@U@bs@V@cs = x) \<and> set x = xs \<and> distinct x} = {ys. P ys}"
lemma bind_map_spmf: "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
lemma rreq_rrep_fresh [simp]: "\<And>hops dip dsn dsk oip osn sip handled. rreq_rrep_fresh crt (Rreq hops dip dsn dsk oip osn sip handled) = (sip \<noteq> oip \<longrightarrow> oip\<in>kD(crt) \<and> (sqn crt oip > osn \<or> (sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val)))" "\<And>hops dip dsn oip sip. rreq_rrep_fresh crt (Rrep hops dip dsn oip sip) = (sip \<noteq> dip \<longrightarrow> dip\<in>kD(crt) \<and> sqn crt dip = dsn \<and> the (dhops crt dip) = hops \<and> the (flag crt dip) = val)" "\<And>dests sip. rreq_rrep_fresh crt (Rerr dests sip) = True" "\<And>d dip. rreq_rrep_fresh crt (Newpkt d dip) = True" "\<And>d dip sip. rreq_rrep_fresh crt (Pkt d dip sip) = True"
lemma point_surj [simp]: "Point (abscissa M) (ordinate M) = M"
lemma execute_of_list [execute_simps]: "execute (of_list xs) h = Some (alloc xs h)"
lemma CondReds2T: assumes e_steps: "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>" and e\<^sub>1_steps: "P \<turnstile> \<langle>e\<^sub>1, s\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2, s\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" (<)(is "(?x, ?z) \<in> (red P)\<^sup>*")
lemma infsum_bounded_linear: assumes \<open>bounded_linear f\<close> assumes \<open>g summable_on S\<close> shows \<open>infsum (f \<circ> g) S = f (infsum g S)\<close>
lemma HNatInfinite_upward_closed: "x \<in> HNatInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> y \<in> HNatInfinite"
lemma cms:"m O s \<subseteq> m"
lemma bpred_eval[simp]: "bpred p\<cdot>(box\<cdot>x)\<cdot>(box\<cdot>y) = (if p x y then TT else FF)"
lemma ineM_L1: assumes "ch \<in> M" and "ch \<in> ins P" and "exprChannel ch E" shows "ineM P M E"
lemma sign_r_pos_0[simp]:"\<not> sign_r_pos 0 (x::real)"
lemma ln_prod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (prod f I) = sum (\<lambda>x. ln(f x)) I" for f :: "'a \<Rightarrow> real"
lemma ideal_Union: fixes I::"nat => 'a set" defines S: "S\<equiv>{I n\|n. n\<in>UNIV}" assumes all_ideal: "\<forall>A\<in>S. ideal A" and inc: "\<forall>n. I(n) \<subseteq> I(n+1)" shows "ideal (\<Union>S)"
lemma toProcess_start: "toProcess sidx p = (ss,a,n,l,idx,start,r) \<Longrightarrow> \<exists>s. start = Index s \<and> s < IArray.length ss"
lemma image_diff_atMost: "(\<lambda>i. (n::nat) - i) ` {..n} = {..n}" (is "?l = ?r")
lemma Some_Sup: "A \<noteq> {} \<Longrightarrow> Some (\<Squnion>A) = \<Squnion>(Some ` A)"
lemma no_plus_overflow_uint_size: "x \<le> x + y \<longleftrightarrow> uint x + uint y < 2 ^ size x" for x y :: "'a::len word"
lemma MGT_lemma: assumes MGT_Calls: "\<forall>p \<in> dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta> \<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s=Z \<and> \<Gamma>\<turnstile>\<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>({Stuck} \<union> Fault ` (-F)) \<and> \<Gamma>\<turnstile>(Call p)\<down>Normal s} (Call p) {t. \<Gamma>\<turnstile>\<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t}, {t. \<Gamma>\<turnstile>\<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}" shows "\<And>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s=Z \<and> \<Gamma>\<turnstile>\<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>({Stuck} \<union> Fault ` (-F)) \<and> \<Gamma>\<turnstile>c\<down>Normal s} c {t. \<Gamma>\<turnstile>\<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile>\<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
lemma subst_atmss_id_subst[simp]: "AAA \<cdot>ass id_subst = AAA"
lemma continuous_on_imp_set_integrable_cbox: fixes h :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space" assumes "continuous_on (cbox a b) h" shows "set_integrable lborel (cbox a b) h"
lemma continuous_on_infnorm[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
lemma linepath_le_1: fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1"
lemma cont_at': "(f \<longlongrightarrow> f x) (at' x) \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"
lemma \<Gamma>\<^sub>v_Var_image: "\<Gamma>\<^sub>v ` X = \<Gamma> ` Var ` X"
lemma n_add_distr: "n x + (n y \<cdot> n z) = (n x + n y) \<cdot> (n x + n z)"
lemma card_differenceset_singleton_mem_eq: assumes "a \<in> G" and "A \<subseteq> G" shows "card A = card (differenceset A {a})"
lemma "(make_foo a b) \<lparr> field_1 := y \<rparr> = make_foo y b"
lemma lnth_lappend[simp]: assumes "lfinite xs" and "\<not> lnull ys" shows "lnth (xs @\<^sub>l ys) (the_enat (llength xs)) = lhd ys"
lemma state_cover_transition_converges : assumes "observable M" and "is_state_cover_assignment M V" and "t \<in> transitions M" and "t_source t \<in> reachable_states M" shows "converge M ((V (t_source t)) @ [(t_input t,t_output t)]) (V (t_target t))"

lemma ifbd_inj_iff: "Sup_pres \<phi> \<Longrightarrow> Sup_pres \<psi> \<Longrightarrow> (bd\<^sup>-\<^sub>\<F> \<phi> = bd\<^sup>-\<^sub>\<F> \<psi>) = (\<phi> = \<psi>)"

lemma measurable_count_space_insert[measurable (raw)]: "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"

lemma inverse_of_nat_tendsto_zero: "(\<lambda>x. inverse (of_nat x :: 'a :: real_normed_div_algebra)) \<longlonglongrightarrow> 0"

lemma map_entry_parametric [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B) (\<lambda>k f m. (case m k of None \<Rightarrow> m | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m | Some v \<Rightarrow> m (k \<mapsto> (f v))))"

lemma safe_only_owner_enter_normal_aux[simp]: "\<lbrakk> s : reach; safe s r; (k',roomk s r) \<in> cards s g \<rbrakk> \<Longrightarrow> owns s r = Some g"

lemma min_Suc_gt[simp]: "a<b \<Longrightarrow> min (Suc a) b = Suc a" "a<b \<Longrightarrow> min b (Suc a) = Suc a"

lemma H_loop: "\<^bold>{P\<^bold>} X \<^bold>{P\<^bold>} \<Longrightarrow> \<^bold>{P\<^bold>} (LOOP X INV I) \<^bold>{P\<^bold>}"

lemma (in deutsch) adjoint_of_deutsch_transform: shows "(U\<^sub>f)\<^sup>\<dagger> = U\<^sub>f"

lemma atd_lem: "take n xs = t \<Longrightarrow> drop n xs = d \<Longrightarrow> xs = t @ d"

lemma field_differentiable_compose: "f field_differentiable at z \<Longrightarrow> g field_differentiable at (f z) \<Longrightarrow> (g o f) field_differentiable at z"

lemma two: "(2::nat) = Suc 1"

lemma reduce_below_abs_0_case_m1: assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n" and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))" and Aaj: "A $$ (a,0) \<noteq> 0" and mn: "m\<ge>n" assumes "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x" and "m\<noteq>a" and "D>0" shows "reduce_below_abs a (xs @ [m]) D A $$ (m,0) = 0"

lemma "(t \<in> B h \<longrightarrow> ins x t \<in> Bp h) \<and> (t \<in> U h \<longrightarrow> ins x t \<in> T h)"

lemma "Oc\<up>(0+1) = [Oc]"

lemma (in uprio_pop) autoref_prio_pop_min[autoref_rules]: "\<And>Re Ra. \<lbrakk>PREFER_id Re; PREFER_id Ra \<rbrakk> \<Longrightarrow> (\<lambda>s. RETURN (pop s),prio_pop_min)\<in>\<langle>Re,Ra\<rangle>rel\<rightarrow>\<langle>\<langle>Re,\<langle>Ra,\<langle>Re,Ra\<rangle>rel\<rangle>prod_rel\<rangle>prod_rel\<rangle>nres_rel"

lemma bin_split_minus1 [simp]: "bin_split n (- 1) = (- 1, (take_bit :: nat \<Rightarrow> int \<Rightarrow> int) n (- 1))"

lemma prv_eqv_prv: assumes "\<phi> \<in> fmla" and "\<chi> \<in> fmla" assumes "prv \<phi>" and "prv (eqv \<phi> \<chi>)" shows "prv \<chi>"

lemma circuit_in_elim: assumes "circuit_in \<E> C\<^sub>1" "circuit_in \<E> C\<^sub>2" "C\<^sub>1 \<noteq> C\<^sub>2" "x \<in> C\<^sub>1 \<inter> C\<^sub>2" shows "\<exists>C\<^sub>3 \<subseteq> (C\<^sub>1 \<union> C\<^sub>2) - {x}. circuit_in \<E> C\<^sub>3"

lemma post_meas0_index_0_alice: assumes "state 1 \<phi>" and "\<alpha> = \<phi> $$ (0,0)" and "\<beta> = \<phi> $$ (1,0)" shows "post_meas0 3 (alice \<phi>) 0 = mat_of_cols_list 8 [[\<alpha>/sqrt(2), \<beta>/sqrt(2), \<beta>/sqrt(2), \<alpha>/sqrt(2), 0, 0, 0, 0]]"

lemma in_singleton:" m \<^bold>\<in> Abs_nat_int{n} \<longrightarrow> m = n"

lemma infinite_times_eqpoll_self: assumes "infinite A" shows "A \<times> A \<approx> A"

lemma progress_le: "progress \<sigma> Map.empty \<phi> j \<le> j"

lemma ordst_trans: assumes As1: "ordst X Y" and As2: "ordst Y Z" shows "ordst X Z"

lemma wlp_skip_eq: "wlp SKIP Q s = Q s"

lemma strong_del_point_incidence_wf: "incidence_system (del_point p) (str_del_point_blocks p)"

lemma SplitGuards_sound: assumes valid_c1: "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,A" assumes valid_c2: "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c\<^sub>2 UNIV,UNIV" assumes c: "(c\<^sub>1 \<inter>\<^sub>g c\<^sub>2) = Some c" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"

lemma ImplLeadsto_simple: "\<And>F G. \<turnstile> F \<longrightarrow> G \<Longrightarrow> \<turnstile> F \<leadsto> G"

lemma vsubset_vdiff: assumes "A \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> B"

lemma qp_D3_post: "is_quantum_predicate (adjoint exM0 * P' * exM0 + adjoint exM1 * Q * exM1)"

lemma lsl_top [simp]: "\<nu> \<top> = (\<top>::'a::unital_quantale)"

lemma fixbound_in: "Q \<in> fixbound \<Q> x \<Longrightarrow> Q \<in> \<Q>"

lemma tree_edge_disc: "(v,w) \<in> tree_edges s \<Longrightarrow> \<delta> s v < \<delta> s w"

lemma elementsButlastTrailIsButlastElementsTrail [simp]: shows "elements (butlast M) = butlast (elements M)"

lemma (in field) npepow_correct: "in_carrier xs \<Longrightarrow> peval xs (npepow e n) = peval xs (PExpr2 (PPow e n))"

lemma abupd_def2 [simp]: "abupd f (x,s) = (f x,s)"

lemma dom_empty [simp]: "dom empty = {}"

lemma "when" [simp]: "P \<Longrightarrow> (a when P) = a" "\<not> P \<Longrightarrow> (a when P) = 0"

lemma rbt_lookup_from_in_tree: assumes "rbt_sorted t1" "rbt_sorted t2" and "\<And>v. (k, v) \<in> set (entries t1) \<longleftrightarrow> (k, v) \<in> set (entries t2)" shows "rbt_lookup t1 k = rbt_lookup t2 k"

lemma fun_eq_on_cong: "fun_eq_on f h A \<Longrightarrow> fun_eq_on g h A \<Longrightarrow> fun_eq_on f g A"

lemma class_add_widens: "\<lbrakk> P \<turnstile> Ts [\<le>] Ts'; \<not> is_class P C \<rbrakk> \<Longrightarrow> (class_add P (C, cdec)) \<turnstile> Ts [\<le>] Ts'"

lemma alphas_abs_eqvt: shows "(bs, x) \<approx>abs_set (cs, y) \<Longrightarrow> (p \<bullet> bs, p \<bullet> x) \<approx>abs_set (p \<bullet> cs, p \<bullet> y)" and "(bs, x) \<approx>abs_res (cs, y) \<Longrightarrow> (p \<bullet> bs, p \<bullet> x) \<approx>abs_res (p \<bullet> cs, p \<bullet> y)" and "(ds, x) \<approx>abs_lst (es, y) \<Longrightarrow> (p \<bullet> ds, p \<bullet> x) \<approx>abs_lst (p \<bullet> es, p \<bullet> y)"

lemma scalingI[intro]: "\<lbrakk> \<And>P c x. \<lbrakk> sound P; 0 \<le> c \<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x \<rbrakk> \<Longrightarrow> scaling t"

lemma nonzero_fstI[intro, simp]: "fst x \<noteq> 0 \<Longrightarrow> x \<noteq> 0" and nonzero_sndI[intro, simp]: "snd x \<noteq> 0 \<Longrightarrow> x \<noteq> 0"

lemma cpx_length_of_vec_of_list [simp]: "\<parallel>vec_of_list l\<parallel> = sqrt(\<Sum>i<length l. (cmod (l ! i))\<^sup>2)"

lemma OclInt1_non_null [simp,code_unfold]: "(\<one> \<doteq> null) = false"

lemma ND0aux3[rule_format]: "AllowPortFromTo x y p \<in> set b \<Longrightarrow> x \<in> set (net_list_aux b)"

lemma semialg_intersect: assumes "A \<in> semialg_sets n" assumes "B \<in> semialg_sets n" shows "(A \<inter> B) \<in> semialg_sets n "

lemma box_import_shunting: "-p * -q \<le> |x](-r) \<longleftrightarrow> -q \<le> |-p * x](-r)"

lemma big_step_CS: "(c,s) \<Rightarrow> t \<Longrightarrow> t \<in> post(CS c)"

lemma nn_set_integral_cong: assumes "AE x in M. f x = g x" shows "(\<integral>\<^sup>+x \<in> A. f x \<partial>M) = (\<integral>\<^sup>+x \<in> A. g x \<partial>M)"

lemma rule_\<alpha>_type[simp]: "(N, \<alpha>) \<in> \<RR> \<Longrightarrow> is_sentence \<alpha>"

lemma weakPsiCongE: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" and \<Psi>' :: 'b assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> P \<approx> Q" and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q" and "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"

lemma liftE_def2: "liftE f = (\<lambda>s. ((\<lambda>(v,s'). (Inr v, s')) (fst (f s)), snd (f s)))"

lemma HMA_V_diff [transfer_rule]: "(HMA_V ===> HMA_V ===> HMA_V) (-) (-)"

lemma dvd_associated2: fixes a::"'a::idom" assumes ab: "a dvd b" and ba: "b dvd a" and a: "a\<noteq>0" shows "\<exists>u. u dvd 1 \<and> a = u*b"

lemma has_bochner_integral_imp_has_integral': "has_bochner_integral lborel (\<lambda>x. indicator S x *\<^sub>R f x) I \<Longrightarrow> (f has_integral (I :: 'b :: euclidean_space)) S"

lemma value_class_inhabitants: "(\<forall>x. typeof x = CClassT typeId \<longrightarrow> P x) = (\<forall> a. P (objV typeId a))" (is "(\<forall> x. ?A x) = ?B")

lemma Assign_is_state_update_before: "ASSIGN x e = state_update_before (\<lambda> (s, s') . s' = (update x (\<lambda>_. (e s))) s) Skip"

lemma expand_prop_all: assumes "expand_assm_incoming n_ns \<and> expand_name_ident (snd n_ns)" (is "?Q n_ns") shows "expand n_ns \<le> SPEC (expand_rslt_all \<xi> n_ns)" (is "_ \<le> SPEC (?P n_ns)")

lemma subst_simps [simp]: "subst x t x = t" "subst x (Var x) = Var"

lemma enforce_option_bot [simp]: "enforce_option \<bottom> = (\<lambda>_. None)"

lemma path_vertices_shift: assumes "path v v' pth" shows "map fst (map fst pth)@[v'] = v#map fst (map snd pth)"

lemma extends_subst_cong_lit: "extends_subst \<sigma> \<tau> \<Longrightarrow> vars_lit L \<subseteq> dom \<sigma> \<Longrightarrow> L \<cdot>lit subst_of_map Var \<sigma> = L \<cdot>lit subst_of_map Var \<tau>"

lemma signed_drop_bit_word_numeral [simp]: \<open>signed_drop_bit (numeral n) (numeral k) = (word_of_int (drop_bit (numeral n) (signed_take_bit (LENGTH('a) - 1) (numeral k))) :: 'a::len word)\<close>

lemma l2norm_lmult: "f square_integrable S \<Longrightarrow> l2norm S (\<lambda>x. c * f x) = \<bar>c\<bar> * l2norm S f"

lemma open_chart_last_domain: "open (Collect chart_last_domainP)"

lemma mono_ndet_ref: " \<lbrakk>P \<sqsubseteq> P'; S \<sqsubseteq> S'\<rbrakk> \<Longrightarrow> (P \<sqinter> S) \<sqsubseteq> (P' \<sqinter> S')"

lemma parts_insert_subset_Un: "X \<in> G \<Longrightarrow> parts(insert X H) \<subseteq> parts G \<union> parts H"

lemma evaluate_iff_sym: "evaluate True env st e r \<longleftrightarrow> (eval env e st = r)" "evaluate_list True env st es r' \<longleftrightarrow> (eval_list env es st = r')" "evaluate_match True env st v pes v' r \<longleftrightarrow> (eval_match env v pes v' st = r)"

lemma insert_if: "finite A \<Longrightarrow> F g (insert x A) = (if x \<in> A then F g A else g x \<^bold>* F g A)"

lemma valid_UV_lists_alt: assumes "P = (\<lambda>x. (\<exists>as bs cs. as@U@bs@V@cs = x) \<and> set x = xs \<and> distinct x)" shows "{x. (\<exists>as bs cs. as@U@bs@V@cs = x) \<and> set x = xs \<and> distinct x} = {ys. P ys}"

lemma bind_map_spmf: "map_spmf f p \<bind> g = p \<bind> g \<circ> f"

lemma rreq_rrep_fresh [simp]: "\<And>hops dip dsn dsk oip osn sip handled. rreq_rrep_fresh crt (Rreq hops dip dsn dsk oip osn sip handled) = (sip \<noteq> oip \<longrightarrow> oip\<in>kD(crt) \<and> (sqn crt oip > osn \<or> (sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val)))" "\<And>hops dip dsn oip sip. rreq_rrep_fresh crt (Rrep hops dip dsn oip sip) = (sip \<noteq> dip \<longrightarrow> dip\<in>kD(crt) \<and> sqn crt dip = dsn \<and> the (dhops crt dip) = hops \<and> the (flag crt dip) = val)" "\<And>dests sip. rreq_rrep_fresh crt (Rerr dests sip) = True" "\<And>d dip. rreq_rrep_fresh crt (Newpkt d dip) = True" "\<And>d dip sip. rreq_rrep_fresh crt (Pkt d dip sip) = True"

lemma point_surj [simp]: "Point (abscissa M) (ordinate M) = M"

lemma execute_of_list [execute_simps]: "execute (of_list xs) h = Some (alloc xs h)"

lemma CondReds2T: assumes e_steps: "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>" and e\<^sub>1_steps: "P \<turnstile> \<langle>e\<^sub>1, s\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2, s\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" (*<*)(is "(?x, ?z) \<in> (red P)\<^sup>*")

lemma infsum_bounded_linear: assumes \<open>bounded_linear f\<close> assumes \<open>g summable_on S\<close> shows \<open>infsum (f \<circ> g) S = f (infsum g S)\<close>

lemma HNatInfinite_upward_closed: "x \<in> HNatInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> y \<in> HNatInfinite"

lemma cms:"m O s \<subseteq> m"

lemma bpred_eval[simp]: "bpred p\<cdot>(box\<cdot>x)\<cdot>(box\<cdot>y) = (if p x y then TT else FF)"

lemma ineM_L1: assumes "ch \<in> M" and "ch \<in> ins P" and "exprChannel ch E" shows "ineM P M E"

lemma sign_r_pos_0[simp]:"\<not> sign_r_pos 0 (x::real)"

lemma ln_prod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (prod f I) = sum (\<lambda>x. ln(f x)) I" for f :: "'a \<Rightarrow> real"

lemma ideal_Union: fixes I::"nat => 'a set" defines S: "S\<equiv>{I n|n. n\<in>UNIV}" assumes all_ideal: "\<forall>A\<in>S. ideal A" and inc: "\<forall>n. I(n) \<subseteq> I(n+1)" shows "ideal (\<Union>S)"

lemma toProcess_start: "toProcess sidx p = (ss,a,n,l,idx,start,r) \<Longrightarrow> \<exists>s. start = Index s \<and> s < IArray.length ss"

lemma image_diff_atMost: "(\<lambda>i. (n::nat) - i) ` {..n} = {..n}" (is "?l = ?r")

lemma Some_Sup: "A \<noteq> {} \<Longrightarrow> Some (\<Squnion>A) = \<Squnion>(Some ` A)"

lemma no_plus_overflow_uint_size: "x \<le> x + y \<longleftrightarrow> uint x + uint y < 2 ^ size x" for x y :: "'a::len word"

lemma MGT_lemma: assumes MGT_Calls: "\<forall>p \<in> dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta> \<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s=Z \<and> \<Gamma>\<turnstile>\<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>({Stuck} \<union> Fault ` (-F)) \<and> \<Gamma>\<turnstile>(Call p)\<down>Normal s} (Call p) {t. \<Gamma>\<turnstile>\<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t}, {t. \<Gamma>\<turnstile>\<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}" shows "\<And>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s=Z \<and> \<Gamma>\<turnstile>\<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>({Stuck} \<union> Fault ` (-F)) \<and> \<Gamma>\<turnstile>c\<down>Normal s} c {t. \<Gamma>\<turnstile>\<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile>\<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"

lemma subst_atmss_id_subst[simp]: "AAA \<cdot>ass id_subst = AAA"

lemma continuous_on_imp_set_integrable_cbox: fixes h :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space" assumes "continuous_on (cbox a b) h" shows "set_integrable lborel (cbox a b) h"

lemma continuous_on_infnorm[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

lemma linepath_le_1: fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1"

lemma cont_at': "(f \<longlongrightarrow> f x) (at' x) \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"

lemma \<Gamma>\<^sub>v_Var_image: "\<Gamma>\<^sub>v ` X = \<Gamma> ` Var ` X"

lemma n_add_distr: "n x + (n y \<cdot> n z) = (n x + n y) \<cdot> (n x + n z)"

lemma card_differenceset_singleton_mem_eq: assumes "a \<in> G" and "A \<subseteq> G" shows "card A = card (differenceset A {a})"

lemma "(make_foo a b) \<lparr> field_1 := y \<rparr> = make_foo y b"

lemma lnth_lappend[simp]: assumes "lfinite xs" and "\<not> lnull ys" shows "lnth (xs @\<^sub>l ys) (the_enat (llength xs)) = lhd ys"

lemma state_cover_transition_converges : assumes "observable M" and "is_state_cover_assignment M V" and "t \<in> transitions M" and "t_source t \<in> reachable_states M" shows "converge M ((V (t_source t)) @ [(t_input t,t_output t)]) (V (t_target t))"