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

Statement: stringlengths 7 24.3k
lemma wf_imethdsD: "\<lbrakk>im \<in> imethds G I sig;wf_prog G; is_iface G I\<rbrakk> \<Longrightarrow> \<not>is_static im \<and> accmodi im = Public"
lemma llrg_linear_sys: "llrg \<R> \<Longrightarrow> linear_sys \<R>"
lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"
lemma final_lemma6: "(\<Inter>v \<in> V. \<Inter>w \<in> V. (reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter> nmsg_eq 0 (v,w)) \<subseteq> final"
lemma height_0_rel: "height t = 0 \<Longrightarrow> \<exists>r. t = Relation r"
lemma "(Abs_perm id :: nat perm) \<bullet> Suc 0 = Suc 0"
lemma connected_card_eq_iff_nontrivial: fixes S :: "'a::metric_space set" shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"
lemma rel_envT_eq [relator_eq]: "rel_envT (=) (=) = (=)"
lemma sublist_equal_lengths: fixes l1 l2 assumes "subseq l1 l2" "(length l1 = length l2)" shows "(l1 = l2)"
lemma ht_hash_distinct: "ht_hash l \<Longrightarrow> \<forall>i j . i\<noteq>j \<and> i < length l \<and> j < length l \<longrightarrow> set (l!i) \<inter> set (l!j) = {}"
lemma sup_pres_multl: "sup_pres (\<lambda>(z::'a::proto_quantale). x \<cdot> z)"
lemma update_arg_wf_tuples' [elim]: "\<And>n hops nhip pre. Suc 0 \<le> n \<Longrightarrow> update_arg_wf (n, kno, val, hops, nhip, pre)"
lemma div_op_SOME: "is_div_op (\<lambda>a b. (SOME k. k * b = a))"
lemma tau_at_0: "\<tau> i 0 \<down>= i"
lemma Cond': "\<lbrakk>P \<subseteq> {s. (b \<subseteq> P1) \<and> (- b \<subseteq> P2)};\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P1 c1 Q,A; \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P2 c2 Q,A\<rbrakk> \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P (Cond b c1 c2) Q,A"
lemma (in is_functor) is_functor_ntcf_id_components: shows "ntcf_id \<FF>\<lparr>NTMap\<rparr> = \<BB>\<lparr>CId\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr>" and "ntcf_id \<FF>\<lparr>NTDom\<rparr> = \<FF>" and "ntcf_id \<FF>\<lparr>NTCod\<rparr> = \<FF>" and "ntcf_id \<FF>\<lparr>NTDGDom\<rparr> = \<AA>" and "ntcf_id \<FF>\<lparr>NTDGCod\<rparr> = \<BB>"
lemma "((((sep_true imp p0) imp ((p0 ** p0) \<longrightarrow>* ((sep_true imp p0) (p0 p0)))) imp (p1 \<longrightarrow>* (((sep_true imp p0) imp ((p0 ** p0) \<longrightarrow>* (((sep_true imp p0) p0) p0))) ** p1)))) (h::'a::heap_sep_algebra)"
lemma vconverse_inject[simp]: assumes "vbrelation r" and "vbrelation s" shows "r\<inverse>\<^sub>\<circ> = s\<inverse>\<^sub>\<circ> \<longleftrightarrow> r = s"
lemma gbinomial_int_mono: assumes "0 \<le> x" and "x \<le> (y::int)" shows "x gchoose k \<le> y gchoose k"
lemma solution_upd1: "c \<noteq> 0 \<Longrightarrow> solution (A(p:=(\<lambda>j. A p j / c))) n x = solution A n x"
lemma correctly_synchronized_compP [simp]: "correctly_synchronized (compP f P) = correctly_synchronized P"
lemma max_mix_is_density: assumes "0 < n" shows "density_operator (max_mix_density n)"
lemma interaction_bounded_by_case_prod [interaction_bound]: "(\<And>a b. x = (a, b) \<Longrightarrow> interaction_bounded_by consider (f a b) (n a b)) \<Longrightarrow> interaction_bounded_by consider (case_prod f x) (case_prod n x)"
lemma dbm_int_guard_abstr: assumes "valid_abstraction A X k" "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "dbm_int (abstr g (\<lambda>i j. \<infinity>) v) n"
lemma eventually_going_to_at_bot_linorder: fixes f :: "'a \<Rightarrow> 'b :: linorder" shows "eventually P (f going_to at_bot within A) \<longleftrightarrow> (\<exists>C. \<forall>x\<in>A. f x \<le> C \<longrightarrow> P x)"
lemma iso_fSup: fixes F :: "('a::order \<Rightarrow> 'b::complete_lattice) set" shows "(\<forall>f \<in> F. mono f) \<Longrightarrow> mono (\<Squnion>F)"
lemma isTrue_neg_neg: assumes "\<phi> \<in> fmla" "Fvars \<phi> = {}" and "isTrue (neg (neg \<phi>))" shows "isTrue \<phi>"
lemma thinish: "ThinishChamberComplex X"
lemma E_thm01: "Lift \<in> (stopped \<inter> atFloor n) LeadsTo (opened \<inter> atFloor n)"
lemma trimmed_pos_real_iff [simp]: "trimmed_pos (x :: real) \<longleftrightarrow> x > 0"
lemma one_dim_iso_adjoint[simp]: \<open>one_dim_iso (A) = (one_dim_iso A)\<close>
lemma incls_coherent: assumes "par f f'" and "incl f" and "incl f'" shows "f = f'"
lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x" for m :: "'a::linordered_field"
lemma run_put_writerT [simp]: "run_writer (put s m) = put s (run_writer m)"
lemma \<open>(1705 :: nat) << 3 = 13640\<close>
lemma fresh_asTerm_qFresh: assumes "qGood qX" shows "fresh xs x (asTerm qX) = qFresh xs x qX"
lemma nxt_alw: "nxt (alw P) s \<Longrightarrow> alw (nxt P) s"
lemma split_vars_ground_vars: assumes "ground_mctxt C" and "num_holes C = length xs" shows "split_vars (fill_holes C (map Var xs)) = (C, xs)"
lemma diff_add_assoc: "b \<le> c \<Longrightarrow> c - (c - b) = c - c + (b::nat)"
lemma shiftD: "x \<^bold>\<in> shift f delta \<Longrightarrow> \<exists>u. u \<^bold>\<in> f \<and> x = \<langle>delta @+ hfst u, hsnd u\<rangle>"
lemma Ch_range_disjoint: assumes "h \<noteq> h'" shows "Ch h A \<inter> Ch h' A = {}"
lemma variance_prod_pmf_slice: fixes f :: "'a \<Rightarrow> real" assumes "i \<in> I" "finite I" assumes "integrable (measure_pmf (M i)) (\<lambda>\<omega>. f \<omega>^2)" shows "prob_space.variance (Pi_pmf I d M) (\<lambda>\<omega>. f (\<omega> i)) = prob_space.variance (M i) f"
lemma length_concat_ge: assumes "as \<in> lists (- {[]})" shows "length (concat as) \<ge> length as"
lemma parabolic_isometry_inv: assumes "parabolic_isometry f" shows "parabolic_isometry (inv f)"
lemma scount_eq_0D: "scount P \<omega> = 0 \<Longrightarrow> alw (not P) \<omega>"
lemma older_seniors_finite: "finite (older_seniors x n)"
lemma bezout_coefficients: assumes "bezout_coefficients a b = (x, y)" shows "x * a + y * b = gcd a b"
lemma obtain_t_subset_points: obtains T where "T \<subseteq> \<V>" "card T = \<t>" "finite T"
lemma RM_subset : "RM M2 M1 \<Omega> V m i \<subseteq> C M2 M1 \<Omega> V m i"
lemma KRP_fresh_iff [simp]: "a \<sharp> KRP v x x' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> x \<and> a \<sharp> x'"
theorem Lp_Lq_duality: fixes p q::ennreal assumes "f \<in> space\<^sub>N (\<LL> p M)" "1/p + 1/q = 1" "p = \<infinity> \<Longrightarrow> sigma_finite_measure M" shows "bdd_above ((\<lambda>g. (\<integral>x. f x * g x \<partial>M))`{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1})" "Norm (\<LL> p M) f = (SUP g\<in>{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1}. (\<integral>x. f x * g x \<partial>M))"
lemma sc_threshold_2_3_tfff: "boolfunc_from_sc 4 sc_threshold_2_3 (a(0:=True,1:=False,2:=False,3:=False)) = False"
lemma rebase_id[simp]: "@(@(x::'b mod_ring) :: 'a mod_ring) = x"
lemma all_leq_Max: assumes "x \<le>\<^sub>v y" and "x \<noteq> []" shows "\<forall>xi \<in> set x. xi \<le> Max (set y)"
lemma simpneq_nb[simp]: assumes "SORT_CONSTRAINT('a::field_char_0)" shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
lemma (in Category) MapsToId: assumes "X \<in> obj" shows "id X \<approx>> id X"
lemma [autoref_rules]: "(bs_insert,insert)\<in>nat_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
lemma "a \<in> set ls \<Longrightarrow> b \<in> set (List_replace1 a b ls)"
lemma fold_fun_chain: "g x = (g ^^ 1) x" "(g ^^ m) ((g ^^ n) x) = (g ^^ (m+n)) x"
lemma len_sub_int:"len v ts c \<le> ext v"
lemma (in ceuclidean_space) SOME_CBasis: "(SOME i. i \<in> CBasis) \<in> CBasis"
lemma processed_alt: "processed (a, b) xs ps \<longleftrightarrow> ((a \<in> set xs) \<and> (b \<in> set xs) \<and> (a, b) \<notin>\<^sub>p set ps)"
lemma staticSecret_parts_agent: "\<lbrakk>m \<in> parts (knows C evs); m \<in> staticSecret A\<rbrakk> \<Longrightarrow> A=C \<or> (\<exists>D E X. Says D E X \<in> set evs \<and> m \<in> parts{X}) \<or> (\<exists>Y. Notes C Y \<in> set evs \<and> m \<in> parts{Y})"
lemma valid_and_I : "\<tau> \<Turnstile> \<upsilon> (x) \<Longrightarrow> \<tau> \<Turnstile> \<upsilon> (y) \<Longrightarrow> \<tau> \<Turnstile> \<upsilon> (x and y)"
lemma precedes_append_left_iff: assumes "x \<notin> set ys" shows "x \<preceq> y in (ys @ xs) \<longleftrightarrow> x \<preceq> y in xs" (is "?lhs = ?rhs")
lemma ctxt_comp_n_funas [simp]: "(f, v) \<in> funas_ctxt (C^n) \<Longrightarrow> (f, v) \<in> funas_ctxt C"
lemma bounded_antilinear_from_conjugate_space[simp]: \<open>bounded_antilinear from_conjugate_space\<close>
lemma nAct_active[simp]: fixes n::nat and n'::nat and t::"(cnf llist)" and c::'id assumes "enat i < llength t" and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "\<langle>c #\<^bsub>Suc i\<^esub> t\<rangle> = eSuc (\<langle>c #\<^bsub>i\<^esub> t\<rangle>)"
lemma (in t2_space) LIMSEQ_Uniq: "\<exists>\<^sub>\<le>\<^sub>1l. X \<longlonglongrightarrow> l"
lemma glist_delete_id_impl: "(glist_delete (=), \<lambda>x s. s-{x}) \<in> Id\<rightarrow>br set distinct \<rightarrow> br set distinct"
lemma closest_pair_rec_dist: assumes "1 < length xs" "distinct xs" "sorted_fst xs" "(ys, c\<^sub>0, c\<^sub>1) = closest_pair_rec xs" shows "sparse (dist c\<^sub>0 c\<^sub>1) (set xs)"
lemma position_Left_only_subst: assumes "list_all (\<lambda>x. x = Left) p" and "position_of (subst \<rho> w) (p @ [d])" and "num_args (subst \<rho> w) = num_args w" shows "position_of w (p @ [d])"
lemma "eq\<cdot>Nothing\<cdot>x = isNothing\<cdot>x"
lemma mult_two_impl1[elim]: assumes "a * 2 = 2 * b" shows "(a::enat) = b"
lemma float_mult_r0: "x * float (0, e) = float (0, 0)"
lemma mod_subset_add: "\<lbrakk>(c1,c2) \<in> mod_subset p X; (d1,d2) \<in> mod_subset p X\<rbrakk> \<Longrightarrow> (c1+d1, c2+d2) \<in> mod_subset p X"
lemma std_apart_apart': assumes "x \<in> vars\<^sub>l (l \<cdot>\<^sub>l (\<lambda>x. Var (y@x)))" shows "\<exists>x'. x = y@x'"
lemma Conjs_pull_out: "Conjs Q (xys @ (x, y) # xys') \<triangleq> Conjs (Conj Q (x \<approx> y)) (xys @ xys')"
lemma mcont_fun_lub_Sup: "\<lbrakk> Complete_Partial_Order.chain (fun_ord (\<le>)) M; \<forall>f\<in>M. mcont lub ord Sup (\<le>) f \<rbrakk> \<Longrightarrow> mcont lub (\<sqsubseteq>) Sup (\<le>) (fun_lub Sup M)"
lemma vars\<^sub>s\<^sub>s\<^sub>t_Nil[simp]: "vars\<^sub>s\<^sub>s\<^sub>t [] = {}"
lemma "\<exists>x. \<forall>y::'a::linordered_field. y < 2 \<longrightarrow> 2 * (y - x) \<le> 0"
lemma maximal_Seq_term: fixes r::"'s prog" and s::"'s prog" assumes mr: "maximal (wp r)" and ws: "well_def s" and ts: "(\<lambda>s. 1) \<tturnstile> wp s (\<lambda>s. 1)" shows "(\<lambda>s. 1) \<tturnstile> wp (r ;; s) (\<lambda>s. 1)"
lemma image_mset_subset_mono: "M \<subset># N \<Longrightarrow> image_mset f M \<subset># image_mset f N"
lemma graphE [elim?]: assumes "(x, y) \<in> graph F f" obtains "x \<in> F" and "y = f x"
lemma SeqCTerm_imp_wf_dbtm: assumes "SeqCTerm vf s k x" shows "\<exists>t::dbtm. wf_dbtm t \<and> x = \<lbrakk>quot_dbtm t\<rbrakk>e"
lemma mod_mat_one: "mat_mod (1\<^sub>m n) = (1\<^sub>m n)"
lemma test_suite_to_input_sequences_pass : fixes T :: "('b::linorder \<times> 'c::linorder) prefix_tree" assumes "finite_tree T" and "(L M = L M') \<longleftrightarrow> (L M \<inter> set T = L M' \<inter> set T)" shows "(L M = L M') \<longleftrightarrow> ({io \<in> L M . (\<exists> xs \<in> list.set (test_suite_to_input_sequences T) . \<exists> xs' \<in> list.set (prefixes xs) . input_portion io = xs')} = {io \<in> L M' . (\<exists> xs \<in> list.set (test_suite_to_input_sequences T) . \<exists> xs' \<in> list.set (prefixes xs) . input_portion io = xs')})"
lemma preal_add_le_cancel_left [simp]: "(t + (r::preal) \<le> t + s) = (r \<le> s)"
lemma word_leI: "(\<And>n. \<lbrakk>n < size (u::'a::len word); bit u n \<rbrakk> \<Longrightarrow> bit (v::'a::len word) n) \<Longrightarrow> u <= v"
lemma casts_casts_eq_result: fixes s :: state assumes casts:"P \<turnstile> T casts v to v'" and casts':"P \<turnstile> T casts v to w'" and type:"is_type P T" and wte:"P,E \<turnstile> e :: T'" and leq:"P \<turnstile> T' \<le> T" and eval:"P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h,l)\<rangle>" and sconf:"P,E \<turnstile> s \<surd>" and wf:"wf_C_prog P" shows "v' = w'"
lemma listrel_reflcl_if_listrel1: "(xs,ys) \<in> listrel1 r \<Longrightarrow> (xs,ys) \<in> listrel(r\<^sup>*)"
lemma [code]: "toy_instance_single.truncate = toy_instance_single_rec (co3 K toy_instance_single.make)"
lemma asymmetric_conv_closed: "asymmetric x \<Longrightarrow> asymmetric (x\<^sup>T)"
lemma adj_0[simp]: \<open>0* = 0\<close>
lemma "(3::nat) * x + t < 0 \<and> (2 * x + y \<noteq> 17)"
lemma LIMSEQ_neg_powr: assumes s: "s < 0" shows "(%x. (real x) powr s) \<longlonglongrightarrow> 0"
lemma \<Delta>\<Phi>_pass2: "hs \<noteq> Leaf \<Longrightarrow> \<Phi> (pass\<^sub>2 hs) - \<Phi> hs \<le> log 2 (size hs)"
lemma le_Integer_bottom_iff [simp]: fixes x y :: Integer shows "le\<cdot>x\<cdot>y = \<bottom> \<longleftrightarrow> x = \<bottom> \<or> y = \<bottom>"
lemma Graph4: "\<lbrakk>T \<in> Reach E; Roots\<subseteq>Blacks M; I\<le>length E; T<length M; R<length E; \<forall>i<I. \<not>BtoW(E!i,M); R<I; M!fst(E!R)=Black; M!T\<noteq>Black\<rbrakk> \<Longrightarrow> (\<exists>r. I\<le>r \<and> r<length E \<and> BtoW(E[R:=(fst(E!R),T)]!r,M))"
lemma trans_ile_iadd1: "i \<le> (j::enat) \<Longrightarrow> i \<le> j + m"

lemma wf_imethdsD: "\<lbrakk>im \<in> imethds G I sig;wf_prog G; is_iface G I\<rbrakk> \<Longrightarrow> \<not>is_static im \<and> accmodi im = Public"

lemma llrg_linear_sys: "llrg \<R> \<Longrightarrow> linear_sys \<R>"

lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"

lemma final_lemma6: "(\<Inter>v \<in> V. \<Inter>w \<in> V. (reachable v <==> {s. (root,v) \<in> REACHABLE}) \<inter> nmsg_eq 0 (v,w)) \<subseteq> final"

lemma height_0_rel: "height t = 0 \<Longrightarrow> \<exists>r. t = Relation r"

lemma "(Abs_perm id :: nat perm) \<bullet> Suc 0 = Suc 0"

lemma connected_card_eq_iff_nontrivial: fixes S :: "'a::metric_space set" shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"

lemma rel_envT_eq [relator_eq]: "rel_envT (=) (=) = (=)"

lemma sublist_equal_lengths: fixes l1 l2 assumes "subseq l1 l2" "(length l1 = length l2)" shows "(l1 = l2)"

lemma ht_hash_distinct: "ht_hash l \<Longrightarrow> \<forall>i j . i\<noteq>j \<and> i < length l \<and> j < length l \<longrightarrow> set (l!i) \<inter> set (l!j) = {}"

lemma sup_pres_multl: "sup_pres (\<lambda>(z::'a::proto_quantale). x \<cdot> z)"

lemma update_arg_wf_tuples' [elim]: "\<And>n hops nhip pre. Suc 0 \<le> n \<Longrightarrow> update_arg_wf (n, kno, val, hops, nhip, pre)"

lemma div_op_SOME: "is_div_op (\<lambda>a b. (SOME k. k * b = a))"

lemma tau_at_0: "\<tau> i 0 \<down>= i"

lemma Cond': "\<lbrakk>P \<subseteq> {s. (b \<subseteq> P1) \<and> (- b \<subseteq> P2)};\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P1 c1 Q,A; \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P2 c2 Q,A\<rbrakk> \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P (Cond b c1 c2) Q,A"

lemma (in is_functor) is_functor_ntcf_id_components: shows "ntcf_id \<FF>\<lparr>NTMap\<rparr> = \<BB>\<lparr>CId\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr>" and "ntcf_id \<FF>\<lparr>NTDom\<rparr> = \<FF>" and "ntcf_id \<FF>\<lparr>NTCod\<rparr> = \<FF>" and "ntcf_id \<FF>\<lparr>NTDGDom\<rparr> = \<AA>" and "ntcf_id \<FF>\<lparr>NTDGCod\<rparr> = \<BB>"

lemma "((((sep_true imp p0) imp ((p0 ** p0) \<longrightarrow>* ((sep_true imp p0) ** (p0 ** p0)))) imp (p1 \<longrightarrow>* (((sep_true imp p0) imp ((p0 ** p0) \<longrightarrow>* (((sep_true imp p0) ** p0) ** p0))) ** p1)))) (h::'a::heap_sep_algebra)"

lemma vconverse_inject[simp]: assumes "vbrelation r" and "vbrelation s" shows "r\<inverse>\<^sub>\<circ> = s\<inverse>\<^sub>\<circ> \<longleftrightarrow> r = s"

lemma gbinomial_int_mono: assumes "0 \<le> x" and "x \<le> (y::int)" shows "x gchoose k \<le> y gchoose k"

lemma solution_upd1: "c \<noteq> 0 \<Longrightarrow> solution (A(p:=(\<lambda>j. A p j / c))) n x = solution A n x"

lemma correctly_synchronized_compP [simp]: "correctly_synchronized (compP f P) = correctly_synchronized P"

lemma max_mix_is_density: assumes "0 < n" shows "density_operator (max_mix_density n)"

lemma interaction_bounded_by_case_prod [interaction_bound]: "(\<And>a b. x = (a, b) \<Longrightarrow> interaction_bounded_by consider (f a b) (n a b)) \<Longrightarrow> interaction_bounded_by consider (case_prod f x) (case_prod n x)"

lemma dbm_int_guard_abstr: assumes "valid_abstraction A X k" "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "dbm_int (abstr g (\<lambda>i j. \<infinity>) v) n"

lemma eventually_going_to_at_bot_linorder: fixes f :: "'a \<Rightarrow> 'b :: linorder" shows "eventually P (f going_to at_bot within A) \<longleftrightarrow> (\<exists>C. \<forall>x\<in>A. f x \<le> C \<longrightarrow> P x)"

lemma iso_fSup: fixes F :: "('a::order \<Rightarrow> 'b::complete_lattice) set" shows "(\<forall>f \<in> F. mono f) \<Longrightarrow> mono (\<Squnion>F)"

lemma isTrue_neg_neg: assumes "\<phi> \<in> fmla" "Fvars \<phi> = {}" and "isTrue (neg (neg \<phi>))" shows "isTrue \<phi>"

lemma thinish: "ThinishChamberComplex X"

lemma E_thm01: "Lift \<in> (stopped \<inter> atFloor n) LeadsTo (opened \<inter> atFloor n)"

lemma trimmed_pos_real_iff [simp]: "trimmed_pos (x :: real) \<longleftrightarrow> x > 0"

lemma one_dim_iso_adjoint[simp]: \<open>one_dim_iso (A*) = (one_dim_iso A)*\<close>

lemma incls_coherent: assumes "par f f'" and "incl f" and "incl f'" shows "f = f'"

lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x" for m :: "'a::linordered_field"

lemma run_put_writerT [simp]: "run_writer (put s m) = put s (run_writer m)"

lemma \<open>(1705 :: nat) << 3 = 13640\<close>

lemma fresh_asTerm_qFresh: assumes "qGood qX" shows "fresh xs x (asTerm qX) = qFresh xs x qX"

lemma nxt_alw: "nxt (alw P) s \<Longrightarrow> alw (nxt P) s"

lemma split_vars_ground_vars: assumes "ground_mctxt C" and "num_holes C = length xs" shows "split_vars (fill_holes C (map Var xs)) = (C, xs)"

lemma diff_add_assoc: "b \<le> c \<Longrightarrow> c - (c - b) = c - c + (b::nat)"

lemma shiftD: "x \<^bold>\<in> shift f delta \<Longrightarrow> \<exists>u. u \<^bold>\<in> f \<and> x = \<langle>delta @+ hfst u, hsnd u\<rangle>"

lemma Ch_range_disjoint: assumes "h \<noteq> h'" shows "Ch h A \<inter> Ch h' A = {}"

lemma variance_prod_pmf_slice: fixes f :: "'a \<Rightarrow> real" assumes "i \<in> I" "finite I" assumes "integrable (measure_pmf (M i)) (\<lambda>\<omega>. f \<omega>^2)" shows "prob_space.variance (Pi_pmf I d M) (\<lambda>\<omega>. f (\<omega> i)) = prob_space.variance (M i) f"

lemma length_concat_ge: assumes "as \<in> lists (- {[]})" shows "length (concat as) \<ge> length as"

lemma parabolic_isometry_inv: assumes "parabolic_isometry f" shows "parabolic_isometry (inv f)"

lemma scount_eq_0D: "scount P \<omega> = 0 \<Longrightarrow> alw (not P) \<omega>"

lemma older_seniors_finite: "finite (older_seniors x n)"

lemma bezout_coefficients: assumes "bezout_coefficients a b = (x, y)" shows "x * a + y * b = gcd a b"

lemma obtain_t_subset_points: obtains T where "T \<subseteq> \<V>" "card T = \<t>" "finite T"

lemma RM_subset : "RM M2 M1 \<Omega> V m i \<subseteq> C M2 M1 \<Omega> V m i"

lemma KRP_fresh_iff [simp]: "a \<sharp> KRP v x x' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> x \<and> a \<sharp> x'"

theorem Lp_Lq_duality: fixes p q::ennreal assumes "f \<in> space\<^sub>N (\<LL> p M)" "1/p + 1/q = 1" "p = \<infinity> \<Longrightarrow> sigma_finite_measure M" shows "bdd_above ((\<lambda>g. (\<integral>x. f x * g x \<partial>M))`{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1})" "Norm (\<LL> p M) f = (SUP g\<in>{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1}. (\<integral>x. f x * g x \<partial>M))"

lemma sc_threshold_2_3_tfff: "boolfunc_from_sc 4 sc_threshold_2_3 (a(0:=True,1:=False,2:=False,3:=False)) = False"

lemma rebase_id[simp]: "@(@(x::'b mod_ring) :: 'a mod_ring) = x"

lemma all_leq_Max: assumes "x \<le>\<^sub>v y" and "x \<noteq> []" shows "\<forall>xi \<in> set x. xi \<le> Max (set y)"

lemma simpneq_nb[simp]: assumes "SORT_CONSTRAINT('a::field_char_0)" shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"

lemma (in Category) MapsToId: assumes "X \<in> obj" shows "id X \<approx>> id X"

lemma [autoref_rules]: "(bs_insert,insert)\<in>nat_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"

lemma "a \<in> set ls \<Longrightarrow> b \<in> set (List_replace1 a b ls)"

lemma fold_fun_chain: "g x = (g ^^ 1) x" "(g ^^ m) ((g ^^ n) x) = (g ^^ (m+n)) x"

lemma len_sub_int:"len v ts c \<le> ext v"

lemma (in ceuclidean_space) SOME_CBasis: "(SOME i. i \<in> CBasis) \<in> CBasis"

lemma processed_alt: "processed (a, b) xs ps \<longleftrightarrow> ((a \<in> set xs) \<and> (b \<in> set xs) \<and> (a, b) \<notin>\<^sub>p set ps)"

lemma staticSecret_parts_agent: "\<lbrakk>m \<in> parts (knows C evs); m \<in> staticSecret A\<rbrakk> \<Longrightarrow> A=C \<or> (\<exists>D E X. Says D E X \<in> set evs \<and> m \<in> parts{X}) \<or> (\<exists>Y. Notes C Y \<in> set evs \<and> m \<in> parts{Y})"

lemma valid_and_I : "\<tau> \<Turnstile> \<upsilon> (x) \<Longrightarrow> \<tau> \<Turnstile> \<upsilon> (y) \<Longrightarrow> \<tau> \<Turnstile> \<upsilon> (x and y)"

lemma precedes_append_left_iff: assumes "x \<notin> set ys" shows "x \<preceq> y in (ys @ xs) \<longleftrightarrow> x \<preceq> y in xs" (is "?lhs = ?rhs")

lemma ctxt_comp_n_funas [simp]: "(f, v) \<in> funas_ctxt (C^n) \<Longrightarrow> (f, v) \<in> funas_ctxt C"

lemma bounded_antilinear_from_conjugate_space[simp]: \<open>bounded_antilinear from_conjugate_space\<close>

lemma nAct_active[simp]: fixes n::nat and n'::nat and t::"(cnf llist)" and c::'id assumes "enat i < llength t" and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "\<langle>c #\<^bsub>Suc i\<^esub> t\<rangle> = eSuc (\<langle>c #\<^bsub>i\<^esub> t\<rangle>)"

lemma (in t2_space) LIMSEQ_Uniq: "\<exists>\<^sub>\<le>\<^sub>1l. X \<longlonglongrightarrow> l"

lemma glist_delete_id_impl: "(glist_delete (=), \<lambda>x s. s-{x}) \<in> Id\<rightarrow>br set distinct \<rightarrow> br set distinct"

lemma closest_pair_rec_dist: assumes "1 < length xs" "distinct xs" "sorted_fst xs" "(ys, c\<^sub>0, c\<^sub>1) = closest_pair_rec xs" shows "sparse (dist c\<^sub>0 c\<^sub>1) (set xs)"

lemma position_Left_only_subst: assumes "list_all (\<lambda>x. x = Left) p" and "position_of (subst \<rho> w) (p @ [d])" and "num_args (subst \<rho> w) = num_args w" shows "position_of w (p @ [d])"

lemma "eq\<cdot>Nothing\<cdot>x = isNothing\<cdot>x"

lemma mult_two_impl1[elim]: assumes "a * 2 = 2 * b" shows "(a::enat) = b"

lemma float_mult_r0: "x * float (0, e) = float (0, 0)"

lemma mod_subset_add: "\<lbrakk>(c1,c2) \<in> mod_subset p X; (d1,d2) \<in> mod_subset p X\<rbrakk> \<Longrightarrow> (c1+d1, c2+d2) \<in> mod_subset p X"

lemma std_apart_apart': assumes "x \<in> vars\<^sub>l (l \<cdot>\<^sub>l (\<lambda>x. Var (y@x)))" shows "\<exists>x'. x = y@x'"

lemma Conjs_pull_out: "Conjs Q (xys @ (x, y) # xys') \<triangleq> Conjs (Conj Q (x \<approx> y)) (xys @ xys')"

lemma mcont_fun_lub_Sup: "\<lbrakk> Complete_Partial_Order.chain (fun_ord (\<le>)) M; \<forall>f\<in>M. mcont lub ord Sup (\<le>) f \<rbrakk> \<Longrightarrow> mcont lub (\<sqsubseteq>) Sup (\<le>) (fun_lub Sup M)"

lemma vars\<^sub>s\<^sub>s\<^sub>t_Nil[simp]: "vars\<^sub>s\<^sub>s\<^sub>t [] = {}"

lemma "\<exists>x. \<forall>y::'a::linordered_field. y < 2 \<longrightarrow> 2 * (y - x) \<le> 0"

lemma maximal_Seq_term: fixes r::"'s prog" and s::"'s prog" assumes mr: "maximal (wp r)" and ws: "well_def s" and ts: "(\<lambda>s. 1) \<tturnstile> wp s (\<lambda>s. 1)" shows "(\<lambda>s. 1) \<tturnstile> wp (r ;; s) (\<lambda>s. 1)"

lemma image_mset_subset_mono: "M \<subset># N \<Longrightarrow> image_mset f M \<subset># image_mset f N"

lemma graphE [elim?]: assumes "(x, y) \<in> graph F f" obtains "x \<in> F" and "y = f x"

lemma SeqCTerm_imp_wf_dbtm: assumes "SeqCTerm vf s k x" shows "\<exists>t::dbtm. wf_dbtm t \<and> x = \<lbrakk>quot_dbtm t\<rbrakk>e"

lemma mod_mat_one: "mat_mod (1\<^sub>m n) = (1\<^sub>m n)"

lemma test_suite_to_input_sequences_pass : fixes T :: "('b::linorder \<times> 'c::linorder) prefix_tree" assumes "finite_tree T" and "(L M = L M') \<longleftrightarrow> (L M \<inter> set T = L M' \<inter> set T)" shows "(L M = L M') \<longleftrightarrow> ({io \<in> L M . (\<exists> xs \<in> list.set (test_suite_to_input_sequences T) . \<exists> xs' \<in> list.set (prefixes xs) . input_portion io = xs')} = {io \<in> L M' . (\<exists> xs \<in> list.set (test_suite_to_input_sequences T) . \<exists> xs' \<in> list.set (prefixes xs) . input_portion io = xs')})"

lemma preal_add_le_cancel_left [simp]: "(t + (r::preal) \<le> t + s) = (r \<le> s)"

lemma word_leI: "(\<And>n. \<lbrakk>n < size (u::'a::len word); bit u n \<rbrakk> \<Longrightarrow> bit (v::'a::len word) n) \<Longrightarrow> u <= v"

lemma casts_casts_eq_result: fixes s :: state assumes casts:"P \<turnstile> T casts v to v'" and casts':"P \<turnstile> T casts v to w'" and type:"is_type P T" and wte:"P,E \<turnstile> e :: T'" and leq:"P \<turnstile> T' \<le> T" and eval:"P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h,l)\<rangle>" and sconf:"P,E \<turnstile> s \<surd>" and wf:"wf_C_prog P" shows "v' = w'"

lemma listrel_reflcl_if_listrel1: "(xs,ys) \<in> listrel1 r \<Longrightarrow> (xs,ys) \<in> listrel(r\<^sup>*)"

lemma [code]: "toy_instance_single.truncate = toy_instance_single_rec (co3 K toy_instance_single.make)"

lemma asymmetric_conv_closed: "asymmetric x \<Longrightarrow> asymmetric (x\<^sup>T)"

lemma adj_0[simp]: \<open>0* = 0\<close>

lemma "(3::nat) * x + t < 0 \<and> (2 * x + y \<noteq> 17)"

lemma LIMSEQ_neg_powr: assumes s: "s < 0" shows "(%x. (real x) powr s) \<longlonglongrightarrow> 0"

lemma \<Delta>\<Phi>_pass2: "hs \<noteq> Leaf \<Longrightarrow> \<Phi> (pass\<^sub>2 hs) - \<Phi> hs \<le> log 2 (size hs)"

lemma le_Integer_bottom_iff [simp]: fixes x y :: Integer shows "le\<cdot>x\<cdot>y = \<bottom> \<longleftrightarrow> x = \<bottom> \<or> y = \<bottom>"

lemma Graph4: "\<lbrakk>T \<in> Reach E; Roots\<subseteq>Blacks M; I\<le>length E; T<length M; R<length E; \<forall>i<I. \<not>BtoW(E!i,M); R<I; M!fst(E!R)=Black; M!T\<noteq>Black\<rbrakk> \<Longrightarrow> (\<exists>r. I\<le>r \<and> r<length E \<and> BtoW(E[R:=(fst(E!R),T)]!r,M))"

lemma trans_ile_iadd1: "i \<le> (j::enat) \<Longrightarrow> i \<le> j + m"