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

Statement: stringlengths 7 24.3k
lemma updates_append_drop [simp]: "size xs = size ys \<Longrightarrow> updates (xs @ zs) ys al = updates xs ys al"
lemma h_method_via_pair_framework_2_completeness_and_finiteness : assumes "observable M" and "observable I" and "minimal M" and "size I \<le> m" and "m \<ge> size_r M" and "inputs I = inputs M" and "outputs I = outputs M" shows "(L M = L I) \<longleftrightarrow> (L M \<inter> set (h_method_via_pair_framework_2 M m c) = L I \<inter> set (h_method_via_pair_framework_2 M m c))" and "finite_tree (h_method_via_pair_framework_2 M m c)"
lemma link_tree_invar: "\<lbrakk>tree_invar t1; tree_invar t2; rank t1 = rank t2\<rbrakk> \<Longrightarrow> tree_invar (link t1 t2)"
lemma wand_ent_self: "P \<Longrightarrow>\<^sub>A Q -* (Q * P)"
lemma Spy_see_shrK [simp]: "evs \<in> kerbV \<Longrightarrow> (Key (shrK A) \<in> parts (spies evs)) = (A \<in> bad)"
lemma g_sng_impl: "set_sng \<alpha> invar g_sng"
lemma new_tv_Cons[simp]: "new_tv n (x#l) = (new_tv n x \<and> new_tv n l)"
lemma TensorDiag_in_Hom: assumes "Diag t" and "Diag u" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<in> Hom (Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u) (Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u)"
lemma Lxy_rev: "rev (Lxy qs S) = Lxy (rev qs) S"
lemma all_in_list: "all_in_list (policy2list Policy) (Nets_List Policy)"
lemma finTrace_imp_MtransT: "finTrace tr \<Longrightarrow> iconfig tr \<rightarrow>*t fstate tr"
lemma goGS_rel_sv[relator_props,intro!,simp]: "single_valued goGS_rel"
lemma modal_neq: fixes A :: "('a,'b) form" and ps :: "('a,'b) form list" shows "A \<noteq> Modal M [A]" and "ps \<noteq> [Modal M ps]"
lemma synth_parts_trans: assumes "A \<subseteq> synth (parts B)" and "B \<subseteq> synth (parts C)" shows "A \<subseteq> synth (parts C)"
lemma dim_row_take_cols[simp]: shows "dim_row (take_cols A ls) = dim_row A"
lemma (in encoding) enc_reflects_pred_iff_source_target_rel_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = (\<exists>Rel. (\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_preserves_pred Rel Pred)"
lemma fds_divisor_count: "fds divisor_count = fds_zeta ^ 2"
lemma fic_ImpR_elim: assumes a: "fic (ImpR (x).<a>.M b) c" shows "b=c \<and> b\<sharp>[a].M"
lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x"
lemma circline_intersection_at_most_2_points: assumes "H1 \<noteq> H2" shows "finite (circline_intersection H1 H2) \<and> card (circline_intersection H1 H2) \<le> 2"
lemma (in jozsa) jozsa_transform_times_\<psi>\<^sub>1_is_\<psi>\<^sub>2: shows "U\<^sub>f * (\<psi>\<^sub>1 n) = \<psi>\<^sub>2"
lemma correctness_applicative: assumes distinct: "\<And>n. pure (assert distinct) \<diamondop> symbols n = symbols n" shows "State_Monad.return dlabels \<diamondop> label_tree t = symbols (leaves t)"
lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: division_ring fps)"
lemma closure_of_eq_empty_gen: "X closure_of S = {} \<longleftrightarrow> disjnt (topspace X) S"
lemma emeasure_lborel_Ico[simp]: assumes [simp]: "l \<le> u" shows "emeasure lborel {l ..< u} = ennreal (u - l)"
lemma set_comprehension_list_comprehension: "set [f i . i <- [x..<a]] = {f i \|i. i \<in> {x..<a}}"
lemma iso_bij_betwn_block_sets_inv: "bij_betw ((`) (inv_into \<V> \<pi>)) (set_mset \<B>') (set_mset \<B>)"
lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
lemma proj_poly_syz_lift_poly_syz: assumes "i < n" shows "proj_poly_syz n (lift_poly_syz n p i) = p"
lemma lfinite_lconst[simp]: "\<not> lfinite (lconst a)"
lemma cond_spmf_spmf_of_set: "cond_spmf (spmf_of_set A) B = spmf_of_set (A \<inter> B)" if "finite A"
lemma bc_mt_corresp_Dup: " bc_mt_corresp [Dup] dupST (T # ST, LT) cG rT mxr (Suc 0)"
lemma vintersection_vunion_left: "(A \<union>\<^sub>\<circ> B) \<inter>\<^sub>\<circ> C = (A \<inter>\<^sub>\<circ> C) \<union>\<^sub>\<circ> (B \<inter>\<^sub>\<circ> C)"
lemma heap_copy_loc_progress: assumes hconf: "hconf h" and alconfa: "P,h \<turnstile> a@al : T" and alconfa': "P,h \<turnstile> a'@al : T" shows "\<exists>v h'. heap_copy_loc a a' al h ([ReadMem a al v, WriteMem a' al v]) h' \<and> P,h \<turnstile> v :\<le> T \<and> hconf h'"
lemma isometryD[simp]: "isometry U \<Longrightarrow> U* o\<^sub>C\<^sub>L U = id_cblinfun"
lemma G_ctor_o_fold: "ctor_fold_G s o ctor_G = s o map_pre_G id (ctor_fold_G s)"
lemma LUNIT_simps [simp]: assumes "B.ide f" shows "B.arr (LUNIT f)" and "src\<^sub>B (LUNIT f) = src\<^sub>B f" and "trg\<^sub>B (LUNIT f) = trg\<^sub>B f" and "B.dom (LUNIT f) = TRG f \<star>\<^sub>B f" and "B.cod (LUNIT f) = f"
lemma enforce_spmf_top [simp]: "enforce_spmf \<top> = id"
lemma wf_list_graph_iff_wf_graph: "wf_graph (list_graph_to_graph G) \<longleftrightarrow> wf_list_graph_axioms G"
lemma homeomorphic_maps_nsphere_euclidean_sphere: fixes B :: "'n::euclidean_space set" assumes B: "independent B" and orth: "pairwise orthogonal B" and n: "card B = n" and "n \<noteq> 0" and 1: "\<And>u. u \<in> B \<Longrightarrow> norm u = 1" obtains f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n::euclidean_space" and g where "homeomorphic_maps (nsphere(n - 1)) (top_of_set (sphere 0 1 \<inter> span B)) f g"
lemma (in encoding) trans_source_target_relation_impl_fully_abstract: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" and SRel :: "('procS \<times> 'procS) set" and TRel :: "('procT \<times> 'procT) set" assumes enc: "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel" and srel: "SRel = {(S1, S2). (SourceTerm S1, SourceTerm S2) \<in> Rel}" and trel: "TRel = {(T1, T2). (TargetTerm T1, TargetTerm T2) \<in> Rel}" and trans: "trans Rel" shows "fully_abstract SRel TRel"
lemma prime_elem_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) prime_elem_m prime_elem"
lemma \<open>drop_bit 3 (1 :: int) = 0\<close>
lemma shadow_root_ptr_casts_commute [simp]: "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some shadow_root_ptr \<longleftrightarrow> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = ptr"
lemma harry_meas_False_end [simp]: "harry_meas (xs @ [False]) = harry_meas xs"
lemma tm_weak_copy_correct_pre: assumes "0 < x" shows "\<lbrace>inv_begin1 x\<rbrace> tm_weak_copy \<lbrace>inv_end0 x\<rbrace>"
lemma While_sound_aux [rule_format]: "\<lbrakk> pre \<inter> - b \<subseteq> post; \<Turnstile> P sat [pre \<inter> b, rely, guar, pre]; \<forall>s. (s, s) \<in> guar; stable pre rely; stable post rely; x \<in> cptn_mod \<rbrakk> \<Longrightarrow> \<forall>s xs. x=(Some(While b P),s)#xs \<longrightarrow> x\<in>assum(pre, rely) \<longrightarrow> x \<in> comm (guar, post)"
lemma hnfSummandsRemove: fixes P :: pi and Q :: pi assumes "P \<in> summands Q" and "uhnf Q" shows "(summands Q) - {P' \| P'. P' \<in> summands Q \<and> P' \<equiv>\<^sub>e P} = (summands Q) - {P}"
lemma bot_pres_multr: "bot_pres (\<lambda>(z::'a::proto_near_quantale). z \<cdot> y)"
lemma gval_fold_cons: "gval (fold gAnd (g # gs) (Bc True)) s = gval g s \<and>? gval (fold gAnd gs (Bc True)) s"
lemma \<nu>_improving_iff: "\<nu>_improving v d \<longleftrightarrow> d \<in> D\<^sub>R \<and> (\<forall>d' \<in> D\<^sub>R. \<forall>s. L d' v s \<le> L d v s)"
lemma uGraph_from_list_invar_subset: "uGraph_from_list_invar L \<Longrightarrow> set L'\<subseteq> set L \<Longrightarrow> distinct L' \<Longrightarrow> uGraph_from_list_invar L'"
lemma prm_compose_push: shows "\<pi> \<diamondop> [a \<leftrightarrow> b] = [\<pi> $ a \<leftrightarrow> \<pi> $ b] \<diamondop> \<pi>"
lemma gmctxtex_onp_rel_mono: "\<L> \<subseteq> \<R> \<Longrightarrow> gmctxtex_onp P \<L> \<subseteq> gmctxtex_onp P \<R>"
lemma density_unique_real: fixes f f'::"_ \<Rightarrow> real" assumes M[measurable]: "integrable M f" "integrable M f'" assumes density_eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>x \<in> A. f x \<partial>M) = (\<integral>x \<in> A. f' x \<partial>M)" shows "AE x in M. f x = f' x"
lemma exists_bezout_extended: assumes S: "finite S" and ne: "S \<noteq> {}" shows "\<exists>f d. (\<Sum>a\<in>S. f a * a) = d \<and> (\<forall>a\<in>S. d dvd a) \<and> (\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"
lemma unfold_ind: fixes P :: "('s \<rightarrow> 'a LList) \<Rightarrow> bool" assumes "adm P" and "P \<bottom>" and "\<And>u. P u \<Longrightarrow> P (unfoldF\<cdot>h\<cdot>u)" shows "P (unfold\<cdot>h)"
lemma seq_nhop_quality_increases': shows "opaodv i \<Turnstile> (otherwith ((=)) {i} (orecvmsg (\<lambda>\<sigma> m. msg_fresh \<sigma> m \<and> msg_zhops m)), other quality_increases {i} \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<sigma>, _). \<forall>dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> (rt (\<sigma> i)) \<sqsubset>\<^bsub>dip\<^esub> (rt (\<sigma> nhip)))" (is "_ \<Turnstile> (?S i, _ \<rightarrow>) _")
lemma pigeonhole: assumes "finite T" "S \<subseteq> \<Union>T" "card T < card S" shows "\<exists>x \<in> S. \<exists>y \<in> S. \<exists>X \<in> T. x \<noteq> y \<and> x \<in> X \<and> y \<in> X"
lemma prime_odd_int: "prime p \<Longrightarrow> p > (2::int) \<Longrightarrow> odd p"
lemma Pi_pmf_return_pmf' [simp]: assumes "finite A" shows "Pi_pmf A dflt (\<lambda>_. return_pmf dflt) = return_pmf (\<lambda>_. dflt)"
lemma Phi_conv_map: assumes "is_direct_product w x" and "is_direct_product y z" shows "is_map ((\<Phi> w x y z)\<^sup>\<smile>)"
lemma limit_mat_denote_while_n: assumes wc: "well_com (While M S)" and dr: "\<rho> \<in> carrier_mat d d" and pdor: "partial_density_operator \<rho>" shows "limit_mat (matrix_sum d (\<lambda>k. denote_while_n (M 0) (M 1) (denote S) k \<rho>)) (denote (While M S) \<rho>) d"
lemma normalize1_dverts_app_bfr_cntr_rnks': assumes "v \<in> dverts (normalize1 t)" and "v \<notin> dverts t" shows "\<exists>U\<in>dverts t. \<exists>V\<in>dverts t. U @ V = v \<and> before U V \<and> rank (rev V) \<le> rank (rev U) \<and> (\<forall>xs \<in> dverts t. (\<exists>y\<in>set xs. \<not> (\<exists>x'\<in>set V. x' \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y) \<and> (\<exists>x\<in>set U. x \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y) \<and> xs \<noteq> U) \<longrightarrow> rank (rev V) \<le> rank (rev xs))"
lemma is_runs2sigs_upd_resp_none [simp]: "\<lbrakk> Rb \<notin> dom runz \<rbrakk> \<Longrightarrow> is_runs2sigs (runz(Rb \<mapsto> (Resp, [A, B], []))) = is_runs2sigs runz"
lemma frameImpClosed: fixes F :: "'b frame" and \<Phi> :: 'c and p :: "name prm" assumes "F \<turnstile>\<^sub>F \<Phi>" shows "(p \<bullet> F) \<turnstile>\<^sub>F (p \<bullet> \<Phi>)"
lemma map: "\<not> free_in x p \<Longrightarrow> semantics e f g p \<longleftrightarrow> semantics (e(x := v)) f g p"
theorem "\<not>(turing_computable_partial (chi_fun K1))"
lemma eq_\<I>_converterD_WT: assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<surd> \<longleftrightarrow> \<I>,\<I>' \<turnstile>\<^sub>C conv2 \<surd>"
lemma assumes "Even n" shows "P1 n" and "P2 n"
lemma pcompose_diff: "pcompose (p - q) r = pcompose p r - pcompose q r" for p q r :: "'a::comm_ring poly"
lemma add_prop: assumes "PROP (T)" shows "A ==> PROP (T)"
lemma pullback_arr_cod: assumes "arr f" shows "inverse_arrows \<p>\<^sub>1[f, cod f] \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>" and "inverse_arrows \<p>\<^sub>0[cod f, f] \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>"
lemma "\<not> evasive 4 proj_2_n3"
lemma fls_base_factor_of_int [simp]: "fls_base_factor (of_int i :: 'a::ring_1 fls) = (of_int i :: 'a fls)"
lemma cf_comp_ArrMap_vrange: assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> ((\<GG> \<circ>\<^sub>C\<^sub>F \<FF>)\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
lemma after_fun1: "\|~ f<x>` = f<x`>"
lemma semiring_transfer[transfer_rule]: assumes[transfer_rule]: "bi_unique A" "right_total A" shows "((A ===> A ===> A) ===> (A ===> A ===> A) ===> (=)) (semiring_ow (Collect (Domainp A))) class.semiring"
lemma A4b: "cod(x\<cdot>y) \<cong> cod(x\<cdot>(cod y))"
lemma read_actions_not_write_actions: "\<lbrakk> a \<in> read_actions E; a \<in> write_actions E \<rbrakk> \<Longrightarrow> False"
lemma Exit_in_obs_slice_node:"(_Exit_) \<in> obs n' (PDG_BS S) \<Longrightarrow> (_Exit_) \<in> S"
theorem soundnessForUNSAT: fixes F0 :: Formula and decisionVars :: "Variable set" and state0 :: State and state :: State assumes "isInitialState state0 F0" and "(state0, state) \<in> transitionRelation F0 decisionVars" "getConflictFlag state = True" and "getC state = []" shows "\<not> satisfiable F0"
lemma [code]: "Gcd (Set t) = (Gcd\<^sub>f\<^sub>i\<^sub>n (Set t) :: nat)"
lemma "\<bar> ln 2 - 544531980202654583340825686620847 / 785593587443817081832229725798400 \<bar> < (inverse (2^51) :: real)"
lemma support_on_nonneg_sum_subset: "support_on X (\<lambda>x. \<Sum>i\<in>S. f i x) \<subseteq> (\<Union>i\<in>S. support_on X (f i))" for f::"_\<Rightarrow>_\<Rightarrow>_::ordered_comm_monoid_add"
lemma prm_unit_commutes: fixes a b :: 'a shows "[a \<leftrightarrow> b] = [b \<leftrightarrow> a]"
lemma subset_smaller_list: shows "\<forall> a x . set (a#x) \<subseteq> Z \<longrightarrow> set x \<subseteq> Z"
lemma rel_fun_conversep: includes lifting_syntax shows "(A^--1 ===> B^--1) = (A ===> B)^--1"
lemma "steps0 (1, [], [Oc, Oc]) tm_semi_id_eq0 2 = (1, [], [Oc, Oc])"
lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1" for x :: "'a::{real_normed_field,banach}"
lemma Inter_FullL[simp]: "Inter Full r = r"
lemma next_fun4: "\|~ \<circle>f<x,y,z,zz> = f<\<circle>x,\<circle>y,\<circle>z,\<circle>zz>"
lemma (in graph) init_loop_1_a[simp]: "\<Turnstile> Init {\| Q1_a \|} Loop"
lemma keys_uminus: "keys (- p) = keys p"
lemma height_0 : assumes "height t = 0" shows "t = empty"
lemma bbw_oa_dist: "(x AND y) OR z = (x OR z) AND (y OR z)" for x y z :: int
lemma (in srules) seval_wellformed: assumes "rs, \<Gamma> \<turnstile>\<^sub>s t \<down> u" "wellformed t" "wellformed_env \<Gamma>" shows "wellformed u"
lemma [code]: "ExcessTable g = List.map_filter (\<lambda>v. let e = ExcessAt g v in if 0 < e then Some (v, e) else None)"
lemma pp_prod_iff: "w \<in> X\<cdot>Y \<longleftrightarrow> (\<exists>u v. w = pp_fusion u v \<and> u \<in> X \<and> v \<in> Y \<and> pp_last u = pp_first v)"
lemma to_mregex_progress: "safe_regex m g r \<Longrightarrow> to_mregex r = (mr, \<phi>s) \<Longrightarrow> progress_regex \<sigma> P r j = (if \<phi>s = [] then j else (MIN \<phi>\<in>set \<phi>s. progress \<sigma> P \<phi> j))"

lemma updates_append_drop [simp]: "size xs = size ys \<Longrightarrow> updates (xs @ zs) ys al = updates xs ys al"

lemma h_method_via_pair_framework_2_completeness_and_finiteness : assumes "observable M" and "observable I" and "minimal M" and "size I \<le> m" and "m \<ge> size_r M" and "inputs I = inputs M" and "outputs I = outputs M" shows "(L M = L I) \<longleftrightarrow> (L M \<inter> set (h_method_via_pair_framework_2 M m c) = L I \<inter> set (h_method_via_pair_framework_2 M m c))" and "finite_tree (h_method_via_pair_framework_2 M m c)"

lemma link_tree_invar: "\<lbrakk>tree_invar t1; tree_invar t2; rank t1 = rank t2\<rbrakk> \<Longrightarrow> tree_invar (link t1 t2)"

lemma wand_ent_self: "P \<Longrightarrow>\<^sub>A Q -* (Q * P)"

lemma Spy_see_shrK [simp]: "evs \<in> kerbV \<Longrightarrow> (Key (shrK A) \<in> parts (spies evs)) = (A \<in> bad)"

lemma g_sng_impl: "set_sng \<alpha> invar g_sng"

lemma new_tv_Cons[simp]: "new_tv n (x#l) = (new_tv n x \<and> new_tv n l)"

lemma TensorDiag_in_Hom: assumes "Diag t" and "Diag u" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<in> Hom (Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u) (Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u)"

lemma Lxy_rev: "rev (Lxy qs S) = Lxy (rev qs) S"

lemma all_in_list: "all_in_list (policy2list Policy) (Nets_List Policy)"

lemma finTrace_imp_MtransT: "finTrace tr \<Longrightarrow> iconfig tr \<rightarrow>*t fstate tr"

lemma goGS_rel_sv[relator_props,intro!,simp]: "single_valued goGS_rel"

lemma modal_neq: fixes A :: "('a,'b) form" and ps :: "('a,'b) form list" shows "A \<noteq> Modal M [A]" and "ps \<noteq> [Modal M ps]"

lemma synth_parts_trans: assumes "A \<subseteq> synth (parts B)" and "B \<subseteq> synth (parts C)" shows "A \<subseteq> synth (parts C)"

lemma dim_row_take_cols[simp]: shows "dim_row (take_cols A ls) = dim_row A"

lemma (in encoding) enc_reflects_pred_iff_source_target_rel_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = (\<exists>Rel. (\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_preserves_pred Rel Pred)"

lemma fds_divisor_count: "fds divisor_count = fds_zeta ^ 2"

lemma fic_ImpR_elim: assumes a: "fic (ImpR (x).<a>.M b) c" shows "b=c \<and> b\<sharp>[a].M"

lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x"

lemma circline_intersection_at_most_2_points: assumes "H1 \<noteq> H2" shows "finite (circline_intersection H1 H2) \<and> card (circline_intersection H1 H2) \<le> 2"

lemma (in jozsa) jozsa_transform_times_\<psi>\<^sub>1_is_\<psi>\<^sub>2: shows "U\<^sub>f * (\<psi>\<^sub>1 n) = \<psi>\<^sub>2"

lemma correctness_applicative: assumes distinct: "\<And>n. pure (assert distinct) \<diamondop> symbols n = symbols n" shows "State_Monad.return dlabels \<diamondop> label_tree t = symbols (leaves t)"

lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: division_ring fps)"

lemma closure_of_eq_empty_gen: "X closure_of S = {} \<longleftrightarrow> disjnt (topspace X) S"

lemma emeasure_lborel_Ico[simp]: assumes [simp]: "l \<le> u" shows "emeasure lborel {l ..< u} = ennreal (u - l)"

lemma set_comprehension_list_comprehension: "set [f i . i <- [x..<a]] = {f i |i. i \<in> {x..<a}}"

lemma iso_bij_betwn_block_sets_inv: "bij_betw ((`) (inv_into \<V> \<pi>)) (set_mset \') (set_mset \)"

lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"

lemma proj_poly_syz_lift_poly_syz: assumes "i < n" shows "proj_poly_syz n (lift_poly_syz n p i) = p"

lemma lfinite_lconst[simp]: "\<not> lfinite (lconst a)"

lemma cond_spmf_spmf_of_set: "cond_spmf (spmf_of_set A) B = spmf_of_set (A \<inter> B)" if "finite A"

lemma bc_mt_corresp_Dup: " bc_mt_corresp [Dup] dupST (T # ST, LT) cG rT mxr (Suc 0)"

lemma vintersection_vunion_left: "(A \<union>\<^sub>\<circ> B) \<inter>\<^sub>\<circ> C = (A \<inter>\<^sub>\<circ> C) \<union>\<^sub>\<circ> (B \<inter>\<^sub>\<circ> C)"

lemma heap_copy_loc_progress: assumes hconf: "hconf h" and alconfa: "P,h \<turnstile> a@al : T" and alconfa': "P,h \<turnstile> a'@al : T" shows "\<exists>v h'. heap_copy_loc a a' al h ([ReadMem a al v, WriteMem a' al v]) h' \<and> P,h \<turnstile> v :\<le> T \<and> hconf h'"

lemma isometryD[simp]: "isometry U \<Longrightarrow> U* o\<^sub>C\<^sub>L U = id_cblinfun"

lemma G_ctor_o_fold: "ctor_fold_G s o ctor_G = s o map_pre_G id (ctor_fold_G s)"

lemma LUNIT_simps [simp]: assumes "B.ide f" shows "B.arr (LUNIT f)" and "src\<^sub>B (LUNIT f) = src\<^sub>B f" and "trg\<^sub>B (LUNIT f) = trg\<^sub>B f" and "B.dom (LUNIT f) = TRG f \<star>\<^sub>B f" and "B.cod (LUNIT f) = f"

lemma enforce_spmf_top [simp]: "enforce_spmf \<top> = id"

lemma wf_list_graph_iff_wf_graph: "wf_graph (list_graph_to_graph G) \<longleftrightarrow> wf_list_graph_axioms G"

lemma homeomorphic_maps_nsphere_euclidean_sphere: fixes B :: "'n::euclidean_space set" assumes B: "independent B" and orth: "pairwise orthogonal B" and n: "card B = n" and "n \<noteq> 0" and 1: "\<And>u. u \<in> B \<Longrightarrow> norm u = 1" obtains f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n::euclidean_space" and g where "homeomorphic_maps (nsphere(n - 1)) (top_of_set (sphere 0 1 \<inter> span B)) f g"

lemma (in encoding) trans_source_target_relation_impl_fully_abstract: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" and SRel :: "('procS \<times> 'procS) set" and TRel :: "('procT \<times> 'procT) set" assumes enc: "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel" and srel: "SRel = {(S1, S2). (SourceTerm S1, SourceTerm S2) \<in> Rel}" and trel: "TRel = {(T1, T2). (TargetTerm T1, TargetTerm T2) \<in> Rel}" and trans: "trans Rel" shows "fully_abstract SRel TRel"

lemma prime_elem_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) prime_elem_m prime_elem"

lemma \<open>drop_bit 3 (1 :: int) = 0\<close>

lemma shadow_root_ptr_casts_commute [simp]: "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some shadow_root_ptr \<longleftrightarrow> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = ptr"

lemma harry_meas_False_end [simp]: "harry_meas (xs @ [False]) = harry_meas xs"

lemma tm_weak_copy_correct_pre: assumes "0 < x" shows "\<lbrace>inv_begin1 x\<rbrace> tm_weak_copy \<lbrace>inv_end0 x\<rbrace>"

lemma While_sound_aux [rule_format]: "\<lbrakk> pre \<inter> - b \<subseteq> post; \<Turnstile> P sat [pre \<inter> b, rely, guar, pre]; \<forall>s. (s, s) \<in> guar; stable pre rely; stable post rely; x \<in> cptn_mod \<rbrakk> \<Longrightarrow> \<forall>s xs. x=(Some(While b P),s)#xs \<longrightarrow> x\<in>assum(pre, rely) \<longrightarrow> x \<in> comm (guar, post)"

lemma hnfSummandsRemove: fixes P :: pi and Q :: pi assumes "P \<in> summands Q" and "uhnf Q" shows "(summands Q) - {P' | P'. P' \<in> summands Q \<and> P' \<equiv>\<^sub>e P} = (summands Q) - {P}"

lemma bot_pres_multr: "bot_pres (\<lambda>(z::'a::proto_near_quantale). z \<cdot> y)"

lemma gval_fold_cons: "gval (fold gAnd (g # gs) (Bc True)) s = gval g s \<and>? gval (fold gAnd gs (Bc True)) s"

lemma \<nu>_improving_iff: "\<nu>_improving v d \<longleftrightarrow> d \<in> D\<^sub>R \<and> (\<forall>d' \<in> D\<^sub>R. \<forall>s. L d' v s \<le> L d v s)"

lemma uGraph_from_list_invar_subset: "uGraph_from_list_invar L \<Longrightarrow> set L'\<subseteq> set L \<Longrightarrow> distinct L' \<Longrightarrow> uGraph_from_list_invar L'"

lemma prm_compose_push: shows "\<pi> \<diamondop> [a \<leftrightarrow> b] = [\<pi> $ a \<leftrightarrow> \<pi> $ b] \<diamondop> \<pi>"

lemma gmctxtex_onp_rel_mono: "\<L> \<subseteq> \<R> \<Longrightarrow> gmctxtex_onp P \<L> \<subseteq> gmctxtex_onp P \<R>"

lemma density_unique_real: fixes f f'::"_ \<Rightarrow> real" assumes M[measurable]: "integrable M f" "integrable M f'" assumes density_eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>x \<in> A. f x \<partial>M) = (\<integral>x \<in> A. f' x \<partial>M)" shows "AE x in M. f x = f' x"

lemma exists_bezout_extended: assumes S: "finite S" and ne: "S \<noteq> {}" shows "\<exists>f d. (\<Sum>a\<in>S. f a * a) = d \<and> (\<forall>a\<in>S. d dvd a) \<and> (\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"

lemma unfold_ind: fixes P :: "('s \<rightarrow> 'a LList) \<Rightarrow> bool" assumes "adm P" and "P \<bottom>" and "\<And>u. P u \<Longrightarrow> P (unfoldF\<cdot>h\<cdot>u)" shows "P (unfold\<cdot>h)"

lemma seq_nhop_quality_increases': shows "opaodv i \<Turnstile> (otherwith ((=)) {i} (orecvmsg (\<lambda>\<sigma> m. msg_fresh \<sigma> m \<and> msg_zhops m)), other quality_increases {i} \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<sigma>, _). \<forall>dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> (rt (\<sigma> i)) \<sqsubset>\<^bsub>dip\<^esub> (rt (\<sigma> nhip)))" (is "_ \<Turnstile> (?S i, _ \<rightarrow>) _")

lemma pigeonhole: assumes "finite T" "S \<subseteq> \<Union>T" "card T < card S" shows "\<exists>x \<in> S. \<exists>y \<in> S. \<exists>X \<in> T. x \<noteq> y \<and> x \<in> X \<and> y \<in> X"

lemma prime_odd_int: "prime p \<Longrightarrow> p > (2::int) \<Longrightarrow> odd p"

lemma Pi_pmf_return_pmf' [simp]: assumes "finite A" shows "Pi_pmf A dflt (\<lambda>_. return_pmf dflt) = return_pmf (\<lambda>_. dflt)"

lemma Phi_conv_map: assumes "is_direct_product w x" and "is_direct_product y z" shows "is_map ((\<Phi> w x y z)\<^sup>\<smile>)"

lemma limit_mat_denote_while_n: assumes wc: "well_com (While M S)" and dr: "\<rho> \<in> carrier_mat d d" and pdor: "partial_density_operator \<rho>" shows "limit_mat (matrix_sum d (\<lambda>k. denote_while_n (M 0) (M 1) (denote S) k \<rho>)) (denote (While M S) \<rho>) d"

lemma normalize1_dverts_app_bfr_cntr_rnks': assumes "v \<in> dverts (normalize1 t)" and "v \<notin> dverts t" shows "\<exists>U\<in>dverts t. \<exists>V\<in>dverts t. U @ V = v \<and> before U V \<and> rank (rev V) \<le> rank (rev U) \<and> (\<forall>xs \<in> dverts t. (\<exists>y\<in>set xs. \<not> (\<exists>x'\<in>set V. x' \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y) \<and> (\<exists>x\<in>set U. x \<rightarrow>\<^sup>+\<^bsub>T\<^esub> y) \<and> xs \<noteq> U) \<longrightarrow> rank (rev V) \<le> rank (rev xs))"

lemma is_runs2sigs_upd_resp_none [simp]: "\<lbrakk> Rb \<notin> dom runz \<rbrakk> \<Longrightarrow> is_runs2sigs (runz(Rb \<mapsto> (Resp, [A, B], []))) = is_runs2sigs runz"

lemma frameImpClosed: fixes F :: "'b frame" and \<Phi> :: 'c and p :: "name prm" assumes "F \<turnstile>\<^sub>F \<Phi>" shows "(p \<bullet> F) \<turnstile>\<^sub>F (p \<bullet> \<Phi>)"

lemma map: "\<not> free_in x p \<Longrightarrow> semantics e f g p \<longleftrightarrow> semantics (e(x := v)) f g p"

theorem "\<not>(turing_computable_partial (chi_fun K1))"

lemma eq_\_converterD_WT: assumes "\,\' \<turnstile>\<^sub>C conv1 \<sim> conv2" shows "\,\' \<turnstile>\<^sub>C conv1 \<surd> \<longleftrightarrow> \,\' \<turnstile>\<^sub>C conv2 \<surd>"

lemma assumes "Even n" shows "P1 n" and "P2 n"

lemma pcompose_diff: "pcompose (p - q) r = pcompose p r - pcompose q r" for p q r :: "'a::comm_ring poly"

lemma add_prop: assumes "PROP (T)" shows "A ==> PROP (T)"

lemma pullback_arr_cod: assumes "arr f" shows "inverse_arrows \\<^sub>1[f, cod f] \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>" and "inverse_arrows \\<^sub>0[cod f, f] \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>"

lemma "\<not> evasive 4 proj_2_n3"

lemma fls_base_factor_of_int [simp]: "fls_base_factor (of_int i :: 'a::ring_1 fls) = (of_int i :: 'a fls)"

lemma cf_comp_ArrMap_vrange: assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> ((\<GG> \<circ>\<^sub>C\<^sub>F \<FF>)\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"

lemma after_fun1: "|~ f<x>` = f<x`>"

lemma semiring_transfer[transfer_rule]: assumes[transfer_rule]: "bi_unique A" "right_total A" shows "((A ===> A ===> A) ===> (A ===> A ===> A) ===> (=)) (semiring_ow (Collect (Domainp A))) class.semiring"

lemma A4b: "cod(x\<cdot>y) \<cong> cod(x\<cdot>(cod y))"

lemma read_actions_not_write_actions: "\<lbrakk> a \<in> read_actions E; a \<in> write_actions E \<rbrakk> \<Longrightarrow> False"

lemma Exit_in_obs_slice_node:"(_Exit_) \<in> obs n' (PDG_BS S) \<Longrightarrow> (_Exit_) \<in> S"

theorem soundnessForUNSAT: fixes F0 :: Formula and decisionVars :: "Variable set" and state0 :: State and state :: State assumes "isInitialState state0 F0" and "(state0, state) \<in> transitionRelation F0 decisionVars" "getConflictFlag state = True" and "getC state = []" shows "\<not> satisfiable F0"

lemma [code]: "Gcd (Set t) = (Gcd\<^sub>f\<^sub>i\<^sub>n (Set t) :: nat)"

lemma "\<bar> ln 2 - 544531980202654583340825686620847 / 785593587443817081832229725798400 \<bar> < (inverse (2^51) :: real)"

lemma support_on_nonneg_sum_subset: "support_on X (\<lambda>x. \<Sum>i\<in>S. f i x) \<subseteq> (\<Union>i\<in>S. support_on X (f i))" for f::"_\<Rightarrow>_\<Rightarrow>_::ordered_comm_monoid_add"

lemma prm_unit_commutes: fixes a b :: 'a shows "[a \<leftrightarrow> b] = [b \<leftrightarrow> a]"

lemma subset_smaller_list: shows "\<forall> a x . set (a#x) \<subseteq> Z \<longrightarrow> set x \<subseteq> Z"

lemma rel_fun_conversep: includes lifting_syntax shows "(A^--1 ===> B^--1) = (A ===> B)^--1"

lemma "steps0 (1, [], [Oc, Oc]) tm_semi_id_eq0 2 = (1, [], [Oc, Oc])"

lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1" for x :: "'a::{real_normed_field,banach}"

lemma Inter_FullL[simp]: "Inter Full r = r"

lemma next_fun4: "|~ \<circle>f<x,y,z,zz> = f<\<circle>x,\<circle>y,\<circle>z,\<circle>zz>"

lemma (in graph) init_loop_1_a[simp]: "\<Turnstile> Init {| Q1_a |} Loop"

lemma keys_uminus: "keys (- p) = keys p"

lemma height_0 : assumes "height t = 0" shows "t = empty"

lemma bbw_oa_dist: "(x AND y) OR z = (x OR z) AND (y OR z)" for x y z :: int

lemma (in srules) seval_wellformed: assumes "rs, \<Gamma> \<turnstile>\<^sub>s t \<down> u" "wellformed t" "wellformed_env \<Gamma>" shows "wellformed u"

lemma [code]: "ExcessTable g = List.map_filter (\<lambda>v. let e = ExcessAt g v in if 0 < e then Some (v, e) else None)"

lemma pp_prod_iff: "w \<in> X\<cdot>Y \<longleftrightarrow> (\<exists>u v. w = pp_fusion u v \<and> u \<in> X \<and> v \<in> Y \<and> pp_last u = pp_first v)"

lemma to_mregex_progress: "safe_regex m g r \<Longrightarrow> to_mregex r = (mr, \<phi>s) \<Longrightarrow> progress_regex \<sigma> P r j = (if \<phi>s = [] then j else (MIN \<phi>\<in>set \<phi>s. progress \<sigma> P \<phi> j))"