Statement:
stringlengths 7
24.3k
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lemma marked_deletions_sweep_loop_free[simp]:
notes fun_upd_apply[simp]
shows
"\<lbrakk> mut_m.marked_deletions m s; mut_m.reachable_snapshot_inv m s; no_grey_refs s; white r s \<rbrakk>
\<Longrightarrow> mut_m.marked_deletions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := None)\<rparr>))"
|
lemma thetaSAtm_Sbis:
assumes "compatAtm atm"
shows "thetaSAtm atm \<subseteq> Sbis"
|
lemma a_comm_var: "ad x \<cdot> ad y \<le> ad y \<cdot> ad x"
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lemma older_seniors_older:
"y \<in> older_seniors x n \<Longrightarrow> y < x"
|
lemma iterate_to_set_correct :
assumes ins_dj_OK: "set_ins_dj \<alpha> invar ins_dj"
assumes emp_OK: "set_empty \<alpha> invar emp"
assumes it: "set_iterator it S0"
shows "\<alpha> (iterate_to_set emp ins_dj it) = S0 \<and> invar (iterate_to_set emp ins_dj it)"
|
lemma to_fun_ring_hom:
assumes "a \<in> carrier R"
shows "(\<lambda>p. to_fun p a) \<in> ring_hom P R"
|
lemma element_ptr_kinds_simp [simp]:
"element_ptr_kinds (Heap (fmupd (cast element_ptr) element (the_heap h))) =
{|element_ptr|} |\<union>| element_ptr_kinds h"
|
lemma indis_sym[sym]: "s \<approx> s' \<Longrightarrow> s' \<approx> s"
|
lemma sum_sumj_eq2: "i<I ==> sum I s = c s i + sumj I i s"
|
lemma el_loc_ok_update: "\<lbrakk> \<B> e n; V < n \<rbrakk> \<Longrightarrow> el_loc_ok e (xs[V := v]) = el_loc_ok e xs"
and els_loc_ok_update: "\<lbrakk> \<B>s es n; V < n \<rbrakk> \<Longrightarrow> els_loc_ok es (xs[V := v]) = els_loc_ok es xs"
|
lemma length_assoc_list_of_array [simp]:
"length (assoc_list_of_array a) = array_length a"
|
lemma (in wf_digraph) adj_mk_symmetric_eq:
"symmetric G \<Longrightarrow> parcs (mk_symmetric G) = arcs_ends G"
|
lemma is_ground_lit_is_ground_on_var:
assumes ground_lit: "is_ground_lit (subst_lit L \<sigma>)" and v_in_L: "v \<in> vars_lit L"
shows "is_ground_atm (\<sigma> v)"
|
lemma m2f_by_from_m2f :
"(m2f_by g f xs) = g (m2f f xs)"
|
lemma iso_finfun_uminus [code_unfold]:
fixes A :: "'a pred\<^sub>f"
shows "- ($) A = ($) (- A)"
|
lemma squareE [elim]:
"\<lbrakk> (s,t) \<Turnstile> [A]_v; A (s,t) \<Longrightarrow> B (s,t); v t = v s \<Longrightarrow> B (s,t) \<rbrakk> \<Longrightarrow> B (s,t)"
|
lemma af\<^sub>F_semantics_rtl:
assumes
"\<forall>i. \<exists>j>i. af\<^sub>F \<phi> (F\<^sub>n \<phi>) (w [0 \<rightarrow> j]) \<sim> true\<^sub>n"
shows
"\<forall>i. \<exists>j. af (F\<^sub>n \<phi>) (w [i \<rightarrow> j]) \<sim>\<^sub>L true\<^sub>n"
|
lemma validTrans_step_srcOf_actOf_tgtOf:
"validTrans trn \<Longrightarrow> step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn)"
|
lemma proj2_incident_iff_Col:
assumes "p \<noteq> q" and "proj2_incident p l" and "proj2_incident q l"
shows "proj2_incident r l \<longleftrightarrow> proj2_Col p q r"
|
lemma [simp]:
"size_exp' (Handle e pes) = Suc (size_exp' e + size_list (size_prod size size_exp') pes)"
|
lemma coeff_finite_fourier_poly:
assumes "n < length ws"
defines "k \<equiv> length ws"
shows "coeff (finite_fourier_poly ws) n =
(1/k) * (\<Sum>m < k. ws ! m * unity_root k (-n*m))"
|
lemma ccompatible1:
fixes X k fixes c :: real
defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}"
assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X"
shows "ccompatible \<R> (EQ x c)"
|
lemma ntsmcf_tdghm_smcf_ntsmcf_comp[slicing_commute]:
"smcf_dghm \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M ntsmcf_tdghm \<NN> = ntsmcf_tdghm (\<HH> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F\<^sub>-\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)"
|
lemma min_union_finite:
"finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<union>\<^sub>m B)"
|
lemma decompV_decomp:
assumes "validFrom s tr"
and "reach s"
shows "decompV (V tr) nid = lV nid (decomp tr nid)"
|
lemma Zp_square_root_criterion:
assumes "p \<noteq> 2"
assumes "a \<in> carrier Zp"
assumes "b \<in> carrier Zp"
assumes "val_Zp b \<ge> val_Zp a"
assumes "a \<noteq> \<zero>"
assumes "b \<noteq> \<zero>"
shows "\<exists>y \<in> carrier Zp. a[^](2::nat) \<oplus> \<p>\<otimes>b[^](2::nat) = (y [^]\<^bsub>Zp\<^esub> (2::nat))"
|
lemma
interpret_floatariths_fresh_eqI:
assumes "\<And>i. fresh_floatariths ea i \<or> (i < length ys \<and> i < length zs \<and> ys ! i = zs ! i)"
shows "interpret_floatariths ea ys = interpret_floatariths ea zs"
|
lemma defensive_move_exists_for_Even:
assumes [intro]:"position p"
shows "winning_position_Odd p \<or> (\<exists> m. move_defensive_by_Even m p)" (is "?w \<or> ?d")
|
lemma nth_append_singl[simp]:
"i < length al \<Longrightarrow> (al @ [a]) ! i = al ! i"
|
lemma Checkcast_correct:
"\<lbrakk> wt_jvm_prog G phi;
method (G,C) sig = Some (C,rT,maxs,maxl,ins,et);
ins!pc = Checkcast D;
wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) et pc;
Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ;
G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd>;
fst (exec_instr (ins!pc) G hp stk loc C sig pc frs) = None \<rbrakk>
\<Longrightarrow> G,phi \<turnstile>JVM state'\<surd>"
|
lemma uminus_add_in_Ker_eq_eq_im:
"g\<in>G \<Longrightarrow> h\<in>G \<Longrightarrow> (-g + h \<in> Ker) = (T g = T h)"
|
lemma lift_spmf_bind_spmf: "lift_spmf (p \<bind> f) = lift_spmf p \<bind> (\<lambda>x. lift_spmf (f x))"
|
lemma global_oinvariantI [intro]:
assumes init: "\<And>\<sigma> p. (\<sigma>, p) \<in> init A \<Longrightarrow> P \<sigma>"
and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U; P \<sigma>; U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P \<sigma>'"
and step: "\<And>\<sigma> p a \<sigma>' p'.
\<lbrakk> (\<sigma>, p) \<in> oreachable A S U;
P \<sigma>;
((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A;
S \<sigma> \<sigma>' a \<rbrakk> \<Longrightarrow> P \<sigma>'"
shows "A \<Turnstile> (S, U \<rightarrow>) (\<lambda>(\<sigma>, _). P \<sigma>)"
|
lemma nonpos_Reals_one_I [simp]: "1 \<notin> \<real>\<^sub>\<le>\<^sub>0"
|
lemma "w \<le>p w \<Longrightarrow> \<^bold>|w\<^bold>| = 2 \<Longrightarrow> w \<in> lists {a,b} \<Longrightarrow> hd w = a \<Longrightarrow> w = [a, b] \<or> w = [a, a]"
|
lemma awalk_verts_dom_if_uneq: "\<lbrakk>u\<noteq>v; awalk u p v\<rbrakk> \<Longrightarrow> \<exists>x. x \<rightarrow>\<^bsub>G\<^esub> v \<and> x \<in> set (awalk_verts u p)"
|
lemma sys_block_sizes_uniform [simp]: "sys_block_sizes = {\<k>}"
|
lemma hfs_valid_None_Cons:
assumes "hfs_valid_None ainfo uinfo p" "p = hf1 # hf2 # post"
shows "hfs_valid_None ainfo (upd_uinfo uinfo hf2) (hf2 # post)"
|
lemma "(\<Lambda> x. [x]) = (\<Lambda> z. z : [])"
|
lemma Object_neq_SXcpt [simp]: "Object \<noteq> SXcpt xn"
|
lemma sigma_le_sets:
assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"
|
lemma step_in_valid_step: "knights_path b ps \<Longrightarrow> step_in ps s\<^sub>i s\<^sub>j \<Longrightarrow> valid_step s\<^sub>i s\<^sub>j"
|
lemma bound_typ_inst_gen [simp]:
"free_tv(t::typ) \<subseteq> free_tv(A) \<Longrightarrow> bound_typ_inst S (gen A t) = t"
|
lemma smult_unique_scalars :
fixes a::'f
assumes vs: "basis_for V vs" and v: "v \<in> V"
defines as: "as \<equiv> (THE cs. length cs = length vs \<and> v = cs \<bullet>\<cdot> vs)"
and bs: "bs \<equiv> (THE cs. length cs = length vs \<and> a \<cdot> v = cs \<bullet>\<cdot> vs)"
shows "bs = map ((*) a) as"
|
lemma typ_of_axiom: "wf_theory thy \<Longrightarrow> t \<in> axioms thy \<Longrightarrow> typ_of t = Some propT"
|
lemma invalid_assn_mono: "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt B x y
\<Longrightarrow> hn_invalid A x y \<Longrightarrow>\<^sub>t hn_invalid B x y"
|
lemma padic_add_comm:
assumes "prime p"
shows " \<And>x y.
x \<in> carrier (padic_int p) \<Longrightarrow>
y \<in> carrier (padic_int p) \<Longrightarrow>
x \<oplus>\<^bsub>padic_int p\<^esub> y = y \<oplus>\<^bsub>padic_int p\<^esub> x"
|
lemma chain_sup_const[simp]:
"chain_sup (\<lambda> x. S) = S"
|
lemma segment_a'_a_ne: "segment G.face_cycle_succ a' a \<noteq> {}"
|
lemma octo_of_real_mult [simp]: "octo_of_real (x * y) = octo_of_real x * octo_of_real y"
|
lemma unambigous_prefix_routing_strong_mono:
assumes lpfx: "is_longest_prefix_routing (rr#rtbl)"
assumes uam: "unambiguous_routing (rr#rtbl)"
assumes e:"rr' \<in> set rtbl"
assumes ne: "routing_match rr' = routing_match rr"
shows "routing_rule_sort_key rr' > routing_rule_sort_key rr"
|
lemma from_hma\<^sub>v_inj[simp]: "from_hma\<^sub>v x = from_hma\<^sub>v y \<longleftrightarrow> x = y"
|
lemma "Sup \<circ> atom_map = (id::'a::complete_atomic_boolean_algebra \<Rightarrow> 'a)"
|
lemma z_lemma_R:
fixes I:: "nat list * nat list"
fixes sign:: "rat list"
assumes consistent: "sign \<in> set (characterize_consistent_signs_at_roots p qs)"
assumes welldefined1: "list_constr (fst I) (length qs)"
assumes welldefined2: "list_constr (snd I) (length qs)"
shows "(z_R I sign = 1) \<or> (z_R I sign = 0) \<or> (z_R I sign = -1)"
|
theorem jvm_typesafe:
assumes wf: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
and start: "wf_start_state P C M vs"
and exec: "P \<turnstile> JVM_start_state P C M vs -\<triangleright>ttas\<rightarrow>\<^bsub>jvm\<^esub>* s'"
shows "s' \<in> correct_jvm_state \<Phi>"
|
lemma rank_of_image:
assumes "finite S"
shows "(\<lambda>x. rank_of x S) ` S = {0..<card S}"
|
lemma negate_negate_dfa: "negate_dfa (negate_dfa A) = A"
|
lemma insert_node_ok:
assumes "known_ptr parent" and "type_wf h"
assumes "parent |\<in>| object_ptr_kinds h"
assumes "\<not>is_character_data_ptr_kind parent"
assumes "is_document_ptr parent \<Longrightarrow> h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r []"
assumes "is_document_ptr parent \<Longrightarrow> \<not>is_character_data_ptr_kind node"
assumes "known_ptr (cast node)"
shows "h \<turnstile> ok (a_insert_node parent node ref)"
|
lemma eval_terms_fv_fo_terms_set: "\<sigma> \<odot> ts = \<sigma>' \<odot> ts \<Longrightarrow> n \<in> fv_fo_terms_set ts \<Longrightarrow> \<sigma> n = \<sigma>' n"
|
lemma clop_iso: "clop f \<Longrightarrow> mono f"
|
lemma emeasure_T_state_Nil:
"T (s, o\<^sub>0) {\<omega> \<in> space S. V [] as \<omega>} = 1"
|
lemma HEndPhase2_valueChosen2:
assumes act: "HEndPhase2 s s' q"
and asm4: "\<forall>d\<in>D. b \<le> bal(disk s d p)
\<and>(\<forall>q.( phase s q = 1
\<and> b \<le>mbal(dblock s q)
\<and> hasRead s q d p
) \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal(block br)))" (is "?P s")
shows "?P s'"
|
lemma pl_1[axiom]:
"[[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<phi>)]]"
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lemma fv_ik_subset_fv_st[simp]: "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
|
lemma num_params_polymul:
shows "num_params (p1 *\<^sub>p p2) \<le> max (num_params p1) (num_params p2)"
|
lemma minSetOfComponentsTestL2p3:
"minSetOfComponents level2 {data1, data10, data11} = {sS1, sS2, sS3}"
|
lemma map_def: "Applicative_DNEList.map = map_fun id (map_fun list_of_dnelist Abs_dnelist) (\<lambda>f xs. remdups (list.map f xs))"
|
lemma bin_num: "bin0 = 0" "bin1 = 1"
|
lemma set_disconnected_nodes_typess_preserved:
assumes "w \<in> set_disconnected_nodes_locs object_ptr"
assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'"
shows "type_wf h = type_wf h'"
|
lemma finite_fold_fold_keys:
assumes "comp_fun_commute f"
shows "Finite_Set.fold f A (Set t) = fold_keys f t A"
|
lemma rbl_add_carry_Cons:
"(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) =
xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)"
|
lemma FP_weakest:
"(\<And>B. F \<in> stable (A Int B)) \<Longrightarrow> A <= FP F"
|
lemma ereal_minus_less:
fixes x y z :: ereal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
|
lemma max_spec_preserves_length:
"max_spec G C (mn, pTs) = {((md,rT),pTs')} \<Longrightarrow> length pTs = length pTs'"
|
lemma pdevs_val_minus: "pdevs_val (\<lambda>i. e i - f i) xs = pdevs_val e xs - pdevs_val f xs"
|
lemma is_node_ptr_kind_cast [simp]: "is_node_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr)"
|
lemma p_add: "p x y (n + m) = (\<integral>\<^sup>+ w. p x w n * p w y m \<partial>count_space UNIV)"
|
lemma miles_to_feet: "mile = 5280 *\<^sub>Q foot"
|
lemma (in aGroup) ring_nsum_zeroTr:"(\<forall>j \<le> (n::nat). f j \<in> carrier A) \<and>
(\<forall>j \<le> n. f j = \<zero>) \<longrightarrow> nsum A f n = \<zero>"
|
lemma closed_sum_left_subset: \<open>0 \<in> B \<Longrightarrow> A \<subseteq> A +\<^sub>M B\<close> for A B :: "_::monoid_add"
|
lemma fst_gb_schema_incr:
"fst ` set (gb_schema_incr sel ap ab compl upd (b0 # bs) data) =
(let (gs, n, data') = add_indices (gb_schema_incr sel ap ab compl upd bs data, data) (0, data);
b = (fst b0, n, snd b0); data'' = upd gs b data' in
fst ` set (gb_schema_aux sel ap ab compl gs (count_rem_components (b # gs), Suc n, data'')
(ab gs [] [b] (Suc n, data'')) (ap gs [] [] [b] (Suc n, data'')))
)"
|
lemma monic_factorization_uniqueness:
fixes P::"'a poly set"
assumes finite_P: "finite P"
and PQ: "\<Prod>P = \<Prod>Q"
and P: "P \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"
and finite_Q: "finite Q"
and Q: "Q \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"
shows "P = Q"
|
lemma ogets_NF_wp [wp]:
"ovalidNF (\<lambda>s. P (f s) s) (ogets f) P"
|
lemma induced_arrow_self:
shows "induced_arrow a \<chi> = a"
|
lemma coprime_pderiv_imp_rsquarefree:
assumes "coprime (p :: 'a :: field_char_0 poly) (pderiv p)"
shows "rsquarefree p"
|
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
|
lemma Gromov_extension_quasi_isometry_boundary_to_boundary:
fixes f::"'a::Gromov_hyperbolic_space_geodesic \<Rightarrow> 'b::Gromov_hyperbolic_space_geodesic"
assumes "lambda C-quasi_isometry f"
"x \<in> Gromov_boundary"
shows "(Gromov_extension f) x \<in> Gromov_boundary"
|
lemma listsum2_lemma: "\<lbrakk>length xs = vl; n \<le> vl\<rbrakk> \<Longrightarrow>
rec_exec (rec_listsum2 vl n) xs = listsum2 xs n"
|
lemma strategy_proofI:
assumes "\<And>Pd Pd' Ch d y. \<lbrakk> mechanism_domain Pd Ch; mechanism_domain (Pd(d:=Pd')) Ch; d \<in> ds;
y \<in> \<phi> (Pd(d := Pd')) Ch ds; y \<in> Field (Pd d);
\<forall>x\<in>dX (\<phi> Pd Ch ds) d. x \<noteq> y \<and> (x, y) \<in> Pd d \<rbrakk> \<Longrightarrow> False"
shows "strategy_proof ds \<phi>"
|
lemma equivalence_plus_closed:
"equivalence x \<Longrightarrow> equivalence (x\<^sup>+)"
|
lemma mix_of_preferred_is_preferred:
assumes "p \<succeq>[\<R>] w"
assumes "q \<succeq>[\<R>] w"
assumes "\<alpha> \<in> {0..1}"
shows "mix_pmf \<alpha> p q \<succeq>[\<R>] w"
|
lemma expected_value_is_utility_function:
assumes fnt: "finite outcomes" and "outcomes \<noteq> {}"
assumes "x \<in> lotteries_on outcomes" and "y \<in> lotteries_on outcomes"
assumes "ordinal_utility (lotteries_on outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
shows "measure_pmf.expectation x u \<ge> measure_pmf.expectation y u \<longleftrightarrow> x \<succeq>[\<R>] y" (is "?L \<longleftrightarrow> ?R")
|
lemma prime_power_eq_one_iff [simp]: "prime p \<Longrightarrow> p ^ n = 1 \<longleftrightarrow> n = 0"
|
lemma ex_is_poincare_line_points':
assumes i12: "i1 \<in> circline_set H \<inter> unit_circle_set"
"i2 \<in> circline_set H \<inter> unit_circle_set"
"i1 \<noteq> i2"
assumes a: "a \<in> circline_set H" "a \<notin> unit_circle_set"
shows "\<exists> b. b \<noteq> i1 \<and> b \<noteq> i2 \<and> b \<noteq> a \<and> b \<noteq> inversion a \<and> b \<in> circline_set H"
|
lemma RP_state_repetition_distribution_productF :
assumes "OFSM M2"
and "OFSM M1"
and "(card (nodes M2) * m) \<le> length xs"
and "card (nodes M1) \<le> m"
and "vs@xs \<in> L M2 \<inter> L M1"
and "is_det_state_cover M2 V"
and "V'' \<in> Perm V M1"
shows "\<exists> q \<in> nodes M2 . card (RP M2 q vs xs V'') > m"
|
lemma first_baseE:
assumes H1: "basevars v" and H2: "\<And>x. v (first x) = c \<Longrightarrow> Q"
shows "Q"
|
lemma Hermite_of_upt_row_preserves_zero_rows:
fixes A::"'a::{bezout_ring_div,normalization_semidom,unique_euclidean_ring}^'cols::{mod_type}^'rows::{mod_type}"
assumes i: "is_zero_row i A"
and e: "echelon_form A" and a: "ass_function ass" and r: "res_function res" and k: "k \<le> nrows A"
shows "is_zero_row i (Hermite_of_upt_row_i A k ass res)"
|
lemma reduction_word:
assumes "q \<in> nodes" "run v q"
obtains u w
where
"R.run w q"
"v =\<^sub>I u" "u \<preceq>\<^sub>I w"
"lproject visible (llist_of_stream u) = lproject visible (llist_of_stream w)"
|
lemma ipset_from_cidr_base_wellforemd: fixes base:: "'i::len word"
assumes "mask (LENGTH('i) - l) AND base = 0"
shows "ipset_from_cidr base l = {base .. base OR mask (LENGTH('i) - l)}"
|
lemma square_integrable_iff_lspace:
assumes "S \<in> sets lebesgue"
shows "f square_integrable S \<longleftrightarrow> f \<in> lspace (lebesgue_on S) 2" (is "?lhs = ?rhs")
|
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