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lemma raise_trap_result : "snd (raise_trap t s) = False" |
lemma lift_CFGExit_wf:
assumes wf:"CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc
get_return_edges procs Main Exit Def Use ParamDefs ParamUses"
and pd:"Postdomination sourcenode targetnode kind valid_edge Entry get_proc
get_return_edges procs Main Exit"
shows "CFGExit_wf src trg knd
(lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry
(lift_get_proc get_proc Main)
(lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind)
procs Main NewExit (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L)
(lift_ParamDefs ParamDefs) (lift_ParamUses ParamUses)" |
lemma top_higher [simp, intro]:
"x \<in> carrier L \<Longrightarrow> x \<sqsubseteq> \<top>" |
lemma eval_hom_is_SA_hom:
assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
shows "ring_hom_ring (SA n) Q\<^sub>p (eval_hom a)" |
lemma list_middle_eq:
"length xs = length ys \<Longrightarrow> hd xs = hd ys \<Longrightarrow> last xs = last ys
\<Longrightarrow> butlast (tl xs) = butlast (tl ys) \<Longrightarrow> xs = ys" |
lemma isNSCont_isCont: "isNSCont f a \<Longrightarrow> isCont f a" |
lemma const_replace_closedD:
assumes "const_replace_closed c U" "p \<in> poss_of_term (constT c) s" "(s, t) \<in> U"
shows "(s[p \<leftarrow> u], t) \<in> U \<or> (\<exists> q. q \<in> poss_of_term (constT c) t \<and> (s[p \<leftarrow> u], t[q \<leftarrow> u]) \<in> U)" |
lemma hoare3a_sound_GC: "\<turnstile>\<^sub>3\<^sub>a {P} c { Q} \<Longrightarrow> \<Turnstile>\<^sub>3\<^sub>' {P} c { Q ** sep_true}" |
lemma (in Torder) inc_segment_segment:"\<lbrakk>b \<in> carrier D;
a \<in> segment D b\<rbrakk> \<Longrightarrow> segment (Iod D (segment D b)) a = segment D a" |
lemma prefixes_tl_only_01:
assumes "prefixes t j \<down>= b"
shows "\<forall>x>0. e_nth b x = 0 \<or> e_nth b x = 1" |
lemma \<pi>_mono [intro]: "x \<le> y \<Longrightarrow> \<pi> x \<le> \<pi> y"
and \<theta>_mono [intro]: "x \<le> y \<Longrightarrow> \<theta> x \<le> \<theta> y"
and primes_M_mono [intro]: "x \<le> y \<Longrightarrow> \<MM> x \<le> \<MM> y" |
lemma subcls1_induct_aux:
"\<lbrakk> is_class P C; wf_prog wf_md P; Q Object;
\<And>C D fs ms.
\<lbrakk> C \<noteq> Object; is_class P C; class P C = Some (D,fs,ms) \<and>
wf_cdecl wf_md P (C,D,fs,ms) \<and> P \<turnstile> C \<prec>\<^sup>1 D \<and> is_class P D \<and> Q D\<rbrakk> \<Longrightarrow> Q C \<rbrakk>
\<Longrightarrow> Q C"
(*<*)
(is "?A \<Longrightarrow> ?B \<Longrightarrow> ?C \<Longrightarrow> PROP ?P \<Longrightarrow> _") |
lemma nextss_Cons: "nextss g (x#xs) = set (nexts g x) \<union> nextss g xs" |
lemma FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m_rec: "FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i = (\<lambda> s. \<box> e \<in> (Tr\<^sub>f i s) \<rightarrow> FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i (Up\<^sub>f i s e))" |
lemma conjugation_is_size_invariant:
assumes fin:"finite (carrier G)"
assumes P:"P \<in> subgroups_of_size p"
assumes g:"g \<in> carrier G"
shows "conjugation_action p g P \<in> subgroups_of_size p" |
lemma irrefl_less_nmultiset:
fixes X :: "'a nmultiset"
shows "X < X \<Longrightarrow> False" |
lemma (in Semilat) plus_list_ub1 [rule_format]:
"\<lbrakk> set xs \<subseteq> A; set ys \<subseteq> A; size xs = size ys \<rbrakk>
\<Longrightarrow> xs [\<sqsubseteq>\<^bsub>r\<^esub>] xs [\<squnion>\<^bsub>f\<^esub>] ys" |
lemma bitAND_single_digit:
fixes x c :: nat
assumes "2 ^ c \<le> x"
assumes "x < 2 ^ Suc c"
shows "nth_bit x c = 1" |
lemma skl1_obs_iagreement_Resp [iff]: "oreach skl1 \<subseteq> l1_iagreement_Resp" |
lemma compP2_compP1_convs:
"is_type (compP2 (compP1 P)) = is_type P"
"is_class (compP2 (compP1 P)) = is_class P"
"sc.addr_loc_type (compP2 (compP1 P)) = sc.addr_loc_type P"
"sc.conf (compP2 (compP1 P)) = sc.conf P" |
lemma bst_wrt_le_if_bst: "bst t \<Longrightarrow> bst_wrt (\<le>) t" |
lemma jpTraces_step_length_inv:
"{ t \<in> jpTraces jp . tLength t = Suc n }
= { t \<leadsto> s |eact aact t s. t \<in> { t \<in> jpTraces jp . tLength t = n }
\<and> eact \<in> set (envAction (tLast t))
\<and> (\<forall>a. aact a \<in> set (actJP jp t a))
\<and> s = envTrans eact aact (tLast t) }" |
lemma cong_r: "H \<in> normalfilters \<Longrightarrow> cong H = cong_r H" |
lemma "\<FF> \<F> \<Longrightarrow> \<forall>a b. (a \<^bold>\<and> \<^bold>\<not>a) \<^bold>\<rightarrow> b \<^bold>\<approx> \<^bold>\<top>" |
lemma inv_free_Abelian_group [simp]: "Poly_Mapping.keys x \<subseteq> S \<Longrightarrow> inv\<^bsub>free_Abelian_group S\<^esub> x = -x" |
lemma non_contractible_space_nsphere: "\<not> (contractible_space(nsphere n))" |
lemma restrict_map_inv[simp]: "f |` (- dom f) = Map.empty" |
lemma nat2Nat_Nat2nat[simp]: "Elem n Nat \<Longrightarrow> nat2Nat (Nat2nat n) = n" |
lemma less_4_cases:
"(x::word64) < 4 \<Longrightarrow> x=0 \<or> x=1 \<or> x=2 \<or> x=3" |
lemma reflects_right_adjoint:
assumes "C.ide g" and "D.is_right_adjoint (F g)"
shows "C.is_right_adjoint g" |
lemma minus_closed'[simp]: "\<forall>x\<in>U\<^sub>M. \<forall>y\<in>U\<^sub>M. x -\<^sub>M y \<in> U\<^sub>M" |
lemma sum_of_2_squares_nat_mult [intro]:
assumes "sum_of_2_squares_nat x" "sum_of_2_squares_nat y"
shows "sum_of_2_squares_nat (x * y)" |
lemma "\<nu>\<^sup>\<natural> (x \<cdot> y) = \<nu>\<^sup>\<natural> x \<cdot> \<nu>\<^sup>\<natural> y" |
lemma PosPropertiesNecExist: "\<lfloor>\<^bold>\<forall>Y. \<P> Y \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<exists>\<^sup>E Y\<rfloor>" |
lemma (in is_ntcf) inv_ntcf_NTMap_vdomain[cat_cs_simps]:
"\<D>\<^sub>\<circ> (inv_ntcf \<NN>\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>" |
lemma dm_downset_var: "dm {x} = \<down>(x::'a::preorder)" |
lemma face_ofD: "\<lbrakk>T face_of S; x \<in> open_segment a b; a \<in> S; b \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> a \<in> T \<and> b \<in> T" |
lemma sym_joinable: "sym (joinable R)" |
lemma count_list_upt [simp]: "count_list [a..<b] x = (if a \<le> x \<and> x < b then 1 else 0)" |
lemma rational_approximation:
assumes "e > 0"
obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e" |
lemma finite_dimensional_vector_space_with_overloaded[ud_with]:
"finite_dimensional_vector_space =
finite_dimensional_vector_space_with (+) 0 (-) uminus" |
lemma bin_cat_inj: "(concat_bit n b a) = concat_bit n d c \<longleftrightarrow> a = c \<and> take_bit n b = take_bit n d" |
lemma lookup_tabulate2: "Mapping.lookup (Mapping.tabulate xs f) x = Some y \<Longrightarrow> y = f x" |
lemma (in Semilat) plus_list_lub [rule_format]:
shows "\<forall>xs ys zs. set xs <= A \<longrightarrow> set ys <= A \<longrightarrow> set zs <= A
\<longrightarrow> size xs = n & size ys = n \<longrightarrow>
xs <=[r] zs & ys <=[r] zs \<longrightarrow> xs +[f] ys <=[r] zs" |
lemma sum_list_length_card_dom_map_of_concat:
assumes "ht_hash l"
assumes "ht_distinct l"
shows "sum_list (map length l) = card (dom (map_of (concat l)))" |
lemma weak_All_transfer2 [transfer_rule]: "right_total R \<Longrightarrow> ((R ===> (=)) ===> (\<longrightarrow>)) All All" |
lemma union_add_left_zmset[simp]: "add_zmset a A + B = add_zmset a (A + B)" |
lemma apply_cltn2_line_injective:
assumes "apply_cltn2_line l C = apply_cltn2_line m C"
shows "l = m" |
lemma orbit_nonempty:
"orbit f s \<noteq> {}" |
lemma Ide_implies_Can:
shows "Ide t \<Longrightarrow> Can t" |
lemma nsetTr1:"\<lbrakk>j \<in> nset a b; j \<noteq> a\<rbrakk> \<Longrightarrow> j \<in> nset (Suc a) b" |
lemma norm_cblinfun_bound: "0 \<le> b \<Longrightarrow> (\<And>x. norm (cblinfun_apply f x) \<le> b * norm x) \<Longrightarrow> norm f \<le> b" |
lemma IC_rate2:
"rho_bound2 (\<lambda> p t. IC p i t)" |
lemma pop_refine:
assumes A: "(s,(p,D,pE))\<in>GS_rel"
assumes B: "p \<noteq> []" "pE \<inter> last p \<times> UNIV = {}"
shows "pop_impl s \<le> \<Down>GS_rel (RETURN (pop (p,D,pE)))" |
lemma vec_of_basis_enum_to_inverse:
fixes \<psi> :: "'a::one_dim"
shows "vec_of_basis_enum (inverse \<psi>) = vec_of_list [inverse (vec_index (vec_of_basis_enum \<psi>) 0)]" |
lemma dghm_cn_cov_comp_ObjMap_vdomain[dg_cn_cs_simps]:
assumes "\<GG> : \<BB> \<^sub>D\<^sub>G\<mapsto>\<mapsto>\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
shows "\<D>\<^sub>\<circ> ((\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>" |
lemma LV_FROMNTIMES_3:
shows "LV (From r (Suc n)) [] =
(\<lambda>(v,vs). Stars (v#vs)) ` (LV r [] \<times> (Stars -` (LV (From r n) [])))" |
lemma min_shortest_path: assumes "u \<in> V" "v \<in> V" "u \<noteq> v"
shows "shortest_path u v > 0" |
lemma ahm_lookup_impl:
assumes bhc: "is_bounded_hashcode Id (=) bhc"
shows "(ahm_lookup (=) bhc, op_map_lookup) \<in> Id \<rightarrow> ahm_map_rel' bhc \<rightarrow> Id" |
lemma le_iff_le_nth_eucl: "x \<le> y \<longleftrightarrow> (\<forall>i<DIM('a). (x $\<^sub>e i) \<le> (y $\<^sub>e i))"
for x::"'a::executable_euclidean_space" |
lemma A23_DAcc_level1: "DAcc level1 sA23 = {sA32}" |
lemma zero_vector_7:
"zero_vector x \<longleftrightarrow> (\<forall>y . x * top = x * y)" |
lemma union_single_eq_diff_zmset: "add_zmset x M = N \<Longrightarrow> M = N - {#x#}\<^sub>z" |
lemma this_loc: "mtx_pointwise_cmpop_gen_impl N M f g assn A get_impl fi gi" |
lemma max_list_set: "max_list xs = (if set xs = {} then 0 else (THE x. x \<in> set xs \<and> (\<forall> y \<in> set xs. y \<le> x)))" |
lemma length_unique_prefix:
"al1 \<le> al \<Longrightarrow> al2 \<le> al \<Longrightarrow> length al1 = length al2 \<Longrightarrow> al1 = al2" |
lemma imp_OO_imp [simp]: "(\<longrightarrow>) OO (\<longrightarrow>) = (\<longrightarrow>)" |
lemma obsf_resource_of_oracle [simp]:
"obsf_resource (resource_of_oracle oracle s) = resource_of_oracle (obsf_oracle oracle) (OK s)" |
lemma diff_add_zmset:
fixes M N Q :: "'a zmultiset"
shows "M - (N + Q) = M - N - Q" |
lemma maddux_142: "x\<^sup>\<smile> ; z \<cdot> y = 0 \<longleftrightarrow> z ; y\<^sup>\<smile> \<cdot> x = 0" |
lemma CI1: "Cl_1 \<phi> \<Longrightarrow> Int_1 \<phi>\<^sup>d" |
lemma is_pseudonatural_equivalence:
shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
(H o F) HoF.cmp (H o G) HoG.cmp map\<^sub>0 map\<^sub>1" |
lemma size_sub_fm [simp]: \<open>size (sub_fm s p) = size p\<close> |
lemma R3_idem2: "R3 (R3 x) = R3 x" |
lemma binsert_eqvt [eqvt]: "p \<bullet> (binsert x B) = binsert (p \<bullet> x) (p \<bullet> B)" |
lemma (in ring_hom_cring) hom_zero [simp]: "h \<zero> = \<zero>\<^bsub>S\<^esub>" |
lemma "wordinterval_intersection
(RangeUnion (RangeUnion (WordInterval (1::32 word) 3) (WordInterval 8 10))
(WordInterval 1 3)) (WordInterval 1 3) =
WordInterval 1 3" |
lemma take_Suc: "take (Suc n) xs = (if xs = [] then [] else hd xs # take n (tl xs))" |
lemma gen_isSng[autoref_rules_raw]:
assumes PRIO_TAG_GEN_ALGO
assumes "GEN_OP sizea op_map_size_abort (Id \<rightarrow> (\<langle>Rk,Rv\<rangle>Rm) \<rightarrow> Id)"
shows "(gen_isSng sizea,op_map_isSng)
\<in> \<langle>Rk,Rv\<rangle>Rm \<rightarrow> Id" |
lemma cont_applyI [cont_intro]:
assumes cont: "cont luba orda lubb ordb g"
shows "cont (fun_lub luba) (fun_ord orda) lubb ordb (\<lambda>f. g (f x))" |
lemma rbl_succ2_simps:
"rbl_succ2 b [] = []"
"rbl_succ2 b (x # xs) = (b \<noteq> x) # rbl_succ2 (x \<and> b) xs" |
lemma subset_inext_closed: "\<lbrakk> n \<in> B; A \<subseteq> B \<rbrakk> \<Longrightarrow> inext n A \<in> B" |
lemma "\<exists>f::(('a \<times> 's) set[('a \<times> 's) set] \<Rightarrow> 'a \<Rightarrow> 's set['s set]). bij f" |
lemma cons_set_cons_eq: "a#l \<in> b\<cdot>S = (a=b & l\<in>S)" |
lemma reduce_basis_cost:
"result (reduce_basis_cost fs_init) = LLL_Impl.reduce_basis \<alpha> fs_init" (is ?g1)
"cost (reduce_basis_cost fs_init) \<le> initial_gso_cost + body_cost * num_loops" (is ?g2) |
lemma in_results_gpv_sub_gvps: "\<lbrakk> x \<in> results_gpv \<I> gpv'; gpv' \<in> sub_gpvs \<I> gpv \<rbrakk> \<Longrightarrow> x \<in> results_gpv \<I> gpv" |
lemma drop_yields_results_implies_nbound:
shows "drop n x \<noteq> [] \<longrightarrow> n < length x" |
lemma dom_comp_subset: "g ` dom (f \<circ> g) \<subseteq> dom f" |
lemma iIN_pred_insert_conv: "
0 < n \<Longrightarrow> insert (n - Suc 0) [n\<dots>,d] = [n - Suc 0\<dots>,Suc d]" |
lemma rbt_map_entry_inv2: "inv2 (rbt_map_entry k f t) = inv2 t" "bheight (rbt_map_entry k f t) = bheight t" |
lemma empty_gta_lang:
"gta_lang Q (TA {||} {||}) = {}" |
lemma set_of_idx_code[code]:
fixes t :: "('a :: ccompare, 'b set) mapping_rbt"
shows "set_of_idx (RBT_Mapping t) =
(case ID CCOMPARE('a) of None \<Rightarrow> Code.abort (STR ''set_of_idx RBT_Mapping: ccompare = None'') (\<lambda>_. set_of_idx (RBT_Mapping t))
| Some _ \<Rightarrow> \<Union>(snd ` set (RBT_Mapping2.entries t)))" |
lemma part_UNpart:
assumes cl: "properL cl" and
P: "\<And> n. n < length cl \<Longrightarrow> part {..< brn (cl!n)} (P n)"
shows "part {..< brnL cl (length cl)} (UNpart cl P)"
(is "part ?J ?Q") |
lemma sinits_bottom_iff[simp]: "(sinits\<cdot>xs = \<bottom>) \<longleftrightarrow> (xs = \<bottom>)" |
lemma [simp]: "y \<le> x \<squnion> (y \<squnion> z)" |
lemma face_cycles_dv: "H.face_cycle_sets = face_cycle_sets" |
lemma "(not ((A \<longrightarrow>* (not (A ** B))) and (((not A) \<longrightarrow>* (not B)) and B)))
(h::'a::heap_sep_algebra)" |
lemma irrelevant_set_bit[simp]:
fixes p m n :: nat
assumes "n \<le> m"
shows "(p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n" |
lemma sqrt_int_floor_code[code]: "sqrt_int_floor x = (if x \<ge> 0 then sqrt_int_floor_pos x else - sqrt_int_ceiling_pos (- x))" |
lemma is_interval_path_connected_1:
fixes s :: "real set"
shows "is_interval s \<longleftrightarrow> path_connected s" |
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