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lemma substitute_not_carrier:
assumes "x \<notin> carrier t"
assumes "\<And> x'. x \<notin> carrier (f x')"
shows "x \<notin> carrier (substitute f T t)" |
lemma nextI[intro]:
assumes "P (\<sigma> |\<^sub>s Suc 0)"
shows "(\<circle>P) \<sigma>" |
lemma list2_induct[case_names NilNil Cons1 Cons2]:
assumes NN: "P [] []"
and CN: "\<And>x xs ys. P xs ys \<Longrightarrow> P (x # xs) ys"
and NC: "\<And>xs y ys. P xs ys \<Longrightarrow> P xs (y # ys)"
shows "P xs ys" |
lemma sup_loc_trans [intro?, trans]:
"\<lbrakk>P \<turnstile> a [\<le>\<^sub>\<top>] b; P \<turnstile> b [\<le>\<^sub>\<top>] c\<rbrakk> \<Longrightarrow> P \<turnstile> a [\<le>\<^sub>\<top>] c" |
lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C" |
lemma NotifyWatchesLoopConflictFlagEffect:
fixes literal :: Literal and Wl :: "nat list" and newWl :: "nat list" and state :: State
assumes
"InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" and
"\<forall> (c::nat). c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" and
"InvariantConsistent (getM state)"
"\<forall> (c::nat). c \<in> set Wl \<longrightarrow> Some literal = (getWatch1 state c) \<or> Some literal = (getWatch2 state c)"
"literalFalse literal (elements (getM state))"
"uniq Wl"
shows
"let state' = notifyWatches_loop literal Wl newWl state in
getConflictFlag state' =
(getConflictFlag state \<or>
(\<exists> clause. clause \<in> set Wl \<and> clauseFalse (nth (getF state) clause) (elements (getM state))))" |
lemma shadow_root_different_get_put [simp]:
"shadow_root_ptr \<noteq> shadow_root_ptr' \<Longrightarrow>
get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr (put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr' shadow_root h) = get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h" |
theorem HPhase0Read_HInv1:
assumes inv1: "HInv1 s"
and act: "HPhase0Read s s' p d"
shows "HInv1 s'" |
lemma is_walk_rev: "is_walk xs \<longleftrightarrow> is_walk (rev xs)" |
lemma seq_comp_divergences:
"weakly_sequential P \<Longrightarrow>
divergences (P ; Q) = {}" |
lemma fresh_context:
fixes \<Gamma> :: "Ctxt"
and a :: "name"
assumes "a\<sharp>\<Gamma>"
shows "\<not>(\<exists>\<tau>::ty. (a,\<tau>)\<in>set \<Gamma>)" |
lemma PO_m2_refines_trans_m1 [iff]:
"{R12 \<inter>
UNIV \<times> (m2_inv4_inon_secret \<inter> m2_inv3_msg2)}
(trans m1), (trans m2)
{> R12}" |
lemma bdd_all_is_node_subsetbdd:
assumes "list_all (bdd_all (nfa_is_node A)) (fst A)"
and "nfa_is_node A q"
shows "bdd_all (nfa_is_node A) (subsetbdd (fst A) q (nfa_emptybdd (length q)))" |
lemma MSB_map_index'_SNOC[simp]:
"MSB Is \<le> n \<Longrightarrow> MSB (map_index' i (snoc n bs) Is) =
(if (\<forall>i \<in> {i ..< i + length Is}. \<not> bs ! i) then MSB Is else Suc n)" |
lemma rel_filter_neg_distr_cond'_eq: "rel_filter_neg_distr_cond' (=) (=)" |
lemma conjl2_strict: "x \<lhd> \<bottom> = \<bottom>" |
lemma smult_cancel[simp]:
fixes c :: "'a :: idom"
shows "smult c f = smult c g \<longleftrightarrow> c = 0 \<or> f = g" |
lemma sumPres:
fixes P :: pi
and Q :: pi
and R :: pi
assumes PSimQ: "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
and RelRel': "Rel \<subseteq> Rel'"
and C: "Id \<subseteq> Rel'"
shows "P \<oplus> R \<leadsto>\<guillemotleft>Rel'\<guillemotright> Q \<oplus> R" |
lemma gauss_int_norm_pos_iff [simp]: "gauss_int_norm z > 0 \<longleftrightarrow> z \<noteq> 0" |
lemma arr_cod [simp]:
assumes "arr f"
shows "arr (cod f)" |
lemma
\<open>(- 1705 :: 16 word) >> Suc (Suc (Suc 0)) = 7978\<close> |
lemma imp_self: "AX0 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> F" |
lemma le_less_TC_trans [trans]: "\<lbrakk>x \<sqsubseteq> y; y \<sqsubset> z\<rbrakk> \<Longrightarrow> x \<sqsubset> z" |
lemma Psi_imp_not_Prob0:
assumes "\<Phi> \<subseteq> S" "\<Psi> \<subseteq> S" shows "s \<in> \<Psi> \<Longrightarrow> s \<notin> Prob0 \<Phi> \<Psi>" |
lemma loc_srj_3: "\<lbrakk>nat_to_sch (sch_to_nat sch1) = sch1; nat_to_sch (sch_to_nat sch2) = sch2\<rbrakk>
\<Longrightarrow> nat_to_sch (c_pair 4 (c_pair (sch_to_nat sch1) (sch_to_nat sch2))) = Comp_op sch1 sch2" |
lemma Spy_see_priK [simp]:
"evs \<in> zg ==> (Key (priK A) \<in> parts (spies evs)) = (A \<in> bad)" |
lemma sorted_wrt_linorder_index_le_iff:
assumes "linorder_on A R" "set xs \<subseteq> A" "sorted_wrt R xs"
"x \<in> set xs" "y \<in> set xs"
shows "index xs x \<le> index xs y \<longleftrightarrow> (x,y) \<in> R" |
lemma rename_ensures:
"bij h ==> (rename h F \<in> (h`A) ensures (h`B)) = (F \<in> A ensures B)" |
lemma in_set_real_minus_interval[intro, simp]:
"x - y \<in>\<^sub>r X - Y" if "x \<in>\<^sub>r X" "y \<in>\<^sub>r Y" |
lemma bc_mt_corresp_New: "\<lbrakk>is_class cG cname \<rbrakk>
\<Longrightarrow> bc_mt_corresp [New cname] (pushST [Class cname]) (ST, LT) cG rT mxr (Suc 0)" |
lemma inj_map_fset_cong:
shows "inj f \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T" |
lemma frameImpI:
fixes F :: "'b frame"
and \<phi> :: 'c
and A\<^sub>F :: "name list"
and \<Psi>\<^sub>F :: 'b
assumes "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"
and "A\<^sub>F \<sharp>* \<phi>"
and "\<Psi>\<^sub>F \<turnstile> \<phi>"
shows "F \<turnstile>\<^sub>F \<phi>" |
lemma foldr_conv_fold [code_abbrev]: "foldr f xs = fold f (rev xs)" |
lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})" |
lemma e_val: "isval v \<Longrightarrow> \<exists> v'. v' \<in> E v \<rho>" |
lemma inext_empty: "inext n {} = n" |
lemma tendsto_0_le:
"(f \<longlongrightarrow> 0) F \<Longrightarrow> eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F \<Longrightarrow> (g \<longlongrightarrow> 0) F" |
lemma bouncing_ball_dyn: "g < 0 \<Longrightarrow> h \<ge> 0 \<Longrightarrow>
(\<lambda>s. s$1 = h \<and> s$2 = 0) \<le>
|LOOP (
(EVOL (\<phi> g) (\<lambda> s. s$1 \<ge> 0) T) ;
(IF (\<lambda> s. s$1 = 0) THEN (2 ::= (\<lambda>s. - s$2)) ELSE skip))
INV (\<lambda>s. 0 \<le> s$1 \<and>2 * g * s$1 = 2 * g * h + s$2 * s$2)]
(\<lambda>s. 0 \<le> s$1 \<and> s$1 \<le> h)" |
lemma distinct7_rot_CW:
assumes "distinct [A,B,C,D,E,F,G]"
shows "distinct [C,A,B,F,D,E,G]" |
lemma \<Phi>_simp:
assumes "D.ide y" and "C.ide x"
shows "S.arr (\<Phi> (y, x))"
and "\<Phi> (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(\<phi>D (y, G x) o \<phi> y o \<psi>C (F y, x))" |
lemma res_ri1: "y \<cdot> (y \<rightarrow> x) \<le> x" |
lemma test_suite_from_io_tree_refined[code] :
fixes M :: "('a,'b :: ccompare, 'c :: ccompare) fsm"
and m :: "(('b\<times>'c), ('b\<times>'c) prefix_tree) mapping_rbt"
shows "test_suite_from_io_tree M q (MPT (RBT_Mapping m))
= (case ID CCOMPARE(('b \<times> 'c)) of
None \<Rightarrow> Code.abort (STR ''test_suite_from_io_tree RBT_set: ccompare = None'') (\<lambda>_ . test_suite_from_io_tree M q (MPT (RBT_Mapping m))) |
Some _ \<Rightarrow> MPT (Mapping.tabulate (map (\<lambda>((x,y),t) . ((x,y),h_obs M q x y \<noteq> None)) (RBT_Mapping2.entries m)) (\<lambda> ((x,y),b) . case h_obs M q x y of None \<Rightarrow> Prefix_Tree.empty | Some q' \<Rightarrow> test_suite_from_io_tree M q' (case RBT_Mapping2.lookup m (x,y) of Some t' \<Rightarrow> t'))))" |
lemma source_all_outarcs_T:
"\<lbrakk>undirected_tree G; tail G e = root; e \<in> arcs G\<rbrakk> \<Longrightarrow> e \<in> arcs T" |
lemma integral_continuous_on_param:
fixes f::"'a::topological_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
assumes cont_fx: "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). f x t)"
shows "continuous_on U (\<lambda>x. integral (cbox a b) (f x))" |
lemma (in is_tiny_functor) tiny_cf_ntcf_id_is_tiny_ntcf'[cat_small_cs_intros]:
assumes "\<FF>' = \<FF>" and "\<GG>' = \<FF>"
shows "ntcf_id \<FF> : \<FF>' \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG>' : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" |
lemma icard_atMost_int: "icard {..(u::int)} = \<infinity>" |
lemma ln_upper_9_pos:
assumes "1 \<le> x" shows "ln(x) \<le> ln_upper_9 x" |
lemma joinable_sym: "(x, y) \<in> joinable R \<Longrightarrow> (y, x) \<in> joinable R" |
lemma exp_dvd_iff_exp_udvd:
\<open>2 ^ n dvd w \<longleftrightarrow> 2 ^ n udvd w\<close> for v w :: \<open>'a::len word\<close> |
lemma delete_lhs_var: assumes norm: "\<triangle> t" and t: "t = t1 @ (x,p) # t2"
and t': "t' = t1 @ t2"
and tv: "tv = (\<lambda> v. upd x (p \<lbrace> \<langle>v\<rangle> \<rbrace>) v)"
and x_as: "x \<notin> atom_var ` snd ` set as"
shows "\<triangle> t'" \<comment> \<open>new tableau is normalized\<close>
"\<langle>w\<rangle> \<Turnstile>\<^sub>t t' \<Longrightarrow> \<langle>tv w\<rangle> \<Turnstile>\<^sub>t t" \<comment> \<open>solution of new tableau is translated to solution of old tableau\<close>
"(I, \<langle>w\<rangle>) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s set as \<Longrightarrow> (I, \<langle>tv w\<rangle>) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s set as" \<comment> \<open>solution translation also works for bounds\<close>
"v \<Turnstile>\<^sub>t t \<Longrightarrow> v \<Turnstile>\<^sub>t t'" \<comment> \<open>solution of old tableau is also solution for new tableau\<close> |
lemma sub_ss_equality:
assumes "sub_set_system \<U> \<A> \<V> \<B>"
and "sub_set_system \<V> \<B> \<U> \<A>"
shows "\<U> = \<V>" and "\<A> = \<B>" |
lemma conc':
"\<lbrakk> \<turnstile> \<Gamma>\<^sub>1,\<S>,P { c } \<Gamma>',\<S>',P';
\<Gamma>\<^sub>1 = (\<Gamma>\<^sub>2(x \<mapsto> t));
x \<in> dom \<Gamma>\<^sub>2;
type_equiv (the (\<Gamma>\<^sub>2 x)) P t;
type_wellformed t;
type_stable \<S> t \<rbrakk> \<Longrightarrow>
\<turnstile> \<Gamma>\<^sub>2,\<S>,P { c } \<Gamma>',\<S>',P'" |
lemma no_defection_def:
"no_defection s round_votes r =
(\<forall>r' < r. \<forall>a Q v. quorum_for Q v (votes s r') \<and> a \<in> Q \<longrightarrow> round_votes a \<in> {None, Some v})" |
lemma weakResPres:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"
and Q :: "('a, 'b, 'c) psi"
and x :: name
and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"
assumes PSimQ: "\<Psi> \<rhd> P \<leadsto><Rel> Q"
and "eqvt Rel'"
and "x \<sharp> \<Psi>"
and "Rel \<subseteq> Rel'"
and C1: "\<And>\<Psi>' R S y. \<lbrakk>(\<Psi>', R, S) \<in> Rel; y \<sharp> \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<leadsto><Rel'> \<lparr>\<nu>x\<rparr>Q" |
lemma simple_match_and_SomeD: "simple_match_and m1 m2 = Some m \<Longrightarrow>
simple_matches m p \<longleftrightarrow> (simple_matches m1 p \<and> simple_matches m2 p)" |
lemma relation_of_Chaos: "relation_of Chaos = (R(false \<turnstile> true))" |
lemma cat_Par_is_iso_arrI[intro]:
assumes "T : A \<mapsto>\<^bsub>cat_Par \<alpha>\<^esub> B"
and "v11 (T\<lparr>ArrVal\<rparr>)"
and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = A"
and "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = B"
shows "T : A \<mapsto>\<^sub>i\<^sub>s\<^sub>o\<^bsub>cat_Par \<alpha>\<^esub> B" |
lemma sorted_nths_atLeastAtMost_0: "\<lbrakk> m \<le> n; sorted (nths xs {0..n}) \<rbrakk> \<Longrightarrow> sorted (nths xs {0..m})" |
lemma pred_envirI [Pure.intro!, intro!]: "(\<And>x. p (f x)) \<Longrightarrow> pred_envir p f" |
lemma nprv_ptrmE_uni:
"\<sigma> \<in> ptrm n \<Longrightarrow> nprv F (subst \<sigma> t1 out) \<Longrightarrow> nprv F (subst \<sigma> t2 out) \<Longrightarrow>
nprv (insert (eql t1 t2) F) \<psi> \<Longrightarrow>
F \<subseteq> fmla \<Longrightarrow> finite F \<Longrightarrow> \<psi> \<in> fmla \<Longrightarrow> t1 \<in> trm \<Longrightarrow> t2 \<in> trm
\<Longrightarrow> nprv F \<psi>" |
lemma termMOD_igVarIPresIGWls: "igVarIPresIGWls termMOD" |
lemma degree_unit_factor[simp]: "degree (unit_factor f) = 0" |
lemma Eps_psimp[nitpick_psimp]:
"\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x" |
lemma Assign_complete:
assumes v: "ir_valid P (x1 ::= x2) c' Q"
assumes q: "Q t t'"
shows "\<exists>s'. (\<exists>v. P (t(x1 := v)) s' \<and> t x1 = aval x2 (t(x1 := v))) \<and> (c', s') \<Rightarrow> t'" |
lemma Key_notin_guardK: "X \<in> guardK n Ks \<Longrightarrow> X \<noteq> Key n" |
lemma msubstltpos:
assumes nz: "Ipoly vs c > 0"
and l: "islin (Lt (CNP 0 a b))"
shows "Ifm vs (x#bs) (msubstltpos c t a b) =
Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Lt (CNP 0 a b))" |
lemma gc_to_red: "chain (\<leadsto>GC) Ns \<Longrightarrow> chain (\<rhd>L) Ns" |
lemma card_before: "distinct xs \<Longrightarrow> x : set xs \<Longrightarrow> card (before x xs) = index xs x" |
lemma H_assign: "P = (\<lambda>s. Q (\<chi> j. ((($) s)(x := (e s))) j)) \<Longrightarrow> \<^bold>{P\<^bold>} (x ::= e) \<^bold>{Q\<^bold>}" |
lemma (in aGroup) mem_sum_subgs:"\<lbrakk>A +> H; A +> K; h \<in> H; k \<in> K\<rbrakk> \<Longrightarrow>
h \<plusminus> k \<in> H \<minusplus> K" |
lemma set_fold1_code [code]:
fixes rbt :: "'a :: {ccompare, lattice} set_rbt"
and dxs :: "'b :: {ceq, lattice} set_dlist" shows
set_fold1_Complement[set_complement_code]:
"set_fold1 f (Complement A) = Code.abort (STR ''set_fold1: Complement'') (\<lambda>_. set_fold1 f (Complement A))"
and "set_fold1 f (Collect_set P) = Code.abort (STR ''set_fold1: Collect_set'') (\<lambda>_. set_fold1 f (Collect_set P))"
and "set_fold1 f (Set_Monad (x # xs)) = fold (semilattice_set_apply f) xs x" (is "?Set_Monad")
and
"set_fold1 f' (DList_set dxs) =
(case ID CEQ('b) of None \<Rightarrow> Code.abort (STR ''set_fold1 DList_set: ceq = None'') (\<lambda>_. set_fold1 f' (DList_set dxs))
| Some _ \<Rightarrow> if DList_Set.null dxs then Code.abort (STR ''set_fold1 DList_set: empty set'') (\<lambda>_. set_fold1 f' (DList_set dxs))
else DList_Set.fold (semilattice_set_apply f') (DList_Set.tl dxs) (DList_Set.hd dxs))"
(is "?DList_set")
and
"set_fold1 f'' (RBT_set rbt) =
(case ID CCOMPARE('a) of None \<Rightarrow> Code.abort (STR ''set_fold1 RBT_set: ccompare = None'') (\<lambda>_. set_fold1 f'' (RBT_set rbt))
| Some _ \<Rightarrow> if RBT_Set2.is_empty rbt then Code.abort (STR ''set_fold1 RBT_set: empty set'') (\<lambda>_. set_fold1 f'' (RBT_set rbt))
else RBT_Set2.fold1 (semilattice_set_apply f'') rbt)"
(is "?RBT_set") |
lemma CallRedsThrowParams:
assumes e_steps: "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1,b\<^sub>1\<rangle>"
and es_steps: "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs\<^sub>1 @ throw a # es\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>"
shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>2,b\<^sub>2\<rangle>"
(*<*)(is "(?x, ?z) \<in> (red P)\<^sup>*") |
lemma subst_idem_support: "subst_idem \<theta> \<Longrightarrow> \<theta> supports \<theta> \<circ>\<^sub>s \<delta>" |
lemma zmult_div_pos_le:"
\<lbrakk> (0::int) \<le> a; 0 \<le> b; b \<le> c \<rbrakk> \<Longrightarrow> a * b div c \<le> a" |
lemma min_per_min: assumes "w \<le>p r\<^sup>\<omega>" shows "\<pi> w \<le>p r" |
lemma lemma3: "T a ws zs \<Longrightarrow> good ws \<Longrightarrow> good zs" |
lemma ex_ncol_cop:
assumes "D \<noteq> E"
shows "\<exists> F. Coplanar A B C F \<and> \<not> Col D E F" |
lemma sync_withI:
"\<lbrakk> P,E \<turnstile> a \<le>so a'; P \<turnstile> (action_tid E a, action_obs E a) \<leadsto>sw (action_tid E a', action_obs E a') \<rbrakk>
\<Longrightarrow> P,E \<turnstile> a \<le>sw a'" |
lemma measurable_trace_at': "(\<lambda>((s, j), \<omega>). trace_at s \<omega> j) \<in> ((count_space UNIV \<Otimes>\<^sub>M borel) \<Otimes>\<^sub>M T) \<rightarrow>\<^sub>M count_space UNIV" |
lemma JVM_rnd:
assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
shows
"sc_scheduler
JVM_final (sc_mexec P) convert_RA
(JVM_rnd.random_scheduler P) (pick_wakeup_via_sel (\<lambda>s P. rm_sel s (\<lambda>(k,v). P k v))) (\<lambda>_ _. True)
(sc_jvm_state_invar P \<Phi>)" |
lemma detAB_Znm:
assumes A: "A \<in> carrier_mat n m"
and B: "B \<in> carrier_mat m n"
shows "det (A*B) = (\<Sum>(f, \<pi>)\<in>Z n m. signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i)))" |
lemma dg_Rel_is_arr_ArrValE:
assumes "T : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B" and "ab \<in>\<^sub>\<circ> T\<lparr>ArrVal\<rparr>"
obtains a b
where "ab = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>)" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>)" |
theorem CK_nf_real_card:
shows "card ((\<lambda> f. f RRR) ` {f . CK_nf f}) = 14" |
lemma gauss_cnj_diff [simp]:
\<open>cnj (x - y) = cnj x - cnj y\<close> |
lemma kronecker_one:
shows "kronecker_product ((1\<^sub>m x)::'a :: ring_1 mat) (1\<^sub>m y) = 1\<^sub>m (x*y)" |
lemma map_rsuml_plus_oracle: includes lifting_syntax shows
"(id ---> rsuml ---> (map_spmf (map_prod lsumr id))) (oracle1 \<oplus>\<^sub>O (oracle2 \<oplus>\<^sub>O oracle3)) =
((oracle1 \<oplus>\<^sub>O oracle2) \<oplus>\<^sub>O oracle3)" |
lemma final_lemma2:
"E\<noteq>{}
==> (\<Inter>v \<in> V. \<Inter>e \<in> E. {s. ((s \<in> reachable v) = ((root,v) \<in> REACHABLE))}
\<inter> nmsg_eq 0 e) \<subseteq> final" |
lemma take_bv_prepend: "take n (bv_prepend n b x) = bv_prepend n b []" |
lemma "(x::real) < y ==> \<not> y < x" |
lemma col__image_spec:
assumes "Col A B X"
shows "X X ReflectL A B" |
lemma Tag_Nonce: "Tag X \<noteq> Nonce X'" |
lemma leg_in_hom [intro]:
shows "\<guillemotleft>leg0 : apex \<rightarrow> src\<guillemotright>"
and "\<guillemotleft>leg1 : apex \<rightarrow> trg\<guillemotright>" |
lemma shift_zero_id[simp]: "shift 0 c \<tau> = \<tau>" |
lemma fds_of_real_higher_deriv [simp]:
"(fds_deriv ^^ n) (fds_of_real f) = fds_of_real ((fds_deriv ^^ n) f)" |
lemma fps_conv_radius_sin [simp]:
fixes c :: "'a :: {banach, real_normed_field, field_char_0}"
shows "fps_conv_radius (fps_sin c) = \<infinity>" |
lemma progress_le_ts:
assumes "\<And>t. t \<in> set ts \<Longrightarrow> t \<in> tfin"
shows "progress phi ts \<le> length ts" |
lemma path_lverts_merge_sup_aux:
assumes "list_dtree (Node r xs)" and "t1 \<in> fst ` fset xs" and "a \<in> dlverts t1"
and "ffold (merge_f r xs) [] xs = (v1, e1) # ys"
shows "path_lverts t1 a \<subseteq> path_lverts (dtree_from_list v1 ys) a" |
lemma map_lc_map_lc[simp]: "map_lc p1 (map_lc p2 f) = map_lc (p1 \<circ> p2) f" |
lemma Rep_formula_sup:
"Rep_formula (x \<squnion> y) = Rep_formula x \<union> Rep_formula y" |
lemma convex_on_realI:
assumes "connected A"
and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
shows "convex_on A f" |
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