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lemma compE\<^sub>2_not_Nil[simp]: "compE\<^sub>2 e \<noteq> []" |
lemma nat_to_pr_lm_5: "c_fst n mod 7 = 5 \<Longrightarrow> nat_to_pr n x = (c_f_pair (nat_to_pr (c_fst (c_snd n))) (nat_to_pr (c_snd (c_snd n)))) x" |
lemma uc_unc [simp]: "x \<parallel> U \<parallel> x \<parallel> U = x \<parallel> U" |
lemma coherent_Unity:
shows "coherent \<^bold>\<I>" |
lemma success_map_entryI [success_intros]:
"i < length h a \<Longrightarrow> success (map_entry i f a) h" |
lemma get_ancestors_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_get_ancestors_wf [instances]:
"l_get_ancestors_wf heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel known_ptr known_ptrs type_wf
get_ancestors get_ancestors_locs get_child_nodes get_parent" |
lemma ZN_atLeastAtMost [transfer_rule]:
"(ZN ===> ZN ===> rel_set ZN) atLeastAtMost atLeastAtMost" |
lemma path_image_semicircle_Im_le:
assumes "r \<ge> 0"
shows "path_image (part_circlepath c r pi (2 * pi)) =
sphere c r \<inter> {s. Im s \<le> Im c}" |
lemma project_enc_extend:
fixes x I
defines "n \<equiv> length I"
defines "z \<equiv> \<lambda>n. (any, replicate n False)"
defines "I' \<equiv> Inr (positions_in_row (x @- sconst (z (Suc n))) 0) # I"
assumes wf: "wf_interp w I"
assumes enc: "fin_cut_same (z n) (map \<pi> x) @ replicate m (z n) \<in> enc (w, I)"
assumes nonempty: "\<forall>(_, x) \<in> set x. x \<noteq> []"
shows "x \<in> enc (w, I')" |
lemma \<phi>_in_terms_of_\<eta>o:
assumes "D.ide y" and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>"
shows "\<phi> y f = G f \<cdot>\<^sub>D \<eta>o y" |
lemma gen_gctxt_closure_sound:
fixes \<A> :: "('q, 'f) reg"
assumes "q\<^sub>c |\<notin>| \<Q>\<^sub>r \<A>" and "q\<^sub>f |\<notin>| \<Q>\<^sub>r \<A>" and "q\<^sub>c |\<notin>| fin \<A>" and "q\<^sub>c \<noteq> q\<^sub>f"
shows "\<L> (gen_ctxt_closure_reg \<F> \<A> q\<^sub>c q\<^sub>f) = {C\<langle>s\<rangle>\<^sub>G | C s. funas_gctxt C \<subseteq> fset \<F> \<and> s \<in> \<L> \<A>}" |
lemma vector_choose_dist:
assumes "0 \<le> c"
obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c" |
lemma psubstT_zer[simp]:
assumes "snd ` (set txs) \<subseteq> var" and "fst ` (set txs) \<subseteq> trm"
shows "psubstT zer txs = zer" |
lemma reachable_constraints_wf:
assumes P:
"\<forall>T \<in> set P. wellformed_transaction T"
"\<forall>T \<in> set P. wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s' arity (trms_transaction T)"
and A: "A \<in> reachable_constraints P"
shows "wf\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)" |
lemma ideal_of_UN: "\<I> (\<Union> (A ` J)) = (\<Inter>j\<in>J. \<I> (A j))" |
lemma gen_subalgebra_sigma_sets:
assumes "G \<subseteq> sets M"
and "sigma_algebra (space M) G"
shows "sets (gen_subalgebra M G) = G" |
lemma intersect_tsI [intro!]: "\<lbrakk>c \<in> A k; c \<in> B k\<rbrakk> \<Longrightarrow> c \<in> (A \<Rightarrow>\<inter> B) k" |
lemma merge_root_if_contr:
"\<lbrakk>\<And>r1 t2 e2. is_subtree (Node r1 {|(t2,e2)|}) t1 \<Longrightarrow> rank (rev r1) \<le> rank (rev (Dtree.root t2));
is_subtree (Node v {|(t2,e2)|}) (merge t1); rank (rev (Dtree.root t2)) < rank (rev v)\<rbrakk>
\<Longrightarrow> Node v {|(t2,e2)|} = merge t1" |
lemma IF1a: "Fr_1b \<phi> \<Longrightarrow> Int_1a(\<I>\<^sub>F \<phi>)" |
lemma (in Ring) ideal_n_prod_primeTr:"prime_ideal R P \<Longrightarrow>
(\<forall>k \<le> n. ideal R (J k)) \<longrightarrow> (ideal_n_prod R n J \<subseteq> P) \<longrightarrow>
(\<exists>i \<le> n. (J i) \<subseteq> P)" |
lemma virtual_memop_step_simulates_direct_memop_step:
assumes step:
"(is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is', \<theta>', x', m', \<D>', \<O>', \<R>', \<S>')"
shows "(is, \<theta>, x, m, \<D>, \<O>, (), \<S>) \<rightarrow>\<^sub>v (is', \<theta>', x', m', \<D>', \<O>', (), \<S>')" |
lemma even_square_cong_4_nat: "even (x::nat) \<Longrightarrow> [x ^ 2 = 0] (mod 4)" |
lemma op5_left5: "\<lbrakk>us \<noteq> \<bottom>; v \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> op5\<cdot>pat\<cdot>(left5\<cdot>pat\<cdot>us)\<cdot>v = left5\<cdot>pat\<cdot>(us :@ [:v:])" |
lemma normalize_ports_generic:
assumes n: "normalized_nnf_match m"
and normalize_pos: "\<And>m. normalized_nnf_match m \<Longrightarrow> \<not> has_disc_negated disc False m \<Longrightarrow>
match_list \<gamma> (normalize_pos m) a p \<longleftrightarrow> matches \<gamma> m a p"
and rewrite_neg: "\<And>m. normalized_nnf_match m \<Longrightarrow>
matches \<gamma> (rewrite_neg m) a p = matches \<gamma> m a p"
and noNeg: "\<And>m. normalized_nnf_match m \<Longrightarrow> \<not> has_disc_negated disc False (rewrite_neg m)"
shows
"match_list \<gamma> (normalize_ports_generic normalize_pos rewrite_neg m) a p \<longleftrightarrow> matches \<gamma> m a p" |
lemma round_diff_minimal: "\<bar>z - of_int (round z)\<bar> \<le> \<bar>z - of_int m\<bar>"
for z :: "'a::floor_ceiling" |
lemma (in order) descending_chain_list_distinct:
assumes "descending_chain_list S"
shows "distinct S" |
lemma f_nth_set:
"\<lbrakk> f (xs ! n) = v; n < length xs \<rbrakk> \<Longrightarrow> v \<in> f ` set xs" |
lemma rrstep_basicI [intro]:
"(l, r) \<in> \<R> \<Longrightarrow> (l, r) \<in> rrstep \<R>" |
lemma gtt_of_gtt_rel_impl_sound:
"gtt_of_gtt_rel_impl \<F> Rs g = Some g' \<Longrightarrow> gtt_of_gtt_rel \<F> Rs g = Some g'' \<Longrightarrow> agtt_lang g' = agtt_lang g''" |
lemma SFAssInitThrowReds:
assumes e\<^sub>2_steps: "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2,l\<^sub>2,sh\<^sub>2),False\<rangle>"
and cF: "P \<turnstile> C has F,Static:t in D"
and nDone: "\<nexists>sfs. sh\<^sub>2 D = Some (sfs, Done)"
and INIT_steps: "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>2,l\<^sub>2,sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D}:=e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
(*<*)(is "(?x, ?z) \<in> (red P)\<^sup>*") |
lemma has_laplaceI:
assumes "((\<lambda>t. exp (t *\<^sub>R - s) * f t) has_integral L) {0..}"
shows "(f has_laplace L) s" |
lemma in_ofail [simp]:
"ofail s \<noteq> Some v" |
lemma Says_Kas_message_form:
"\<lbrakk> Says Kas A \<lbrace>Crypt K \<lbrace>Key authK, Agent Peer, Ta\<rbrace>, authTicket\<rbrace>
\<in> set evs;
evs \<in> kerbV \<rbrakk>
\<Longrightarrow> authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys \<and>
authTicket = (Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace>) \<and>
K = shrK A \<and> Peer = Tgs" |
lemma cwiseext_compat_cons: "cwiseext gt ys xs \<Longrightarrow> cwiseext gt (x # ys) (x # xs)" |
lemma cf_cn_cov_lcomp_ArrMap_app[cat_cs_simps]:
assumes "category \<alpha> \<CC>"
and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
and "f \<in>\<^sub>\<circ> op_cat \<AA>\<lparr>Arr\<rparr>"
and "g \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
shows "cf_cn_cov_lcomp \<CC> \<SS> \<FF>\<lparr>ArrMap\<rparr>\<lparr>f, g\<rparr>\<^sub>\<bullet> = \<SS>\<lparr>ArrMap\<rparr>\<lparr>\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>, g\<rparr>\<^sub>\<bullet>" |
lemma conjugation_of_cycle:
assumes "cycle cs" and "bij p"
shows "p \<circ> (cycle_of_list cs) \<circ> (inv p) = cycle_of_list (map p cs)" |
lemma ANR_path_component_ANR:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<Longrightarrow> ANR(path_component_set S x)" |
lemma All_surj_conv: assumes "surj f" shows "(\<forall>x. P (f x)) \<longleftrightarrow> (\<forall>y. P y)" |
lemma power_down_even_nonneg: "even n \<Longrightarrow> 0 \<le> power_down p x n" |
lemma all_intro: "(\<And>a. P a x) \<Longrightarrow> (\<^bold>\<forall>a. P a) x" |
lemma split_face_edges_or: "(f12, f21) = split_face oldF ram1 ram2 newVertexList \<Longrightarrow> pre_split_face oldF ram1 ram2 newVertexList \<Longrightarrow> (a, b) \<in> edges oldF \<Longrightarrow> (a,b) \<in> edges f12 \<or> (a,b) \<in> edges f21" |
lemma intersect_mult_set_block_subset_iff_2:
assumes "blv \<in># \<B>" and "p \<subseteq> blv" and "\<Lambda> \<ge> 2" and "card p = 2"
shows "p \<in># \<Sum>\<^sub>#{# mset_set {y .y \<subseteq> blv \<inter> b2 \<and> card y = 2} .b2 \<in># (\<B> - {#blv#})#}" |
lemma real_arch_inverse: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)" |
lemma w_values_vs_type_all_start_heap_obs:
assumes wf: "wf_syscls P"
shows "w_values P (vs_type_all P) (map snd (lift_start_obs h.start_tid h.start_heap_obs)) = vs_type_all P"
(is "?lhs = ?rhs") |
lemma equiv_subseteq_in_sym:
"\<lbrakk> r `` X \<subseteq> X; (x, y) \<in> r; y \<in> X; equiv Y r; X \<subseteq> Y \<rbrakk> \<Longrightarrow> x \<in> X" |
lemma mem_cong[rule_format]:
"x \<in> set list \<longrightarrow> (f x \<in> set (map f list))" |
lemma idmap_bij:"bij_to (idmap A) A A" |
lemma weak_barbed_correspondence_simulation_and_closures:
fixes Rel :: "('proc \<times> 'proc) set"
and CWB :: "('proc, 'barbs) calculusWithBarbs"
assumes corrSim: "weak_barbed_correspondence_simulation Rel CWB"
shows "weak_barbed_correspondence_simulation (Rel\<^sup>=) CWB"
and "weak_barbed_correspondence_simulation (Rel\<^sup>+) CWB"
and "weak_barbed_correspondence_simulation (Rel\<^sup>*) CWB" |
lemma xconf_raise_if: "xconf h x \<Longrightarrow> xconf h (raise_if b xcn x)" |
lemma image_rev_conc[simp]: "rev ` (A @@ B) = rev ` B @@ rev ` A" |
lemma (in Ring) prime_ideal_cont1:"\<lbrakk>ideal R A; \<forall>i \<le> (n::nat).
prime_ideal R (P i); A \<subseteq> \<Union> {X. (\<exists>i \<le> n. X = (P i))} \<rbrakk> \<Longrightarrow>
\<exists>i\<le> n. A\<subseteq>(P i)" |
lemma e_less_eq_lem1:
"\<lbrakk>\<not> e_less_eq e a;e_less_eq e (a + b)\<rbrakk> \<Longrightarrow> e_less_eq e b" |
lemma epair_eqD: "epair e = Some (x,y) \<Longrightarrow> (x\<noteq>y \<and> e={x,y})" |
lemma invariant_iff_pos_invariant_and_compl_pos_invariant:
shows "invariant M \<longleftrightarrow> positively_invariant M \<and> positively_invariant (X-M)" |
lemma (in into_set) homf_preserves_comp:
assumes f: "f \<in> Ar"
and g: "g \<in> Ar"
and fg: "Cod f = Dom g"
shows "Hom(A,_)\<^bsub>\<a>\<^esub> (g \<bullet> f) = (Hom(A,_)\<^bsub>\<a>\<^esub> g) \<odot> (Hom(A,_)\<^bsub>\<a>\<^esub> f)" |
theorem banach_fix:\<comment> \<open>TODO: rename to \<open>Banach_fix\<close>\<close>
assumes s: "complete s" "s \<noteq> {}"
and c: "0 \<le> c" "c < 1"
and f: "f ` s \<subseteq> s"
and lipschitz: "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
shows "\<exists>!x\<in>s. f x = x" |
lemma topfloat_simps:
"sign (topfloat::('e, 'f)float) = 0"
"exponent (topfloat::('e, 'f)float) = emax TYPE(('e, 'f)float) - 1"
"fraction (topfloat::('e, 'f)float) = 2 ^ (fracwidth TYPE(('e, 'f)float)) - 1"
and bottomfloat_simps:
"sign (bottomfloat::('e, 'f)float) = 1"
"exponent (bottomfloat::('e, 'f)float) = emax TYPE(('e, 'f)float) - 1"
"fraction (bottomfloat::('e, 'f)float) = 2 ^ (fracwidth TYPE(('e, 'f)float)) - 1" |
lemma mcont_bind_spmf [cont_intro]:
assumes f: "mcont luba orda lub_spmf (ord_spmf (=)) f"
and g: "\<And>y. mcont luba orda lub_spmf (ord_spmf (=)) (g y)"
shows "mcont luba orda lub_spmf (ord_spmf (=)) (\<lambda>x. bind_spmf (f x) (\<lambda>y. g y x))" |
lemma omit_oob:
assumes \<open>length ps \<le> v\<close>
shows \<open>omit ({v} \<union> xs) ps = omit xs ps\<close> |
lemma (in encoding) rel_with_target_impl_reflC_transC_TRel_is_strong_reduction_simulation:
fixes TRel :: "('procT \<times> 'procT) set"
and Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"
assumes sim: "strong_reduction_simulation Rel (STCal Source Target)"
and target: "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"
and trel: "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel
\<longrightarrow> (T1, T2) \<in> TRel\<^sup>*"
shows "strong_reduction_simulation (TRel\<^sup>*) Target" |
lemma funpow_dist_0:
assumes "x = y" shows "funpow_dist f x y = 0" |
lemma gcard_image: "inj_on f X \<Longrightarrow> gcard (f ` X) = gcard X" |
lemma rel_inv_client_map_drop_map: "(rel o inv client_map o drop_map i o inv sysOfClient) =
rel o sub i o client" |
lemma contracting_inverse:
assumes \<open>dilating f sub r\<close>
shows \<open>contracting (dil_inverse f) r sub f\<close> |
lemma factor_bound_smult: assumes f: "f \<noteq> 0" and d: "d \<noteq> 0"
and dvd: "g dvd smult d f" and deg: "degree g \<le> n"
shows "\<bar>coeff g k\<bar> \<le> \<bar>d\<bar> * factor_bound f n" |
lemma inj_Init[simp]: "inj_on Init A" |
theorem moebius_inverse:
assumes "a * d \<noteq> b * c" "c * z + d \<noteq> 0"
shows "moebius d (-b) (-c) a (moebius a b c d z) = z" |
lemma prefixes_length_mono: "e_length (prefixes j t) < e_length (prefixes j (Suc t))" |
lemma mergeable_envs_listset: "mergeable_envs (length As) (listset As)" |
lemma concat_last_suf: "ws \<noteq> \<epsilon> \<Longrightarrow> last ws \<le>s concat ws" |
lemma a_supdist_var: "ad (x + y) \<le> ad x" |
lemma strategy_attracts_viaI [intro]:
assumes "\<And>P. vmc_path G P v0 p \<sigma> \<Longrightarrow> visits_via P A W"
shows "strategy_attracts_via p \<sigma> v0 A W" |
lemma poly_vars: assumes eq: "\<And> w. w \<in> poly_vars p \<Longrightarrow> f w = g w"
shows "poly_subst f p = poly_subst g p" |
lemma br'_rcm_aux:
assumes A: "(Q,W,rcm)\<in>br'_invar \<delta>" "q\<in>W"
shows "{lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> Suc 0}
= {lhs r | r. r\<in>\<delta> \<and> q\<in>set (rhsq r) \<and> set (rhsq r) \<subseteq> (Q - (W-{q}))}" |
lemma (in Corps) Pseq_decompos:"\<lbrakk>valuation K v; PolynRg R (Vr K v) X;
F n \<in> carrier R; deg R (Vr K v) X (F n) \<le> an (Suc d)\<rbrakk>
\<Longrightarrow> F n = ((Pseql\<^bsub>R X K v d\<^esub> F) n) \<plusminus>\<^bsub>R\<^esub> ((Pseqh\<^bsub>R X K v d\<^esub> F) n)" |
lemma augment_impl_refine[sepref_fr_rules]:
"(uncurry2 (augment_imp N), uncurry2 (PR_CONST augment_cf_impl))
\<in> (asmtx_assn N id_assn)\<^sup>d *\<^sub>a (is_path)\<^sup>k *\<^sub>a (pure Id)\<^sup>k \<rightarrow>\<^sub>a asmtx_assn N id_assn" |
lemma jkbpTn_jkbpCn_represents: "jkbpT\<^bsub>n\<^esub> jkbpC = jkbpC\<^bsub>n\<^esub>" |
lemma \<R>\<^sub>GI [intro]:
assumes
bij_\<alpha>: "bij \<alpha>" and
none_case: "\<And>r. s r = None \<Longrightarrow> s' (\<alpha> r) = None" and
some_case: "\<And>r \<sigma> \<tau> e. s r = Some (\<sigma>,\<tau>,e) \<Longrightarrow> s' (\<alpha> r) = Some (\<R>\<^sub>L \<alpha> \<beta> (\<sigma>,\<tau>,e))"
shows
"\<R>\<^sub>G \<alpha> \<beta> s = s'" |
lemma cong_respects_seq\<^sub>P:
assumes "seq T U" and "T \<^sup>*\<sim>\<^sup>* T'" and "U \<^sup>*\<sim>\<^sup>* U'"
shows "seq T' U'" |
lemma MatchOr: "matches \<gamma> (MatchOr m1 m2) a p \<longleftrightarrow> matches \<gamma> m1 a p \<or> matches \<gamma> m2 a p" |
lemma CACH_BodiesDerivable[rule_format]:
"\<lbrakk> mbody_is C m (par, code, l); compileProg F; TP_MST \<Sigma>; TP \<Sigma> F\<rbrakk>
\<Longrightarrow> \<exists> n . MST\<down>(C,m) = Some(mkSPEC(Cachera n) emp) \<and>
deriv [] C m l (Cachera n)" |
lemma hm_exch_length[simp]: "hm_length (hm_exch hm i j) = hm_length hm" |
lemma "\<lfloor>\<^bold>\<box>\<^bold>\<exists>\<^sup>E (Q::\<up>\<langle>\<zero>\<rangle>) \<^bold>\<leftrightarrow> ((\<lambda>X. \<^bold>\<box>\<^bold>\<exists>\<^sup>E \<^bold>\<down>X) Q)\<rfloor>" |
lemma sd_4r_correct:
assumes "s\<^sub>o - s\<^sub>e > safe_distance_4r"
assumes "other.s \<delta> \<le> u_max"
assumes "\<delta> \<le> other.t_stop"
assumes "a\<^sub>o > a\<^sub>e"
shows "no_collision_react {0..}" |
lemma oreach_pi_translation:
"pi`(oreach S) = oreach (S(| obs := pi o (obs S) |))" |
lemma orthogonal_matrix_intro:
fixes A::"real^'n^'n"
assumes p: "(pairwise orthogonal (columns A))"
and n: "\<forall>i. norm (column i A) = 1"
and c: "card (columns A) = ncols A" (*We need that premise to avoid the case that column i A = column j A when i \<noteq> j*)
shows "orthogonal_matrix A" |
lemma hull_hull [simp]: "S hull (S hull s) = S hull s" |
lemma restriction_to_subgroup_is_hom :
fixes H :: "'g set"
assumes subgrp: "Group.Subgroup G H"
shows "FGModuleHom H smult V smult' T" |
lemma symcl_converse:
"(A\<^sup>\<leftrightarrow>)\<inverse> = A\<^sup>\<leftrightarrow>" |
lemma (in is_ft_dghm) ft_dghm_op_dghm_is_ft_dghm:
"op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> op_dg \<BB>" |
lemma Some_Inf:
"Some (\<Sqinter>A) = \<Sqinter>(Some ` A)" |
lemma filter_zmset_conclude_predicate: "0 < zcount {# x \<in>#\<^sub>z M. P x #} x \<Longrightarrow> 0 < zcount M x \<Longrightarrow> P x" |
lemma shift3[simp]:
assumes n: "n < length cl" and I: "I \<subseteq> {..< brn (cl!n)}"
and ii: "ii \<in> shift cl n I"
shows "ii < brnL cl (length cl)" |
lemma B_theorem_3: "h i = \<Theta> i * f i" "h i = ff (lead_coeff (H i))" |
lemma clique_edges_outside :
assumes "uwellformed G" and "finite (uverts G)"
and p2: "2 \<le> p" and pn: "p \<le> card (uverts G)" and n: "n = card(uverts G)"
and C: "uclique C G (p-1)" and C_max: "(\<forall>C q. uclique C G q \<longrightarrow> q \<le> p-1)"
and IH: "\<And>G y. y < n \<Longrightarrow> finite (uverts G) \<Longrightarrow> uwellformed G \<Longrightarrow> \<forall>C p'. uclique C G p' \<longrightarrow> p' < p
\<Longrightarrow> 2 \<le> p \<Longrightarrow> card (uverts G) = y \<Longrightarrow> real (card (uedges G)) \<le> (1 - 1 / real (p - 1)) * real (y\<^sup>2) / 2"
shows "card {e \<in> uedges G. e \<subseteq> uverts G - uverts C} \<le> (1 - 1 / (p-1)) * (n - p + 1) ^ 2 / 2" |
lemma quot_one_finiteI [intro]:
shows "finite (UNIV // \<approx>{[]})" |
lemma impl_OrdSum_first: "Abs_OrdSum (x, 1) l\<rightarrow> Abs_OrdSum (y, 1) = Abs_OrdSum (x l\<rightarrow> y, 1)" |
lemma lenlex_append2 [simp]:
assumes "irrefl R"
shows "(us @ xs, us @ ys) \<in> lenlex R \<longleftrightarrow> (xs, ys) \<in> lenlex R" |
lemma vcg_prop[intro]:
"\<lbrace>\<langle>P\<rangle>\<rbrace> c" |
lemma change_eval_neq:
fixes x y:: "real"
assumes "((aEvalUni (NeqUni (a,b,c)) x \<noteq> aEvalUni (NeqUni (a,b,c)) y) \<and> x < y)"
shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)" |
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