instruct stringclasses 1
value | input stringlengths 25 4.64k | output stringlengths 17 413 |
|---|---|---|
Given a lemma, please prove it. | lemma small_step_terminated:
"s \<rightarrow>\<^bsub>x\<^esub> s' \<Longrightarrow> atCs (fst s) = {} \<Longrightarrow> atCs (fst s') = {}" | by (induct pred: small_step) auto |
Given a lemma, please prove it. | theorem HEndPhase2_HInv3:
"\<lbrakk> HEndPhase2 s s' p; HInv3 s \<rbrakk> \<Longrightarrow> HInv3 s'" | by(auto simp add: HInv3_def dest: EndPhase2_HInv3) |
Given a lemma, please prove it. | lemma retract_of_closed:
fixes S :: "'a :: t2_space set"
shows "\<lbrakk>closed T; S retract_of T\<rbrakk> \<Longrightarrow> closed S" | by (metis closedin_retract closedin_closed_eq) |
Given a lemma, please prove it. | lemma fmap_below_add_restrI:
assumes "x f|` (-S) \<sqsubseteq> y f|` (-S)"
and "x f|` S \<sqsubseteq> z f|` S"
shows "x \<sqsubseteq> y ++\<^bsub>S\<^esub> z" | using assms
by (auto intro!: fun_belowI dest:env_restr_belowD simp add: lookup_override_on_eq) |
Given a lemma, please prove it. | lemma ground_subst_no_var[intro]: "ground (subst_range s) \<Longrightarrow> x \<notin> range_vars s" | using ground_range_vars[of s]
by blast |
Given a lemma, please prove it. | lemma HPhase0Read_maxBalInp:
assumes act: "HPhase0Read s s' q d"
and asm3: "maxBalInp s b v"
shows "maxBalInp s' b v" | proof(auto simp add: maxBalInp_def) fix bk assume bk: "bk \<in> allBlocks s'" and b_bal: "b\<le> bal bk" from subsetD[OF HPhase0Read_allBlocks[OF act] bk] asm3 b_bal show "inp bk = v"
by(auto simp add: maxBalInp_def)
qed |
Given a lemma, please prove it. | lemma outside_frontier_eq_complement_closure:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
assumes "bounded S" "convex S"
shows "outside(frontier S) = - closure S" | by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure outside_subset_convex subset_antisym) |
Given a lemma, please prove it. | lemma AssertionsImplyMethInvariants:
"\<lbrakk> G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I; A s0 s\<rbrakk> \<Longrightarrow> I s0 s (heap s)" | by (drule AssertionsImplyInvariantsAux, fast) |
Given a lemma, please prove it. | lemma Star_string:
assumes "s \<in> star A"
shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)" | using assms
by (metis in_star_iff_concat subsetD) |
Given a lemma, please prove it. | lemma is_subseq_closed:
assumes "s \<in> closed_seqs Zp"
assumes "is_subseq_of s s'"
shows "s' \<in> closed_seqs Zp" | apply(rule closed_seqs_memI)
using is_subseq_ind assms closed_seqs_memE
by metis |
Given a lemma, please prove it. | lemma condition_split:
"P (condition C a b s) = ((((C s) \<longrightarrow> P (a s)) \<and> (\<not> (C s) \<longrightarrow> P (b s))))" | apply (clarsimp simp: condition_def) done |
Given a lemma, please prove it. | lemma siterate_eq_scons[iff]: "siterate f s = a ## w \<longleftrightarrow> s = a \<and> siterate f (f s) = w" | using siterate.ctr stream.inject
by metis |
Given a lemma, please prove it. | theorem while_rule_lemma:
assumes invariant: "!!s. P s ==> b s ==> P (c s)"
and terminate: "!!s. P s ==> \<not> b s ==> Q s"
and wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
shows "P s \<Longrightarrow> Q (while b c s)" | using wf
apply (induct s)
apply simp
apply (subst while_unfold)
apply (simp add: invariant terminate) done |
Given a lemma, please prove it. | lemma matchI[intro?]:
assumes "validTrans (Trans s1 a1 ou1 s1')"
and "consume (Trans s1 a1 ou1 s1') vl1 vl1'" and "\<gamma> (Trans s a ou s') = \<gamma> (Trans s1 a1 ou1 s1')"
and "\<gamma> (Trans s a ou s') \<Longrightarrow> g (Trans s a ou s') = g (Trans s1 a1 ou1 s1')"
and "\<Delta> s' vl' s1' vl1'"
shows "match \<D... | unfolding match_def
using assms
by auto |
Given a lemma, please prove it. | lemma (in poly_set_iteratei) dres_ne_bot_iterate[refine_transfer]:
assumes A: "\<And>x s. f x s \<noteq> dSUCCEED"
shows "iteratei r c (\<lambda>x s. dbind s (f x)) (dRETURN s) \<noteq> dSUCCEED" | unfolding iteratei_def it_to_list_def it_to_it_def
apply (rule dres_foldli_ne_bot)
by (simp_all add: A) |
Given a lemma, please prove it. | lemma list_all_add_6E[elim]: "list_all (\<lambda>(acn, s). s \<le> Suc q) xs
\<Longrightarrow> list_all (\<lambda>(acn, s). s \<le> q + (2 * n + 6)) xs" | by(auto simp: list_all_length) |
Given a lemma, please prove it. | lemma Abs_type':
assumes H: "\<Gamma> \<turnstile> (\<lambda>x:S. s) : U \<rightarrow> U'"
and H': "x \<sharp> \<Gamma>"
obtains S'
where "\<Gamma> \<turnstile> U <: S"
and "(VarB x S) # \<Gamma> \<turnstile> s : S'"
and "\<Gamma> \<turnstile> S' <: U'" | using H subtype_reflexivity_from_typing [OF H] H'
by (rule Abs_type) |
Given a lemma, please prove it. | theorem invertible_is_bijective:
assumes dom: "\<alpha> \<in> S \<rightarrow>\<^sub>E S"
shows "invertible \<alpha> \<longleftrightarrow> bij_betw \<alpha> S S" | proof - from dom interpret map \<alpha> S S
by unfold_locales show ?thesis
by (auto simp add: bij_betw_iff_has_inverse invertible_def)
qed |
Given a lemma, please prove it. | lemma convex_rel_interior_iff2:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "S \<noteq> {}"
shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" | using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
by auto |
Given a lemma, please prove it. | lemma simplicial_chain_diff:
"\<lbrakk>simplicial_chain p S c1; simplicial_chain p S c2\<rbrakk> \<Longrightarrow> simplicial_chain p S (c1 - c2)" | unfolding simplicial_chain_def
by (meson UnE keys_diff subset_iff) |
Given a lemma, please prove it. | lemma additiveI[intro]:
"\<lbrakk> \<And>P Q s. \<lbrakk> sound P; sound Q \<rbrakk> \<Longrightarrow> t (\<lambda>s. P s + Q s) s = t P s + t Q s \<rbrakk> \<Longrightarrow>
additive t" | unfolding additive_def
by(blast) |
Given a lemma, please prove it. | lemma pre_zeta_aux_eq_pre_zeta:
assumes "Re s > -2 * real n" and "a > 0"
shows "pre_zeta_aux n a s = pre_zeta a s" | unfolding pre_zeta_def
proof (intro pre_zeta_aux_eq') from assms show "- 2 * real (nat (1 - \<lceil>Re s / 2\<rceil>)) < Re s"
by linarith
qed (insert assms, simp_all) |
Given a lemma, please prove it. | lemma homeomorphic_to_disc_imp_simply_connected:
assumes "S = {} \<or> S homeomorphic ball (0::complex) 1"
shows "simply_connected S" | using assms homeomorphic_simply_connected_eq convex_imp_simply_connected
by auto |
Given a lemma, please prove it. | lemma continuous_on_components_open:
fixes S :: "'a::real_normed_vector set"
assumes "open S "
"\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
shows "continuous_on S f" | using continuous_on_components open_imp_locally_connected assms
by blast |
Given a lemma, please prove it. | lemma exhaustive_fusion_code:
"exhaustive_fusion g s =
(list.has_next g s \<and>
(let (x, s') = list.next g s
in \<not> proper_interval None (Some x) \<and> exhaustive_above_fusion g x s'))" | unfolding exhaustive_fusion_def exhaustive_above_fusion_def
by(subst (1) list.unfoldr.simps)(simp add: split_beta) |
Given a lemma, please prove it. | lemma countable_image_inj_Int_vimage:
"\<lbrakk>inj_on f S; countable A\<rbrakk> \<Longrightarrow> countable (S \<inter> f -` A)" | by (meson countable_image_inj_on countable_subset image_subset_iff_subset_vimage inf_le2 inj_on_Int) |
Given a lemma, please prove it. | lemma fbox_test[simp]: "(\<lambda>s. ( |\<questiondown>P?] Q) s) = (\<lambda>s. P s \<longrightarrow> Q s)" | unfolding fbox_def test_def
by simp \<comment> \<open> Assignments \<close> |
Given a lemma, please prove it. | lemma colours_flip_fM[simp]:
"fl \<noteq> sys_fM s \<Longrightarrow> black b (s(sys := (s sys)\<lparr>fM := fl, mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>)) \<longleftrightarrow> white b s \<and> \<not>grey b s" | unfolding black_def white_def
by (simp add: fun_upd_apply) |
Given a lemma, please prove it. | lemma length_tl_list_minus_butlast_gt_zero:
assumes "length s < length t" and "strict_prefix (butlast s) t" and "length s > 0"
shows "length (tl (t - (butlast s))) > 0" | using assms
by (metis Nitpick.size_list_simp(2) butlast_snoc hd_Cons_tl length_butlast length_greater_0_conv length_tl less_trans nat_neq_iff strict_prefix_minus_not_empty prefix_order.dual_order.strict_implies_order prefix_concat_minus) |
Given a lemma, please prove it. | lemma assumes "s \<in> S" and "fair rs"
shows trim_in_R: "shd (trim rs s) \<in> R"
and trim_enabled: "enabled (shd (trim rs s)) s"
and trim_fair: "fair (trim rs s)" | unfolding trim_alt[OF assms] minWait_def
using LeastI_ex[OF minWait_ex[OF assms]] sdrop_fair[OF assms(2)] conjunct1[OF assms(2)[unfolded fair_def]]
by simp_all (metis subsetD snth_sset) |
Given a lemma, please prove it. | lemma SpecIf:
"\<lbrakk>P \<subseteq> {s. (b s \<longrightarrow> f s \<in> Q) \<and> (\<not> b s \<longrightarrow> g s \<in> Q \<and> h s \<in> Q)}\<rbrakk> \<Longrightarrow>
\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F\<^esub> P Spec (if_rel b f g h) Q,A" | apply (rule Spec)
apply (auto simp add: if_rel_def) done |
Given a lemma, please prove it. | lemma rfn_tool12_2:"\<lbrakk>0 < (s::nat); (i::nat) mod s = s - Suc 0\<rbrakk> \<Longrightarrow>
(Suc i) mod s = 0" | apply (insert div_mult_mod_eq [THEN sym, of "i" "s"])
apply (insert add_Suc_right [THEN sym, of "i div s * s" "i mod s"])
apply simp done |
Given a lemma, please prove it. | lemma wlp_strengthen_modset: "wlp \<pi> c Q s \<Longrightarrow> wlp \<pi> c (\<lambda>s'. Q s' \<and> modifies (lhsv \<pi> c) s' s) s" | unfolding wlp_def
by (blast intro: modifies_lhsv) |
Given a lemma, please prove it. | lemma streams_sset:
assumes "s \<in> streams A"
shows "sset s \<subseteq> A" | proof fix x assume "x \<in> sset s" from this \<open>s \<in> streams A\<close> show "x \<in> A"
by (induct s) (auto intro: streams_shd streams_stl)
qed |
Given a lemma, please prove it. | lemma Astack_restr_stack_below:
"Astack (restr_stack V S) \<sqsubseteq> Astack S" | by (induction V S rule: restr_stack.induct) auto |
Given a lemma, please prove it. | lemma closure_inside_subset:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closure(inside S) \<subseteq> S \<union> inside S" | by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside) |
Given a lemma, please prove it. | lemma hmac_trans_lkr:
"hmac X K \<in> parts (extr (bad s') (ik s') (chan s')) \<Longrightarrow>
K \<notin> synth (analz (extr (bad s) (ik s) (chan s))) \<Longrightarrow>
s \<in> l2_inv3 \<Longrightarrow>
(s, s') \<in> l2_lkr_others A \<union> l2_lkr_after A \<Longrightarrow>
hmac X K \<in> parts (extr (b... | apply (auto simp add: l2_defs dest!: parts_monotone [OF _ extr_insert_bad])
apply (auto intro: parts_monotone intro!:hmac_trans_lkr_aux) done |
Given a lemma, please prove it. | lemma dependent_biggerset_general:
"(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S" | using independent_bound_general[of S]
by (metis linorder_not_le) |
Given a lemma, please prove it. | lemma peek_st_split [simp]: "(\<lambda>s.. \<lambda>Y s'. P s Y s') = (\<lambda>Y s. P (store s) Y s)" | apply (rule ext)
apply (rule ext)
apply (simp (no_asm)) done |
Given a lemma, please prove it. | lemma deg_fun_superset:
fixes A::"'a set"
assumes "supp_fun s \<subseteq> A" and "finite A"
shows "deg_fun s = (\<Sum>x\<in>A. s x)" | unfolding deg_fun_def
proof (rule sum.mono_neutral_cong_left, fact, fact, rule) fix x assume "x \<in> A - supp_fun s" hence "x \<notin> supp_fun s"
by simp thus "s x = 0"
by (simp add: supp_fun_def)
qed rule |
Given a lemma, please prove it. | lemma dfoldli_simps[simp]:
"dfoldli [] c f s = dRETURN s"
"dfoldli (x#ls) c f s =
(if c s then do { s\<leftarrow>f x s; dfoldli ls c f s} else dRETURN s)" | apply (subst dfoldli.simps, simp)+ done |
Given a lemma, please prove it. | lemma card_insert_Suc:
"card (insert x S) \<le> Suc (card S)" | by (metis Suc_n_not_le_n card.infinite card_insert_if finite_insert linear) |
Given a lemma, please prove it. | lemma wlp_while_rl[vcg_comb_rules]:
assumes "\<paragraph>''invar-initial'' \<Longrightarrow> I s\<^sub>0"
assumes "\<And>s. \<lbrakk> \<paragraph>''invar-pres''; modifies modset s s\<^sub>0; I s; bval b s \<rbrakk> \<Longrightarrow> wlp \<pi> c I s"
assumes "\<And>s. \<lbrakk> \<paragraph>''invar-post''; mo... | using wlp_whileI_modset[of I, OF _ assms(2)] assms(1,3,4) unfolding WHILE_annotI_def GOAL_INDICATION_def
by auto |
Given a lemma, please prove it. | lemma Phase1or2ReadThen_HInv3_hasRead:
"\<lbrakk> \<not>hasRead s pp dd qq;
Phase1or2ReadThen s s' p d q;
pp\<noteq>p \<or> qq\<noteq>q \<or> dd\<noteq>d\<rbrakk>
\<Longrightarrow> \<not>hasRead s' pp dd qq" | by(auto simp add: hasRead_def Phase1or2ReadThen_def) |
Given a lemma, please prove it. | lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S" | by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI) |
Given a lemma, please prove it. | lemma homeomorphism_on_homeomorphic:
assumes "homeomorphism_on S f"
shows "S homeomorphic (f`S)" | using assms unfolding homeomorphism_on_def homeomorphic_def
by auto |
Given a lemma, please prove it. | lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s" | unfolding tagged_division_of_def tagged_partial_division_of_def
by auto |
Given a lemma, please prove it. | lemma stack_set_def:
"set (stack s) = dom (discovered s) - dom (finished s)" | using basic_invar
by (simp add: basic_invar_def) |
Given a lemma, please prove it. | lemma Spair_induced_automorph:
"s\<in>S \<Longrightarrow> s = case_prod Abs_induced_automorph (Spair s)" | using Spair_def someI_ex[of "\<lambda>fg. fg \<in> fundfoldpairs \<and> s = case_prod Abs_induced_automorph fg" ]
by auto |
Given a lemma, please prove it. | lemma hd_stack_in_tj_stack:
"stack s \<noteq> [] \<Longrightarrow> hd (stack s) \<in> set (tj_stack s)" | using stack_ss_tj_stack hd_in_set
by auto |
Given a lemma, please prove it. | lemma ms_of_greek_cons:
"ms_of_greek (x # s) = {# adj_msog [] s (x,[]) #} + image_mset (adj_msog [x] []) (ms_of_greek s)" | using msog_append[of "[x]" s]
by (auto simp add: adj_msog_def ms_of_greek_def accent.splits) |
Given a lemma, please prove it. | lemma F_hiding: "F(P \\ A) = {(s,X). \<exists> t. s = trace_hide t (ev`A) \<and> (t,X \<union> (ev`A)) \<in> F P} \<union>
{(s,X). s \<in> div_hide P A}" | by (subst F_def, simp only: hiding_def Rep_Abs_Hiding FAILURES_def fst_conv) |
Given a lemma, please prove it. | lemma piecewise_C1_differentiable_on_imp_continuous_on:
assumes "f piecewise_C1_differentiable_on s"
shows "continuous_on s f" | using assms
by (auto simp add: piecewise_C1_differentiable_on_def) |
Given a lemma, please prove it. | lemma borel_measurableI:
fixes f :: "'a \<Rightarrow> 'x::topological_space"
assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
shows "f \<in> borel_measurable M" | unfolding borel_def
proof (rule measurable_measure_of, simp_all) fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
using assms[of S]
by simp
qed |
Given a lemma, please prove it. | lemma connected_openin:
"connected S \<longleftrightarrow>
\<not>(\<exists>E1 E2. openin (top_of_set S) E1 \<and>
openin (top_of_set S) E2 \<and>
S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})" | unfolding connected_def openin_open disjoint_iff_not_equal
by blast |
Given a lemma, please prove it. | lemma
assumes forward: "\<And>s g p v. A g p v s \<Longrightarrow> AA g p s "
shows "\<And>xv yv P s y x s. (A g x yv ** A g y yv ** P) s \<Longrightarrow> (AA g y ** sep_True) s" | by (sep_cancel add: forward) |
Given a lemma, please prove it. | lemma step_Stuck_prop:
assumes step: "\<Gamma> \<turnstile> (c, s) \<rightarrow> (c', s')"
shows "s=Stuck \<Longrightarrow> s'=Stuck" | using step
by (induct) auto |
Given a lemma, please prove it. | lemma back_edges_tree_pathes_disjoint:
"back_edges s \<inter> (tree_edges s)\<^sup>+ = {}" | using back_edge_disc tree_path_disc
by force |
Given a lemma, please prove it. | lemma HT_mods_cong[named_ss vcg_bb cong]:
assumes "vs = vs'"
assumes "P=P'"
assumes "c=c'"
assumes "\<And>s\<^sub>0 s. modifies vs s s\<^sub>0 \<Longrightarrow> Q s\<^sub>0 s = Q' s\<^sub>0 s"
shows "HT_mods \<pi> vs P c Q = HT_mods \<pi> vs' P' c' Q'" | unfolding HT_mods_def HT_def
using assms
by (auto intro: wp_conseq) |
Given a lemma, please prove it. | lemma HPhase0Read_HInv4b_p:
"HPhase0Read s s' p d \<Longrightarrow> HInv4b s' p" | by(auto simp add: Phase0Read_def HInv4b_def) |
Given a lemma, please prove it. | lemma connected_complement_homeomorphic_interval:
fixes S :: "'a::euclidean_space set"
assumes "S homeomorphic cbox a b" "2 \<le> DIM('a)"
shows "connected(-S)" | using assms path_connected_complement_homeomorphic_interval path_connected_imp_connected
by blast |
Given a lemma, please prove it. | lemma ENR_from_Un_Int:
fixes S :: "'a::euclidean_space set"
assumes "closed S" "closed T" "ENR(S \<union> T)" "ENR(S \<inter> T)"
shows "ENR S" | by (meson ENR_from_Un_Int_gen assms closed_subset sup_ge1 sup_ge2) |
Given a lemma, please prove it. | lemma compact_simplex: "n simplex S \<Longrightarrow> compact S" | unfolding simplex
using finite_imp_compact_convex_hull
by blast |
Given a lemma, please prove it. | lemma absolutely_integrable_negligible:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "negligible S"
shows "f absolutely_integrable_on S" | using assms
by (simp add: absolutely_integrable_on_def integrable_negligible) |
Given a lemma, please prove it. | lemma V_lemma2: "Mutex \<in> {s. n s = 2} LeadsTo {s. ~ p s}" | apply (rule LeadsTo_Diff [OF LeadsTo_weaken_L Int_lower2 [THEN subset_imp_LeadsTo]])
apply (rule LeadsTo_Trans [OF V_F2 V_F3], auto) done |
Given a lemma, please prove it. | lemma resolution_blocks_subset: "S \<in># \<P> \<Longrightarrow> S \<subseteq># \<B>" | using partition_on_mset_subsets resolutionD1
by auto |
Given a lemma, please prove it. | lemma restrictA_noop: "domA \<Gamma> \<subseteq> S \<Longrightarrow> restrictA S \<Gamma> = \<Gamma>" | unfolding restrict_eq
by (induction \<Gamma>) auto |
Given a lemma, please prove it. | lemma inorder_node44: "height n > 0 \<Longrightarrow>
inorder (tree\<^sub>d (node44 l a m b n c r)) = inorder l @ a # inorder m @ b # inorder n @ c # inorder (tree\<^sub>d r)" | by(induct l a m b n c r rule: node44.induct) auto |
Given a lemma, please prove it. | lemma inorder_del:
"t \<in> T h \<Longrightarrow> sorted1(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)" | by(induction h arbitrary: t) (auto simp: del_list_simps inorder_n2 inorder_split_min[OF UnI1] inorder_split_min[OF UnI2] split: option.splits) |
Given a lemma, please prove it. | lemma inorder_del:
"sorted(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)" | by(induction x t rule: del.induct) (auto simp: del_list_simps inorder_baldL inorder_baldR split_minD split: prod.splits) |
Given a lemma, please prove it. | lemma inorder_update: "t \<in> T h \<Longrightarrow>
sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)" | by(simp add: update_def inorder_upd inorder_tree) |
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