instruct stringclasses 1
value | input stringlengths 25 4.64k | output stringlengths 17 413 |
|---|---|---|
Given a lemma, please prove it. | lemma pmdl_comp_red_monic_basis_subset: "pmdl (set (comp_red_monic_basis xs)) \<subseteq> pmdl (set xs)" | unfolding set_comp_red_monic_basis pmdl_image_monic
by (fact pmdl_comp_red_basis_subset) |
Given a lemma, please prove it. | lemma linord_of_listP_domain:
assumes "linord_of_listP x y xs"
shows "x \<in> set xs \<and> y \<in> set xs" | using assms
by (induct xs) auto |
Given a lemma, please prove it. | lemma list_of_oalist_tc_of_list_id:
assumes "tc.oalist_inv_raw xs"
shows "list_of_oalist_tc (OAlist_tc xs) = xs" | using assms
by (simp add: list_of_oalist_OAlist_tc tc.sort_oalist_id) |
Given a lemma, please prove it. | lemma remdups_adj_rev[simp]: "remdups_adj (rev xs) = rev (remdups_adj xs)" | by (induct xs rule: remdups_adj.induct, simp_all add: remdups_adj_append_two) |
Given a lemma, please prove it. | lemma
mtf2_forward_effect3': "q \<in> set xs \<Longrightarrow> distinct xs \<Longrightarrow> index xs q - n \<le> index xs x \<Longrightarrow> index xs x < index xs q
\<Longrightarrow> index (mtf2 n q xs) (xs!index xs x) = Suc (index xs (xs!index xs x)) \<and> index xs q - n < index (mtf2 n q xs) (xs!index xs x... | using mtf2_forward_effect3[where xs=xs and i="index xs x"]
by fast |
Given a lemma, please prove it. | lemma (in comm_monoid_add) sum_list_addf:
"(\<Sum>x\<leftarrow>xs. f x + g x) = sum_list (map f xs) + sum_list (map g xs)" | by (induct xs) (simp_all add: algebra_simps) |
Given a lemma, please prove it. | lemma stkOk_r_rewrite [simp]: "\<And>x. x \<notin> set xs \<Longrightarrow>
stkOk c l (r(x := g)) iL iR (Ref x) xs = stkOk c l r iL iR (Ref x) xs" | apply (induct xs)
apply (auto simp:eq_sym_conv) done |
Given a lemma, please prove it. | lemma insort_insert_key_code [code]:
"insort_insert_key f x xs =
(if List.member (map f xs) (f x) then xs else insort_key f x xs)" | by(simp add: insort_insert_key_def List.member_def split del: if_split) |
Given a lemma, please prove it. | lemma some_gcd_ff_list_greatest: "(\<forall>x \<in> set xs. divides_ff d x) \<Longrightarrow> divides_ff d (some_gcd_ff_list xs)" | using some_gcd_ff_list[of xs] unfolding gcd_ff_list_def
by auto |
Given a lemma, please prove it. | lemma fold_max_le:
fixes x::"'a::linorder"
shows "x \<in> set xs \<Longrightarrow> x \<le> fold max xs z" | by (induct xs arbitrary: x z) (auto intro: order_trans[OF _ fold_max_le_self]) |
Given a lemma, please prove it. | lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow>
dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))" | by (metis add.commute nth_append_length_plus takeWhile_dropWhile_id) |
Given a lemma, please prove it. | theorem f_Exec_Stream_drop:"
(f_Exec_Comp_Stream trans_fun xs c) \<up> n =
f_Exec_Comp_Stream trans_fun (xs \<up> n)
(f_Exec_Comp trans_fun (xs \<down> n) c)" | apply (case_tac "length xs \<le> n", simp)
apply (rule subst[OF append_take_drop_id, of _ n xs])
apply (simp add: f_Exec_Stream_append del: append_take_drop_id) done |
Given a lemma, please prove it. | lemma interpret_others:
"interpret (Neg(ExQ (Neg f))) xs = (\<forall>x. interpret f (x#xs))"
"interpret (Or (Neg f1) f2) xs = (interpret f1 xs \<longrightarrow> interpret f2 xs)" | by(simp_all add:interpret_def) |
Given a lemma, please prove it. | lemma of_weak_ranking_imp_in_set:
assumes "of_weak_ranking xs a b"
shows "a \<in> \<Union>(set xs)" "b \<in> \<Union>(set xs)" | using assms
by (fastforce elim!: of_weak_ranking.cases)+ |
Given a lemma, please prove it. | lemma removeAll_induct:
assumes "\<And>xs. (\<And>x. x \<in> set xs \<Longrightarrow> P (removeAll x xs)) \<Longrightarrow> P xs"
shows "P xs" | by (induct xs rule:length_induct, rule assms) auto |
Given a lemma, please prove it. | theorem set2_tllist_induct[consumes 1, case_names find step]:
assumes "x \<in> set2_tllist xs" and "\<And>xs. is_TNil xs \<Longrightarrow> P (terminal xs) xs"
and "\<And>xs y. \<lbrakk>\<not> is_TNil xs; y \<in> set2_tllist (ttl xs); P y (ttl xs)\<rbrakk> \<Longrightarrow> P y xs"
shows "P x xs" | using assms
by(induct)(fastforce simp del: tllist.disc(1) iff: tllist.disc(1), auto) |
Given a lemma, please prove it. | theorem mset_quicksort [simp]: "mset (quicksort R xs) = mset xs" | by (induction R xs rule: quicksort.induct) (simp_all) |
Given a lemma, please prove it. | theorem VSGeneralBlocksLimited: "\<forall>xs. eval (VSGeneralBlocksLimited \<phi>) xs = eval \<phi> xs" | unfolding VSGeneralBlocksLimited_def Unpower_def opt_group_def
using QE_dnf_eval[OF gen_qe_eval_augment opt] opt VSLuckiestBlocks
by fastforce |
Given a lemma, please prove it. | lemma lconjby_Abs_freelist_relator_freeword:
"\<lbrakk> rs\<in>R; xs\<in>lists S \<rbrakk> \<Longrightarrow> lconjby (Abs_freelist xs) (relator_freeword rs) \<in> Q" | using proper_signed_list_map_uniform_snd
by (force intro: lconj_relator_freeword_R) |
Given a lemma, please prove it. | lemma list_emb_Nil2 [simp]:
assumes "list_emb P xs []" shows "xs = []" | using assms
by (cases rule: list_emb.cases) auto |
Given a lemma, please prove it. | lemma llength_ltakeWhile_all: "llength (ltakeWhile P xs) = llength xs \<longleftrightarrow> ltakeWhile P xs = xs" | by(auto intro: lprefix_llength_eq_imp_eq lprefix_ltakeWhile) |
Given a lemma, please prove it. | lemma scalar_product_CONS: "length xs = length (bs :: bool list) \<Longrightarrow>
scalar_product (map_index (\<lambda>i n. 2 * n + bs ! i) xs) is =
scalar_product bs is + 2 * scalar_product xs is" | by (induct "is" arbitrary: bs xs) (auto split: list.splits simp: algebra_simps) |
Given a lemma, please prove it. | lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)" | by (cases n, simp, cases xs, auto) |
Given a lemma, please prove it. | lemma in_acquired_reads_no_pending_write_outstanding_write:
"\<And>A. a \<in> acquired_reads False xs A \<Longrightarrow> outstanding_refs (is_volatile_Write\<^sub>s\<^sub>b) xs \<noteq> {}" | apply (induct xs)
apply simp
apply (auto split: memref.splits)
apply auto done |
Given a lemma, please prove it. | lemma list_head_length_one:
assumes "hd xs = x"
and "length xs = 1"
shows "xs = [x]" | using assms
by(metis One_nat_def Suc_length_conv hd_Cons_tl length_0_conv list.sel(3)) |
Given a lemma, please prove it. | lemma memEqvt[eqvt]:
fixes p :: "name prm"
and x :: "'a::pt_name"
and xs :: "('a::pt_name) list"
shows "(p \<bullet> (x mem xs)) = ((p \<bullet> x) mem (p \<bullet> xs))" | by(induct xs) (auto simp add: pt_bij[OF pt_name_inst, OF at_name_inst] eqvts) |
Given a lemma, please prove it. | lemma sscan_smap[simp]: "sscan f (smap g xs) a = sscan (f \<circ> g) xs a" | by (coinduction arbitrary: xs a) (auto) |
Given a lemma, please prove it. | lemma distinct_ext:
assumes "distinct xs" "a \<notin> set xs"
shows "distinct (extup a xs)" "distinct (extdown a xs)" | using assms set_ext
apply (induction xs arbitrary: a)
apply auto
apply (metis eq_iff insert_iff subset_iff)+ done |
Given a lemma, please prove it. | lemma Sup_upto_llist_subset_Sup_llist: "Sup_upto_llist Xs j \<subseteq> Sup_llist Xs" | unfolding Sup_llist_def Sup_upto_llist_def
by auto |
Given a lemma, please prove it. | lemma ldropn_llist_of [simp]: "ldropn n (llist_of xs) = llist_of (drop n xs)" | proof(induct n arbitrary: xs) case Suc thus ?case
by(cases xs) simp_all
qed simp |
Given a lemma, please prove it. | lemma length_greaters_less [intro]:
assumes "x \<in> set xs"
shows "length (greaters R x xs) < length xs" | using assms
by (induction xs) (auto simp: greaters_Cons intro: le_less_trans) |
Given a lemma, please prove it. | lemma e2xs_xs [simp]:
"e2xs (encode_config ((f, xs, ls) # ss, rv)) = list_encode xs" | using e2xs_def e2frame_frame encode_frame
by force |
Given a lemma, please prove it. | lemma foldr_plus_zero_le: "foldr (+) xs (0::'lbl) \<le> foldr (+) xs a" | by (induct xs) (simp_all add: plus_mono) |
Given a lemma, please prove it. | lemma dropWhile_eq_hd_conv: "dropWhile P xs = hd xs # rest \<longleftrightarrow> xs \<noteq> [] \<and> rest = tl xs \<and> \<not> P (hd xs)" | by (metis append_Nil append_is_Nil_conv dropWhile_eq_Cons_conv list.sel(1) neq_Nil_conv takeWhile_dropWhile_id takeWhile_eq_Nil_conv list.sel(3)) |
Given a lemma, please prove it. | lemma set_tries_of_list[simp]:
"set_tries(tries_of_list key xs) = set xs" | by(simp add: tries_of_list_def set_insert_tries) |
Given a lemma, please prove it. | lemma index_vec_of_list: "i<length xs \<Longrightarrow> (vec_of_list xs) $ i = xs ! i" | by (metis vec.abs_eq index_vec vec_of_list.abs_eq) |
Given a lemma, please prove it. | lemma faulty1:
assumes a: "t\<mapsto>(x#xs),t'"
shows "\<not>(x\<sharp>t') \<Longrightarrow> \<not>(x\<sharp>bind xs t')" | using a
by (nominal_induct t xs'\<equiv>"x#xs" t' avoiding: xs rule: unbind.strong_induct) (simp_all add: free_variable) |
Given a lemma, please prove it. | lemma take_in_listsetI: "xs \<in> listset XS \<Longrightarrow> take n xs \<in> listset (take n XS)" | by (induction XS arbitrary: xs n) (auto simp: take_Cons listset_Cons_mem_conv set_Cons_def split: nat.splits) |
Given a lemma, please prove it. | lemma nth_rule:
"\<lbrakk>i < length xs\<rbrakk> \<Longrightarrow> <a \<mapsto>\<^sub>a xs> Array.nth a i <\<lambda>r. a \<mapsto>\<^sub>a xs * \<up>(r = xs ! i)>" | unfolding hoare_triple_def snga_assn_def
apply (simp add: Let_def Abs_assn_inverse)
apply (auto elim!: run_elims simp: Let_def new_addrs_def Array.get_def Array.set_def Array.alloc_def relH_def in_range.simps Array.length_def ) done |
Given a lemma, please prove it. | lemma sset_simp[simp]:
shows "sset \<bottom> = {}"
and "sset [::] = {}"
and "\<lbrakk>x \<noteq> \<bottom>; xs \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> sset (x :# xs) = insert x (sset xs)" | unfolding sset_def
by (auto elim: slistmem.cases intro: slistmem.intros) |
Given a lemma, please prove it. | lemma msetext_dersh_compat_list:
assumes y_gt_x: "gt y x"
shows "msetext_dersh gt (xs @ y # xs') (xs @ x # xs')" | unfolding msetext_dersh_def Let_def
proof (intro exI conjI) show "mset (xs @ x # xs') = mset (xs @ y # xs') - {#y#} + {#x#}"
by auto
qed (auto intro: y_gt_x) |
Given a lemma, please prove it. | lemma trunc_err_pdevsE:
assumes "e \<in> UNIV \<rightarrow> {-1 .. 1}"
obtains err where
"\<bar>err\<bar> \<le> tdev' p (trunc_err_pdevs p xs)"
"pdevs_val e (trunc_pdevs p xs) = pdevs_val e xs + err" | using trunc_bound_pdevsE[of e p xs]
by (auto simp: trunc_bound_pdevs_def assms) |
Given a lemma, please prove it. | lemma set_ass_list_to_single_list [simp]:
"set (ass_list_to_single_list xs) = {x. \<exists>n. (x, n) \<in> set xs \<and> n > 0}" | by (induct xs rule: ass_list_to_single_list.induct) auto |
Given a lemma, please prove it. | lemma iprefix_take_eq_iprefix_take_ex: "
(f \<Down> length xs = xs) = (\<exists>n. f \<Down> n = xs)" | apply (rule iffI)
apply (rule_tac x="length xs" in exI, assumption)
apply clarsimp done |
Given a lemma, please prove it. | lemma terminal_transfer [transfer_rule]:
"(pcr_tllist A (=) ===> (=)) (\<lambda>(xs, b). if lfinite xs then b else undefined) terminal" | by(force simp add: cr_tllist_def pcr_tllist_def terminal_tllist_of_llist dest: llist_all2_lfiniteD) |
Given a lemma, please prove it. | lemma listI: "\<lbrakk> size xs = n; set xs \<subseteq> A \<rbrakk> \<Longrightarrow> xs \<in> list n A" | apply (unfold list_def)
apply blast done |
Given a lemma, please prove it. | lemma lift_prod_list: "lift (prod_list xs) = prod_list (map lift xs)" | by (induction xs) (simp_all add: lift_mult) |
Given a lemma, please prove it. | lemma sorted_wrt_hd_drop_less_drop:
assumes "sorted_wrt f xs" "\<And>x. f x x"
shows "\<forall>x \<in> set (drop n xs). f (hd (drop n xs)) x" | using assms sorted_wrt_drop sorted_wrt_hd_less
by blast |
Given a lemma, please prove it. | lemma pmf_of_set_code_aux:
assumes "A \<noteq> {}" "set xs = A" "distinct xs"
shows "mapping_of_pmf (pmf_of_set A) = Mapping.tabulate xs (\<lambda>_. 1 / real (length xs))" | using assms
by (intro mapping_of_pmfI, subst pmf_of_set) (auto simp: lookup_tabulate distinct_card) |
Given a lemma, please prove it. | lemma exec_xcpt: "G \<turnstile> xs -st-> xs' \<Longrightarrow> gx xs' = None \<Longrightarrow> gx xs = None"
(is "?H1 \<Longrightarrow> ?H2 \<Longrightarrow> ?T") | proof- assume h1: ?H1 assume h2: ?H2 from h1 h2 eval_evals_exec_xcpt show "?T"
by simp
qed |
Given a lemma, please prove it. | lemma linord_of_list_trans:
assumes "distinct xs"
shows "trans (linord_of_list xs)" | using assms unfolding linord_of_list_def
by (induct xs) (auto intro!: transI dest: linord_of_listP_domain elim: transE) |
Given a lemma, please prove it. | lemma set_swap[simp]:
"\<lbrakk> i < size xs; j < size xs \<rbrakk> \<Longrightarrow> set(xs[i := xs!j, j := xs!i]) = set xs" | by(simp add: set_conv_nth nth_list_update) metis |
Given a lemma, please prove it. | lemma nonsimple_length_gt_1: "xs \<noteq> [] \<Longrightarrow> hd xs \<noteq> last xs \<Longrightarrow> length xs > 1" | by (metis length_0_conv less_one nat_neq_iff singleton_list_hd_last) |
Given a lemma, please prove it. | lemma remdups_adj_Nil_iff[simp]: "remdups_adj xs = [] \<longleftrightarrow> xs = []" | by (induct xs rule: remdups_adj.induct, simp_all) |
Given a lemma, please prove it. | lemma natlist_trivial_1: "natpermute n 1 = {[n]}" | proof - have "\<lbrakk>length xs = 1; n = sum_list xs\<rbrakk> \<Longrightarrow> xs = [sum_list xs]" for xs
by (cases xs) auto then show ?thesis
by (auto simp add: natpermute_def)
qed |
Given a lemma, please prove it. | lemma map_list_update_id:
"f (xs ! pc) = f instr \<Longrightarrow> map f (xs[pc := instr]) = map f xs" | using list_update_id map_update
by metis |
Given a lemma, please prove it. | lemma luckiestFind_eval' : "
(\<exists>xs. (length xs = var + 1) \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) =
(\<exists>xs. (length xs = var + 1) \<and> eval (luckiestFind var L F) (xs @ \<Gamma>))" | apply(rule step_converter[of luckiestFind var L F \<Gamma>])
using luckiestFind_eval
by blast |
Given a lemma, please prove it. | lemma successively_rev [simp]: "successively P (rev xs) \<longleftrightarrow> successively (\<lambda>x y. P y x) xs" | by (induction xs rule: remdups_adj.induct) (auto simp: successively_append_iff successively_Cons) |
Given a lemma, please prove it. | lemma path_from_to_first': "v \<leadsto>(xs @ x # xs')\<leadsto> w \<Longrightarrow> v \<notin> set xs'" | by (metis path_from_toE append_eq_append_conv2 distinct.simps(2) hd_append list.exhaust_sel list.sel(3) list.set_sel(1,2) list.simps(3) path_disjoint self_append_conv) |
Given a lemma, please prove it. | lemma list_all' [iff]:
"list_all' P xs = (\<forall>n < size xs. P (xs!n) n)" | by (unfold list_all'_def) (simp add: list_all'_rec) |
Given a lemma, please prove it. | lemma filter_by_key_merge_all_sequences [simp]:
"[x\<leftarrow>merge_all (sequences xs) . key x = k] = [x\<leftarrow>xs . key x = k]" | using sorted_sequences [of xs]
by simp |
Given a lemma, please prove it. | lemma compact_LCons_iff [simp]:
"ccpo.compact lSup (\<sqsubseteq>) (LCons x xs) \<longleftrightarrow> ccpo.compact lSup (\<sqsubseteq>) xs" | by(blast intro: compact_LConsI compact_LConsD) |
Given a lemma, please prove it. | lemma llength_ltakeWhile_eq_infinity':
"llength (ltakeWhile P xs) = \<infinity> \<longleftrightarrow> \<not> lfinite xs \<and> (\<forall>x\<in>lset xs. P x)" | by (metis lfinite_ltakeWhile llength_eq_infty_conv_lfinite) |
Given a lemma, please prove it. | lemma foldri_foldr :
"foldri xs (\<lambda>_. True) f \<sigma> = foldr (\<lambda>x \<sigma>. f x \<sigma>) xs \<sigma>" | by (induct xs arbitrary: \<sigma>) simp_all |
Given a lemma, please prove it. | lemma (in field) npeneg_correct: "peval xs (npeneg e) = peval xs (PExpr1 (PNeg e))" | by (cases e rule: pexpr_cases) (simp_all add: npeneg_def) |
Given a lemma, please prove it. | lemma shift_snth_less[simp]: "p < length xs \<Longrightarrow> (xs @- s) !! p = xs ! p" | by (induct p arbitrary: xs) (auto simp: hd_conv_nth nth_tl) |
Given a lemma, please prove it. | lemma UN_set_permutations_of_set [simp]:
"finite A \<Longrightarrow> (\<Union>xs\<in>permutations_of_set A. set xs) = A" | using finite_distinct_list
by (auto simp: permutations_of_set_def) |
Given a lemma, please prove it. | lemma real_of_3_remdups_equal_3[simp]: "real_of_3 ` set (remdups_gen equal_3 xs) = real_of_3 ` set xs" | by (induct xs, auto simp: equal_3) |
Given a lemma, please prove it. | lemma Lxx1: "xs \<in> Lxx x y \<Longrightarrow> length xs \<ge> 2" | apply(rule LxxI[where P="(\<lambda>x y qs. length qs \<ge> 2)"])
apply(auto)
by(auto simp: conc_def) |
Given a lemma, please prove it. | lemma bounded_by_update_var:
assumes "bounded_by xs vs"
and "vs ! i = Some ivl"
and bnd: "x \<in>\<^sub>r ivl"
shows "bounded_by (xs[i := x]) vs" | using assms
using nth_list_update
by (cases "i < length xs") (force simp: bounded_by_def split: option.splits)+ |
Given a lemma, please prove it. | lemma G'_steps_V_last:
"V (last xs)" if "G'.steps xs" "V (hd xs)" | using that
by induction (auto dest: E'_V2) |
Given a lemma, please prove it. | lemma insert_before_list_subset: "set xs \<subseteq> set (insert_before_list x ref xs)" | apply(induct x ref xs rule: insert_before_list.induct)
by(auto) |
Given a lemma, please prove it. | lemma alw_ev_holds_mp:
"alw (holds P) xs \<Longrightarrow> ev (holds Q) xs \<Longrightarrow> ev (holds (\<lambda>x. P x \<and> Q x)) xs" | by (subst ev_cong, assumption) auto |
Given a lemma, please prove it. | lemma af_abs_equiv:
"foldl \<up>af \<psi> (xs @ [x]) = \<up>step (foldl \<up>af\<^sub>\<UU> (\<up>Unf \<psi>) xs) x" | unfolding af_unfold af_opt_unfold
by (induction xs arbitrary: x \<psi> rule: rev_induct) simp+ |
Given a lemma, please prove it. | lemma successively_map: "successively P (map f xs) \<longleftrightarrow> successively (\<lambda>x y. P (f x) (f y)) xs" | by (induction xs rule: induct_list012) auto |
Given a lemma, please prove it. | lemma ltl_lSup [simp]: "ltl (lSup A) = lSup (ltl ` (A \<inter> {xs. \<not> lNone xs}))" | by(cases "\<forall>xs\<in>A. lNone xs")(auto 4 3 simp add: lSup_def intro: llist.expand) |
Given a lemma, please prove it. | lemma hd_exp_inverse:
"xs \<noteq> MSLNil \<Longrightarrow> ms_aux_hd_exp (inverse_ms_aux xs) = map_option uminus (ms_aux_hd_exp xs)" | by (cases xs) (auto simp: Let_def hd_exp_basis hd_exp_powser inverse_ms_aux.simps) |
Given a lemma, please prove it. | lemma toplevel_summands_PLUS_strong:
"\<lbrakk>xs \<noteq> []; list_all (\<lambda>x. \<not>(\<exists>r s. x = Plus r s)) xs\<rbrakk> \<Longrightarrow> toplevel_summands (PLUS xs) = set xs" | by (induct xs rule: list_singleton_induct) auto |
Given a lemma, please prove it. | lemma rga_ops_rem_last:
assumes "rga_ops (xs @ [x])"
shows "rga_ops xs" | using assms crdt_ops_rem_last rga_ops_def
by blast |
Given a lemma, please prove it. | lemma unstream_stream: "unstream stream xs = xs" | by(induction xs)(auto simp add: stream.rep_eq) |
Given a lemma, please prove it. | lemma Kdelta_in_Zinf:"\<lbrakk>j \<le> (Suc n); k \<le> (Suc n)\<rbrakk> \<Longrightarrow>
z *\<^sub>a (\<delta>\<^bsub>j k\<^esub>) \<in> Z\<^sub>\<infinity>" | apply (simp add:Kronecker_delta_def)
apply (simp add:z_in_aug_inf Zero_in_aug_inf)
apply (simp add:asprod_n_0 Zero_in_aug_inf) done |
Given a lemma, please prove it. | lemma gds_init_refine: "gds_init gds
\<le> SPEC (\<lambda>s. gen_rwof s \<and> gds_is_empty_stack gds s)" | apply (rule SPEC_rule_conj_leofI1)
apply (rule rwof_init[OF nofail])
apply (rule init_empty_stack) done |
Given a lemma, please prove it. | lemma analz_image_Key [simp]: "analz (Key`N) = Key`N" | apply auto
apply (erule analz.induct, auto) done |
Given a lemma, please prove it. | lemma monotone_lmerge: "monotone (rel_prod lprefix lprefix) lprefix (case_prod lmerge)" | apply(rule llist.fixp_preserves_mono2[OF lmerge_mono lmerge_conv_fixp])
apply(erule conjE|rule allI conjI cont_intro|simp|erule allE, erule llist.mono2mono)+ done |
Given a lemma, please prove it. | lemma ft_tailR_impl: "al_tailR ft_\<alpha> ft_invar ft_tailR" | apply unfold_locales
apply (auto simp add: ft_defs FingerTree.tailR_correct FingerTree.empty_correct) done |
Given a lemma, please prove it. | lemma filter_oseq_idemp: "filter_oseq p (filter_oseq p s) = filter_oseq p s" | apply (simp add: filter_oseq_def)
apply (rule ext)
apply (case_tac "s i")
apply simp_all done |
Given a lemma, please prove it. | lemma all2_less_lemma [simp]:
"rec_eval (rec_all2_less f) [x, y1, y2] = (if (\<forall>z < x. 0 < rec_eval f [z, y1, y2]) then 1 else 0)" | apply(auto simp add: Let_def rec_all2_less_def)
apply(metis nat_less_le)+ done |
Given a lemma, please prove it. | lemma l2_inv7_step1:
"{l2_inv7} l2_step1 Ra A B {> l2_inv7}" | apply (auto simp add: PO_hoare_defs l2_defs intro!: l2_inv7I)
apply (auto intro: synth_analz_increasing) done |
Given a lemma, please prove it. | lemma wf_ciD3_ci_app:
"\<lbrakk> ci_app ci ins P h stk loc C M pc frs; ins_jump_ok ins pc' \<rbrakk> \<Longrightarrow> ci_app ci ins P h stk loc C M pc' frs" | apply(cases ci)
apply(simp add: Abs_check_instr_inverse)
apply(erule (2) wf_ciD3) done |
Given a lemma, please prove it. | lemma "[2, mod 10, 4] = {2, 12, 22, 32, 42}" | apply (simp only: iMODb_conv)
apply (simp add: iT_defs iT_Plus_def iT_Mult_def)
apply fastforce done |
Given a lemma, please prove it. | lemma strict_mono_cancel_le:
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
shows "strict_mono f \<Longrightarrow> (f x \<le> f y) = (x \<le> y)" | apply (auto simp add: order_le_less)
apply (simp add: strict_mono_cancel_less)
apply (simp add: strict_mono_cancel_eq)
apply (simp add: strict_monoD) done |
Given a lemma, please prove it. | lemma equiv_class_member:
assumes "x \<in> A"
and "\<approx>A `` {x} = \<approx>A `` {y}"
shows "y \<in> A" | using assms
apply(simp)
apply(simp add: str_eq_def)
apply(metis append_Nil2) done |
Given a lemma, please prove it. | lemma mor_image1: "mor B1 B2 s1 s2 B1' B2' s1' s2' f g \<Longrightarrow> f ` B1 \<subseteq> B1'" | apply (tactic \<open>dtac @{context} @{thm iffD1[OF mor_def]} 1\<close>)
apply (erule conjE)+
apply (rule image_subsetI)
apply (erule bspec)
apply assumption done |
Given a lemma, please prove it. | lemma list_strict_asc_distinct: "list_strict_asc (xs::'a::preorder list) \<Longrightarrow> distinct xs" | apply (rule_tac ord="(<)" in list_ord_distinct)
apply (unfold irrefl_def list_strict_asc_def trans_def)
apply (blast intro: less_trans)+ done |
Given a lemma, please prove it. | lemma hs_ins_impl: "imp_set_ins is_hashset hs_ins" | apply unfold_locales
apply (sep_auto heap: hm_update_rule simp: hs_ins_def is_hashset_def) done |
Given a lemma, please prove it. | lemma and_neq_0_is_nth:
\<open>x AND y \<noteq> 0 \<longleftrightarrow> x !! n\<close> if \<open>y = 2 ^ n\<close> for x y :: \<open>'a::len word\<close> | apply (simp add: bit_eq_iff bit_simps)
using that
apply (simp add: bit_simps not_le)
apply transfer
apply auto done |
Given a lemma, please prove it. | lemma const_le_unat:
"\<lbrakk> b < 2 ^ LENGTH('a); of_nat b \<le> a \<rbrakk> \<Longrightarrow> b \<le> unat (a :: 'a :: len word)" | apply (simp add: word_le_def)
apply (simp only: uint_nat zle_int)
apply transfer
apply (simp add: take_bit_nat_eq_self) done |
Given a lemma, please prove it. | lemma AddSA_HAInitValue_IFF:
"\<lbrakk> States SA \<inter> HAStates HA = {};
S \<in> HAStates HA;
(HAInitValue HA) = X \<rbrakk> \<Longrightarrow> (HAInitValue (HA [++] (S, SA))) = X" | apply (subst AddSA_HAInitValue)
apply auto done |
Given a lemma, please prove it. | lemma order_widen [intro,simp]:
"wf_prog m P \<Longrightarrow> order (subtype P)" | apply (unfold Semilat.order_def lesub_def)
apply (auto intro: widen_trans widen_antisym) done |
Given a lemma, please prove it. | lemma hta_\<alpha>_is_ta[simp, intro!]: "tree_automaton (hta_\<alpha> H)" | apply unfold_locales
apply (unfold hta_\<alpha>_def)
apply auto done |
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