UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
lemma marked_deletions_sweep_loop_free[simp]: notes fun_upd_apply[simp] shows "\<lbrakk> mut_m.marked_deletions m s; mut_m.reachable_snapshot_inv m s; no_grey_refs s; white r s \<rbrakk> \<Longrightarrow> mut_m.marked_deletions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := None)\<rparr>))"
lemma thetaSAtm_Sbis: assumes "compatAtm atm" shows "thetaSAtm atm \<subseteq> Sbis"
lemma a_comm_var: "ad x \<cdot> ad y \<le> ad y \<cdot> ad x"
lemma older_seniors_older: "y \<in> older_seniors x n \<Longrightarrow> y < x"
lemma iterate_to_set_correct : assumes ins_dj_OK: "set_ins_dj \<alpha> invar ins_dj" assumes emp_OK: "set_empty \<alpha> invar emp" assumes it: "set_iterator it S0" shows "\<alpha> (iterate_to_set emp ins_dj it) = S0 \<and> invar (iterate_to_set emp ins_dj it)"
lemma to_fun_ring_hom: assumes "a \<in> carrier R" shows "(\<lambda>p. to_fun p a) \<in> ring_hom P R"
lemma element_ptr_kinds_simp [simp]: "element_ptr_kinds (Heap (fmupd (cast element_ptr) element (the_heap h))) = {\|element_ptr\|} \|\<union>\| element_ptr_kinds h"
lemma indis_sym[sym]: "s \<approx> s' \<Longrightarrow> s' \<approx> s"
lemma sum_sumj_eq2: "i<I ==> sum I s = c s i + sumj I i s"
lemma el_loc_ok_update: "\<lbrakk> \<B> e n; V < n \<rbrakk> \<Longrightarrow> el_loc_ok e (xs[V := v]) = el_loc_ok e xs" and els_loc_ok_update: "\<lbrakk> \<B>s es n; V < n \<rbrakk> \<Longrightarrow> els_loc_ok es (xs[V := v]) = els_loc_ok es xs"
lemma length_assoc_list_of_array [simp]: "length (assoc_list_of_array a) = array_length a"
lemma (in wf_digraph) adj_mk_symmetric_eq: "symmetric G \<Longrightarrow> parcs (mk_symmetric G) = arcs_ends G"
lemma is_ground_lit_is_ground_on_var: assumes ground_lit: "is_ground_lit (subst_lit L \<sigma>)" and v_in_L: "v \<in> vars_lit L" shows "is_ground_atm (\<sigma> v)"
lemma m2f_by_from_m2f : "(m2f_by g f xs) = g (m2f f xs)"
lemma iso_finfun_uminus [code_unfold]: fixes A :: "'a pred\<^sub>f" shows "- ($) A = ($) (- A)"
lemma squareE [elim]: "\<lbrakk> (s,t) \<Turnstile> [A]_v; A (s,t) \<Longrightarrow> B (s,t); v t = v s \<Longrightarrow> B (s,t) \<rbrakk> \<Longrightarrow> B (s,t)"
lemma af\<^sub>F_semantics_rtl: assumes "\<forall>i. \<exists>j>i. af\<^sub>F \<phi> (F\<^sub>n \<phi>) (w [0 \<rightarrow> j]) \<sim> true\<^sub>n" shows "\<forall>i. \<exists>j. af (F\<^sub>n \<phi>) (w [i \<rightarrow> j]) \<sim>\<^sub>L true\<^sub>n"
lemma validTrans_step_srcOf_actOf_tgtOf: "validTrans trn \<Longrightarrow> step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn)"
lemma proj2_incident_iff_Col: assumes "p \<noteq> q" and "proj2_incident p l" and "proj2_incident q l" shows "proj2_incident r l \<longleftrightarrow> proj2_Col p q r"
lemma [simp]: "size_exp' (Handle e pes) = Suc (size_exp' e + size_list (size_prod size size_exp') pes)"
lemma coeff_finite_fourier_poly: assumes "n < length ws" defines "k \<equiv> length ws" shows "coeff (finite_fourier_poly ws) n = (1/k) * (\<Sum>m < k. ws ! m * unity_root k (-n*m))"
lemma ccompatible1: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r \|I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "ccompatible \<R> (EQ x c)"
lemma ntsmcf_tdghm_smcf_ntsmcf_comp[slicing_commute]: "smcf_dghm \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M ntsmcf_tdghm \<NN> = ntsmcf_tdghm (\<HH> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F\<^sub>-\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)"
lemma min_union_finite: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<union>\<^sub>m B)"
lemma decompV_decomp: assumes "validFrom s tr" and "reach s" shows "decompV (V tr) nid = lV nid (decomp tr nid)"
lemma Zp_square_root_criterion: assumes "p \<noteq> 2" assumes "a \<in> carrier Zp" assumes "b \<in> carrier Zp" assumes "val_Zp b \<ge> val_Zp a" assumes "a \<noteq> \<zero>" assumes "b \<noteq> \<zero>" shows "\<exists>y \<in> carrier Zp. a[^](2::nat) \<oplus> \<p>\<otimes>b[^](2::nat) = (y [^]\<^bsub>Zp\<^esub> (2::nat))"
lemma interpret_floatariths_fresh_eqI: assumes "\<And>i. fresh_floatariths ea i \<or> (i < length ys \<and> i < length zs \<and> ys ! i = zs ! i)" shows "interpret_floatariths ea ys = interpret_floatariths ea zs"
lemma defensive_move_exists_for_Even: assumes [intro]:"position p" shows "winning_position_Odd p \<or> (\<exists> m. move_defensive_by_Even m p)" (is "?w \<or> ?d")
lemma nth_append_singl[simp]: "i < length al \<Longrightarrow> (al @ [a]) ! i = al ! i"
lemma Checkcast_correct: "\<lbrakk> wt_jvm_prog G phi; method (G,C) sig = Some (C,rT,maxs,maxl,ins,et); ins!pc = Checkcast D; wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) et pc; Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ; G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd>; fst (exec_instr (ins!pc) G hp stk loc C sig pc frs) = None \<rbrakk> \<Longrightarrow> G,phi \<turnstile>JVM state'\<surd>"
lemma uminus_add_in_Ker_eq_eq_im: "g\<in>G \<Longrightarrow> h\<in>G \<Longrightarrow> (-g + h \<in> Ker) = (T g = T h)"
lemma lift_spmf_bind_spmf: "lift_spmf (p \<bind> f) = lift_spmf p \<bind> (\<lambda>x. lift_spmf (f x))"
lemma global_oinvariantI [intro]: assumes init: "\<And>\<sigma> p. (\<sigma>, p) \<in> init A \<Longrightarrow> P \<sigma>" and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U; P \<sigma>; U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P \<sigma>'" and step: "\<And>\<sigma> p a \<sigma>' p'. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U; P \<sigma>; ((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A; S \<sigma> \<sigma>' a \<rbrakk> \<Longrightarrow> P \<sigma>'" shows "A \<Turnstile> (S, U \<rightarrow>) (\<lambda>(\<sigma>, _). P \<sigma>)"
lemma nonpos_Reals_one_I [simp]: "1 \<notin> \<real>\<^sub>\<le>\<^sub>0"
lemma "w \<le>p w \<Longrightarrow> \<^bold>\|w\<^bold>\| = 2 \<Longrightarrow> w \<in> lists {a,b} \<Longrightarrow> hd w = a \<Longrightarrow> w = [a, b] \<or> w = [a, a]"
lemma awalk_verts_dom_if_uneq: "\<lbrakk>u\<noteq>v; awalk u p v\<rbrakk> \<Longrightarrow> \<exists>x. x \<rightarrow>\<^bsub>G\<^esub> v \<and> x \<in> set (awalk_verts u p)"
lemma sys_block_sizes_uniform [simp]: "sys_block_sizes = {\<k>}"
lemma hfs_valid_None_Cons: assumes "hfs_valid_None ainfo uinfo p" "p = hf1 # hf2 # post" shows "hfs_valid_None ainfo (upd_uinfo uinfo hf2) (hf2 # post)"
lemma "(\<Lambda> x. [x]) = (\<Lambda> z. z : [])"
lemma Object_neq_SXcpt [simp]: "Object \<noteq> SXcpt xn"
lemma sigma_le_sets: assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"
lemma step_in_valid_step: "knights_path b ps \<Longrightarrow> step_in ps s\<^sub>i s\<^sub>j \<Longrightarrow> valid_step s\<^sub>i s\<^sub>j"
lemma bound_typ_inst_gen [simp]: "free_tv(t::typ) \<subseteq> free_tv(A) \<Longrightarrow> bound_typ_inst S (gen A t) = t"
lemma smult_unique_scalars : fixes a::'f assumes vs: "basis_for V vs" and v: "v \<in> V" defines as: "as \<equiv> (THE cs. length cs = length vs \<and> v = cs \<bullet>\<cdot> vs)" and bs: "bs \<equiv> (THE cs. length cs = length vs \<and> a \<cdot> v = cs \<bullet>\<cdot> vs)" shows "bs = map ((*) a) as"
lemma typ_of_axiom: "wf_theory thy \<Longrightarrow> t \<in> axioms thy \<Longrightarrow> typ_of t = Some propT"
lemma invalid_assn_mono: "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt B x y \<Longrightarrow> hn_invalid A x y \<Longrightarrow>\<^sub>t hn_invalid B x y"
lemma padic_add_comm: assumes "prime p" shows " \<And>x y. x \<in> carrier (padic_int p) \<Longrightarrow> y \<in> carrier (padic_int p) \<Longrightarrow> x \<oplus>\<^bsub>padic_int p\<^esub> y = y \<oplus>\<^bsub>padic_int p\<^esub> x"
lemma chain_sup_const[simp]: "chain_sup (\<lambda> x. S) = S"
lemma segment_a'_a_ne: "segment G.face_cycle_succ a' a \<noteq> {}"
lemma octo_of_real_mult [simp]: "octo_of_real (x * y) = octo_of_real x * octo_of_real y"
lemma unambigous_prefix_routing_strong_mono: assumes lpfx: "is_longest_prefix_routing (rr#rtbl)" assumes uam: "unambiguous_routing (rr#rtbl)" assumes e:"rr' \<in> set rtbl" assumes ne: "routing_match rr' = routing_match rr" shows "routing_rule_sort_key rr' > routing_rule_sort_key rr"
lemma from_hma\<^sub>v_inj[simp]: "from_hma\<^sub>v x = from_hma\<^sub>v y \<longleftrightarrow> x = y"
lemma "Sup \<circ> atom_map = (id::'a::complete_atomic_boolean_algebra \<Rightarrow> 'a)"
lemma z_lemma_R: fixes I:: "nat list * nat list" fixes sign:: "rat list" assumes consistent: "sign \<in> set (characterize_consistent_signs_at_roots p qs)" assumes welldefined1: "list_constr (fst I) (length qs)" assumes welldefined2: "list_constr (snd I) (length qs)" shows "(z_R I sign = 1) \<or> (z_R I sign = 0) \<or> (z_R I sign = -1)"
theorem jvm_typesafe: assumes wf: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" and start: "wf_start_state P C M vs" and exec: "P \<turnstile> JVM_start_state P C M vs -\<triangleright>ttas\<rightarrow>\<^bsub>jvm\<^esub>* s'" shows "s' \<in> correct_jvm_state \<Phi>"
lemma rank_of_image: assumes "finite S" shows "(\<lambda>x. rank_of x S) ` S = {0..<card S}"
lemma negate_negate_dfa: "negate_dfa (negate_dfa A) = A"
lemma insert_node_ok: assumes "known_ptr parent" and "type_wf h" assumes "parent \|\<in>\| object_ptr_kinds h" assumes "\<not>is_character_data_ptr_kind parent" assumes "is_document_ptr parent \<Longrightarrow> h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r []" assumes "is_document_ptr parent \<Longrightarrow> \<not>is_character_data_ptr_kind node" assumes "known_ptr (cast node)" shows "h \<turnstile> ok (a_insert_node parent node ref)"
lemma eval_terms_fv_fo_terms_set: "\<sigma> \<odot> ts = \<sigma>' \<odot> ts \<Longrightarrow> n \<in> fv_fo_terms_set ts \<Longrightarrow> \<sigma> n = \<sigma>' n"
lemma clop_iso: "clop f \<Longrightarrow> mono f"
lemma emeasure_T_state_Nil: "T (s, o\<^sub>0) {\<omega> \<in> space S. V [] as \<omega>} = 1"
lemma HEndPhase2_valueChosen2: assumes act: "HEndPhase2 s s' q" and asm4: "\<forall>d\<in>D. b \<le> bal(disk s d p) \<and>(\<forall>q.( phase s q = 1 \<and> b \<le>mbal(dblock s q) \<and> hasRead s q d p ) \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal(block br)))" (is "?P s") shows "?P s'"
lemma pl_1[axiom]: "[[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<phi>)]]"
lemma fv_ik_subset_fv_st[simp]: "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
lemma num_params_polymul: shows "num_params (p1 *\<^sub>p p2) \<le> max (num_params p1) (num_params p2)"
lemma minSetOfComponentsTestL2p3: "minSetOfComponents level2 {data1, data10, data11} = {sS1, sS2, sS3}"
lemma map_def: "Applicative_DNEList.map = map_fun id (map_fun list_of_dnelist Abs_dnelist) (\<lambda>f xs. remdups (list.map f xs))"
lemma bin_num: "bin0 = 0" "bin1 = 1"
lemma set_disconnected_nodes_typess_preserved: assumes "w \<in> set_disconnected_nodes_locs object_ptr" assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
lemma finite_fold_fold_keys: assumes "comp_fun_commute f" shows "Finite_Set.fold f A (Set t) = fold_keys f t A"
lemma rbl_add_carry_Cons: "(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) = xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)"
lemma FP_weakest: "(\<And>B. F \<in> stable (A Int B)) \<Longrightarrow> A <= FP F"
lemma ereal_minus_less: fixes x y z :: ereal shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
lemma max_spec_preserves_length: "max_spec G C (mn, pTs) = {((md,rT),pTs')} \<Longrightarrow> length pTs = length pTs'"
lemma pdevs_val_minus: "pdevs_val (\<lambda>i. e i - f i) xs = pdevs_val e xs - pdevs_val f xs"
lemma is_node_ptr_kind_cast [simp]: "is_node_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)"
lemma p_add: "p x y (n + m) = (\<integral>\<^sup>+ w. p x w n * p w y m \<partial>count_space UNIV)"
lemma miles_to_feet: "mile = 5280 *\<^sub>Q foot"
lemma (in aGroup) ring_nsum_zeroTr:"(\<forall>j \<le> (n::nat). f j \<in> carrier A) \<and> (\<forall>j \<le> n. f j = \<zero>) \<longrightarrow> nsum A f n = \<zero>"
lemma closed_sum_left_subset: \<open>0 \<in> B \<Longrightarrow> A \<subseteq> A +\<^sub>M B\<close> for A B :: "_::monoid_add"
lemma fst_gb_schema_incr: "fst ` set (gb_schema_incr sel ap ab compl upd (b0 # bs) data) = (let (gs, n, data') = add_indices (gb_schema_incr sel ap ab compl upd bs data, data) (0, data); b = (fst b0, n, snd b0); data'' = upd gs b data' in fst ` set (gb_schema_aux sel ap ab compl gs (count_rem_components (b # gs), Suc n, data'') (ab gs [] [b] (Suc n, data'')) (ap gs [] [] [b] (Suc n, data''))) )"
lemma monic_factorization_uniqueness: fixes P::"'a poly set" assumes finite_P: "finite P" and PQ: "\<Prod>P = \<Prod>Q" and P: "P \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}" and finite_Q: "finite Q" and Q: "Q \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}" shows "P = Q"
lemma ogets_NF_wp [wp]: "ovalidNF (\<lambda>s. P (f s) s) (ogets f) P"
lemma induced_arrow_self: shows "induced_arrow a \<chi> = a"
lemma coprime_pderiv_imp_rsquarefree: assumes "coprime (p :: 'a :: field_char_0 poly) (pderiv p)" shows "rsquarefree p"
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
lemma Gromov_extension_quasi_isometry_boundary_to_boundary: fixes f::"'a::Gromov_hyperbolic_space_geodesic \<Rightarrow> 'b::Gromov_hyperbolic_space_geodesic" assumes "lambda C-quasi_isometry f" "x \<in> Gromov_boundary" shows "(Gromov_extension f) x \<in> Gromov_boundary"
lemma listsum2_lemma: "\<lbrakk>length xs = vl; n \<le> vl\<rbrakk> \<Longrightarrow> rec_exec (rec_listsum2 vl n) xs = listsum2 xs n"
lemma strategy_proofI: assumes "\<And>Pd Pd' Ch d y. \<lbrakk> mechanism_domain Pd Ch; mechanism_domain (Pd(d:=Pd')) Ch; d \<in> ds; y \<in> \<phi> (Pd(d := Pd')) Ch ds; y \<in> Field (Pd d); \<forall>x\<in>dX (\<phi> Pd Ch ds) d. x \<noteq> y \<and> (x, y) \<in> Pd d \<rbrakk> \<Longrightarrow> False" shows "strategy_proof ds \<phi>"
lemma equivalence_plus_closed: "equivalence x \<Longrightarrow> equivalence (x\<^sup>+)"
lemma mix_of_preferred_is_preferred: assumes "p \<succeq>[\<R>] w" assumes "q \<succeq>[\<R>] w" assumes "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> p q \<succeq>[\<R>] w"
lemma expected_value_is_utility_function: assumes fnt: "finite outcomes" and "outcomes \<noteq> {}" assumes "x \<in> lotteries_on outcomes" and "y \<in> lotteries_on outcomes" assumes "ordinal_utility (lotteries_on outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "measure_pmf.expectation x u \<ge> measure_pmf.expectation y u \<longleftrightarrow> x \<succeq>[\<R>] y" (is "?L \<longleftrightarrow> ?R")
lemma prime_power_eq_one_iff [simp]: "prime p \<Longrightarrow> p ^ n = 1 \<longleftrightarrow> n = 0"
lemma ex_is_poincare_line_points': assumes i12: "i1 \<in> circline_set H \<inter> unit_circle_set" "i2 \<in> circline_set H \<inter> unit_circle_set" "i1 \<noteq> i2" assumes a: "a \<in> circline_set H" "a \<notin> unit_circle_set" shows "\<exists> b. b \<noteq> i1 \<and> b \<noteq> i2 \<and> b \<noteq> a \<and> b \<noteq> inversion a \<and> b \<in> circline_set H"
lemma RP_state_repetition_distribution_productF : assumes "OFSM M2" and "OFSM M1" and "(card (nodes M2) * m) \<le> length xs" and "card (nodes M1) \<le> m" and "vs@xs \<in> L M2 \<inter> L M1" and "is_det_state_cover M2 V" and "V'' \<in> Perm V M1" shows "\<exists> q \<in> nodes M2 . card (RP M2 q vs xs V'') > m"
lemma first_baseE: assumes H1: "basevars v" and H2: "\<And>x. v (first x) = c \<Longrightarrow> Q" shows "Q"
lemma Hermite_of_upt_row_preserves_zero_rows: fixes A::"'a::{bezout_ring_div,normalization_semidom,unique_euclidean_ring}^'cols::{mod_type}^'rows::{mod_type}" assumes i: "is_zero_row i A" and e: "echelon_form A" and a: "ass_function ass" and r: "res_function res" and k: "k \<le> nrows A" shows "is_zero_row i (Hermite_of_upt_row_i A k ass res)"
lemma reduction_word: assumes "q \<in> nodes" "run v q" obtains u w where "R.run w q" "v =\<^sub>I u" "u \<preceq>\<^sub>I w" "lproject visible (llist_of_stream u) = lproject visible (llist_of_stream w)"
lemma ipset_from_cidr_base_wellforemd: fixes base:: "'i::len word" assumes "mask (LENGTH('i) - l) AND base = 0" shows "ipset_from_cidr base l = {base .. base OR mask (LENGTH('i) - l)}"
lemma square_integrable_iff_lspace: assumes "S \<in> sets lebesgue" shows "f square_integrable S \<longleftrightarrow> f \<in> lspace (lebesgue_on S) 2" (is "?lhs = ?rhs")

lemma marked_deletions_sweep_loop_free[simp]: notes fun_upd_apply[simp] shows "\<lbrakk> mut_m.marked_deletions m s; mut_m.reachable_snapshot_inv m s; no_grey_refs s; white r s \<rbrakk> \<Longrightarrow> mut_m.marked_deletions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := None)\<rparr>))"

lemma thetaSAtm_Sbis: assumes "compatAtm atm" shows "thetaSAtm atm \<subseteq> Sbis"

lemma a_comm_var: "ad x \<cdot> ad y \<le> ad y \<cdot> ad x"

lemma older_seniors_older: "y \<in> older_seniors x n \<Longrightarrow> y < x"

lemma iterate_to_set_correct : assumes ins_dj_OK: "set_ins_dj \<alpha> invar ins_dj" assumes emp_OK: "set_empty \<alpha> invar emp" assumes it: "set_iterator it S0" shows "\<alpha> (iterate_to_set emp ins_dj it) = S0 \<and> invar (iterate_to_set emp ins_dj it)"

lemma to_fun_ring_hom: assumes "a \<in> carrier R" shows "(\<lambda>p. to_fun p a) \<in> ring_hom P R"

lemma element_ptr_kinds_simp [simp]: "element_ptr_kinds (Heap (fmupd (cast element_ptr) element (the_heap h))) = {|element_ptr|} |\<union>| element_ptr_kinds h"

lemma indis_sym[sym]: "s \<approx> s' \<Longrightarrow> s' \<approx> s"

lemma sum_sumj_eq2: "i<I ==> sum I s = c s i + sumj I i s"

lemma el_loc_ok_update: "\<lbrakk> \<B> e n; V < n \<rbrakk> \<Longrightarrow> el_loc_ok e (xs[V := v]) = el_loc_ok e xs" and els_loc_ok_update: "\<lbrakk> \<B>s es n; V < n \<rbrakk> \<Longrightarrow> els_loc_ok es (xs[V := v]) = els_loc_ok es xs"

lemma length_assoc_list_of_array [simp]: "length (assoc_list_of_array a) = array_length a"

lemma (in wf_digraph) adj_mk_symmetric_eq: "symmetric G \<Longrightarrow> parcs (mk_symmetric G) = arcs_ends G"

lemma is_ground_lit_is_ground_on_var: assumes ground_lit: "is_ground_lit (subst_lit L \<sigma>)" and v_in_L: "v \<in> vars_lit L" shows "is_ground_atm (\<sigma> v)"

lemma m2f_by_from_m2f : "(m2f_by g f xs) = g (m2f f xs)"

lemma iso_finfun_uminus [code_unfold]: fixes A :: "'a pred\<^sub>f" shows "- ($) A = ($) (- A)"

lemma squareE [elim]: "\<lbrakk> (s,t) \<Turnstile> [A]_v; A (s,t) \<Longrightarrow> B (s,t); v t = v s \<Longrightarrow> B (s,t) \<rbrakk> \<Longrightarrow> B (s,t)"

lemma af\<^sub>F_semantics_rtl: assumes "\<forall>i. \<exists>j>i. af\<^sub>F \<phi> (F\<^sub>n \<phi>) (w [0 \<rightarrow> j]) \<sim> true\<^sub>n" shows "\<forall>i. \<exists>j. af (F\<^sub>n \<phi>) (w [i \<rightarrow> j]) \<sim>\<^sub>L true\<^sub>n"

lemma validTrans_step_srcOf_actOf_tgtOf: "validTrans trn \<Longrightarrow> step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn)"

lemma proj2_incident_iff_Col: assumes "p \<noteq> q" and "proj2_incident p l" and "proj2_incident q l" shows "proj2_incident r l \<longleftrightarrow> proj2_Col p q r"

lemma [simp]: "size_exp' (Handle e pes) = Suc (size_exp' e + size_list (size_prod size size_exp') pes)"

lemma coeff_finite_fourier_poly: assumes "n < length ws" defines "k \<equiv> length ws" shows "coeff (finite_fourier_poly ws) n = (1/k) * (\<Sum>m < k. ws ! m * unity_root k (-n*m))"

lemma ccompatible1: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "ccompatible \<R> (EQ x c)"

lemma ntsmcf_tdghm_smcf_ntsmcf_comp[slicing_commute]: "smcf_dghm \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M ntsmcf_tdghm \<NN> = ntsmcf_tdghm (\<HH> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F\<^sub>-\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)"

lemma min_union_finite: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<union>\<^sub>m B)"

lemma decompV_decomp: assumes "validFrom s tr" and "reach s" shows "decompV (V tr) nid = lV nid (decomp tr nid)"

lemma Zp_square_root_criterion: assumes "p \<noteq> 2" assumes "a \<in> carrier Zp" assumes "b \<in> carrier Zp" assumes "val_Zp b \<ge> val_Zp a" assumes "a \<noteq> \<zero>" assumes "b \<noteq> \<zero>" shows "\<exists>y \<in> carrier Zp. a[^](2::nat) \<oplus> \<p>\<otimes>b[^](2::nat) = (y [^]\<^bsub>Zp\<^esub> (2::nat))"

lemma interpret_floatariths_fresh_eqI: assumes "\<And>i. fresh_floatariths ea i \<or> (i < length ys \<and> i < length zs \<and> ys ! i = zs ! i)" shows "interpret_floatariths ea ys = interpret_floatariths ea zs"

lemma defensive_move_exists_for_Even: assumes [intro]:"position p" shows "winning_position_Odd p \<or> (\<exists> m. move_defensive_by_Even m p)" (is "?w \<or> ?d")

lemma nth_append_singl[simp]: "i < length al \<Longrightarrow> (al @ [a]) ! i = al ! i"

lemma Checkcast_correct: "\<lbrakk> wt_jvm_prog G phi; method (G,C) sig = Some (C,rT,maxs,maxl,ins,et); ins!pc = Checkcast D; wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) et pc; Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ; G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd>; fst (exec_instr (ins!pc) G hp stk loc C sig pc frs) = None \<rbrakk> \<Longrightarrow> G,phi \<turnstile>JVM state'\<surd>"

lemma uminus_add_in_Ker_eq_eq_im: "g\<in>G \<Longrightarrow> h\<in>G \<Longrightarrow> (-g + h \<in> Ker) = (T g = T h)"

lemma lift_spmf_bind_spmf: "lift_spmf (p \<bind> f) = lift_spmf p \<bind> (\<lambda>x. lift_spmf (f x))"

lemma global_oinvariantI [intro]: assumes init: "\<And>\<sigma> p. (\<sigma>, p) \<in> init A \<Longrightarrow> P \<sigma>" and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U; P \<sigma>; U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P \<sigma>'" and step: "\<And>\<sigma> p a \<sigma>' p'. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U; P \<sigma>; ((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A; S \<sigma> \<sigma>' a \<rbrakk> \<Longrightarrow> P \<sigma>'" shows "A \<Turnstile> (S, U \<rightarrow>) (\<lambda>(\<sigma>, _). P \<sigma>)"

lemma nonpos_Reals_one_I [simp]: "1 \<notin> \<real>\<^sub>\<le>\<^sub>0"

lemma "w \<le>p w \<Longrightarrow> \<^bold>|w\<^bold>| = 2 \<Longrightarrow> w \<in> lists {a,b} \<Longrightarrow> hd w = a \<Longrightarrow> w = [a, b] \<or> w = [a, a]"

lemma awalk_verts_dom_if_uneq: "\<lbrakk>u\<noteq>v; awalk u p v\<rbrakk> \<Longrightarrow> \<exists>x. x \<rightarrow>\<^bsub>G\<^esub> v \<and> x \<in> set (awalk_verts u p)"

lemma sys_block_sizes_uniform [simp]: "sys_block_sizes = {\<k>}"

lemma hfs_valid_None_Cons: assumes "hfs_valid_None ainfo uinfo p" "p = hf1 # hf2 # post" shows "hfs_valid_None ainfo (upd_uinfo uinfo hf2) (hf2 # post)"

lemma "(\<Lambda> x. [x]) = (\<Lambda> z. z : [])"

lemma Object_neq_SXcpt [simp]: "Object \<noteq> SXcpt xn"

lemma sigma_le_sets: assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"

lemma step_in_valid_step: "knights_path b ps \<Longrightarrow> step_in ps s\<^sub>i s\<^sub>j \<Longrightarrow> valid_step s\<^sub>i s\<^sub>j"

lemma bound_typ_inst_gen [simp]: "free_tv(t::typ) \<subseteq> free_tv(A) \<Longrightarrow> bound_typ_inst S (gen A t) = t"

lemma smult_unique_scalars : fixes a::'f assumes vs: "basis_for V vs" and v: "v \<in> V" defines as: "as \<equiv> (THE cs. length cs = length vs \<and> v = cs \<bullet>\<cdot> vs)" and bs: "bs \<equiv> (THE cs. length cs = length vs \<and> a \<cdot> v = cs \<bullet>\<cdot> vs)" shows "bs = map ((*) a) as"

lemma typ_of_axiom: "wf_theory thy \<Longrightarrow> t \<in> axioms thy \<Longrightarrow> typ_of t = Some propT"

lemma invalid_assn_mono: "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt B x y \<Longrightarrow> hn_invalid A x y \<Longrightarrow>\<^sub>t hn_invalid B x y"

lemma padic_add_comm: assumes "prime p" shows " \<And>x y. x \<in> carrier (padic_int p) \<Longrightarrow> y \<in> carrier (padic_int p) \<Longrightarrow> x \<oplus>\<^bsub>padic_int p\<^esub> y = y \<oplus>\<^bsub>padic_int p\<^esub> x"

lemma chain_sup_const[simp]: "chain_sup (\<lambda> x. S) = S"

lemma segment_a'_a_ne: "segment G.face_cycle_succ a' a \<noteq> {}"

lemma octo_of_real_mult [simp]: "octo_of_real (x * y) = octo_of_real x * octo_of_real y"

lemma unambigous_prefix_routing_strong_mono: assumes lpfx: "is_longest_prefix_routing (rr#rtbl)" assumes uam: "unambiguous_routing (rr#rtbl)" assumes e:"rr' \<in> set rtbl" assumes ne: "routing_match rr' = routing_match rr" shows "routing_rule_sort_key rr' > routing_rule_sort_key rr"

lemma from_hma\<^sub>v_inj[simp]: "from_hma\<^sub>v x = from_hma\<^sub>v y \<longleftrightarrow> x = y"

lemma "Sup \<circ> atom_map = (id::'a::complete_atomic_boolean_algebra \<Rightarrow> 'a)"

lemma z_lemma_R: fixes I:: "nat list * nat list" fixes sign:: "rat list" assumes consistent: "sign \<in> set (characterize_consistent_signs_at_roots p qs)" assumes welldefined1: "list_constr (fst I) (length qs)" assumes welldefined2: "list_constr (snd I) (length qs)" shows "(z_R I sign = 1) \<or> (z_R I sign = 0) \<or> (z_R I sign = -1)"

theorem jvm_typesafe: assumes wf: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" and start: "wf_start_state P C M vs" and exec: "P \<turnstile> JVM_start_state P C M vs -\<triangleright>ttas\<rightarrow>\<^bsub>jvm\<^esub>* s'" shows "s' \<in> correct_jvm_state \<Phi>"

lemma rank_of_image: assumes "finite S" shows "(\<lambda>x. rank_of x S) ` S = {0..<card S}"

lemma negate_negate_dfa: "negate_dfa (negate_dfa A) = A"

lemma insert_node_ok: assumes "known_ptr parent" and "type_wf h" assumes "parent |\<in>| object_ptr_kinds h" assumes "\<not>is_character_data_ptr_kind parent" assumes "is_document_ptr parent \<Longrightarrow> h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r []" assumes "is_document_ptr parent \<Longrightarrow> \<not>is_character_data_ptr_kind node" assumes "known_ptr (cast node)" shows "h \<turnstile> ok (a_insert_node parent node ref)"

lemma eval_terms_fv_fo_terms_set: "\<sigma> \<odot> ts = \<sigma>' \<odot> ts \<Longrightarrow> n \<in> fv_fo_terms_set ts \<Longrightarrow> \<sigma> n = \<sigma>' n"

lemma clop_iso: "clop f \<Longrightarrow> mono f"

lemma emeasure_T_state_Nil: "T (s, o\<^sub>0) {\<omega> \<in> space S. V [] as \<omega>} = 1"

lemma HEndPhase2_valueChosen2: assumes act: "HEndPhase2 s s' q" and asm4: "\<forall>d\<in>D. b \<le> bal(disk s d p) \<and>(\<forall>q.( phase s q = 1 \<and> b \<le>mbal(dblock s q) \<and> hasRead s q d p ) \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal(block br)))" (is "?P s") shows "?P s'"

lemma pl_1[axiom]: "[[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<phi>)]]"

lemma fv_ik_subset_fv_st[simp]: "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"

lemma num_params_polymul: shows "num_params (p1 *\<^sub>p p2) \<le> max (num_params p1) (num_params p2)"

lemma minSetOfComponentsTestL2p3: "minSetOfComponents level2 {data1, data10, data11} = {sS1, sS2, sS3}"

lemma map_def: "Applicative_DNEList.map = map_fun id (map_fun list_of_dnelist Abs_dnelist) (\<lambda>f xs. remdups (list.map f xs))"

lemma bin_num: "bin0 = 0" "bin1 = 1"

lemma set_disconnected_nodes_typess_preserved: assumes "w \<in> set_disconnected_nodes_locs object_ptr" assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"

lemma finite_fold_fold_keys: assumes "comp_fun_commute f" shows "Finite_Set.fold f A (Set t) = fold_keys f t A"

lemma rbl_add_carry_Cons: "(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) = xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)"

lemma FP_weakest: "(\<And>B. F \<in> stable (A Int B)) \<Longrightarrow> A <= FP F"

lemma ereal_minus_less: fixes x y z :: ereal shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"

lemma max_spec_preserves_length: "max_spec G C (mn, pTs) = {((md,rT),pTs')} \<Longrightarrow> length pTs = length pTs'"

lemma pdevs_val_minus: "pdevs_val (\<lambda>i. e i - f i) xs = pdevs_val e xs - pdevs_val f xs"

lemma is_node_ptr_kind_cast [simp]: "is_node_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)"

lemma p_add: "p x y (n + m) = (\<integral>\<^sup>+ w. p x w n * p w y m \<partial>count_space UNIV)"

lemma miles_to_feet: "mile = 5280 *\<^sub>Q foot"

lemma (in aGroup) ring_nsum_zeroTr:"(\<forall>j \<le> (n::nat). f j \<in> carrier A) \<and> (\<forall>j \<le> n. f j = \<zero>) \<longrightarrow> nsum A f n = \<zero>"

lemma closed_sum_left_subset: \<open>0 \<in> B \<Longrightarrow> A \<subseteq> A +\<^sub>M B\<close> for A B :: "_::monoid_add"

lemma fst_gb_schema_incr: "fst ` set (gb_schema_incr sel ap ab compl upd (b0 # bs) data) = (let (gs, n, data') = add_indices (gb_schema_incr sel ap ab compl upd bs data, data) (0, data); b = (fst b0, n, snd b0); data'' = upd gs b data' in fst ` set (gb_schema_aux sel ap ab compl gs (count_rem_components (b # gs), Suc n, data'') (ab gs [] [b] (Suc n, data'')) (ap gs [] [] [b] (Suc n, data''))) )"

lemma monic_factorization_uniqueness: fixes P::"'a poly set" assumes finite_P: "finite P" and PQ: "\<Prod>P = \<Prod>Q" and P: "P \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}" and finite_Q: "finite Q" and Q: "Q \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}" shows "P = Q"

lemma ogets_NF_wp [wp]: "ovalidNF (\<lambda>s. P (f s) s) (ogets f) P"

lemma induced_arrow_self: shows "induced_arrow a \<chi> = a"

lemma coprime_pderiv_imp_rsquarefree: assumes "coprime (p :: 'a :: field_char_0 poly) (pderiv p)" shows "rsquarefree p"

lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"

lemma Gromov_extension_quasi_isometry_boundary_to_boundary: fixes f::"'a::Gromov_hyperbolic_space_geodesic \<Rightarrow> 'b::Gromov_hyperbolic_space_geodesic" assumes "lambda C-quasi_isometry f" "x \<in> Gromov_boundary" shows "(Gromov_extension f) x \<in> Gromov_boundary"

lemma listsum2_lemma: "\<lbrakk>length xs = vl; n \<le> vl\<rbrakk> \<Longrightarrow> rec_exec (rec_listsum2 vl n) xs = listsum2 xs n"

lemma strategy_proofI: assumes "\<And>Pd Pd' Ch d y. \<lbrakk> mechanism_domain Pd Ch; mechanism_domain (Pd(d:=Pd')) Ch; d \<in> ds; y \<in> \<phi> (Pd(d := Pd')) Ch ds; y \<in> Field (Pd d); \<forall>x\<in>dX (\<phi> Pd Ch ds) d. x \<noteq> y \<and> (x, y) \<in> Pd d \<rbrakk> \<Longrightarrow> False" shows "strategy_proof ds \<phi>"

lemma equivalence_plus_closed: "equivalence x \<Longrightarrow> equivalence (x\<^sup>+)"

lemma mix_of_preferred_is_preferred: assumes "p \<succeq>[\<R>] w" assumes "q \<succeq>[\<R>] w" assumes "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> p q \<succeq>[\<R>] w"

lemma expected_value_is_utility_function: assumes fnt: "finite outcomes" and "outcomes \<noteq> {}" assumes "x \<in> lotteries_on outcomes" and "y \<in> lotteries_on outcomes" assumes "ordinal_utility (lotteries_on outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "measure_pmf.expectation x u \<ge> measure_pmf.expectation y u \<longleftrightarrow> x \<succeq>[\<R>] y" (is "?L \<longleftrightarrow> ?R")

lemma prime_power_eq_one_iff [simp]: "prime p \<Longrightarrow> p ^ n = 1 \<longleftrightarrow> n = 0"

lemma ex_is_poincare_line_points': assumes i12: "i1 \<in> circline_set H \<inter> unit_circle_set" "i2 \<in> circline_set H \<inter> unit_circle_set" "i1 \<noteq> i2" assumes a: "a \<in> circline_set H" "a \<notin> unit_circle_set" shows "\<exists> b. b \<noteq> i1 \<and> b \<noteq> i2 \<and> b \<noteq> a \<and> b \<noteq> inversion a \<and> b \<in> circline_set H"

lemma RP_state_repetition_distribution_productF : assumes "OFSM M2" and "OFSM M1" and "(card (nodes M2) * m) \<le> length xs" and "card (nodes M1) \<le> m" and "vs@xs \<in> L M2 \<inter> L M1" and "is_det_state_cover M2 V" and "V'' \<in> Perm V M1" shows "\<exists> q \<in> nodes M2 . card (RP M2 q vs xs V'') > m"

lemma first_baseE: assumes H1: "basevars v" and H2: "\<And>x. v (first x) = c \<Longrightarrow> Q" shows "Q"

lemma Hermite_of_upt_row_preserves_zero_rows: fixes A::"'a::{bezout_ring_div,normalization_semidom,unique_euclidean_ring}^'cols::{mod_type}^'rows::{mod_type}" assumes i: "is_zero_row i A" and e: "echelon_form A" and a: "ass_function ass" and r: "res_function res" and k: "k \<le> nrows A" shows "is_zero_row i (Hermite_of_upt_row_i A k ass res)"

lemma reduction_word: assumes "q \<in> nodes" "run v q" obtains u w where "R.run w q" "v =\<^sub>I u" "u \<preceq>\<^sub>I w" "lproject visible (llist_of_stream u) = lproject visible (llist_of_stream w)"

lemma ipset_from_cidr_base_wellforemd: fixes base:: "'i::len word" assumes "mask (LENGTH('i) - l) AND base = 0" shows "ipset_from_cidr base l = {base .. base OR mask (LENGTH('i) - l)}"

lemma square_integrable_iff_lspace: assumes "S \<in> sets lebesgue" shows "f square_integrable S \<longleftrightarrow> f \<in> lspace (lebesgue_on S) 2" (is "?lhs = ?rhs")