hcju/isabelleps huggingface dataset:
inv_dghm_ObjMap_app: fixes a' :: "V"   and a :: "V" assumes "a' = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "inv_dghm \<FF>\<lparr>ObjMap\<rparr>\<lparr>a'\<rparr> = a"
simps: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" shows "to_ordered_collection a1 a2 = ((\<exists>\<tau>. a1 = Collection \<tau> \<and> a2 = Sequence \<tau>) \<or> (\<exists>\<tau>. a1 = Set \<tau> \<and> a2 = OrderedSet \<tau>) \<or> (\<exists>\<tau>. a1 = OrderedSet \<tau> \<and> a2 = OrderedSet \<tau>) \<or> (\<exists>\<tau>. a1 = Bag \<tau> \<and> a2 = Sequence \<tau>) \<or> (\<exists>\<tau>. a1 = Sequence \<tau> \<and> a2 = Sequence \<tau>))"
vrestr_inclusion: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "A \<subseteq> B" shows "f --` A \<subseteq> f --` B"
bt_augment_v_subset_1: fixes q :: "('c, 'd) pq"   and t :: "('d, 'c) bintree" shows "set |q| \<subseteq> set |bt_augment t q|"
emeasure_cball_aux_integral: fixes n :: "nat" shows "\<integral>\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\<^sup>2) ^ n) \<partial>lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))"
cast_chunk_assemble_id: fixes n :: "nat"   and m :: "nat"   and x :: "'b word" assumes "n = LENGTH('a)"   and "m = LENGTH('b)"   and "n * 2 = m" shows "UCAST('a \<rightarrow> 'b) (UCAST('b \<rightarrow> 'a) x) || (UCAST('a \<rightarrow> 'b) (UCAST('b \<rightarrow> 'a) (x >> n)) << n) = x"
bdd_aboveI: fixes A :: "'b multiset set"   and M :: "'b multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<subseteq># M" shows "subset_mset.bdd_above A"
ninv_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" shows "ninv n f = (\<lambda>y\<in>{i. i \<le> n}. SOME x. x \<le> n \<and> y = f x)"
sup_2_def: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = ((a r\<rightarrow> b) l\<rightarrow> b) \<sqinter> ((b r\<rightarrow> a) l\<rightarrow> a)"
wlsAbs_imp_goodAbs: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "goodAbs A"
real_Gromov_converging_to_boundary: fixes u :: "nat \<Rightarrow> real" shows "Gromov_converging_at_boundary u = ((\<lambda>x. ereal (u x)) \<longlonglongrightarrow> \<infinity> \<or> (\<lambda>x. ereal (u x)) \<longlonglongrightarrow> - \<infinity>)"
inducts: fixes x :: "('state, 'act, 'out) trans list"   and P :: "('state, 'act, 'out) trans list \<Rightarrow> bool" assumes "valid x"   and "\<And>trn. validTrans trn \<Longrightarrow> P [trn]"   and "\<And>trn tr. \<lbrakk>validTrans trn; tgtOf trn = srcOf (hd tr); valid tr; P tr\<rbrakk> \<Longrightarrow> P (trn # tr)" shows "P x"
the_nat_def: shows "the_nat \<equiv> map_fun Word.rep id (nat \<circ> take_bit LENGTH('k))"
open_quat_def: fixes U :: "quat set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
r_univ': fixes f :: "recf"   and x :: "nat" assumes "recfn (e_length x) f" shows "eval r_univ [encode f, x] = eval f (Partial_Recursive.list_decode x)"
oG: fixes x :: "'a" assumes "x \<in> S" shows "open (G x)"
homE_1: fixes CC :: "('b, 'c, 'd) category_scheme"   and A :: "'b"   and B :: "'b"   and f :: "'c" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>"   and "B \<in> Ob\<^bsub>CC\<^esub>"   and "f \<in> Hom\<^bsub>CC\<^esub> A B" shows "Dom\<^bsub>CC\<^esub> f = A"
Elem_SucNat_Nat: fixes N :: "ZF" assumes "Elem N HOLZF.Nat" shows "Elem (SucNat N) HOLZF.Nat"
notStuck_WhileTrueD2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s'"   and "s \<in> b" shows "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> \<Rightarrow>\<notin>{Stuck}"
simps_3: fixes y :: "'c"   and x :: "'c" shows "(Num_arc_delta y \<le> Num_arc_delta x) = (y \<le> x)"
bilinear_sum: fixes h :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'c"   and S :: "'f set"   and g :: "'g \<Rightarrow> 'd"   and T :: "'g set" assumes "bilinear h" shows "h (sum f S) (sum g T) = (\<Sum>(i, j)\<in>S \<times> T. h (f i) (g j))"
prems: shows "a \<sharp> App (Lam [c].s1) t1"
assms_4: fixes \<eta> :: "nat" shows "local.ind_cpa.lossless (\<A> \<eta>)"
lossless_D: shows "\<forall>a. lossless_spmf (D2 a)"
length_le_list_ex: fixes xs :: "'a list"   and n :: "nat" assumes "length xs \<le> n" shows "\<exists>ys. length (xs @ ys) = n"
Rep_ccs_cases: fixes y :: "ccs_Rep" assumes "y \<in> Collect ccs_Rep_set" obtains x :: "ccs" where "y = Rep_ccs x"
g3ip_def: shows "g3ip \<equiv> {(xa, x). g3ipp xa x}"
left_option_def: fixes g :: "ZF"   and opt :: "ZF" shows "left_option g opt \<equiv> Elem opt (Fst g)"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>(\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>a\<rparr>\<rparr> = a"
assms_3: shows "f \<in> borel_measurable (state_measure A \<Gamma>)"
assms_1: shows "f\<lparr>c\<rparr> \<in>\<^sub>\<circ> A c"
scale_eq_0_iff: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b" assumes "vector_space scale" shows "(scale a x = (0::'b)) = (a = (0::'a) \<or> x = (0::'b))"
ps_chains_is_ps_chains2_with_empty_S: shows "ps_chains = ps_chains2 {||}"
round_robin_NoneD: fixes n0 :: "nat"   and \<sigma> :: "'t list \<times> nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<alpha>.round_robin n0 \<sigma> s = None"   and "Round_Robin.round_robin_invar \<sigma> (dom (thr s))" shows "\<alpha>.active_threads s = {}"
quotients_def: shows "comp\<GG>.quotients = map (\<lambda>i. G\<lparr>carrier := \<GG> ! (i + 1)\<rparr> Mod \<GG> ! i) [0..<length \<GG> - 1]"
lcmI: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b c"   and "\<And>d. \<lbrakk>dvd.dvd times a d; dvd.dvd times b d\<rbrakk> \<Longrightarrow> dvd.dvd times c d"   and "normalize c = c" shows "c = lcm a b"
bind_def2: fixes m :: "'c\<cdot>'b"   and k :: "'c \<rightarrow> 'a\<cdot>'b" shows "m \<bind> k = join\<cdot>(fmap\<cdot>k\<cdot>m)"
extend_indets_plus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "extend_indets (p + q) = extend_indets p + extend_indets q"
process_ret_cases: fixes rs :: "'c rule list" shows "process_ret rs = rs \<or> (\<exists>rs\<^sub>1 rs\<^sub>2 m. rs = rs\<^sub>1 @ [Rule m Return] @ rs\<^sub>2 \<and> process_ret rs = rs\<^sub>1 @ process_ret ([Rule m Return] @ rs\<^sub>2))"
evals_hext: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "expr list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<Rightarrow>] \<langle>es',(h', l')\<rangle>" shows "h \<unlhd> h'"
meval: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and P' :: "String.literal \<Rightarrow> nat option"   and db :: "(String.literal, event_data list set list) mapping" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "wf_envs \<sigma> j P P' V db" shows "case meval n (\<tau> \<sigma> j) db \<phi> of (xs, \<phi>\<^sub>n) \<Rightarrow> wf_mformula \<sigma> (Suc j) P' V n R \<phi>\<^sub>n \<phi>' \<and> list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>')) [Monitor.progress \<sigma> P \<phi>' j..<Monitor.progress \<sigma> P' \<phi>' (Suc j)] xs"
2: shows "(0::'b) \<notin> snd ` set xs"
restrict_restrict: fixes m :: "'c \<Rightarrow> 'd option"   and A :: "'c set"   and B :: "'c set" shows "m |` A |` B = m |` (A \<inter> B)"
poly_eval_cong: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and a :: "'a \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'b" assumes "p = q"   and "\<And>x. x \<in> indets q \<Longrightarrow> a x = b x" shows "poly_eval a p = poly_eval b q"
span_base: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "a \<in> S" shows "a \<in> MPoly_Type_Class.punit.pmdl S"
in_sum_vs: fixes W1 :: "'c set"   and W2 :: "'c set" assumes "subspace K W1 V"   and "subspace K W2 V" shows "W1 \<subseteq> subspace_sum W1 W2"
induct: fixes P :: "'b \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "('b \<times> nat) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x t1 a lv t2. \<lbrakk>cmp x a = LT \<Longrightarrow> P x t1; cmp x a = GT \<Longrightarrow> P x t2\<rbrakk> \<Longrightarrow> P x \<langle>t1, (a, lv), t2\<rangle>" shows "P a0 a1"
argmin_pairs: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "(argmin f xs, f (argmin f xs)) = argmin snd (map (\<lambda>x. (x, f x)) xs)"
Domain_project_act: fixes act :: "('c \<times> 'c) set" shows "Domain (project_act h act) = project_set h (Domain act)"
vertical_boundary_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "vertical_boundary twoC \<equiv> {(- 1, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}"
i_take_imp_iprefix: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> n \<sqsubseteq> f"
le_inf_lproject_visible: fixes u :: "'action stream"   and v :: "'action stream" assumes "u \<preceq>\<^sub>I v" shows "lproject visible (llist_of_stream u) \<le> lproject visible (llist_of_stream v)"
fun_left_idem: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c"   and z :: "'d" assumes "comp_fun_idem f" shows "f x (f x z) = f x z"
assms_3: shows "nodes G1 \<inter> nodes G2 = {}"
I: shows "\<forall>n\<in>{..<Suc (Suc (length ts))}. (t_sorted xt \<longrightarrow> t_sorted ((zt # Branch y yl yr # ts) ! n)) \<and> (0 < n \<longrightarrow> (\<exists>y' yl' yr'. (Branch y yl yr # ts) ! (n - Suc 0) = Branch y' yl' yr')) \<and> (let ts' = zt # Branch y yl yr # ts @ [Branch x xt Leaf] in t_multiset ((zt # Branch y yl yr # ts) ! n) = (if n = 0 then {#x#} else {#}) + (if x \<le> t_val ((Branch y yl yr # ts @ [Branch x xt Leaf]) ! n) then t_multiset (t_left (ts' ! Suc n)) else t_multiset (t_right (ts' ! Suc n))))"
sol: shows "(sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} X"
secureI_secureFillI: fixes I :: "OBJ"   and X :: "Var set"   and C :: "CtxtProg" assumes "secure I"   and "LOW X C"   and "LOW X Ctxt_Body"   and "body = Fill Ctxt_Body I" shows "secure (Fill C I)"
finite_imageD: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
totality: fixes ES :: "'a ES_rec"   and E :: "'a set"   and t :: "'a list"   and t' :: "'a list" assumes "EventSystems.total ES E"   and "t \<in> Tr\<^bsub>ES\<^esub>"   and "set t' \<subseteq> E" shows "t @ t' \<in> Tr\<^bsub>ES\<^esub>"
assms_4: shows "unitary_schur_decomposition A es = (B, P, Q)"
before_in_setD1: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "x < y in xs" shows "x \<in> set xs"
flatten_def: fixes p :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c" shows "flatten p = (\<Sum>t\<in>keys p. MPoly_Type_Class.punit.monom_mult (1::'c) t (lookup p t))"
dist_alt: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x + z = y + z"   and "x \<cdot> z = y \<cdot> z" shows "x = y"
Abs1_eq_iff_3: fixes a :: "'b"   and x :: "'a"   and b :: "'b"   and y :: "'a" shows "([[atom a]]lst. x = [[atom b]]lst. y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = (a \<leftrightarrow> b) \<bullet> y \<and> atom a \<sharp> y)"
ins: fixes x :: "'a"   and A :: "'a set" shows "MaxR_opt (insert x A) = (if x \<in> Field r then Some (case MaxR_opt A of None \<Rightarrow> x | Some y \<Rightarrow> maxR x y) else MaxR_opt A)"
nhds_inter: fixes U :: "'a set"   and x :: "'a"   and V :: "'a set" assumes "U \<in> nhds x"   and "V \<in> nhds x" shows "U \<inter> V \<in> nhds x"
cfuncset_def: fixes l :: "'c"   and u :: "'c"   and X :: "'e set" shows "Abstract_Rigorous_Numerics.cfuncset l u X = ({l..u} \<rightarrow> X) \<inter> Collect (continuous_on {l..u})"
isoI: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "C.iso f"
closure_iff: fixes a :: "'a set" shows "(a closed) = (closure a = a)"
conseq: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and k :: "nat"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} c { Q}"   and "\<And>s n. P' (s, n) \<Longrightarrow> P (s, k * n)"   and "\<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n div k)"   and "0 < k" shows "\<turnstile>\<^sub>3\<^sub>' {P'} c { Q'}"
LT_imp_D: fixes r :: "('c \<times> 'c) set"   and ars :: "('d \<times> 'c \<times> 'd) set"   and P :: "('d \<times> ('c \<times> 'd) list) \<times> 'd \<times> ('c \<times> 'd) list" assumes "trans r"   and "wf r"   and "\<forall>p. local_peak ars p \<longrightarrow> (\<exists>\<gamma>1 \<gamma>2 \<gamma>3 \<delta>1 \<delta>2 \<delta>3. LT ars r (fst p, snd p, \<gamma>1, \<gamma>2, \<gamma>3, \<delta>1, \<delta>2, \<delta>3))"   and "peak ars P" shows "\<exists>\<sigma>' \<tau>'. DD ars r (fst P, snd P, \<sigma>', \<tau>')"
shadow_root_same_host: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and host' :: "(_) element_ptr" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs"   and "heap_is_wellformed h"   and "type_wf h"   and "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"   and "h \<turnstile> get_shadow_root host' \<rightarrow>\<^sub>r Some shadow_root_ptr" shows "host = host'"
convex_rel_interior_iff: fixes S :: "'b set"   and z :: "'b" assumes "convex S"   and "S \<noteq> {}" shows "(z \<in> rel_interior S) = (\<forall>x\<in>S. \<exists>e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
m': fixes m' :: "nat" assumes "m' \<in> {0<..<m}" shows "fds_nth f m' = (0::'a)"
strong_induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "'a \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<mapsto> x2"   and "\<And>s s' t y. \<lbrakk>s \<mapsto> s'; \<And>z. P z s s'\<rbrakk> \<Longrightarrow> P y (App s t) (App s' t)"   and "\<And>t t' s y. \<lbrakk>t \<mapsto> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (App s t) (App s t')"   and "\<And>x s t y. \<lbrakk>x \<sharp> y; x \<sharp> s\<rbrakk> \<Longrightarrow> P y (App (\<Lambda> x . t) s) (t[x::=s])"   and "\<And>t t' x y. \<lbrakk>x \<sharp> y; t \<mapsto> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (\<Lambda> x . t) (\<Lambda> x . t')"   and "\<And>x t y. \<lbrakk>x \<sharp> y; x \<sharp> t\<rbrakk> \<Longrightarrow> P y (\<Lambda> x . App t (Var x)) t"   and "\<And>x s s' t y. \<lbrakk>x \<sharp> y; x \<sharp> (s, s'); s \<mapsto> s'; \<And>z. P z s s'\<rbrakk> \<Longrightarrow> P y (s to x in t) (s' to x in t)"   and "\<And>x s t t' y. \<lbrakk>x \<sharp> y; x \<sharp> s; t \<mapsto> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (s to x in t) (s to x in t')"   and "\<And>x s t y. \<lbrakk>x \<sharp> y; x \<sharp> s\<rbrakk> \<Longrightarrow> P y ([s] to x in t) (t[x::=s])"   and "\<And>x s y. \<lbrakk>x \<sharp> y; x \<sharp> s\<rbrakk> \<Longrightarrow> P y (s to x in [Var x]) s"   and "\<And>x y s u t ya. \<lbrakk>y \<sharp> ya; x \<sharp> ya; y \<noteq> x; x \<sharp> (y, s, u); y \<sharp> (s, t)\<rbrakk> \<Longrightarrow> P ya ((s to x in t) to y in u) (s to x in t to y in u)"   and "\<And>s s' y. \<lbrakk>s \<mapsto> s'; \<And>z. P z s s'\<rbrakk> \<Longrightarrow> P y [s] [s']" shows "P z x1 x2"
oalist_inv_raw_filter: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and P :: "'a \<times> 'c \<Rightarrow> bool" assumes "oalist_inv_raw x xs" shows "oalist_inv_raw x (filter P xs)"
quad_part_2_neq: fixes var :: "nat"   and L :: "real list"   and SQ :: "real"   and p :: "real mpoly"   and deg :: "nat"   and sq :: "real mpoly" assumes "var < length L"   and "0 \<le> SQ"   and "MPoly_Type.degree p var = deg"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) sq = SQ" shows "aEval (Neq p) (L[var := sqrt SQ]) = aEval (Neq (quadratic_part_2 var sq p)) (L[var := sqrt SQ])"
between_trans: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "between (b, c) a"   and "between (a, c) d" shows "between (b, c) d"
converge_2: shows "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>'\<^esup> n'"
RETURN_Some_ph_relI: fixes y :: "'e"   and x :: "'f nres"   and B :: "('e \<times> 'f) set" assumes "(RETURN y, x) \<in> \<langle>B\<rangle>nres_rel" shows "(RETURN (Some y), x) \<in> \<langle>\<langle>B\<rangle>phantom_rel\<rangle>nres_rel"
monom_one: shows "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
mono_gamma_c: fixes C1 :: "'av st option acom"   and C2 :: "'av st option acom" assumes "C1 \<le> C2" shows "local.\<gamma>\<^sub>c C1 \<le> local.\<gamma>\<^sub>c C2"
normalize_positive_dst_ports_def: shows "normalize_positive_dst_ports = normalize_positive_ports_step (is_Dst_Ports, dst_ports_sel) Dst_Ports"
inv_def: fixes f :: "'b \<Rightarrow> 'a" shows "Hilbert_Choice.inv f = (\<lambda>y. SOME x. f x = y)"
silent_move2_def: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tl :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "mthr.silent_move2 s tl s' = (redT s tl s' \<and> m\<tau>move s tl s')"
mul_mat_of_cols: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and cs :: "'b vec list" assumes "A \<in> carrier_mat nr n"   and "\<And>j. j < length cs \<Longrightarrow> cs ! j \<in> carrier_vec n" shows "A * mat_of_cols n cs = mat_of_cols nr (map ((*\<^sub>v) A) cs)"
of_int_diff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w - z) = minus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
env_restr_add: fixes S :: "'d set"   and m1 :: "'d \<Rightarrow> 'e"   and m2 :: "'d \<Rightarrow> 'e"   and S2 :: "'d set" shows "(m1 ++\<^bsub>S2\<^esub> m2) f|` S = m1 f|` S ++\<^bsub>S2\<^esub> m2 f|` S"
invrules_3: fixes T' :: "nat \<Rightarrow> nat"   and A' :: "nat \<Rightarrow> nat set" assumes "lb T' A' j" shows "makespan T \<le> 2 * makespan T'"
assms_1: shows "envObsC (es s) = envObsC (es s')"
HFinite_def: shows "HFinite = {x. \<exists>r\<in>\<real>. hnorm x < r}"
ub_arctan_bound': fixes x :: "float"   and prec :: "nat" assumes "0 \<le> real_of_float x" shows "arctan (real_of_float x) \<le> real_of_float (ub_arctan prec x)"
flatten_dom_def: shows "flatten_dom \<equiv> l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_flatten_dom Shadow_DOM.get_child_nodes get_tag_name Shadow_DOM.get_root_node get_host find_slot Shadow_DOM.remove Shadow_DOM.append_child remove_shadow_root"
sop_z_def: fixes R :: "('e, 'f) Ring_scheme"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and z :: "'g" shows "sop_z R s z = (\<forall>r\<in>carrier R. r \<^sub>s\<cdot> z = z)"
next_recv: fixes c :: "('p, 's, 'm) configuration"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and s :: "'s"   and s' :: "'s"   and msg :: "'m"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Recv i p q s s' msg \<mapsto> c' = (can_occur (Recv i p q s s' msg) c \<and> (states c p = s \<and> states c' p = s') \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c i = Msg msg # msgs c' i \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>r. ps c r = ps c' r) \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') \<and> (if snd (cs c i) = Recording then cs c' i = (fst (cs c i) @ [msg], Recording) else cs c i = cs c' i))"
code: fixes ws :: "'b stream list" shows "stranspose ws = map shd ws ## stranspose (map stl ws)"
5: shows "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> fv s \<union> fv t"