hcju/isabelleps huggingface dataset:
discrete: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'b \<times> 'b"   and divmod_step :: "num \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b \<times> 'b"   and a :: "'b"   and b :: "'b" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less a b = less_eq (plus a one) b"
eval_closed_lcl_unchanged: 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 e :: "expr"   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 e' :: "expr"   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 "wwf_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"   and "fv e = {}" shows "l' = l"
assms_2: fixes exp :: "exp"   and var :: "Vars.var" assumes "P1 exp" shows "P1 (App exp var)"
left_neutral: fixes a :: "'a::monoid_add" shows "(0::'a::monoid_add) + (a::'a::monoid_add) = a"
map_try_gpv: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and gpv :: "('d, 'e, 'c) gpv"   and gpv' :: "('d, 'e, 'c) gpv" shows "map_gpv f g (TRY gpv ELSE gpv') = TRY map_gpv f g gpv ELSE map_gpv f g gpv'"
cases: fixes a1 :: "'b sequent \<times> nat"   and a2 :: "'b sequent list \<times> 'b sequent"   and a3 :: "('b sequent list \<times> 'b sequent) set" assumes "lastRule a1 a2 a3" obtains r :: "'b sequent list \<times> 'b sequent" and R :: "('b sequent list \<times> 'b sequent) set" and S :: "'b sequent" and \<Gamma> :: "'b form multiset" and \<Delta> :: "'b form multiset" where "a1 = ( \<Gamma>__ \<Rightarrow>* \<Delta>__, 0)" and "a2 = ra__" and "a3 = Ra__" and "ra__ \<in> Ax" and "Ax \<subseteq> Ra__" and "snd (extendRule Sa__ ra__) = ( \<Gamma>__ \<Rightarrow>* \<Delta>__)"   | r :: "'b sequent list \<times> 'b sequent" and R :: "('b sequent list \<times> 'b sequent) set" and S :: "'b sequent" and \<Gamma> :: "'b form multiset" and \<Delta> :: "'b form multiset" and n :: "nat" where "a1 = ( \<Gamma> \<Rightarrow>* \<Delta>, n + 1)" and "a2 = r" and "a3 = R" and "r \<in> R" and "r \<notin> Ax" and "snd (extendRule S r) = ( \<Gamma> \<Rightarrow>* \<Delta>)" and "\<forall>p\<in>set (fst (extendRule S r)). \<exists>m\<le>n. (p, m) \<in> derivable R*"
simps_6: shows "eval_ternary_And TernaryTrue TernaryUnknown = TernaryUnknown"
tm_add_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t1 :: "taylor_model"   and y :: "real"   and t2 :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e t1"   and "y \<in>\<^sub>i range_tm e t2" shows "x + y \<in>\<^sub>i range_tm e (tm_add t1 t2)"
distinct_5: fixes x1 :: "'a"   and x41 :: "bexp"   and x42 :: "'a"   and x43 :: "'a acom"   and x44 :: "'a"   and x45 :: "'a acom"   and x46 :: "'a" shows "SKIP {x1} \<noteq> IF x41 THEN {x42} x43 ELSE {x44} x45 {x46}"
dg_prod_is_arrD_6: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "pdigraph_base \<alpha> I \<AA>"   and "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "\<D>\<^sub>\<circ> b = I"
cs_eq_is_eq_shifted: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat"   and k' :: "nat"   and n :: "nat"   and n' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "cs\<^bsup>\<pi>\<^esup> (k + n) = cs\<^bsup>\<pi>'\<^esup> (k' + n')" shows "cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> n = cs\<^bsup>\<pi>' \<guillemotleft> k'\<^esup> n'"
pre_oldest_configuration_tokens: fixes q :: "'b"   and n :: "nat" shows "(pre_oldest_tokens q n \<noteq> {}) = (configuration q (Suc n) \<noteq> {})"
simps: fixes prec :: "nat"   and x :: "float" shows "ub_sqrt prec x = (if 0 < x then sqrt_iteration prec prec x else if x < 0 then - lb_sqrt prec (- x) else 0)"
inject_16: fixes x23 :: "v"   and y23 :: "v" shows "(C x23 = C y23) = (x23 = y23)"
simplePath_empty_conv: fixes s :: "nat"   and t :: "nat" shows "cf.isSimplePath s [] t = (s = t)"
worst_chain_pred: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and i :: "nat" assumes "wf {(x, y). gt y x}"   and "bad p x" shows "p (worst_chain p gt i) (worst_chain p gt (Suc i))"
C: fixes r :: "'b" assumes "post x r" shows "\<Phi> r"
float_defs_15: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fcompare a b = (if is_nan a \<or> is_nan b then Und else if is_infinity a \<and> sign a = 1 then if is_infinity b \<and> sign b = 1 then Eq else Lt else if is_infinity a \<and> sign a = 0 then if is_infinity b \<and> sign b = 0 then Eq else Gt else if is_infinity b \<and> sign b = 1 then Gt else if is_infinity b \<and> sign b = 0 then Lt else if valof a < valof b then Lt else if valof a = valof b then Eq else Gt)"
f_Exec_State_Idle_replicate_NoMsg_state: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState c)" shows "localState (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = localState c"
nn_integral_dens_ctxt_measure: fixes \<rho> :: "nat \<Rightarrow> val"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "f \<in> borel_measurable (state_measure (V \<union> V') \<Gamma>)" shows "integral\<^sup>N (dens_ctxt_measure dc_False.\<Y> \<rho>) f = \<integral>\<^sup>+ x. \<delta> (merge V V' (x, \<rho>)) * (if P' (merge V V' (x, \<rho>)) then 1 else 0) * f (merge V V' (x, \<rho>)) \<partial>state_measure V \<Gamma>"
before_const: fixes c :: "'b" shows "\<turnstile> $#c = #c"
atom_var_first: fixes a :: "QDelta atom"   and cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" assumes "a \<in> snd ` set (Atoms (preprocess' cs start))"   and "\<forall>var\<in>vars_constraints (map snd cs). var < start" shows "atom_var a < FirstFreshVariable (preprocess' cs start)"
simps_12: fixes p :: "fm"   and q :: "fm" shows "bound0 (Imp p q) = (bound0 p \<and> bound0 q)"
is_acc_gbg: fixes m :: "'b" shows "gb_graph.is_acc (to_gbg_ext m) = is_acc"
set_lbounds: fixes as :: "atom list" shows "set (lbounds as) = {i. Less (Suc i) 0 \<in> set as}"
pmf_prob_code_unfold: fixes p :: "'c pmf" shows "pmf_prob p = measure_pmf.prob p"
continuous_on_closed_invariant: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
assms_1: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"
free_in_quad: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and A :: "atom" assumes "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d" shows "freeIn var (quadratic_sub var a b c d A)"
abs_eq: fixes x :: "\<nu> option" shows "d\<^sub>\<kappa> (make\<kappa> x) = x"
minus_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.group_add minus plus zero uminus" shows "uminus zero = zero"
nres_relI: fixes c :: "'f nres"   and R :: "('f \<times> 'g) set"   and a :: "'g nres" assumes "c \<le> \<Down> R a" shows "(c, a) \<in> \<langle>R\<rangle>nres_rel"
matches_rule_and_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m' :: "'c match_expr"   and a' :: "action"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule (MatchAnd m m') a'], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m' a'], s\<rangle> \<Rightarrow> t"
fold_mset_empty: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b" shows "fold_mset f s {#} = s"
surjective: fixes r :: "('c, 'd) Ring_scheme" shows "r = \<lparr>carrier = carrier r, pop = (\<plusminus>\<^bsub>r\<^esub>), mop = mop r, zero = \<zero>\<^bsub>r\<^esub>, tp = (\<cdot>\<^sub>r\<^bsub>r\<^esub>), un = 1\<^sub>r\<^bsub>r\<^esub>, \<dots> = Ring.more r\<rparr>"
e: shows "e * K\<^sup>2 \<le> cmod (contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f)"
oc': fixes F :: "char list"   and T :: "ty" assumes "P \<turnstile> C\<^sub>0 has F,Static:T in C\<^sub>0" shows "\<exists>v. sfs F = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T"
StepStatus_empty: fixes ST :: "('d, 'e, 'f) status"   and U :: "'f update" shows "Abs_status (HA ST, Conf ST, {}, U !!! Value ST) = StepStatus ST {} U"
l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs \<equiv> (get_ancestors_si = l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_si get_parent get_host \<and> get_ancestors_si_locs = l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_si_locs get_parent_locs get_host_locs) \<and> get_root_node_si = l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_si get_parent get_host \<and> get_root_node_si_locs = l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_si_locs get_parent_locs get_host_locs"
int_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'a trm" shows "U.int \<xi> T = F.int \<xi> T"
iMODb_iMOD_iIN_conv: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] = [ r, mod m ] \<inter> [r\<dots>,m * c]"
thread_oks_bisim_inv: fixes ts1 :: "'t \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ts2 :: "'t \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and tas1 :: "('t, 'x1, 'm1) new_thread_action list"   and tas2 :: "('t, 'x2, 'm2) new_thread_action list" assumes "\<forall>t. (ts1 t = None) = (ts2 t = None)"   and "list_all2 (nta_bisim bisim) tas1 tas2" shows "thread_oks ts1 tas1 = thread_oks ts2 tas2"
sup_bot_left: fixes x :: "'g multiset" shows "{#} \<union># x = x"
VPow_not_vempty: fixes A :: "V" shows "VPow A \<noteq> 0"
bot: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f \<in> \<Theta>[bot](g)"
class_add_has_field_rev: fixes C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "class_add P (C, cdec) \<turnstile> C\<^sub>0 has F,b:T in D"   and "\<And>D'. P \<turnstile> C\<^sub>0 \<preceq>\<^sup>* D' \<Longrightarrow> D' \<noteq> C" shows "P \<turnstile> C\<^sub>0 has F,b:T in D"
mat_row_carrierI: fixes nr :: "nat"   and nc :: "nat"   and r :: "nat \<Rightarrow> 'a Matrix.vec" shows "mat\<^sub>r nr nc r \<in> carrier_mat nr nc"
hoare_triple_def: fixes P :: "assn"   and c :: "'c Heap"   and Q :: "'c \<Rightarrow> assn" shows "<P> c <Q> \<equiv> \<forall>h as \<sigma> r. (h, as) \<Turnstile> P \<and> run c (Some h) \<sigma> r \<longrightarrow> (let h' = the_state \<sigma>; as' = new_addrs h as h' in \<not> is_exn \<sigma> \<and> (h', as') \<Turnstile> Q r \<and> relH {a. a < lim h \<and> a \<notin> as} h h' \<and> lim h \<le> lim h')"
summable''': shows "summable (\<lambda>n. norm (dirichlet_prod f g n / nat_power n s))"
inters_fin: fixes A :: "'c set"   and x :: "'c event list" shows "finite {(t, u, r). r setinterleaves ((t, u), insert tick (ev ` A)) \<and> (\<exists>v. x = r @ v \<and> front_tickFree v \<and> (tickFree r \<or> v = []))}"
count_roots: assumes "p' \<noteq> 0" shows "sign_changes_neg_inf (sturm_squarefree p) - sign_changes_inf (sturm_squarefree p) = card {x. poly p' x = 0}"
locally_diff_closed: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and t :: "'b set" assumes "locally P S"   and "closedin (top_of_set S) t" shows "locally P (S - t)"
foundation8: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<tau> \<Turnstile> \<delta> x \<or> \<tau> \<Turnstile> x \<triangleq> invalid \<or> \<tau> \<Turnstile> x \<triangleq> null"
partition_permI: fixes p :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter p xs @ filter (Not \<circ> p) xs <~~> xs"
a1: shows "low_equal s1 s2 \<and> t1 = s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr> \<and> t2 = s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>"
submodule_inc_0: fixes H :: "'a set" assumes "submodule R M H" shows "\<zero> \<in> H"
INF_greatest: fixes A :: "'b set"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> \<Sqinter> (f ` A)"
C2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>, P, Q) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q)} \<union> weakBisim"   and "xvec \<sharp>* \<Psi>" shows "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P, \<lparr>\<nu>*xvec\<rparr>Q) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q)} \<union> weakBisim"
rev_conv_rev_it': fixes xs :: "'b list"   and acc :: "'b list" shows "rev xs @ acc = rev_it' acc xs"
pathstart_def: fixes g :: "real \<Rightarrow> 'c" shows "pathstart g = g 0"
add_nonpos_eq_0_iff: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subseteq># {#}"   and "y \<subseteq># {#}" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
prime_multiplicity_squarefree_part: fixes p :: "'b"   and n :: "'b" assumes "prime p" shows "multiplicity p (squarefree_part n) = multiplicity p n mod 2"
lnull_ltakeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lnull (ltakeWhile P xs) = (\<not> lnull xs \<longrightarrow> \<not> P (lhd xs))"
nth_mem: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "xs ! n \<in> list.set xs"
bspec: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
inf_left_commute: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
cprank_max1_prod: fixes A :: "'a tensor"   and B :: "'a tensor" assumes "cprank_max1 A"   and "cprank_max1 B" shows "cprank_max1 (A \<otimes> B)"
\<beta>_boundedness_diag_le: fixes y :: "'a"   and m :: "int"   and x :: "'a" assumes "- int (k y) \<le> m"   and "m \<le> int (k x)"   and "x \<in> X"   and "y \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. u x - u y \<le> real_of_int m}"
assms_2: shows "A \<in> {[x, y], [y, x]}"
Suc_1: fixes queue :: "'a list" assumes "n = length (takeWhile (\<lambda>x. x \<noteq> t0) queue)"   and "t0 \<in> set queue" shows "P queue"
Id_on_iff: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "((x, y) \<in> Id_on A) = (x = y \<and> x \<in> A)"
assms_1: shows "f = vs \<odot> 0\<^sup>\<infinity>"
dconvfactor_def: fixes c :: "'c \<Rightarrow>\<^sub>U 'd"   and d :: "(int, sdim) dimvec" shows "dconvfactor c d = LengthF c ^\<^sub>Z dim_nth d Length \<cdot> MassF c ^\<^sub>Z dim_nth d Mass \<cdot> TimeF c ^\<^sub>Z dim_nth d Time \<cdot> CurrentF c ^\<^sub>Z dim_nth d Current \<cdot> TemperatureF c ^\<^sub>Z dim_nth d Temperature \<cdot> AmountF c ^\<^sub>Z dim_nth d Amount \<cdot> IntensityF c ^\<^sub>Z dim_nth d Intensity"
equivp_vimage2p: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a" assumes "equivp R" shows "equivp (vimage2p f f R)"
finprod_all1: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = (1::'c)" shows "prod f A = (1::'c)"
independent_split_3: fixes Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K (Us @ Vs)" shows "Span K Us \<inter> Span K Vs = {\<zero>}"
fst_snd_matrix_to_iarray_echelon_form_of_column_k: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and i :: "nat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A"   and "i \<le> nrows A" shows "snd (echelon_form_of_column_k bezout (A, i) k) = fst (snd (echelon_form_of_column_k_iarrays (matrix_to_iarray A, i, bezout) k))"
compl_unique: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "inf x y = bot"   and "sup x y = top" shows "uminus x = y"
ub3_ub: shows "real_of_float (rapprox_rat (max prec 1) 1 3) < 1"
ih: fixes y :: "nat" assumes "y < n"   and "\<exists>a. inf_chain (\<lambda>ys xs. length ys \<le> y \<and> length xs \<le> y \<and> ext gt ys xs) a" shows "False"
h2_def: shows "h2 \<equiv> \<lambda>z. (\<Sum>n. f n * (a * z) ^ n) / (\<Sum>n. g n * (a * z) ^ n) + Digamma (1 + z) - Digamma (1 - z)"
croot_via_root_selection: fixes ys :: "complex list"   and n :: "nat"   and x :: "complex" assumes "set ys = {y. y ^ n = x}"   and "n \<noteq> 0" shows "croot n x = arg_min_list (\<lambda>y. (- Re y, - Im y)) ys"
instance_of\<^sub>t_def: fixes t\<^sub>1 :: "fterm"   and t\<^sub>2 :: "fterm" shows "instance_of\<^sub>t t\<^sub>1 t\<^sub>2 = (\<exists>\<sigma>. t\<^sub>1 = t\<^sub>2 \<cdot>\<^sub>t \<sigma>)"
is_antihomography_def: fixes f :: "complex_homo \<Rightarrow> complex_homo" shows "is_antihomography f = (\<exists>f'. is_moebius f' \<and> f = f' \<circ> conjugate)"
filter_min_append_def: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "General.filter_min_append rel xs ys = (let P = \<lambda>zs x. \<not> (\<exists>z\<in>set zs. rel z x); ys1 = filter (P xs) ys in filter (P ys1) xs @ ys1)"
erfc_real_le_cancel: fixes a :: "real"   and b :: "real" shows "(erfc a \<le> erfc b) = (b \<le> a)"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and succs :: "'b \<Rightarrow> 'b list" shows "(a1 [succs]\<rightarrow>* a2) = ((\<exists>g. a1 = g \<and> a2 = g) \<or> (\<exists>g g' g''. a1 = g \<and> a2 = g'' \<and> g' \<in> set (succs g) \<and> g' [succs]\<rightarrow>* g''))"
Quotient_rep_abs: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and r :: "'d" assumes "Quotient R Abs Rep T"   and "R r r" shows "R (Rep (Abs r)) r"
\<rho>eq: shows "loopfp\<tau> \<alpha> (USubst.adjoint \<sigma> I \<omega>) X = game_sem (USubst.adjoint \<sigma> I \<omega>) \<alpha>** X"
base_def: fixes \<alpha> :: "rat" shows "LLL.base \<alpha> = real_of_rat (4 * \<alpha> / (4 + \<alpha>))"
seq_stop_eval\<^sub>w: fixes c' :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem :: "Eg1.var \<Rightarrow> nat" shows "eval_abv\<^sub>A \<langle>Stop ;; c', mds, mem\<rangle>\<^sub>A \<langle>c', mds, mem\<rangle>\<^sub>A"
prems_1: shows "\<forall>z\<in>set (z # zs). \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"
edka_complexity_def: shows "edka_complexity \<equiv> let f = \<lambda>_. 0::'capacity in while\<^sub>T (\<lambda>(f, brk, uu_). \<not> brk) (\<lambda>(f, uu_, itc). find_shortest_augmenting_spec f \<bind> (\<lambda>p. case p of None \<Rightarrow> return (f, True, itc) | Some p \<Rightarrow> let f = NFlow.augment_with_path c f p in return (f, False, itc + 1))) (f, False, 0) \<bind> (\<lambda>(f, uu_, itc). assert (itc < 2 * card V * card E + card V) \<bind> (\<lambda>_. return f))"
SL_def: shows "S \<subseteq> {u \<in> U. w u \<le> real c / 2} \<and> L \<subseteq> {u \<in> U. real c / 2 < w u}"
bi: fixes i :: "nat" assumes "i < n" shows "basis_completion v ! i = (if i = 0 then v else unit_vec n (I (i - 1)))"
rv_gr_iff: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real" assumes "measure_space M" shows "(f \<in> rv M) = (\<forall>a. {w. a < f w} \<in> measurable_sets M)"
sd_distl: fixes Y :: "('a \<times> 'b) set"   and x :: "'a \<times> 'b" assumes "Y \<noteq> {}" shows "sd_prod x (Sup_prod Y) = Sup_prod {sd_prod x y |y. y \<in> Y}"
dsum_pOp_mem: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme"   and X :: "'h \<Rightarrow> 'i"   and Y :: "'h \<Rightarrow> 'i" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_dsumag I A"   and "Y \<in> carr_dsumag I A" shows "prod_pOp I A X Y \<in> carr_dsumag I A"
frontier_of_subset_closedin: fixes X :: "'c topology"   and S :: "'c set" assumes "closedin X S" shows "X frontier_of S \<subseteq> S"
restrict_matrix_def: fixes as :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and bs :: "'c list" shows "as \<langle>f\<rangle> bs = (\<lambda>(i, j). if List.member as i \<and> List.member bs j then f (i, j) else bot)"
preserves_src: fixes \<mu> :: "('a E.term, 'a) arr" assumes "arr \<mu>" shows "src\<^sub>B (DN \<mu>) = DN.map\<^sub>0 (src \<mu>)"
icPar2F: fixes Q' :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'" shows "F \<alpha> (P \<parallel> Q')"