hcju/isabelleps huggingface dataset:
CallRedsFinal: 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 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and M :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and pns' :: "char list list"   and body' :: "expr"   and Ds :: "char list list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and Cs' :: "char list list"   and vs' :: "val list"   and l\<^sub>2' :: "char list \<Rightarrow> val option"   and new_body :: "expr"   and ef :: "expr"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>3 :: "char list \<Rightarrow> val option" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, S)\<rfloor>"   and "P \<turnstile> last Cs has least M = (Ts', T', pns', body') via Ds"   and "P \<turnstile> (C,Cs @\<^sub>p Ds) selects M = (Ts, T, pns, body) via Cs'"   and "length vs = length pns"   and "P \<turnstile> Ts Casts vs to vs' "   and "l\<^sub>2' = [this \<mapsto> Ref (a, Cs'), pns [\<mapsto>] vs']"   and "new_body = (case T' of Class D \<Rightarrow> \<lparr>D\<rparr>body | _ \<Rightarrow> body)"   and "P,E(this \<mapsto> Class (last Cs'), pns [\<mapsto>] Ts) \<turnstile> \<langle>new_body,(h\<^sub>2, l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3)\<rangle>"   and "final ef" shows "P,E \<turnstile> \<langle>Call e None M es,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2)\<rangle>"
assert_rule: fixes R :: "'b \<Rightarrow> bool"   and x :: "'b" shows "<\<up> (R x)> assert R x <\<lambda>r. \<up> (r = x)>"
end_0smult: fixes T :: "'v \<Rightarrow> 'v" assumes "range T \<subseteq> V" shows "(0::'f) \<cdot>\<cdot> T = 0"
HInv4_def: fixes s :: "state" shows "HInv4 s = (\<forall>p. HInv4a s p \<and> HInv4b s p \<and> HInv4c s p \<and> HInv4d s p)"
commute: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) = b \<sqinter> a"
pochhammer_fact: fixes n :: "nat" shows "fact n = pochhammer (1::'a) n"
total_on_def: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
select_convs_2: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sn \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = sn"
comp_assoc: fixes g :: "'c local.term set"   and f :: "'c local.term set"   and h :: "'c local.term set" assumes "g \<cdot> f \<noteq> null"   and "h \<cdot> g \<noteq> null" shows "h \<cdot> g \<cdot> f = (h \<cdot> g) \<cdot> f"
domain_invol: fixes x :: "'a" shows "d (d x) = d x"
skl1_defs_2: shows "skl1_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, signalsInit2 = \<lambda>x. 0, signalsResp2 = \<lambda>x. 0\<rparr>}"
pivot_positions_main: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> nat"   and j :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat nr nc"   and "pivot_fun A f nc"   and "j \<le> f i \<or> nr \<le> i" shows "set (pivot_positions_main A nr nc i j) = {(i', f i') |i'. i \<le> i' \<and> i' < nr} - UNIV \<times> {nc} \<and> distinct (map snd (pivot_positions_main A nr nc i j)) \<and> distinct (map fst (pivot_positions_main A nr nc i j))"
trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "Pareto R x y"   and "Pareto R y z" shows "Pareto R x z"
max_ex: fixes XS :: "'a list set" assumes "XS \<noteq> {}"   and "finite XS" shows "\<exists>xs\<in>XS. \<forall>ys\<in>XS. length ys \<le> length xs"
add_mset_remove_trivial_eq: fixes N :: "'a multiset"   and a :: "'a" shows "(N = add_mset a (remove1_mset a N)) = (a \<in># N)"
enat_ord_simps_5: fixes q :: "enat" shows "(\<infinity> \<le> q) = (q = \<infinity>)"
max_len_def: shows "max_len = Max (cw_len ` L)"
IrrFinGroupRepresentation_axioms: shows "IrrFinGroupRepresentation G (\<cdot>) V"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<squnion> b) \<sqinter> c = a \<sqinter> c \<squnion> b \<sqinter> c"
set_def: fixes t :: "'b trie" shows "Trie_Fun.set t = {xs. isin t xs}"
absorb2: fixes b :: "'a"   and a :: "'a" assumes "b \<preceq> a" shows "a \<sqinter> b = b"
S'_def: shows "S' \<equiv> {y. \<forall>x\<in>S. orthogonal x y}"
group_l_invI: assumes "\<And>x. x \<in> carrier (H\<lparr>carrier := generate H {b}\<rparr>) \<Longrightarrow> \<exists>y\<in>carrier (H\<lparr>carrier := generate H {b}\<rparr>). y \<otimes>\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub> x = \<one>\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub>" shows "Group.group (H\<lparr>carrier := generate H {b}\<rparr>)"
sup_state_def: fixes P :: "'b prog" shows "sup_state P \<equiv> Product.le (Listn.le (subtype P)) (Listn.le (sup_ty_opt P))"
assms_3: shows "H \<simeq>\<^sub>F I"
fx0: fixes x :: "'a" assumes "x \<in> B - A" shows "(0::'b) \<le> f x"
split_at_conv_split_at_it'_2: fixes ts :: "'b list"   and ds :: "'b list"   and n :: "nat"   and xs :: "'b list"   and ts' :: "'b list"   and ds' :: "'b list"   and acc :: "'b list" assumes "(ts, ds) = split_at n xs"   and "(ts', ds') = split_at_it' acc n xs" shows "ds = ds'"
fls_residue_of_int_times_1: fixes i :: "int"   and f :: "'a fls" shows "fls_residue (of_int i * f) = of_int i * fls_residue f"
fds_exp_const: fixes c :: "'a" shows "fds_exp (fds_const c) = fds_const (exp c)"
iter_isolate: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "iteration_finite_conjunctive (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_iteration.fiter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (d::'b::refinement_lattice) \<sqinter> infinite_iteration.infiter seq c = seq (finite_or_infinite_iteration.iter seq nil c) d"
assms_1: shows "InvariantConsistent (getM state @ [(literal, decision)])"
Gram_Schmidt_upt_k_iarrays_def: fixes A :: "real iarray iarray"   and k :: "nat" shows "Gram_Schmidt_upt_k_iarrays A k = foldl Gram_Schmidt_column_k_iarrays A [0..<Suc k]"
simps: fixes a :: "'e list"   and T :: "('e, 'f) TS_scheme" shows "(a \<in> beh T) = (a = [] \<or> (\<exists>s. a = [s] \<and> s \<in> init T) \<or> (\<exists>s b t. a = t # s # b \<and> s # b \<in> beh T \<and> (s, t) \<in> TS.trans T))"
cons_eqvt: fixes pi :: "('e \<times> 'e) list"   and x :: "'d"   and xs :: "'d list" shows "pi \<bullet> (x # xs) = pi \<bullet> x # pi \<bullet> xs"
pseudo_divmod_hom: fixes p :: "'a poly"   and q :: "'a poly" shows "pseudo_divmod (map_poly hom p) (map_poly hom q) = map_prod (map_poly hom) (map_poly hom) (pseudo_divmod p q)"
mono_mono_def: fixes F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'e" shows "mono_mono F = (mono F \<and> (\<forall>f. mono f \<longrightarrow> mono (F f)))"
cone_\<chi>: shows "\<chi> \<in> D.cones a"
diff_diff_add_mset: fixes M :: "'b multiset"   and N :: "'b multiset"   and P :: "'b multiset" shows "M - N - P = M - (N + P)"
sender_ip_valid': fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), a, uu_). anycast (\<lambda>m. not_Pkt m \<longrightarrow> msg_sender m = ip \<xi>) a)"
ReprF: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and n :: "'b" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv enc F encF N ssap"   and "f \<in> F"   and "n \<in> num" shows "Deduct_with_False_Disj_Rename.prveqlPT var subst prv (Generic_Syntax.instInp var subst (encF f) n) (f n)"
set_dropWhileD: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "x \<in> set (dropWhile P xs)" shows "x \<in> set xs"
stutter_extend_en_Some_eq_1: fixes a :: "'b"   and en :: "'d \<Rightarrow> 'b set"   and gc :: "'d" shows "(Some a \<in> stutter_extend_en en gc) = (a \<in> en gc)"
nodes_root_height_upper_bound: fixes k :: "nat"   and t :: "'b btree" assumes "root_order k t"   and "bal t" shows "nodes t * (2 * k) \<le> bound (2 * k) (height t)"
HEndPhase0_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv3 s" shows "HInv3 s'"
subgraph_subset_1: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" assumes "subgraph G\<^sub>1 G\<^sub>2" shows "vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2"
replace_subterm_is_a_function: fixes t :: "'b trm"   and p :: "indices list"   and u :: "'b trm"   and v :: "'b trm" assumes "subterm t p u" shows "\<exists>s. replace_subterm t p v s"
wf_atable_subset: fixes n :: "nat"   and V :: "nat set"   and X :: "'c option list set"   and Y :: "'c option list set" assumes "table n V X"   and "Y \<subseteq> X" shows "table n V Y"
comp_same: fixes x :: "'b" shows "comparator_of x x = Eq"
prefix_elem_to_carriers: fixes xs :: "'evt list"   and i :: "nat"   and x :: "'evt" assumes "xs prefix of i"   and "x \<in> set xs" shows "x \<in> set (history i)"
K'_def: shows "K' = (\<lambda>xa. case xa of None \<Rightarrow> map_pmf Some N | Some x \<Rightarrow> map_pmf Some (K x))"
nhds_def: fixes a :: "'c" shows "nhds a = Inf (principal ` {S. open S \<and> a \<in> S})"
char_approx3: fixes M :: "real measure"   and \<sigma>2 :: "real"   and t :: "real" assumes "real_distribution M"   and "integrable M (\<lambda>x. x)"   and "LINT x|M. x = 0"   and "integrable M power2"   and "LINT x|M. (x - (LINT x|M. x))\<^sup>2 = \<sigma>2" shows "cmod (char M t - complex_of_real (1 - t\<^sup>2 * \<sigma>2 / 2)) \<le> t\<^sup>2 / 6 * (LINT x|M. min (6 * x\<^sup>2) (\<bar>t\<bar> * \<bar>x\<bar> ^ 3))"
defargs: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "profile_bin\<^sub>S\<^sub>t\<^sub>r\<^sub>o\<^sub>n\<^sub>g\<^sub>E\<^sub>q_\<^sub>v_\<^sub>v f"   and "\<tau> \<Turnstile> f x y" shows "\<tau> \<Turnstile> \<upsilon> x \<and> \<tau> \<Turnstile> \<upsilon> y"
predicate2D_vimage2p: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "R \<le> vimage2p f g S"   and "R x y" shows "S (f x) (g y)"
unit_simps_3: shows "cod \<eta> = f\<^sup>* \<star> f"
spmf_rel_map2: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'h spmf"   and g :: "'j \<Rightarrow> 'i"   and y :: "'j spmf" shows "rel_spmf R x (map_spmf g y) = rel_spmf (\<lambda>x y. R x (g y)) x y"
eqvtI: fixes P :: "pi"   and Q :: "pi"   and p :: "(name \<times> name) list" assumes "P \<simeq> Q" shows "p \<bullet> P \<simeq> p \<bullet> Q"
semicontract: fixes T :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "Kohlberg_Neyman_Karlsson T" shows "dist (T x) (T y) \<le> dist x y"
set_diff_eq: fixes A :: "'b set"   and B :: "'b set" shows "A - B = {x \<in> A. x \<notin> B}"
rh1: fixes l :: "'b tree"   and r :: "'b tree" shows "rh l r \<le> 1"
set_mset_filter: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c multiset" shows "set_mset (image_mset f A) = {f a |a. a \<in># A}"
mon_of_entry: fixes p :: "'p" shows "mon_n fg (entry fg p) = mon fg p"
Sup_insert: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and a :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (insert a A) = sup a (Sup A)"
simps_3: fixes I :: "interp"   and \<phi> :: "fml" shows "fml_sem I (! \<phi>) = {\<omega>. \<omega> \<notin> fml_sem I \<phi>}"
sig_p: shows "sig_of_pair p = lt (poly_of_pair p)"
point_ordinate_minus: fixes A :: "point" shows "ordinate (- A) = - ordinate A"
advantage_def: fixes \<A> :: "((bool list \<times> bool list) \<times> 'b, bool list, bool list \<times> bool list) gpv \<times> (bool list \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, bool list, bool list \<times> bool list) gpv)" shows "local.ind_cpa.advantage \<A> = \<bar>spmf (local.ind_cpa.ind_cpa \<A>) True - 1 / 2\<bar>"
simps: fixes a1 :: "'b Heap"   and a2 :: "heap option"   and a3 :: "heap option"   and a4 :: "'b" shows "run a1 a2 a3 a4 = ((\<exists>\<sigma> c r. a1 = c \<and> a2 = \<sigma> \<and> a3 = \<sigma> \<and> a4 = r \<and> is_exn \<sigma>) \<or> (\<exists>\<sigma> c r. a1 = c \<and> a2 = \<sigma> \<and> a3 = None \<and> a4 = r \<and> \<not> is_exn \<sigma> \<and> execute c (the_state \<sigma>) = None) \<or> (\<exists>\<sigma> c r h'. a1 = c \<and> a2 = \<sigma> \<and> a3 = Some h' \<and> a4 = r \<and> \<not> is_exn \<sigma> \<and> execute c (the_state \<sigma>) = Some (r, h')))"
\<pi>_iff: fixes x :: "V"   and k :: "nat"   and y :: "V" assumes "x \<in> \<phi> ` elts (E k)"   and "y \<in> \<phi> ` elts (E k)"   and "k < length \<alpha>s" shows "(\<pi> k x < \<pi> k y) = ((x, y) \<in> VWF)"
inv: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "G \<circ> F = identity_functor.map B"
3: shows "z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))"
sfirst_finite: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(sfirst P \<omega> < \<infinity>) = ev P \<omega>"
max_ultra: fixes F :: "'b set set"   and T :: "'b set set" assumes "Topology.filter F T"   and "carrier\<^bsub>T\<^esub> \<noteq> {}"   and "\<forall>E\<in>Filters\<^bsub>T\<^esub>. F \<subseteq> E \<longrightarrow> F = E" shows "ultra\<^bsub>T\<^esub> F"
get_lower_bound_3: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set"   and V :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "U \<in> I"   and "V \<in> I" shows "direct_lim.get_lower_bound I U V \<subseteq> V"
6: shows "(f \<sqinter> - q)\<^sup>T * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
mono2mono_gfp_eadd: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> enat"   and t' :: "'c \<Rightarrow> enat" assumes "monotone orda (\<lambda>x y. y \<le> x) t"   and "monotone orda (\<lambda>x y. y \<le> x) t'" shows "monotone orda (\<lambda>x y. y \<le> x) (\<lambda>x. t x + t' x)"
Limsup_le_iff: fixes F :: "'c filter"   and X :: "'c \<Rightarrow> 'b"   and C :: "'b" shows "(Limsup F X \<le> C) = (\<forall>y>C. \<forall>\<^sub>F x in F. X x < y)"
distinct_card: fixes xs :: "'g list" assumes "distinct xs" shows "card (set xs) = length xs"
first_half: shows "arr_mset mod_arr l (m - 1) = arr_mset arr l m - {#Min_mset (arr_mset arr l m)#}"
If: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and d :: "com" assumes "C \<turnstile>\<^sub>t {\<lambda>z s. P z s \<and> b s} c {Q}"   and "C \<turnstile>\<^sub>t {\<lambda>z s. P z s \<and> \<not> b s} d {Q}" shows "C \<turnstile>\<^sub>t {P} IF b THEN c ELSE d {Q}"
blaschke_real_preserve_sgn_Im: fixes a :: "complex"   and z :: "complex_homo" assumes "is_real a"   and "cmod a < 1"   and "z \<noteq> \<infinity>\<^sub>h"   and "z \<noteq> inversion (of_complex a)" shows "sgn (Im (to_complex (moebius_pt (blaschke a) z))) = sgn (Im (to_complex z))"
lve_Entry_edge: fixes e :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node"   and E :: "'g"   and valid_edge :: "'l \<Rightarrow> bool"   and src :: "'l \<Rightarrow> 'g"   and trg :: "'l \<Rightarrow> 'g"   and knd :: "'l \<Rightarrow> ('h, 'i, 'j, 'k) edge_kind"   and X :: "'g" assumes "e = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E)" shows "lift_valid_edge valid_edge src trg knd E X e"
sup_neg_inf: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and p :: "'b"   and q :: "'b"   and r :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq p (sup q r) = less_eq (inf p (uminus q)) r"
Gamma_nonpos_Int: fixes z :: "'a" assumes "z \<in> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma z = (0::'a)"
subringI: fixes R :: "('a, 'b) ring_scheme"   and H :: "'a set" assumes "ring R"   and "H \<subseteq> carrier R"   and "\<one>\<^bsub>R\<^esub> \<in> H"   and "\<And>h. h \<in> H \<Longrightarrow> \<ominus>\<^bsub>R\<^esub> h \<in> H"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H\<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> H"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H\<rbrakk> \<Longrightarrow> h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> H" shows "subring H R"
riesz_frechet_representation_existence: fixes f :: "'a \<Rightarrow> complex" assumes "bounded_clinear f" shows "\<exists>t. \<forall>x. f x = t \<bullet>\<^sub>C x"
real_hyp2_B_def: fixes p :: "hyp2"   and q :: "hyp2"   and r :: "hyp2" shows "B\<^sub>K p q r \<equiv> B\<^sub>\<real> hyp2_rep p hyp2_rep q hyp2_rep r"
widen2: fixes narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.wn narrow widen less_eq less" shows "less_eq y (widen x y)"
node_deliverTE': fixes s :: "'c net_state"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(s, i:deliver(d), s') \<in> node_sos S" obtains s :: "'c" and s' :: "'c" and R :: "nat set" where "sb__ = NodeS i s R" and "s'b__ = NodeS i s' R" and "(s, deliver d, s') \<in> S"
minus_1_power_even: fixes k :: "nat" shows "(- (1::'b)) ^ k = (if even k then 1::'b else - (1::'b))"
smult_as_map_poly: fixes a :: "'b" shows "Polynomial.smult a = map_poly ((*) a)"
relation_of_CSP: fixes x :: "('c, 'd) action" shows "is_CSP_process (action.relation_of x)"
f: fixes i :: "'a" assumes "i \<in> I" shows "f i \<in> N \<rightarrow>\<^sub>M M i"
card_of_empty: fixes A :: "'e set" shows "|{}| \<le>o |A|"
monic_mult: fixes p :: "'b poly"   and q :: "'b poly" assumes "monic p"   and "monic q" shows "monic (p * q)"
set_minus_filter_out: fixes xs :: "'c list"   and y :: "'c" shows "set xs - {y} = set (filter (\<lambda>x. x \<noteq> y) xs)"
elims_3: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "\<not> sat_precond_as x xa" obtains s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "\<not> (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
g_growth2: shows "\<exists>C c2. 0 < c2 \<and> C < Min (set bs) \<and> (\<forall>\<^sub>F x in at_top. \<forall>u\<in>{C * x..x}. g' u \<le> c2 * g' x)"
dual_linorder: shows "class.linorder (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "hindered \<Gamma> = (\<exists>f. hindrance \<Gamma> f \<and> current \<Gamma> f \<and> wave \<Gamma> f)"
assms_3: shows "uninst_code.ssa.trivial uses phis g (snd next)"