hcju/isabelleps huggingface dataset:
frameNilStatEq: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<langle>\<epsilon>, \<Psi>\<rangle> \<simeq>\<^sub>F \<langle>\<epsilon>, \<Psi>'\<rangle> = \<Psi> \<simeq> \<Psi>'"
succ_subset: fixes G :: "('a, 'b) graph"   and v :: "'a" assumes "valid_graph G" shows "succ G v \<subseteq> UNIV \<times> nodes G"
inj_adjoint_iff_surj: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "inj (adjoint f) = surj f"
derangements_singleton: fixes x :: "nat" shows "derangements {x} = {}"
set_shadow_root_get_shadow_root: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and shadow_root_ptr_opt :: "(_) shadow_root_ptr option"   and h' :: "(_) heap" assumes "type_wf h"   and "h \<turnstile> set_shadow_root ptr shadow_root_ptr_opt \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r shadow_root_ptr_opt"
casts_casts: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and 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 T :: "ty"   and v :: "val"   and v' :: "val" assumes "wf_prog wf_md P"   and "P \<turnstile> T casts v to v' " shows "P \<turnstile> T casts v' to v' "
borel_measurableI_le: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>y. {x \<in> space M. f x \<le> y} \<in> sets M" shows "f \<in> borel_measurable M"
int_encode_bound: fixes ks :: "int list"   and l :: "int"   and m :: "int" assumes "dioph_is_node ks l m" shows "int_encode m < nat (2 * max \<bar>l\<bar> (sum_list (map abs ks)) + 1)"
fsmult_smult_comm: fixes r :: "('f, 'g) aezfun"   and v :: "'v"   and a :: "'f" assumes "r \<in> FG"   and "v \<in> V" shows "a \<sharp>\<cdot> r \<cdot> v = r \<cdot> a \<sharp>\<cdot> v"
sup_assoc: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" shows "((x \<or> y) \<or> z) = (x \<or> y \<or> z)"
no_self_channel: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option" assumes "distributed_system channel" shows "\<forall>i. \<nexists>p. channel i = Some (p, p)"
add_le_mono2: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "k + i \<le> k + j"
Gramian_determinant_Ints: fixes k :: "nat" assumes "k \<le> length fs"   and "\<And>i j. \<lbrakk>i < n; j < length fs\<rbrakk> \<Longrightarrow> fs ! j $ i \<in> \<int>" shows "d k \<in> \<int>"
drinks2_vend_r2_none: fixes r :: "nat \<Rightarrow>f value option" assumes "r $ 2 = None" shows "possible_steps drinks2 2 r STR ''vend'' [] = {||}"
reduced_row_echelon_form_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "reduced_row_echelon_form A = reduced_row_echelon_form_upt_k A (ncols A)"
mapCollect_def: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option" shows "mapCollect f m = {f k v |k v. m k = Some v}"
Group_RSpan: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list" assumes "RModule R smult M"   and "set ms \<subseteq> M" shows "Group (R_scalar_mult.RSpan R smult ms)"
ID2: fixes f :: "'e \<rightarrow> 'f" shows "f oo ID = f"
Non_Strict_stable2: shows "Non_Strict.stable {(M1, W3), (M2, W2), (M3, W1)}"
image_diff_atLeastAtMost: fixes d :: "'a"   and a :: "'a"   and b :: "'a" shows "(-) d ` {a..b} = {d - b..d - a}"
has_domain_iff_arr: fixes f :: "'a \<times> 'a" shows "(VV.domains f \<noteq> {}) = VV.arr f"
limproc_Sync_F_annex2: fixes Y :: "nat \<Rightarrow> 'c process"   and A :: "'c set"   and a :: "'c event list"   and S :: "'c process" assumes "chain Y"   and "\<forall>t u r. r setinterleaves ((t, u), insert tick (ev ` A)) \<longrightarrow> (\<forall>v. a = r @ v \<longrightarrow> front_tickFree v \<longrightarrow> \<not> tickFree r \<and> v \<noteq> [] \<or> ((\<exists>x. t \<notin> D (Y x)) \<or> u \<notin> T S) \<and> (t \<in> D S \<longrightarrow> (\<exists>x. u \<notin> T (Y x))))" shows "\<exists>x. a \<notin> D (Y x \<lbrakk>A\<rbrakk> S)"
3: fixes c :: "'f"   and z :: "'v" assumes "Fun c [] \<sqsubseteq> \<delta> z" shows "Fun c [] \<sqsubseteq> s \<or> Fun c [] \<sqsubseteq> t"
laplace: fixes s :: "complex" assumes "a < Re s" shows "(f has_laplace 0) s"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "trg\<^sub>B (g \<cdot>\<^sub>B f) = trg\<^sub>B g \<cdot>\<^sub>B trg\<^sub>B f"
map_bset_bsingleton: fixes f :: "'g \<Rightarrow> 'e"   and x :: "'g" shows "map_bset f (bsingleton x) = bsingleton (f x)"
a: shows "set (divisors_nat 48) = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}"
a_subgroup: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme" assumes "additive_subgroup H G" shows "subgroup H (add_monoid G)"
simps_4: fixes uu :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uw :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT ({uu/uv/uw} CONSEQ C) e = preT C e"
fin: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M < \<infinity>"
var_le_var_simps_1: fixes i :: "'a"   and j :: "'a" shows "(var i \<le> var j) = (i = j)"
impl_of: fixes x :: "('b, 'c) alist" shows "alist.impl_of x \<in> {xs. (distinct \<circ> map fst) xs}"
mon_curve_def: fixes m :: "int"   and a :: "int"   and b :: "int"   and p :: "int point" shows "mon_curve m a b p = (case p of Infinity \<Rightarrow> True | Point x y \<Rightarrow> 0 \<le> x \<and> x < m \<and> 0 \<le> y \<and> y < m \<and> y ^^^\<^bsub>m\<^esub> 2 = x ^^^\<^bsub>m\<^esub> 3 ++\<^bsub>m\<^esub> a **\<^bsub>m\<^esub> x ++\<^bsub>m\<^esub> b)"
step: shows "\<forall>ys. order.greater (length Cs) (length ys) \<longrightarrow> (\<forall>x. pgallery (C0 # ys @ [x]) \<longrightarrow> (\<exists>n. (fold fst (map Spair ss) ^^ n) ` x = C0))"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a{b}.P \<longmapsto> Rs" obtains a :: "name" and b :: "name" and P :: "pi" where "aa__ = a \<and> ba__ = b \<and> Pa__ = P" and "Rs = a[b] \<prec> P"
distinct_3: shows "UMinusOp \<noteq> FloorOp"
is_acc_run_limit_alt: fixes G :: "('b, 'c) gb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" assumes "gb_graph G"   and "finite ((g_E G)\<^sup>* `` g_V0 G)" shows "gb_graph.is_acc_run G r = (graph_defs.is_run G r \<and> (\<forall>A\<in>gbg_F G. limit r \<inter> A \<noteq> {}))"
Ord_ord_of: fixes k :: "nat" shows "Ord (ord_of k)"
eq_upair_simps: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "eq_upair (a, b) (c, d) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
starP_star_of: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "(*p* P) (star_of x) = P x"
partition_filter_conv: fixes f :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "List.partition f xs = (filter f xs, filter (Not \<circ> f) xs)"
irreducible_dvd_prod_list: fixes p :: "'c poly"   and as :: "'c poly list" assumes "irreducible p"   and "p dvd prod_list as" shows "Bex (set as) ((dvd) p)"
select_convs_17: 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 more :: "'a" shows "sip \<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, \<dots> = more\<rparr> = sip"
fpxs_of_fls_power: fixes f :: "'b fls"   and n :: "nat" shows "fpxs_of_fls (f ^ n) = fpxs_of_fls f ^ n"
vars_monom: fixes a :: "'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> (0::'b)" shows "vars (monom m (1::'b)) = vars (monom m a)"
l1: shows "l\<^sub>1 \<equiv>\<^sub>L r1"
at_within_Icc_at_right: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order_topology less_eq less open"   and "less a b" shows "topological_space.at_within open a (ord.atLeastAtMost less_eq a b) = topological_space.at_within open a (ord.greaterThan less a)"
enum_D2_def: shows "enum_class.enum = [D1, D2]"
equality: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme" assumes "m r = m r'"   and "l r = l r'"   and "carrots r = carrots r'"   and "hare r = hare r'"   and "tortoise r = tortoise r'"   and "more r = more r'" shows "r = r'"
leg_simps_3: shows "C.arr assoc'_fgh.cod.leg1"
Suc: shows "partn_lst_VWF (\<omega> \<up> (1 + \<nu> * ord_of_nat n)) [ord_of_nat (2 ^ n), \<omega> \<up> (1 + \<nu>)] 2"
P2'_def: shows "P2' = four_block_mat P2 (0\<^sub>m 2 (dim_row A - 2)) (0\<^sub>m (dim_row A - 2) 2) (1\<^sub>m (dim_row A - 2))"
Nil: fixes y :: "'a"   and ys :: "'a list" shows "local.lexordp [] (y # ys)"
vrange_vcomp: fixes r :: "V"   and s :: "V" shows "\<R>\<^sub>\<circ> (r \<circ>\<^sub>\<circ> s) = r `\<^sub>\<circ> \<R>\<^sub>\<circ> s"
e: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state"   and s :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t"   and "s = Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t"
intro: assumes "CARD('a) \<le> CARD('b)" shows "rebase_ge TYPE('a) TYPE('b)"
assms_3: shows "\<forall>s0 rs. length (aoff s0 rs) = length rs"
transpose_code: fixes A :: "(('e, 'd) vec, 'f) vec" shows "($) (Finite_Cartesian_Product.transpose A) = transpose_row A"
not_prfOf_prv_neg_Pf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "\<not> prfOf n \<phi>" shows "bprv (B.neg (psubst Pf [(n, yy), (\<langle>\<phi>\<rangle>, xx)]))"
A1: shows "(a, b) \<in> (SIGMA x:A. {y. x = f y})"
zposs_aposss: fixes n :: "int" shows "(0 < n) = (0 < ant n)"
ipt_tcp_flags_NoMatch: fixes pkt :: "tcp_flag set" shows "\<not> match_tcp_flags ipt_tcp_flags_NoMatch pkt"
pendingSeqs_def: fixes s :: "('a, 'b, 'c) SLin_state" shows "pendingSeqs s \<equiv> {rs. set rs \<subseteq> pendingReqs s}"
coreflexive_conv_closed: fixes x :: "'a" assumes "coreflexive x" shows "coreflexive (x\<^sup>T)"
proper_design_axioms: shows "proper_design \<V> \<B>\<^sup>C"
inj_linear_imp_inv_bounded_linear: fixes f :: "'a \<Rightarrow> 'a" assumes "bounded_linear f"   and "inj f" shows "bounded_linear (inv f)"
collapse_3: fixes val :: "val" assumes "is_RealVal val" shows "RealVal (extract_real val) = val"
dominates_antisymm: fixes g :: "'g"   and n :: "'node"   and n' :: "'node" assumes "invar g"   and "dominates g n n'"   and "dominates g n' n" shows "n = n'"
renaming: fixes xvec :: "name list"   and Tvec :: "'a list"   and p :: "(name \<times> name) list"   and T :: "'b" assumes "length xvec = length Tvec"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "(p \<bullet> xvec) \<sharp>* T" shows "substAssert T xvec Tvec = substAssert (p \<bullet> T) (p \<bullet> xvec) Tvec"
assms_1: shows "x \<in>\<^sub>i range_tm e t1"
that: fixes x :: "real" assumes "{ereal x<..} \<subseteq> A" shows "thesis"
closedin_discrete_topology: fixes U :: "'c set"   and S :: "'c set" shows "closedin (discrete_topology U) S = (S \<subseteq> U)"
arith_simps_4: fixes m :: "num" shows "num.Bit0 m + num.One = num.Bit1 m"
nil: shows "P (fmap_of_list []) (fmap_of_list [])"
conforms_localD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option" assumes "(x, s)\<Colon>\<preceq>(G, L)" shows "G,s\<turnstile>locals s[\<sim>\<Colon>\<preceq>]L"
closure_outside_subset: fixes S :: "'a set" assumes "closed S" shows "closure (outside S) \<subseteq> S \<union> outside S"
hyps_1: fixes v :: "'a vec" assumes "l = dim_vec v" shows "P v"
jointfun_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'h"   and m :: "nat"   and g :: "nat \<Rightarrow> 'h" shows "jointfun n f m g = (\<lambda>i. if i \<le> n then f i else g (sliden (Suc n) i))"
INV_trans_from_refinement: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme"   and K :: "'f set"   and I :: "'f set" assumes "PO_refines R Ta Tc"   and "K \<subseteq> Range R"   and "Range R \<subseteq> I" shows "{K} TS.trans Tc {> I}"
simps_5: fixes x21 :: "'b"   and x22 :: "'b" shows "DenyAllFromTo x21 x22 \<noteq> DenyAll"
inject_3: fixes x31 :: "nat"   and x32 :: "int"   and x33 :: "Cooper.num"   and y31 :: "nat"   and y32 :: "int"   and y33 :: "Cooper.num" shows "(CN x31 x32 x33 = CN y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
wf_operator_effE: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pres, effs, cost)" obtains "distinct (map (\<lambda>(uu_, v, uu_, uu_). v) effs)" and "\<And>epres x vp v. (epres, x, vp, v) \<in> set effs \<Longrightarrow> wf_partial_state epres" and "\<And>epres x vp v. (epres, x, vp, v) \<in> set effs \<Longrightarrow> x < numVars" and "\<And>epres x vp v. (epres, x, vp, v) \<in> set effs \<Longrightarrow> v < numVals x" and "\<And>epres x vp v. (epres, x, vp, v) \<in> set effs \<Longrightarrow> case vp of None \<Rightarrow> True | Some v \<Rightarrow> v < numVals x"
card_of_Card_order: fixes A :: "'b set" shows "Card_order |A|"
left_unique_rel_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "left_unique A"   and "left_unique B" shows "left_unique (rel_gpv A B)"
foldl_rule_aux: fixes I :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and l0 :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" assumes "I \<sigma>0 l0"   and "\<And>l1 l2 x \<sigma>. \<lbrakk>l0 = l1 @ x # l2; I \<sigma> (x # l2)\<rbrakk> \<Longrightarrow> I (f \<sigma> x) l2" shows "I (foldl f \<sigma>0 l0) []"
mem_simps_7: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
dom_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VxVxV.dom f = (if B.arr (fst f) \<and> B.VxV.arr (snd f) then (B.dom (fst f), B.VxV.dom (snd f)) else (B.null, B.VxV.null))"
normalize_quot_aux_5: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "gcd a b * unit_factor b \<noteq> (0::'b)"
sum_distrib_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and r :: "'a" assumes "class.semiring_0 plus zero times" shows "times (comm_monoid_add.sum plus zero f A) r = comm_monoid_add.sum plus zero (\<lambda>n. times (f n) r) A"
sx: shows "sx = \<lbrakk>quot_dbfm (subst_dbfm (trans_tm [] u) (decode_Var v) B')\<rbrakk>e"
valofn_le_imp_exponent_le: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "valofn x \<le> valofn y" shows "IEEE.exponent x \<le> IEEE.exponent y"
not_nofail_iff: fixes S :: "'b nres" shows "(\<not> nofail S) = (S = FAIL)"
lm081: fixes f :: "('c \<times> 'd) set"   and A :: "'c set" assumes "f \<in> injectionsUniverse" shows "Range (Outside' A f) = Range f - f `` A"
fn_fst_conv: fixes f :: "'a \<Rightarrow> 'c" shows "(\<lambda>x. f (fst x)) = (\<lambda>(a, uu_). f a)"
irfd_def: shows "fd\<^sup>-\<^sub>\<R> = \<R> \<circ> fd\<^sup>-\<^sub>\<F>"
vsubstEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var"   and y1 :: "'var" assumes "goodEnv rho"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "rho &[y1 // y]_ys &[z1 // z]_zs = rho &[z1 // z]_zs &[y1 @ys[z1 / z]_zs // y]_ys"
add_poly_l_add_poly_p': fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset"   and q :: "(char list list \<times> int) list"   and q' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> sorted_poly_rel"   and "(q, q') \<in> sorted_poly_rel" shows "add_poly_l (p, q) \<le> \<Down> sorted_poly_rel (add_poly_p' p' q')"
nnf_def: shows "??.DLO.nnf = DLO.nnf"
Cod_in_Obj: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Cod f \<in> {False, True}"
scc_of_eq: fixes u :: "'a"   and v :: "'a" assumes "u \<in> G.scc_of v" shows "G.scc_of u = G.scc_of v"
finite_derivs: fixes r :: "'a rexp" shows "finite {\<guillemotleft>derivs xs r\<guillemotright> |xs. True}"
env_restr_cont: fixes S :: "'a set" shows "cont (env_restr S)"
lt_eqI_keys: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p"   and "\<And>u. u \<in> keys p \<Longrightarrow> u \<preceq>\<^sub>t v" shows "lt p = v"
zero_less_divide_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide one a) = less zero a"
trian: fixes e :: "'b" assumes "e \<in> arcs G" shows "dist (head G e) \<le> dist (tail G e) + ereal (c e)"
sSum_commute: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "H \<minusplus> K = K \<minusplus> H"
arr_iff_in_hom: fixes f :: "'d" shows "D.arr f = \<guillemotleft>f : D.dom f \<rightarrow>\<^sub>D D.cod f\<guillemotright>"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "nat set \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> 'a"   and f5 :: "'b \<Rightarrow> 'a"   and f6 :: "nat \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'a"   and seq_action :: "'b seq_action" shows "P (case seq_action of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = (\<not> ((\<exists>x1. seq_action = broadcast x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. seq_action = groupcast x21 x22 \<and> \<not> P (f2 x21 x22)) \<or> (\<exists>x31 x32. seq_action = unicast x31 x32 \<and> \<not> P (f3 x31 x32)) \<or> (\<exists>x4. seq_action = \<not>unicast x4 \<and> \<not> P (f4 x4)) \<or> (\<exists>x5. seq_action = send x5 \<and> \<not> P (f5 x5)) \<or> (\<exists>x6. seq_action = deliver x6 \<and> \<not> P (f6 x6)) \<or> (\<exists>x7. seq_action = receive x7 \<and> \<not> P (f7 x7)) \<or> seq_action = \<tau>\<^sub>s \<and> \<not> P f8))"
local_lipschitz_def: fixes T :: "'c set"   and X :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "local_lipschitz T X f \<equiv> \<forall>x\<in>X. \<forall>t\<in>T. \<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)"
saturate_sound: fixes \<Gamma> :: "'b inference set"   and C :: "'b literal multiset"   and CC :: "'b literal multiset set"   and I :: "'b set" assumes "sound_inference_system \<Gamma>"   and "C \<in> inference_system.saturate \<Gamma> CC"   and "I \<Turnstile>s CC" shows "I \<Turnstile> C"
folds_plus_simp: fixes rs :: "'b rexp set" assumes "finite rs" shows "lang (\<Uplus>rs) = \<Union> (lang ` rs)"
prems_2: shows "all_acquired (x # sb) \<inter> dom \<S> - dom \<S>' = {}"
wfactors_mult: fixes as :: "'a list"   and a :: "'a"   and bs :: "'a list"   and b :: "'a" assumes "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "wfactors G (as @ bs) (a \<otimes> b)"
contentsATpre_def: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(self.contents@pre()) = (\<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self then \<tau> \<Turnstile> true \<and> \<tau> \<Turnstile> res \<triangleq> if self.boss@pre \<doteq> null then Set{self.salary@pre} else self.boss@pre.contents@pre()->including\<^sub>S\<^sub>e\<^sub>t(self.salary@pre) endif else \<tau> \<Turnstile> res \<triangleq> invalid)"
coeff_zero: fixes p :: "linear_poly"   and x :: "nat" shows "(coeff p x \<noteq> 0) = (x \<in> vars p)"
ma_ge_0: fixes x :: "mini_alg" shows "ge_0 (real_of x) = ma_ge_0 x"
GF_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltln" shows "suffix i w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"
L_zero_space_2: fixes M :: "'b measure" shows "zero_space\<^sub>N (\<LL> 0 M) = borel_measurable M"
assms: shows "integrable (lebesgue_on {- a..a}) f"
c: shows "(i has_integral f b - (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
psubstBinp_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option"   and binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" shows "binp %%[rho] \<equiv> lift (psubstAbs rho) binp"
6: shows "lselect (s - {least s}) (ltake (enat (least s)) w) = <>"
tsp__ncop2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B CTSP P Q" shows "\<not> Coplanar A B C Q"
arr_char: fixes f :: "(bool, unit) arr" shows "arr f = (f = MkIde False \<or> f = MkIde True)"
refl_on_def': fixes A :: "'g set"   and r :: "('g \<times> 'g) set" shows "refl_on A r = ((\<forall>(x, y)\<in>r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
min_empty: fixes w :: "nat \<Rightarrow> real"   and m :: "nat"   and S :: "nat \<Rightarrow> 'b set"   and R :: "'b set"   and k :: "nat" assumes "Set_Cover w m S"   and "S (Set_Cover.min_arg w S R k) \<inter> R = {}" shows "\<forall>i\<in>{1..k}. S i \<inter> R = {}"
fpxs_X_power_conv_shift: fixes r :: "rat" shows "fpxs_X_power r = fpxs_shift (- r) 1"
assms_9: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
distinct_1: fixes x2 :: "loc" shows "Null \<noteq> Addr x2"
simps_2: fixes bl :: "cell list"   and n :: "nat" shows "bl2nat (Bk # bl) n = bl2nat bl (Suc n)"
prefix_slength: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" assumes "prefix\<cdot>xs\<cdot>ys = TT" shows "le\<cdot>(slength\<cdot>xs)\<cdot>(slength\<cdot>ys) = TT"
prems_1: shows "le\<cdot>1\<cdot>n = TT"
height_Unbal_r: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and r :: "'a tree"   and r' :: "'a tree"   and l :: "'a tree"   and y :: "'a" assumes "ins n (d + 1) x r = Unbal r'"   and "bal_i n (height \<langle>l, y, r\<rangle> + d)" shows "height l < height r'"
finite_imp_small: fixes X :: "'a set" assumes "finite X" shows "small X"
walk_middle_edge: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a"   and ys :: "'a list" assumes "walk (xs @ v # w # ys)" shows "v \<rightarrow> w"