hcju/isabelleps huggingface dataset:
read_dcache_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "read_data_cache s addr = read_data_cache (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr"
hv_him: fixes q :: "quat" shows "Hv (HIm q) = Quat 0 (Im1 q) (Im2 q) (Im3 q)"
mult_zero_right: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<odot> mbot = mbot"
lfp_fixpoint: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "f (lfp f) = lfp f"
conjuncts_list_nFAND: fixes \<phi>s :: "('a, 'b) aformula list" assumes "list_all (\<lambda>x. \<not> is_FAnd x) \<phi>s"   and "\<phi>s \<noteq> []" shows "conjuncts_list (nFAND \<phi>s) = \<phi>s"
node_not_in_disconnected_nodes: fixes d :: "(_) document_ptr" assumes "d |\<in>| document_ptr_kinds h3" shows "node \<notin> set |h3 \<turnstile> get_disconnected_nodes d|\<^sub>r"
effect_shiftr_length: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref"   and h :: "heap"   and h' :: "heap"   and u :: "unit" assumes "effect (shiftr a key i) h h' u" shows "Array.length h' a = Array.length h a"
c3: assumes "sqn rt ip = \<pi>\<^sub>2 r"   and "\<pi>\<^sub>5 r < the (dhops rt ip)" shows "P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"
size_mono: fixes m' :: "('c, 'd) mapping"   and m :: "('c, 'e) mapping" assumes "finite (Mapping.keys m')"   and "Mapping.keys m \<subseteq> Mapping.keys m'" shows "Mapping.size m \<le> Mapping.size m'"
nt_zero: fixes a :: "'a" assumes "sfg A a" shows "0\<triangleright>a\<^bsub>A\<^esub> = \<zero>"
zero_sheaf_spec_def: fixes U :: "'a set set" shows "zero_sheaf_spec U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.zero_rel (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> \<one>"
slast_def: shows "slast \<equiv> \<mu> slast. \<Lambda> v. Fixrec.run\<cdot>(match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>\<bottom>) +++ match_scons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(case xs of [::] \<Rightarrow> x | y :# ys \<Rightarrow> slast\<cdot>xs)))"
invariant_def: fixes ES :: "('b list set \<times> 'b trm set) set" shows "invariant ES \<equiv> finite ES \<and> finite_rhs ES \<and> soundness ES \<and> distinctness ES \<and> ardenable_all ES \<and> validity ES"
binop: shows "binop \<int>\<^sub>\<circ> vint_mult"
lprod_2_1: fixes a :: "'a"   and b :: "'a"   and R :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> R" shows "([c, a], [b, c]) \<in> lprod R"
hyps_2: shows "P \<turnstile> Ts Casts vs to vs' "
select_convs_2: fixes carrier :: "'c set"   and pop :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mop :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and tp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and un :: "'c"   and more :: "'d" shows "1\<^sub>r\<^bsub>\<lparr>carrier = carrier, pop = pop, mop = mop, zero = zero, tp = tp, un = un, \<dots> = more\<rparr>\<^esub> = un"
close_eq_M_def: fixes P :: "ast_problem"   and M :: "'a atom formula set" assumes "wf_ast_problem P" shows "close_eq_M M = M \<inter> {Atom (predAtm p xs) |p xs. True} \<union> {Atom (Eq a a) |a. True} \<union> {\<^bold>\<not> (Atom (Eq a b)) |a b. a \<noteq> b}"
reciprocal_add: fixes P :: "'a poly"   and p :: "nat"   and Q :: "'a poly" assumes "degree P \<le> p"   and "degree Q \<le> p" shows "reciprocal_poly p (P + Q) = reciprocal_poly p P + reciprocal_poly p Q"
numeral_float: fixes i :: "num" shows "numeral i \<in> float"
wave_not_RF_IN_zero: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_IN f x = 0"
M_cong_7_mod_12: shows "[M = 7] (mod 12)"
HPhase1or2ReadThen_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
swapAbs_psubstAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "goodAbs A"   and "goodEnv rho" shows "A $[rho] $[z1 \<and> z2]_zs = A $[z1 \<and> z2]_zs $[rho &[z1 \<and> z2]_zs]"
nb: shows "set (filter (\<lambda>(v, a). (\<phi>\<^sub>S \<Psi> s) (v, a) = Some True) (all_possible_assignments_for \<Psi>)) = Set.filter (\<lambda>(v, a). (\<phi>\<^sub>S \<Psi> s) (v, a) = Some True) (\<Union>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
div_mult_self3: 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 b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus (times c b) a) b = plus c (divide a b)"
extend_fun: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and B1 :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "B \<subseteq> B1" shows "f \<in> A \<rightarrow> B1"
subspace_row_space: fixes A :: "(('a, 'b) vec, 'c) vec" shows "vec.subspace (row_space A)"
multl_strict: fixes x :: "'a" shows "x \<cdot> \<bottom> = \<bottom>"
tranclp_trans: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "r\<^sup>+\<^sup>+ x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
assms_1: shows "convert_poly var p (xs' @ x # xs) = Some (a, b, c)"
map_update: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and k :: "nat"   and y :: "'b" shows "map f (xs[k := y]) = (map f xs)[k := f y]"
monotone_def: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
partition_by_median_def: fixes k :: "'a"   and ps :: "(real, 'a) vec list" shows "partition_by_median k ps = (let m = axis_median k ps; (l, r) = partition (\<lambda>p. p $ k \<le> m) ps in (l, m, r))"
NatTransCompDefined_2: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and g :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and h :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "f \<approx>>\<bullet> g"   and "g \<approx>>\<bullet> h" shows "f \<approx>>\<bullet> g \<bullet> h"
coinduct: fixes X :: "'c stream \<Rightarrow> bool"   and x :: "'c stream"   and \<phi> :: "'c stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>xs. x = xs \<and> \<phi> xs \<and> (X (stl xs) \<or> alw \<phi> (stl xs))" shows "alw \<phi> x"
test_meet_comm: fixes p :: "'a"   and q :: "'a" shows "\<iota> p \<cdot> \<iota> q = \<iota> q \<cdot> \<iota> p"
setdist_le: fixes x :: "(real, 'n) vec"   and \<epsilon> :: "real" assumes "x \<in> B' \<epsilon>"   and "0 \<le> \<epsilon>" shows "setdist {x} B \<le> \<epsilon> * C"
transp_tllist_all2: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "transp R"   and "transp S" shows "transp (tllist_all2 R S)"
set_take_subset_set_take: fixes m :: "nat"   and n :: "nat"   and xs :: "'g list" assumes "m \<le> n" shows "set (take m xs) \<subseteq> set (take n xs)"
negligible_UNIV: fixes S :: "'c set" shows "negligible S = (indicat_real S has_integral 0) UNIV"
1: shows "[\<^bold>\<diamond>p \<^bold>& \<^bold>\<box>\<^bold>\<not>q in v]"
subs': shows "P \<turnstile> Ss [\<le>] Us"
fixp_above_outside: fixes a :: "'a" assumes "a \<notin> Field leq" shows "fixp_above a = a"
ideD_3: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "CC.cod a = a"
g_image_filter_def: fixes f :: "'k1 \<times> 'v1 \<Rightarrow> ('k2 \<times> 'v2) option"   and m1 :: "'s1" shows "g_image_filter f m1 \<equiv> m1.iterate m1 (\<lambda>kv \<sigma>. case f kv of None \<Rightarrow> \<sigma> | Some (k', v') \<Rightarrow> m2.update_dj k' v' \<sigma>) (m2.empty ())"
comm_trans: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<cdot> v = v \<cdot> u"   and "w \<cdot> v = v \<cdot> w"   and "v \<noteq> \<epsilon>" shows "u \<cdot> w = w \<cdot> u"
simps_1: fixes bs :: "real list"   and c :: "int" shows "Inum bs (C c) = real_of_int c"
simps_3: fixes v :: "'b tree"   and va :: "'b" shows "link \<langle>v, va, \<langle>\<rangle>\<rangle> = \<langle>v, va, \<langle>\<rangle>\<rangle>"
same_prefix_nil: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs @ ys \<le> xs) = (ys = [])"
vcomp_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "span_vertical_category.vcomp C = (\<lambda>\<nu> \<mu>. if partial_magma.arr (span_vertical_category.vcomp C) (span_vertical_category.vcomp C \<nu> \<mu>) then \<lparr>Chn = C (Chn \<nu>) (Chn \<mu>), Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr> else partial_magma.null (span_vertical_category.vcomp C))"
intros: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and c :: "('b, 'a, 'c) Stmt"   and \<Gamma>' :: "'b \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma> c \<Gamma>'"   and "sifum_types.mds_consistent dma mds \<Gamma>"   and "sifum_types.tyenv_eq dma \<Gamma> mem\<^sub>1 mem\<^sub>2" shows "(((c, mds), mem\<^sub>1), (c, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>1 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>'"
add: fixes q :: "nat \<Rightarrow> 'a"   and p :: "nat \<Rightarrow> 'a" assumes "q \<in> carrier P"   and "p \<in> carrier P"   and "Q p"   and "Q q" shows "Q (p \<oplus>\<^bsub>P\<^esub> q)"
intros: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and p :: "prog"   and \<sigma>' :: "char list \<Rightarrow> nat option" assumes "\<sigma> \<in> states p"   and "body p \<rhd> \<sigma> \<leadsto> Some \<sigma>'" shows "tracesp p \<lparr>initial = \<sigma>, final = \<sigma>'\<rparr>"
se_preserves_finiteness2: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c' :: "('c, 'd) conf" assumes "se c l c'"   and "finite (pred c)" shows "finite (pred c')"
succ_drest_eq_drest_succ: fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" shows "state_succ (fmrestrict_set vs s) (action_proj a vs) = fmrestrict_set vs (state_succ s (action_proj a vs))"
typing_case_Abs: fixes \<Gamma> :: "(name \<times> ty) list"   and T :: "ty"   and x :: "name"   and t :: "trm"   and S :: "ty" assumes "\<Gamma> \<turnstile> (\<lambda>x:T. t) : S"   and "x \<sharp> \<Gamma>" obtains U :: "ty" where "S = T \<rightarrow> U" and "(x, T) # \<Gamma> \<turnstile> t : U"
interleave_Nil1: fixes xs :: "'b list" shows "[] \<otimes> xs = {xs}"
M_1: fixes l :: "nat" shows "M (V l) \<subseteq> closure (V l)"
QR_lemma_02: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.e p q = QR.n p q"
alpha: shows "(qinp, qinp') \<in> alphaInp"
Convm_def: shows "Convm = Conv (all1_matrix nr (output_size' m)) m"
ZMod_imp_zmod: fixes m :: "int"   and a :: "int"   and b :: "int" assumes "ZMod m a = ZMod m b" shows "a mod m = b mod m"
join_right: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "y \<sqsubseteq>\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> y"
Span_contains_spanset_append_right: fixes ms :: "'v list"   and ns :: "'v list" assumes "set ms \<subseteq> V"   and "set ns \<subseteq> V" shows "set ns \<subseteq> RSpan (ms @ ns)"
open_closed: fixes S :: "'b set" shows "open S = closed (- S)"
SourcesA12_L1: shows "Sources level1 sA12 = {}"
path_weight_conv_path: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s :: "'b"   and l1 :: "'a"   and l2 :: "'a" assumes "graph weights"   and "s \<in>\<^sub>A graph.path_weight weights l1 l2" shows "\<exists>xs. graph.path weights l1 l2 xs \<and> s = foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b) \<and> (\<forall>ys. graph.path weights l1 l2 ys \<longrightarrow> \<not> foldr (+) (map (\<lambda>(s, l, t). l) ys) (0::'b) < foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b))"
SV_inv4_inductive_2: shows "{SV_inv4 \<inter> (Vinv1 \<inter> Vinv2)} TS.trans sv_TS {> SV_inv4}"
Rel_match_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f" shows "Rel_match R x y = R x y"
swap_uniform_limit: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c"   and x :: "'b"   and S :: "'b set"   and F :: "'a filter"   and l :: "'c"   and h :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F n in F. (f n \<longlongrightarrow> g n) (at x within S)"   and "(g \<longlongrightarrow> l) F"   and "uniform_limit S f h F"   and "F \<noteq> bot" shows "(h \<longlongrightarrow> l) (at x within S)"
r_neg: fixes x :: "int" assumes "x \<in> UNIV" shows "x + \<ominus>\<^bsub>\<Z>\<^esub> x = 0"
valid_node_cases: fixes n :: "'b LDCFG_node" assumes "CFGExit.valid_node n" obtains "n = NewEntry"   | "n = NewExit"   | "local.CFGExit.inner_node n"
subcls_ObjectD: fixes C :: "qtname"   and G :: "prog" assumes "G\<turnstile>Object\<prec>\<^sub>C C" shows "False"
surjective: fixes r :: "('a, 'b, 'c) Quantity_scheme" shows "r = \<lparr>mag = mag r, dim = Quantity.dim r, \<dots> = Quantity.more r\<rparr>"
assms_1: shows "p \<cdot> s \<le> s \<cdot> p \<cdot> s"
lubmap_belowI: fixes m :: "'e \<Rightarrow> 'f option"   and e :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and u :: "'g" assumes "\<And>k v. m k = Some v \<Longrightarrow> e k v \<sqsubseteq> u" shows "lub (mapCollect e m) \<sqsubseteq> u"
pair2: fixes B :: "'a trm"   and B' :: "'a trm"   and A :: "'a trm" assumes "B \<rightarrow>\<beta> B'" shows "SimplyTyped.Pair A B \<rightarrow>\<beta> SimplyTyped.Pair A B'"
ar: shows "punit.is_auto_reduced (punit.reduced_GB F)"
differentiable_at_imp_differentiable_on: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> s \<Longrightarrow> f differentiable at x" shows "f differentiable_on s"
image_mset_subset_mono: fixes M :: "'a multiset"   and N :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "M \<subset># N" shows "image_mset f M \<subset># image_mset f N"
ide_char: fixes a :: "'c \<times> 'c" shows "C'.VV.ide a = (C'.VV.arr a \<and> C'.VxV.ide a)"
i_2: shows "\<not> annos C2 ! i \<le> annos C1 ! i"
difference_eqvt_tvar: fixes pi :: "(W.tvar \<times> W.tvar) list"   and Ys :: "W.tvar list"   and Xs :: "W.tvar list" shows "pi \<bullet> (Xs - Ys) = pi \<bullet> Xs - pi \<bullet> Ys"
conjugation_pair_conj: fixes y :: "'a" shows "conjugation_pair ((\<lhd>) y) (\<lambda>x. x \<rhd> y)"
fdia_d_simp: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bdia (\<cdot>) ar x (ar (ar y)) = antirange_semiring.bdia (\<cdot>) ar x y"
T_noop: fixes n :: "nat"   and rs :: "'b list"   and s :: "'b list" assumes "n = length rs" shows "T s rs (replicate n (0, [])) = (\<Sum>r\<leftarrow>rs. index s r + 1)"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and W :: "'c set" assumes "VectorSpaceHom smult V smult' T"   and "VectorSpaceIso_axioms V T W" shows "VectorSpaceIso smult V smult' T W"
Lxy_rev: fixes qs :: "'a list"   and S :: "'a set" shows "rev (Lxy qs S) = Lxy (rev qs) S"
wpo_ns_refl: fixes s :: "('f, 'b) Term.term" shows "local.wpo.wpo_ns s s"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a') (\<Phi>'.Y a') (\<Phi>'.Y a')"
c_d_2: shows "cbox c d \<subseteq> ball x e"
iTILL_subset: fixes n' :: "nat"   and n :: "nat" assumes "n' \<le> n" shows "[\<dots>n'] \<subseteq> [\<dots>n]"
compower_stepsD': fixes n :: "nat"   and E :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and s' :: "'a" assumes "(E ^^ n) s s'"   and "0 < n" obtains xs :: "'a list" where "Graph_Defs.steps E (s # xs @ [s'])" and "length xs + 1 = n"
prems_2: shows "f \<in> gens \<rightarrow> carrier G"
has_antiderivative_sequence: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "convex S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_derivative f' n x) (at x within S)"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h" shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
conj: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (Cooper.conj p q) = Ifm bbs bs (And p q)"
pirreducible_imp_not_splitted: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "pirreducible (carrier R) p"   and "degree p \<noteq> 1" shows "\<not> splitted p"
ind: shows "bv_extend (length xs) b (rem_initial b xs) = xs"
assms_4: shows "DD ars r (\<tau>', \<upsilon>, \<upsilon>', \<tau>'')"
assms: shows "((\<lambda>xa. ereal (Busemann_function_at xi (u xa) x)) \<longlongrightarrow> - \<infinity>) F"
linear_on_directional_derivative: fixes k :: "enat"   and a :: "'c" assumes "k \<noteq> 0" shows "linear_on UNIV (manifold_eucl.tangent_space k a) (*\<^sub>R) (*\<^sub>R) (directional_derivative k a)"
not_mod_2_eq_1_eq_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(modulo a (numeral.numeral one plus (num.Bit0 num.One)) \<noteq> one) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero)"
fetch_instr_result_3: fixes s' :: "'b sparc_state" assumes "\<exists>e. fetch_instruction s' = Inl e" shows "\<nexists>v. fetch_instruction s' = Inr v"
e_def: shows "e = dist (g1 1) (g2 0)"
normal_imp_subgroup_normalizer: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "N \<lhd> G\<lparr>carrier := H\<rparr>" shows "subgroup H (G\<lparr>carrier := normalizer G N\<rparr>)"
vec_of_list_list_of_vec: fixes a :: "'c vec" shows "vec_of_list (list_of_vec a) = a"
interior_Icc: fixes a :: "'c"   and b :: "'c" shows "interior {a..b} = {a<..<b}"
extract_def: fixes id :: "'d Monomorphic_Monad.id" shows "extract id = (case id of return_id x \<Rightarrow> x)"
simps_11: fixes Vs :: "char list list"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (e\<^sub>1;; e\<^sub>2) = compE\<^sub>1 Vs e\<^sub>1;; compE\<^sub>1 Vs e\<^sub>2"
assump: shows "n \<^bold>\<in> i \<and> m \<^bold>\<in> k"
bound: fixes t :: "real" assumes "a \<le> t" shows "l < f t"
rbt_bulkload_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "('c \<times> 'd) list" shows "ord.rbt_bulkload less xs = foldr (\<lambda>(k, v). ord.rbt_insert less k v) xs rbt.Empty"
BU: shows "set Bs \<subseteq> set (unit_vecs n)"
silent_move_iff: shows "\<tau>mredT = (\<lambda>s s'. \<exists>tl. redT s tl s' \<and> m\<tau>move s tl s')"
pmf_prob_def: fixes p :: "'b pmf"   and A :: "'b set" shows "pmf_prob p A = measure_pmf.prob p A"
simps_2: fixes c :: "('a, 'b) MWLsCom"   and V :: "('a, 'b) MWLsCom list" shows "PPV (c # V) = PPc c @ PPV V"
wf_sub_rel_of_def: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "wf_sub_rel_of g subs \<equiv> sub_rel_of g subs \<and> wf_sub_rel subs"
IH: fixes ops :: "('a \<times> 'a option) list"   and rest :: "('a \<times> 'a option) list" assumes "insert_ops ops"   and "insert_ops rest"   and "rest = before @ op_list"   and "ops = before @ (oid, ref) # op_list" shows "\<exists>xs ys zs. interp_ins rest = xs @ zs \<and> interp_ins ops = xs @ ys @ zs"
single_subset_iff: fixes a :: "'k"   and M :: "'k multiset" shows "({#a#} \<subseteq># M) = (a \<in># M)"
imp_igWlsBinp: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'm option"   and hA :: "'m \<Rightarrow> 'o"   and MOD' :: "('p, 'q, 'varSort, 'sort, 'r, 's, 't, 'o, 'u) model_scheme" assumes "igWlsBinp MOD delta binp"   and "ipresIGWlsAbs hA MOD MOD'" shows "igWlsBinp MOD' delta (lift hA binp)"
prems_1: shows "\<Gamma> \<turnstile> s is t : T\<^sub>1\<rightarrow>T\<^sub>2"
strict_sorted_imp_sorted: fixes xs :: "'a list" assumes "strict_sorted xs" shows "sorted xs"
path_determines_loc: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and l3 :: "'vtx" assumes "path l1 l2 xs"   and "path l1 l3 xs" shows "l2 = l3"
alg_dvd_triv_left_iff': fixes x :: "'b"   and y :: "'b" assumes "x \<noteq> (0::'b)" shows "(x alg_dvd x * y) = algebraic_int y"
Smith_normal_form_mat_0m: fixes A :: "'a mat"   and m :: "nat" assumes "A \<in> carrier_mat 0 m" shows "Smith_normal_form_mat A"
arrI: fixes f1 :: "'a"   and f2 :: "'a \<times> 'a" assumes "B.arr f1"   and "B.VxV.arr f2" shows "B.VxVxV.arr (f1, f2)"
select_index_3_subsets: shows "{j. select_index 3 0 j} = {4, 5, 6, 7} \<and> {j. j < 8 \<and> \<not> select_index 3 0 j} = {0, 1, 2, 3} \<and> {j. select_index 3 1 j} = {2, 3, 6, 7} \<and> {j. j < 8 \<and> \<not> select_index 3 1 j} = {0, 1, 4, 5}"
decide_univ_lem_helper: fixes fml_struct :: "nat fml"   and polys :: "rat poly list"   and fml :: "rat poly fml" assumes "(fml_struct, polys) = convert fml" shows "(\<forall>x. lookup_sem fml_struct (map (\<lambda>p. rpoly p x) polys)) = decide_universal fml"
mono_inf: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "mono f"   and "mono g" shows "mono (f \<sqinter> g)"
real_cond_exp_cong': fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "\<forall>w\<in>space M. f w = g w"   and "f \<in> borel_measurable M" shows "AEeq M (real_cond_exp M F f) (real_cond_exp M F g)"
Max_wrt_mono: fixes x :: "'a"   and y :: "'a" assumes "le x y"   and "x \<in> Max_wrt le" shows "y \<in> Max_wrt le"
sorted_Cons_iff: fixes x :: "'b"   and xs :: "'b list" shows "sorted_less (x # xs) = ((\<forall>y\<in>set xs. x < y) \<and> sorted_less xs)"
can_occur_def: fixes ev :: "('p, 's, 'm) event"   and c :: "('p, 's, 'm) configuration" shows "can_occur ev c \<equiv> case ev of Trans p s s' \<Rightarrow> states c p = s \<and> trans p s s' | Send i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (p, q) \<and> send i p q s s' msg | Recv i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Msg msg \<and> recv i p q s s' msg | Snapshot p \<Rightarrow> \<not> ps c p \<noteq> None | RecvMarker i p q \<Rightarrow> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Marker"