hcju/isabelleps huggingface dataset:
zero_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
integral_ineq_eq_0_then_AE: fixes f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "AE x in M. f x \<le> g x"   and "integrable M f"   and "integrable M g"   and "integral\<^sup>L M f = integral\<^sup>L M g" shows "AE x in M. f x = g x"
ln_approx_abs: fixes x :: "real"   and n :: "nat" assumes "1 < x" shows "\<bar>ln x - ((\<Sum>k<n. 2 * ((x - 1) / (x + 1)) ^ (2 * k + 1) / real (2 * k + 1)) + ((x - 1) / (x + 1)) ^ (2 * n + 1) / (1 - ((x - 1) / (x + 1))\<^sup>2) / real (2 * n + 1))\<bar> \<le> ((x - 1) / (x + 1)) ^ (2 * n + 1) / (1 - ((x - 1) / (x + 1))\<^sup>2) / real (2 * n + 1)"
plusU_llist_assoc: fixes xs :: "udom\<cdot>llist"   and ys :: "udom\<cdot>llist"   and zs :: "udom\<cdot>llist" shows "plusU\<cdot>(plusU\<cdot>xs\<cdot>ys)\<cdot>zs = plusU\<cdot>xs\<cdot>(plusU\<cdot>ys\<cdot>zs)"
a_get_owner_document_tups_def: shows "local.CD.a_get_owner_document_tups = [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, local.CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, local.CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, local.CD.a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
K5_imp_Gets: fixes A :: "agent"   and B :: "agent"   and servTicket :: "msg"   and servK :: "nat"   and T3 :: "nat"   and evs :: "event list" assumes "Says A B \<lbrace>servTicket, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>authK Ts authTicket T2. Gets A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs \<and> Says A Tgs \<lbrace>authTicket, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"
induct_matrix_elementary_alt: fixes P :: "((real, 'a) vec, 'a) vec \<Rightarrow> bool"   and A :: "((real, 'a) vec, 'a) vec" assumes "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (A ** B)"   and "\<And>A i. row i A = 0 \<Longrightarrow> P A"   and "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = 0) \<Longrightarrow> P A"   and "\<And>m n. m \<noteq> n \<Longrightarrow> P (\<chi>i j. mat 1 $ i $ Fun.swap m n id j)"   and "\<And>m n. m \<noteq> n \<Longrightarrow> P (\<chi>i j. of_bool (i = m \<and> j = n \<or> i = j))" shows "P A"
noleB: fixes t :: "real" assumes "t \<in> {0..1}" shows "cmod (vector_derivative \<gamma> (at t)) \<le> B"
in_zip_imp_size_lt_apps: fixes s :: "('a, 'b) tm"   and y :: "'c"   and ss :: "('a, 'b) tm list"   and ys :: "'c list"   and \<zeta> :: "('a, 'b) hd" assumes "(s, y) \<in> set (zip ss ys)" shows "size s < size (apps (Hd \<zeta>) ss)"
iso_lunit': fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.iso (Span.lunit' f)"
simps_5: fixes b :: "bexp"   and c :: "com"   and X :: "char list set" shows "L (WHILE b DO c) X = vars b \<union> X \<union> L c X"
in_spanI: fixes v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a"   and A :: "'a vec set"   and W :: "'a vec set" assumes "v = lincomb a A"   and "finite A"   and "A \<subseteq> W" shows "v \<in> local.span W"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q, \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q) \<in> Rel"
MGF_nested_Methd: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and A :: "state triple set" assumes "G,insert ({Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}) A\<turnstile>{Normal \<doteq>} In1l (body G C sig)\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}"
star_aci_5: fixes a :: "assn" shows "a * emp = a"
in_idem: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "max x (fMax A) = fMax A"
coef_\<zeta>'_ge_\<zeta>: shows "coef_hd \<zeta>' (length ss') \<ge>\<^sub>p coef_hd \<zeta> (length ss)"
3: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> set ys"   and "b \<in> set ys"   and "a \<noteq> b" shows "set a \<inter> set b = {}"
r_def: shows "r = (case c of LEQ_ns x xa \<Rightarrow> 1 | GEQ_ns linear_poly a \<Rightarrow> - 1)"
lm_correct_21: fixes ma :: "('a, 'b) assoc_list"   and Pa :: "'a \<times> 'b \<Rightarrow> bool" assumes "lm.invar ma" shows "lm.ball ma Pa = (\<forall>u v. lm.\<alpha> ma u = Some v \<longrightarrow> Pa (u, v))"
bd_cts_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "bd_cts t = (\<forall>M. (\<forall>i. M i \<tturnstile> M (Suc i) \<and> sound (M i)) \<longrightarrow> (\<exists>b. \<forall>i. bounded_by b (M i)) \<longrightarrow> t (Sup_exp (range M)) = Sup_exp (range (t \<circ> M)))"
Graph4: fixes T :: "nat"   and E :: "(nat \<times> nat) list"   and M :: "node list"   and I :: "nat"   and R :: "nat" assumes "T \<in> Reach E"   and "Roots \<subseteq> Blacks M"   and "I \<le> length E"   and "T < length M"   and "R < length E"   and "\<forall>i<I. \<not> BtoW (E ! i, M)"   and "R < I"   and "M ! fst (E ! R) = Black"   and "M ! T \<noteq> Black" shows "\<exists>r\<ge>I. r < length E \<and> BtoW (E[R := (fst (E ! R), T)] ! r, M)"
image2pE: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and R :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and fx :: "'b"   and gy :: "'d" assumes "image2p f g R fx gy" obtains x :: "'a" and y :: "'c" where "fx = f x" and "gy = g y" and "R x y"
boundedE: fixes b :: "'e"   and c :: "'e"   and a :: "'e" assumes "sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
hull_inc: fixes x :: "'a"   and S :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "x \<in> S" shows "x \<in> P hull S"
assms: fixes x :: "'a" shows "f\<cdot>x\<cdot>\<bottom> = \<bottom>"
Un_Diff_cancel2: fixes B :: "'a set"   and A :: "'a set" shows "B - A \<union> A = B \<union> A"
llast_llist_of: fixes xs :: "'b list" shows "llast (llist_of xs) = last xs"
rbl_succ2_def: fixes b :: "bool"   and xs :: "bool list" shows "rbl_succ2 b xs = (if b then rbl_succ xs else xs)"
TC_subcs: fixes G :: "('c, 'd) form list"   and c :: "'c"   and s :: "'c FOL_Fitting.term" assumes "infinite UNIV"   and "\<stileturn> G" shows "\<stileturn> subcs c s G"
simps_2: fixes pverts :: "'b set"   and parcs :: "('b \<times> 'b) set"   and more :: "'c" shows "parcs \<lparr>pverts = pverts, parcs = parcs, \<dots> = more\<rparr> = parcs"
rSum2: assumes "\<Psi> \<rhd> Q \<longmapsto> Rs"   and "guarded Q" shows "Prop"
slistmem_bottom1: fixes x :: "'a" shows "\<not> slistmem x \<bottom>"
equal_prefix_states: shows "\<forall>k\<le>i. S (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) k = S (t[j - 1 := t ! j, j := t ! (j - 1)]) k"
assms_1: shows "theta \<subseteq> RetrT (theta \<union> BisT)"
geod_end: fixes n :: "nat" shows "geod n (2 ^ n) = y0"
lspasl_c_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "(heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h3 h0) = (heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> h2 = h3)"
var\<^sub>2: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "x \<notin> domA \<Gamma>"   and "isVal e" shows "(\<Gamma>, e, Upd x # S) \<Rightarrow> ((x, e) # \<Gamma>, e, S)"
equals0I: fixes A :: "'c set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
weak_action_modality_tau: fixes x :: "('idx, 'pred, 'act) formula" shows "\<langle>\<langle>\<tau>\<rangle>\<rangle>x = weak_tau_modality x"
ln_smallo_imp_flat: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and q :: "real"   and p :: "real" assumes "filterlim f at_top at_top"   and "filterlim g at_top at_top"   and "(\<lambda>x. ln (f x)) \<in> o(\<lambda>x. ln (g x))"   and "0 < q" shows "(\<lambda>x. f x powr p) \<in> o(\<lambda>x. g x powr q)"
sub_block_nominals: fixes f :: "'d \<Rightarrow> 'b"   and block :: "('c, 'd) fm list \<times> 'd" shows "block_nominals (sub_block f block) = f ` block_nominals block"
\<phi>_def: shows "\<phi> \<equiv> \<lambda>y. SOME x. x \<in> X \<and> y \<in> B x"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a + b = b)"
len: shows "length (col (mat_multI ze pl ti nr m1 m2) j) = length (map (\<lambda>i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..<nr])"
simps_2: fixes c :: "color"   and l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"
hom_var: fixes f :: "'b \<Rightarrow> 'a"   and i :: "'b" shows "hom f (var i) = f i"
add_left_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "(x + y = x + z) = (y = z)"
supports_perm: fixes p :: "perm" shows "{a. p \<bullet> a \<noteq> a} supports p"
X: shows "X \<subseteq> S1 \<times> S2"
partial_equivI: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" assumes "R \<subseteq> A \<times> A"   and "sym R"   and "trans R" shows "partial_equiv A R"
of_nat_zero: fixes n :: "nat" assumes "CARD('a) dvd n" shows "of_nat n = 0"
eval_prim: fixes \<rho> :: "(nat \<times> bval) list"   and e1 :: "exp"   and n1 :: "nat"   and e2 :: "exp"   and n2 :: "nat"   and n3 :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "\<rho> \<turnstile> e1 \<Down> BNat n1"   and "\<rho> \<turnstile> e2 \<Down> BNat n2"   and "n3 = f n1 n2" shows "\<rho> \<turnstile> EPrim f e1 e2 \<Down> BNat n3"
simps: fixes x :: "char list" shows "E\<^sub>n\<^sub>a\<^sub>t x = (if x = ''x'' then 26 else if x = ''y'' then 5 else 0)"
RangeE: fixes b :: "'c"   and r :: "('d \<times> 'c) set" assumes "b \<in> Range r" obtains a :: "'d" where "(a, b) \<in> r"
lists_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "lists A \<subseteq> lists B"
holomorphic_on_minus: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "f holomorphic_on s" shows "(\<lambda>z. - f z) holomorphic_on s"
rstep: fixes s :: "'a"   and Tr :: "'a \<Rightarrow> 'b set"   and Up :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s\<^sub>0 :: "'a"   and e :: "'b" assumes "s \<in> \<RR> Tr Up s\<^sub>0"   and "e \<in> Tr s" shows "Up s e \<in> \<RR> Tr Up s\<^sub>0"
abs_eq: fixes xa :: "i"   and x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "[make\<o> x in xa] = x dj xa"
arr_RHS: shows "Span.seq (\<Phi>.map (f, g \<star> h)) ((SPN f \<circ> \<Phi>.map (g, h)) \<bullet> Span.assoc (SPN f) (SPN g) (SPN h))"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_sem_equiv: fixes D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and X :: "'var list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F' :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> set X = {}" shows "negchecks_model \<I> (set D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>) X F F' = (\<forall>G\<in>set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' D). ineq_model \<I> X (F @ G))"
is_NoMsg_set_def: fixes A :: "'b set" shows "is_NoMsg A \<equiv> \<forall>x\<in>A. is_NoMsg x"
assms_4: assumes "r\<lparr>a\<rparr> = b" shows "P"
DomainpE: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e" assumes "Domainp r a" obtains b :: "'f" where "r a b"
border_take_\<ff>: fixes s :: "'b list"   and i :: "nat" shows "border (take (\<ff> s i - 1) s) (take i s)"
quotient_of_def: fixes x :: "rat" shows "quotient_of x = (THE pair. x = Rat.Fract (fst pair) (snd pair) \<and> 0 < snd pair \<and> algebraic_semidom_class.coprime (fst pair) (snd pair))"
simps_1: fixes uu :: "'b nres" shows "(uu \<le> FAILi) = True"
\<iota>_in_hom: shows "\<guillemotleft>\<iota>\<^sub>D : \<I>\<^sub>D \<otimes>\<^sub>D \<I>\<^sub>D \<rightarrow>\<^sub>D \<I>\<^sub>D\<guillemotright>"
countable_UN': fixes X :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "countable X"   and "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
tendsto_PInfty: fixes f :: "'a \<Rightarrow> ereal"   and F :: "'a filter" shows "(f \<longlongrightarrow> \<infinity>) F = (\<forall>r. \<forall>\<^sub>F x in F. ereal r < f x)"
zero_oexp: assumes "Field r \<noteq> {}" shows "zero rs.oexp = rs.const"
simps: fixes p :: "real" shows "geometric_spmf p = bernoulli p \<bind> (\<lambda>b. if b then return_spmf 0 else map_spmf ((+) 1) (geometric_spmf p))"
hd_in_set: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "hd xs \<in> set xs"
assms_1: shows "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"
nat_pow_one: fixes n :: "nat" shows "\<one>\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub> [^]\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub> n = \<one>\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub>"
parents_split: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat" assumes "p \<in> grid (child b dir d) {d}" shows "parents d b p = {b} \<union> parents d (child b dir d) p"
inj_semigroup_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_semigroup_mult_hom hom \<equiv> semigroup_mult_hom hom \<and> injective hom"
simplicial_subdivision_of_cell_complex: fixes \<M> :: "'b set set" assumes "finite \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"   and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1" obtains \<T> :: "'b set set" where "simplicial_complex \<T>" and "\<Union> \<T> = \<Union> \<M>" and "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F" and "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) atm" assumes "Ik.satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "Ik.int \<xi>'__ T1 = Ik.int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "intP p (map (Ik.int \<xi>) Tl)"
countable_support: shows "countable {x. M1.prob {x} \<noteq> 0}"
funion_absorb: fixes A :: "'a fset" shows "A |\<union>| A = A"
n_value_idealTr: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat" assumes "valuation K v"   and "0 \<le> n" shows "vp K v \<^bsup>\<diamondsuit>Vr K v n\<^esup> = Vr K v \<diamondsuit>\<^sub>p (Pg K v^\<^bsup>Vr K v n\<^esup>)"
E'': shows "D.internal_equivalence (F f) (F g) ((F g \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi>) (\<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>))"
list_relD: fixes x :: "'a list"   and y :: "'b list"   and a :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" assumes "(x, y) \<in> \<langle>br a I\<rangle>list_rel" shows "y = map a x"
absorb_iff2: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
TE_\<Phi>_def: shows "TE_\<Phi> \<equiv> ProblemIkTpart.tPB wtFsym arOf resOf \<Phi> tpD tpFD"
rep_eq: fixes x :: "('a, 'b) fmap" shows "fset (fmdom x) = dom (fmlookup x)"
filterlim_compose: fixes g :: "'b \<Rightarrow> 'c"   and F3 :: "'c filter"   and F2 :: "'b filter"   and f :: "'d \<Rightarrow> 'b"   and F1 :: "'d filter" assumes "filterlim g F3 F2"   and "filterlim f F2 F1" shows "LIM x F1. g (f x) :> F3"
chainTransitionAppend_2: fixes P :: "pi"   and \<alpha> :: "Early_Semantics.freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P''"   and "(P'', P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
J: fixes j :: "'b" assumes "j \<in> J" shows "ideal (\<aa> j) R (+) (\<cdot>) \<zero> \<one>"
lub_mono: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'e"   and Y :: "nat \<Rightarrow> 'e" assumes "class.cpo below"   and "po.chain below X"   and "po.chain below Y"   and "\<And>i. below (X i) (Y i)" shows "below (po.lub below (range X)) (po.lub below (range Y))"
bddh_quantify_bdd: fixes n :: "nat"   and bdd :: "bool list bdd"   and v :: "nat" assumes "bddh (Suc n) bdd"   and "v \<le> n" shows "bddh n (quantify_bdd v bdd)"
induct: fixes P :: "exp0 \<Rightarrow> bool"   and exp0 :: "exp0" assumes "\<And>x. P x \<Longrightarrow> P (Raise x)"   and "\<And>x1a x2. \<lbrakk>P x1a; \<And>x2a x2aa. \<lbrakk>x2a \<in> set x2; x2aa \<in> Basic_BNFs.snds x2a\<rbrakk> \<Longrightarrow> P x2aa\<rbrakk> \<Longrightarrow> P (Handle x1a x2)"   and "\<And>x. P (Lit x)"   and "\<And>x1a x2. (\<And>x2a. x2a \<in> set x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (Con x1a x2)"   and "\<And>x. P (Var x)"   and "\<And>x1a x2. P x2 \<Longrightarrow> P (Fun x1a x2)"   and "\<And>x1a x2. (\<And>x2a. x2a \<in> set x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (App x1a x2)"   and "\<And>x1a x2 x3a. \<lbrakk>P x2; P x3a\<rbrakk> \<Longrightarrow> P (Log x1a x2 x3a)"   and "\<And>x1a x2 x3a. \<lbrakk>P x1a; P x2; P x3a\<rbrakk> \<Longrightarrow> P (exp0.If x1a x2 x3a)"   and "\<And>x1a x2. \<lbrakk>P x1a; \<And>x2a x2aa. \<lbrakk>x2a \<in> set x2; x2aa \<in> Basic_BNFs.snds x2a\<rbrakk> \<Longrightarrow> P x2aa\<rbrakk> \<Longrightarrow> P (Mat x1a x2)"   and "\<And>x1a x2 x3a. \<lbrakk>P x2; P x3a\<rbrakk> \<Longrightarrow> P (exp0.Let x1a x2 x3a)"   and "\<And>x1a x2. \<lbrakk>\<And>x1aa x1aaa x1aaaa. \<lbrakk>x1aa \<in> set x1a; x1aaa \<in> Basic_BNFs.snds x1aa; x1aaaa \<in> Basic_BNFs.snds x1aaa\<rbrakk> \<Longrightarrow> P x1aaaa; P x2\<rbrakk> \<Longrightarrow> P (Letrec x1a x2)"   and "\<And>x1a x2. P x1a \<Longrightarrow> P (Tannot x1a x2)"   and "\<And>x1a x2. P x1a \<Longrightarrow> P (Lannot x1a x2)" shows "P exp0"
simps: fixes a1 :: "'b list"   and a2 :: "'b list" shows "a1 \<approx>2 a2 = ((\<exists>a b xs. a1 = a # b # xs \<and> a2 = b # a # xs) \<or> a1 = [] \<and> a2 = [] \<or> (\<exists>xs ys. a1 = ys \<and> a2 = xs \<and> xs \<approx>2 ys) \<or> (\<exists>a xs. a1 = a # a # xs \<and> a2 = a # xs) \<or> (\<exists>xs ys a. a1 = a # xs \<and> a2 = a # ys \<and> xs \<approx>2 ys) \<or> (\<exists>xs1 xs2 xs3. a1 = xs1 \<and> a2 = xs3 \<and> xs1 \<approx>2 xs2 \<and> xs2 \<approx>2 xs3))"
Continuous_Disjunctous: fixes H :: "'a upred \<Rightarrow> 'a upred" assumes "Continuous H" shows "Disjunctuous H"
is_rcoset_replistD_set: fixes H :: "'g set"   and gs :: "'g list" assumes "Supgroup.is_rcoset_replist H gs" shows "set gs \<subseteq> H"
simps: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" shows "((a1, a2) \<in> g3mod2) = ((\<exists>A. a1 = [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] \<and> a2 = ( \<Empt> \<Rightarrow>* \<LM> \<diamond> A  \<RM>)) \<or> (\<exists>A. a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] \<and> a2 = ( \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>)))"
evalV_possible: fixes f :: "'a \<aproc>"   and d :: "val"   and \<beta> :: "nat \<Rightarrow> 'a option"   and ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and p :: "lambda" assumes "f \<in> \<aA> d \<beta> ve"   and "d \<in> vals p"   and "ve \<in> smaps_over (vars p \<times> UNIV) (proc_poss p)"   and "\<beta> \<in> maps_over (labels p) UNIV" shows "f \<in> proc_poss p"
not_Err_eq: fixes x :: "'b err" shows "(x \<noteq> Err) = (\<exists>a. x = OK a)"
exec_all_conf: fixes P :: "jvm_prog"   and \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state"   and \<sigma>'' :: "jvm_state" assumes "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"   and "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>''" shows "P \<turnstile> \<sigma>' -jvm\<rightarrow> \<sigma>'' \<or> P \<turnstile> \<sigma>'' -jvm\<rightarrow> \<sigma>'"
oexp_empty2: assumes "Field s = {}" shows "oexp = {(\<lambda>x. undefined, \<lambda>x. undefined)}"
zeros_fg_def: shows "zeros_fg \<equiv> {p. fg p = 0}"
lm051: fixes X :: "'a set"   and x :: "'a" assumes "finite X"   and "Max X < x" shows "x \<notin> X"
exhaust_sel: fixes prog :: "('a, 'b, 'c) prog" obtains "prog = Prog (the_prog prog)"
pmf_prob_compl: fixes p :: "'b pmf"   and A :: "'b set" shows "pmf_prob p (- A) = 1 - pmf_prob p A"
zero_closed: shows "\<zero>\<^bsub>poly_ring R\<^esub> \<in> carrier (poly_ring R)"
comb: fixes vs :: "sterm list"   and name :: "name" assumes "list_all is_value vs"   and "t = name $$ vs"   and "name |\<in>| C" shows "thesis"
has_derivative_inverse: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and f' :: "'c \<Rightarrow> 'b"   and S :: "'c set" assumes "f x \<noteq> (0::'b)"   and "(f has_derivative f') (at x within S)" shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within S)"
nataction_add_eq0_flip: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "(a + b) +^ n = (0::'a)" shows "(b + a) +^ n = (0::'a)"
mk_coeff_carrier: fixes c :: "nat \<Rightarrow> 'a"   and vs :: "'e list"   and w :: "'e" assumes "c \<in> {0..<length vs} \<rightarrow> carrier R" shows "mk_coeff vs c w \<in> carrier R"
fresh_floatariths_map: fixes f :: "'a \<Rightarrow> floatarith"   and xs :: "'a list"   and i :: "nat" shows "fresh_floatariths (map f xs) i = (\<forall>x\<in>set xs. fresh_floatarith (f x) i)"
simps: fixes k_prf :: "'prf_key"   and k_upf :: "'upf_key"   and m :: "bool list" shows "encrypt (k_prf, k_upf) m = spmf_of_set prf_domain \<bind> (\<lambda>x. let c = prf_fun k_prf x [\<oplus>] m; t = upf_fun k_upf (x @ c) in return_spmf (x, c, t))"
connectedin: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>E1 E2. openin X E1 \<and> openin X E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 \<inter> S = {} \<and> E1 \<inter> S \<noteq> {} \<and> E2 \<inter> S \<noteq> {}))"
path_aux1b: fixes y :: "'a" assumes "backward_terminating_path y" shows "(y \<noteq> 0) = (start_points y \<noteq> 0)"
is_p_fun_def: fixes x :: "'a" shows "is_p_fun x \<equiv> x\<^sup>T ; x \<le> 1'"
Grp_iff: fixes B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" shows "BNF_Def.Grp B g x y = (y = g x \<and> x \<in> B)"
nextI: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g" assumes "P (\<sigma> |\<^sub>s Suc 0)" shows "(\<circle>P) \<sigma>"
suminf_split_initial_segment: fixes f :: "nat \<Rightarrow> 'c"   and k :: "nat" assumes "summable f" shows "suminf f = (\<Sum>n. f (n + k)) + sum f {..<k}"
eq_fold: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "Max (insert x A) = Finite_Set.fold max x A"