hcju/isabelleps huggingface dataset:
borel_cantelli_AE1: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "\<And>n. A n \<in> sets M"   and "\<And>n. emeasure M (A n) < \<infinity>"   and "summable (\<lambda>n. Sigma_Algebra.measure M (A n))" shows "AE x in M. \<forall>\<^sub>F n in sequentially. x \<in> space M - A n"
sys_load_def: fixes p :: "'f process_name"   and ract :: "('e, 'h) mem_load_action" shows "sys_load p ract = do_load_action ract \<circ> processors_view_of_memory p"
0: fixes xs :: "'a stream" assumes "\<phi> xs" shows "\<psi> xs"
starfun_ln_HFinite_not_Infinitesimal: fixes x :: "real star" assumes "x \<in> HFinite - Infinitesimal"   and "0 < x" shows "(*f* real_ln) x \<in> HFinite"
simps_1: fixes x :: "'a option" shows "None \<nabla> x = x"
filter_it_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "filter_it P xs = filter_it' [] P xs"
subterm_reduction_applicable_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "subterm_reduction_applicable S t = (is_compound t \<and> (trm_rep (lhs t) S \<noteq> lhs t \<or> trm_rep (rhs t) S \<noteq> rhs t))"
OF_match_eq2: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'f) flow_entry_match list"   and p :: "'e" assumes "check_no_overlap \<gamma> ft" shows "OF_same_priority_match2 \<gamma> ft p = OF_match_linear \<gamma> (sort_descending_key ofe_prio ft) p"
compositionality: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and list :: "'e list" shows "map f (map g list) = map (f \<circ> g) list"
eq_on_subst_other_1: fixes x :: "'a"   and X :: "'a set"   and s :: "'a \<Rightarrow> 'b"   and v :: "'b"   and t :: "'a \<Rightarrow> 'b" assumes "x \<notin> X" shows "(s(x := v) = t on X) = (s = t on X)"
hyps_2: shows "ipoly (p \<circ>\<^sub>p q) x = ipoly p (ipoly q x)"
dom_simp: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.dom f = VxV.dom f"
crel_vs_checkmem: fixes R :: "'v \<Rightarrow> 'v \<Rightarrow> bool"   and param :: "'k"   and s :: "['v]" assumes "is_equality R"   and "Rel (crel_vs R) (dp param) s" shows "Rel (crel_vs R) (dp param) (checkmem param s)"
assms_1: shows "\<guillemotleft>f : \<I> \<rightarrow> \<I>\<guillemotright>"
affine_ineq: fixes x :: "'a"   and v :: "'a"   and u :: "'a" assumes "x \<le> (1::'a)"   and "v \<le> u" shows "v + x * u \<le> u + x * v"
lb_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" shows "lb P B x = (\<forall>y\<in>B. P\<^sup>=\<^sup>= x y)"
append_coeff: fixes p :: "'a list"   and q :: "'a list" shows "local.coeff (p @ q) = (\<lambda>i. if i < length q then local.coeff q i else local.coeff p (i - length q))"
scalar_vec_plus_distrib_left: fixes nr :: "nat"   and u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "vec nr u"   and "vec nr v"   and "vec nr w" shows "scalar_prod (vec_plus u v) w = scalar_prod u w + scalar_prod v w"
fresh_left: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'b" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
assms: fixes a :: "'a set" assumes "a \<in> A" shows "finite a"
A_3: fixes i :: "nat" shows "emeasure M (A i) \<noteq> \<infinity>"
size_17: fixes x61 :: "'b ltln"   and x62 :: "'b ltln" shows "size (x61 or\<^sub>n x62) = size x61 + size x62 + Suc 0"
connected_open_delete: fixes S :: "'a set"   and a :: "'a" assumes "open S"   and "connected S"   and "2 \<le> DIM('a)" shows "connected (S - {a})"
scale_eq_0_iff: fixes a :: "'a"   and x :: "'c" shows "(scaleC a x = (0::'c)) = (a = (0::'a) \<or> x = (0::'c))"
lens_indep_comm: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'c"   and u :: "'a" assumes "x \<bowtie> y" shows "put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "min a b \<le> a"
cfun_belowI: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b" assumes "\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x" shows "f \<sqsubseteq> g"
bnd_xs: shows "xs ! x \<in>\<^sub>r ivlx"
inj_on_vpair: fixes A :: "(V \<times> V) set" shows "inj_on (\<lambda>(x, y). \<langle>x, y\<rangle>) A"
covering: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = \<t>" shows "\<Lambda>\<^sub>t \<le> int (\<B> index ps)"
antisympD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "antisymp r"   and "r a b"   and "r b a" shows "a = b"
ide_trg: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.ide (S.trg \<mu>)"
ring_hom_memE_3: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
offs_pred_next_cons: fixes ns :: "nat list"   and ub :: "nat"   and x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and j :: "nat" assumes "offs_pred ns ub (x # xs) index key mi ma"   and "index key x (length ns) mi ma = i"   and "i < length ns"   and "0 < offs_num (length ns) (x # xs) index key mi ma j" shows "offs_next ns ub (x # xs) index key mi ma j \<le> offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma j"
simps_7: fixes \<phi> :: "atom fm" shows "simpfm (Neg \<phi>) = neg (simpfm \<phi>)"
diff_streams_if: fixes w :: "'b stream"   and x :: "'b stream" assumes "\<exists>n. w !! n \<noteq> x !! n" shows "w \<noteq> x"
rsp: shows "eq_onp (\<lambda>xss. [] \<in> xss \<and> downset xss) UNIV UNIV"
Max_wrt_among_map_relation_bij: fixes f :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "bij f" shows "f ` Max_wrt_among le A = Max_wrt_among (map_relation (inv f) le) (f ` A)"
LLL_invD_10: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> i"
prems_1: shows "length (v1 # m11) = nc \<and> (\<forall>a\<in>set (v1 # m11). vec nr a)"
fi: assumes "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}" shows "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
Liminf_filtermap_ge: fixes f :: "'b \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'d \<Rightarrow> 'e" assumes "inj f" shows "Liminf F (\<lambda>x. g (f x)) \<le> Liminf (filtermap f F) g"
r_neg2: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> (\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> y) = y"
map_sel_2: fixes a :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> []" shows "tl (map f a) = map f (tl a)"
Suc0_mod_subst: fixes m :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "m \<noteq> Suc 0"   and "P (Suc 0)" shows "P (Suc 0 mod m)"
abs_eq: fixes x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "d\<^sub>0 (make\<o> x) = Some x"
is_strongly_scdom_component_safe_impl: shows "is_strongly_scdom_component_safe = local.a_is_strongly_scdom_component_safe"
termdiff_converges: fixes x :: "'a"   and K :: "real"   and c :: "nat \<Rightarrow> 'a" assumes "norm x < K"   and "\<And>x. norm x < K \<Longrightarrow> summable (\<lambda>n. c n * x ^ n)" shows "summable (\<lambda>n. diffs c n * x ^ n)"
assms_2: shows "q \<in> outs_\<I> \<I>"
new_tv_compatible_gen: fixes n :: "nat"   and t :: "typ"   and A :: "type_scheme list" assumes "new_tv n t" shows "new_tv n (gen A t)"
ipurge_fail_aux_t_in_def: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_fail_aux_t_in I D U xs X \<equiv> \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr>"
hd_replicate: fixes n :: "nat"   and x :: "'a" assumes "n \<noteq> 0" shows "hd (x \<up> n) = x"
right_cancel: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "(b + a = c + a) = (b = c)"
distinct_7: fixes x1 :: "agent"   and x5 :: "ephkey" shows "cAgent x1 \<noteq> cEphK x5"
smc_10_is_arr_iff: fixes \<aa> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<^bsub>smc_10 \<aa>\<^esub> \<BB> = False"
preservation: 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 :: "'b \<Rightarrow> 'd"   and c' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'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 "(((c, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "\<exists>\<Gamma>''. 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> \<longrightarrow> sifum_types.mds_consistent dma mds' \<Gamma>'')"
color_paint_Black: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "color (paint Black t) = Black"
ide_cod: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.cod C f)"
psubsetE: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
predicate2I: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
simps_2: fixes t1 :: "trm"   and t2 :: "trm"   and y :: "vrs"   and t' :: "trm" shows "(t1 \<cdot> t2)[y \<mapsto> t'] = t1[y \<mapsto> t'] \<cdot> t2[y \<mapsto> t']"
l_coset_carrier: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <# H"   and "x \<in> carrier G"   and "subgroup H G" shows "y \<in> carrier G"
lcf_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "(p \<otimes>\<^bsub>P\<^esub> q) (deg R (p \<otimes>\<^bsub>P\<^esub> q)) = p (deg R p) \<otimes> q (deg R q)"
simps_54: fixes x6 :: "(char list \<times> trm) list"   and x51 :: "trm"   and x52 :: "POPLmarkRecord.type" shows "Rcd x6 \<noteq> x51 \<bullet>\<^sub>\<tau> x52"
lower_asymptotic_density_le_upper: fixes A :: "nat set" shows "lower_asymptotic_density A \<le> upper_asymptotic_density A"
coeff_pCons_Suc: fixes a :: "'b"   and p :: "'b poly"   and n :: "nat" shows "coeff (pCons a p) (Suc n) = coeff p n"
i_shrink_last_nth': fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "0 < k" shows "(f \<div>\<^sub>l k) n = f (Suc n * k - Suc 0)"
iter0: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_or_infinite_iteration.iter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) \<sqsubseteq> nil"
muij: shows "w \<bullet> u / (u \<bullet> u) = \<mu> i jj"
seqI: fixes g1 :: "'d"   and f1 :: "'d"   and g2 :: "'c"   and f2 :: "'c" assumes "Dop.seq g1 f1 \<and> C.seq g2 f2" shows "DopxC.seq (g1, g2) (f1, f2)"
rep_eq: fixes x :: "'b" shows "(*\<^sub>V) (cblinfun_cinner_right x) = (\<bullet>\<^sub>C) x"
nths_empty: fixes xs :: "'c list" shows "nths xs {} = []"
nat_pow_Suc: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and n :: "nat" assumes "Group.monoid G" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
r_e2lenls_def: shows "r_e2lenls \<equiv> Cn 1 r_length [r_e2ls]"
red_hext_incr: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "hp s \<unlhd> hp s'"
reduction_doesnt_break_things_signs_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set (reduction_signs_R p qs signs subsets (M_mat_R signs subsets))"
\<theta>_3: shows "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p s) \<inter> range_vars \<theta> = {}"
poly_neighbourhood_without_roots: fixes p :: "real poly"   and x\<^sub>0 :: "real" assumes "p \<noteq> 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. poly p x \<noteq> 0"
induct: fixes P :: "'a list list \<Rightarrow> bool"   and a0 :: "'a list list" assumes "\<And>a b xs. P xs \<Longrightarrow> P (a # b # xs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
b3: fixes x :: "'a" assumes "x \<in> (id - projection M) -` {0::'a}" shows "x \<in> M"
T_2: fixes t :: "real"   and n :: "nat" shows "flow0 x (t + T * real n) = flow0 x t"
s1'_def: shows "s1' \<equiv> activate_cond_action1 (ls, (ts1, m1), ws, is) (ls, (ts2, m2), ws, is) ct"
Labels_LAss: fixes V :: "char list"   and e :: "expr" shows "labels (V:=e) 1 Skip"
quant_equivI: fixes x :: "'g['e, 'h]"   and y :: "'g['f, 'h]" assumes "QD('e) = QD('f)"   and "QD('e) = QD('f) \<Longrightarrow> \<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q" shows "x \<cong>\<^sub>Q y"
cong3_conga2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and A'' :: "'a"   and B'' :: "'a"   and C'' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A'' B'' C''" shows "Tarski_neutral_dimensionless.CongA Bet Cong A' B' C' A'' B'' C''"
null_uni_facts_2: fixes v :: "i"   and x :: "\<nu>" shows "[TAO_8_Definitions.Universal (x\<^sup>P) \<^bold>\<rightarrow> \<^bold>\<box>TAO_8_Definitions.Universal (x\<^sup>P) in v]"
supercls_lst_app: fixes P :: "'b prog"   and C :: "char list"   and Cs :: "char list list"   and C' :: "char list" assumes "supercls_lst P (C # Cs)"   and "P \<turnstile> C \<preceq>\<^sup>* C'" shows "supercls_lst P (C' # C # Cs)"
single_valued_tys_lemma: fixes G :: "prog"   and E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and t :: "Term.term"   and T :: "tys"   and T' :: "tys" assumes "\<forall>S T. G\<turnstile>S\<preceq>T \<longrightarrow> G\<turnstile>T\<preceq>S \<longrightarrow> S = T"   and "E,dt\<Turnstile>t\<Colon>T"   and "G = prg E"   and "E,dt\<Turnstile>t\<Colon>T'" shows "T = T'"
SC_discr_imp_SC_BisT: fixes c :: "('test, 'atom) com" assumes "SC_discr c" shows "SC_BisT c"
intro: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and adv :: "real"   and \<A> :: "('y + 'z, 'aa + 'ab) distinguisher_obsf" assumes "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"   and "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv" shows "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A> adv"
pdevs_vals_def: fixes e :: "nat \<Rightarrow> real"   and X :: "'a pdevs list" shows "pdevs_vals e X = map (pdevs_val e) X"
less_zeroE: fixes n :: "nat"   and R :: "bool" assumes "n < 0" shows "R"
proj_monomial: fixes k :: "'k"   and v :: "'t"   and c :: "'c" shows "proj_poly k (monomial c v) = (monomial c (pp_of_term v) when component_of_term v = k)"
monotone_ltakeWhile: fixes P :: "'c \<Rightarrow> bool" shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) (ltakeWhile P)"
D: fixes x :: "'a"   and t :: "'b" assumes "x \<in> S"   and "t \<in> cbox a b" shows "((\<lambda>(x, y). f x y) has_derivative D (x, t)) (at (x, t))"
inj_comm_ring_hom_axioms: shows "inj_comm_ring_hom hom"
tendsto_mult_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> (0::'b)) F"   and "(g \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> (0::'b)) F"
not_matches_add_matchNot_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>add_match (MatchNot m) rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"
inside_convex: fixes S :: "'b set" assumes "convex S" shows "inside S = {}"
ennreal_add_bot: fixes x :: "ennreal" shows "\<bottom> + x = x"
ipurge_tr_aux_union: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and V :: "'f set"   and xs :: "'e list" shows "ipurge_tr_aux I D (U \<union> V) xs = ipurge_tr_aux I D V (ipurge_tr_aux I D U xs)"
formulaTrueExtendWithEntailedClauses: fixes formula :: "Literal list list"   and formula0 :: "Literal list list"   and valuation :: "Literal list" assumes "\<forall>clause. clause el formula \<longrightarrow> formulaEntailsClause formula0 clause"   and "consistent valuation" shows "formulaTrue formula0 valuation \<longrightarrow> formulaTrue formula valuation"
xin: shows "x \<in> (\<lambda>a. wp (prog a) P s) ` S s"
bindU_plusU_llist: fixes xs :: "udom\<cdot>llist"   and ys :: "udom\<cdot>llist"   and f :: "udom \<rightarrow> udom\<cdot>llist" shows "bindU\<cdot>(plusU\<cdot>xs\<cdot>ys)\<cdot>f = plusU\<cdot>(bindU\<cdot>xs\<cdot>f)\<cdot>(bindU\<cdot>ys\<cdot>f)"
inv_i_iso_\<iota>_3: fixes f :: "'a" assumes "\<guillemotleft>f : \<I> \<rightarrow> \<I>\<^sub>1\<guillemotright>"   and "local.iso f"   and "f \<cdot> \<iota> = \<iota>\<^sub>1 \<cdot> (f \<otimes>\<^sub>1 f)" shows "(local.inv i \<cdot> f) \<cdot> \<iota> = \<iota> \<cdot> (local.inv i \<cdot> f \<otimes>\<^sub>1 local.inv i \<cdot> f)"
T_on_rand'_append: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf"   and xs :: "'a list"   and ys :: "'a list" shows "T\<^sub>p_on_rand' A s (xs @ ys) = T\<^sub>p_on_rand' A s xs + T\<^sub>p_on_rand' A (Partial_Cost_Model.config'_rand A s xs) ys"
ui_rv_is_bv: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat" assumes "RedBlack prb"   and "rv \<in> red_vertices prb" shows "fst rv \<in> Graph.vertices (black prb)"
right_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "monoid f z" shows "f a z = a"
wellFoundedTerminationLessState1: fixes Vbl :: "nat set" assumes "finite Vbl" shows "wf (terminationLessState1 Vbl)"
Rep_binding: fixes x :: "binding" shows "Rep_binding x \<in> Collect binding_Rep_set"
Chernoff_on_union: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set" assumes "Chernoff_on A f"   and "f_range_on A f"   and "B \<subseteq> A"   and "C \<subseteq> A" shows "f (B \<union> C) \<subseteq> f B \<union> f C"
1: shows "\<phi> r \<le> \<phi> \<langle>u, b, r\<rangle>"
sees_method_mono2: fixes C' :: "char list"   and C :: "char list"   and P :: "'b prog"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list"   and b' :: "staticb"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'b"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M, b' :  Ts'\<rightarrow>T' = m' in D'" shows "b = b' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
represents_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.represents jkbp envInit envAction envTrans envVal jview T \<equiv> PreEnvironmentJView.jkbpT jkbp envInit envAction envTrans envVal jview T = T"
corestr_in_hom: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "partial_magma.in_hom S (set_category.corestr S \<AA> f) (partial_magma.dom S f) (set_category.img S \<AA> f)"
left_add_zero_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "0\<^sub>m nr nc + A = A"
sel_1: shows "ReZ \<i>\<^sub>\<int> = 0"
rsp: shows "eq_onp (\<lambda>x. x \<in> {(a, b). a \<le> b}) (1::'a, 1::'a) (1::'a, 1::'a)"
fst_poly_pfd_simple: fixes x :: "'b poly"   and cs :: "('b \<times> nat) list" shows "fst (poly_pfd_simple x cs) = fst (partial_fraction_decomposition x (map (\<lambda>(c, n). ([:1::'b, - c:], n)) cs))"
absolutely_integrable_change_of_variables_linear: fixes g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "linear g" shows "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S) = (f absolutely_integrable_on g ` S)"
inf: shows "infinite {x. \<not> token_succeeds x \<and> token_squats x}"
card_subset_ex: fixes A :: "'b set"   and n :: "nat" assumes "finite A"   and "n \<le> card A" shows "\<exists>B\<subseteq>A. card B = n"