hcju/isabelleps huggingface dataset:
smult_r_minus: fixes a :: "'a"   and x :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "x \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> x = \<ominus>\<^bsub>P\<^esub> (a \<odot>\<^bsub>P\<^esub> x)"
Goal: fixes P' :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'"   and "(P', Q') \<in> Rel" shows "thesis"
\<A>\<^sub>G_s: shows "\<A>\<^sub>G s"
mset_unplusm_dist_cases: fixes s :: "'c"   and A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset" assumes "add_mset s A = B + C" obtains "B = add_mset s (B - {#s#})" and "A = B - {#s#} + C"   | "C = add_mset s (C - {#s#})" and "A = B + (C - {#s#})"
sdnets_in_subnets: fixes p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "p \<in> dom (Cp x)"   and "\<not> member DenyAll x" shows "\<exists>(a, b)\<in>sdnets x. a \<in> subnetsOfAdr (src p) \<and> b \<in> subnetsOfAdr (dest p)"
var_closed_1: fixes K :: "'a list set" assumes "subring K (K [X])" shows "X\<^bsub>K [X]\<^esub> \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"
R\<^sub>d_def: shows "R\<^sub>d = {(a, b). (a, b) \<in> R\<^sup>* \<and> (b, a\<^sup>\<bullet>) \<in> R\<^sup>*}"
max_bot2: fixes x :: "'c list" shows "prefix_order.max x [] = x"
FL_valid_eqvt: fixes P :: "'state"   and x :: "('a, 'pred, 'act, 'effect) formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
set_lookup_insert_tries: fixes ks :: "'d list"   and a :: "'c"   and t :: "('d, 'c list) trie"   and ks' :: "'d list" shows "set (lookup_tries (insert_tries ks a t) ks') = (if ks' = ks then insert a (set (lookup_tries t ks')) else set (lookup_tries t ks'))"
ide_char: fixes a :: "'a" shows "A.ide a = (A.arr a \<and> A.dom a = a \<and> A.cod a = a)"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'm" shows "dim_row (mat nr nc f) = nr"
not_is_empty_single: fixes x :: "'a" shows "\<not> is_empty (single x)"
summable_comparison_test: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> real" assumes "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n"   and "summable g" shows "summable f"
Idl_subset_ideal': fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "(Idl {a} \<subseteq> Idl {b}) = (a \<in> Idl {b})"
Tf_top: fixes t :: "(val \<times> val) list" shows "Tf t <:: \<top>"
f1'c_le_f1: shows "f1' / c \<le> f1"
term_of_pair_injective: fixes p :: "'a \<times> 'k"   and q :: "'a \<times> 'k" assumes "term_of_pair p = term_of_pair q" shows "p = q"
prefs_from_table_wfD_6: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list"   and xs :: "'d set list" assumes "prefs_from_table_wf agents alts xss"   and "xs \<in> set (map snd xss)" shows "is_finite_weak_ranking xs"
permutes_swap: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "p permutes S" shows "Fun.swap x y p permutes insert x (insert y S)"
monom_add: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and b :: "'a" shows "monom m (a + b) = monom m a + monom m b"
locallyI: fixes S :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "\<And>w x. \<lbrakk>openin (top_of_set S) w; x \<in> w\<rbrakk> \<Longrightarrow> \<exists>u v. openin (top_of_set S) u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w" shows "locally P S"
vdomain_vsingleton: fixes a :: "V"   and b :: "V" shows "\<D>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}) = set {a}"
representant_exists: fixes A :: "'a set" assumes "A \<in> Partition" shows "\<exists>a\<in>G. a \<in> A \<and> A = Class a"
rev_H'_simps: shows "rev_H' uw = rev_H uw \<and> rev_H' vw = rev_H vw \<or> rev_H' uw = rev_H vw \<and> rev_H' vw = rev_H uw"
ta_hb_consistent_coinduct_append: fixes X :: "('a \<times> ('b, 'a) obs_event action) list \<Rightarrow> ('a \<times> ('b, 'a) obs_event action) llist \<Rightarrow> bool"   and E :: "('a \<times> ('b, 'a) obs_event action) list"   and tobs :: "('a \<times> ('b, 'a) obs_event action) llist"   and P :: "'c prog" assumes "X E tobs"   and "\<And>E tobs. X E tobs \<Longrightarrow> tobs = LNil \<or> (\<exists>tobs' tobs''. tobs = lappend tobs' tobs'' \<and> tobs' \<noteq> LNil \<and> ta_hb_consistent P E tobs' \<and> (lfinite tobs' \<longrightarrow> X (E @ list_of tobs') tobs'' \<or> ta_hb_consistent P (E @ list_of tobs') tobs''))" shows "ta_hb_consistent P E tobs"
rbt_eq_iff: fixes t1 :: "('b, 'c) RBT.rbt"   and t2 :: "('b, 'c) RBT.rbt" shows "(t1 = t2) = (rbt.impl_of t1 = rbt.impl_of t2)"
segment_convex_hull: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b = convex hull {a, b}"
lt_plus_lessI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p + q \<noteq> 0"   and "lt q = lt p"   and "lc q = - lc p" shows "lt (p + q) \<prec>\<^sub>t lt p"
ex: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> sub.diff_fun_space" shows "\<exists>f'. f' \<in> diff_fun_space \<and> (\<forall>x\<in>closure B. f x = f' x)"
L_arr: fixes l :: "'a \<Rightarrow> 'b"   and \<chi> :: "'a \<Rightarrow> 'j \<Rightarrow> 'b"   and a :: "'a" assumes "\<forall>a. A.ide a \<longrightarrow> limit_cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (\<lambda>j. D (j, a)) (l a) (\<chi> a)"   and "A.arr a" shows "(\<exists>!f. P l \<chi> a f) \<and> P l \<chi> a (L l \<chi> a)"
produces_imp_Pos_in_lits: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset"   and A :: "'a" assumes "production N C = {A}" shows "Pos A \<in># C"
intros_3: fixes ck :: "bool"   and s1 :: "v sem_env"   and env :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and err :: "v error_result" assumes "evaluate ck s1 env e (s2, Rerr err)" shows "evaluate ck s1 env (Raise e) (s2, Rerr err)"
init: fixes C :: "'a eclause"   and S :: "'a eclause set" assumes "C \<in> S" shows "derivable_ecl C S"
assms: shows "(if b then c1 else c2, s) \<rightarrow>t s'"
iT_Div_0_card_if: fixes I :: "nat set" shows "card (I \<oslash> 0) = (if I = {} then 0 else Suc 0)"
AsSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and p :: "'e set" shows "assertionsR \<Gamma> \<Theta> Q A (AnnExpr p) Skip p"
lazy_channel_recv_f'_def: shows "lazy_channel_recv_f' = (\<lambda>s q. case s of (flg, cstate.Collect m, None, as) \<Rightarrow> return_spmf (Some m, flg, cstate.Fail, None, as) | (flg, cstate.Collect m, Some (a', m'), as) \<Rightarrow> case as m' of None \<Rightarrow> spmf_of_set (rnd \<eta>) \<bind> (\<lambda>a. return_spmf (None, flg \<or> a = a', cstate.Fail, None, as(m' \<mapsto> a))) | Some a \<Rightarrow> return_spmf (if a = a' then Some m' else None, flg, cstate.Fail, None, as) | (flg, _, None, as) \<Rightarrow> return_spmf (None, s) | (flg, _, Some (a', m'), as) \<Rightarrow> case as m' of None \<Rightarrow> spmf_of_set (rnd \<eta>) \<bind> (\<lambda>a. return_spmf (None, flg \<or> a = a', cstate.Fail, None, as(m' \<mapsto> a))) | Some a \<Rightarrow> return_spmf (if a = a' then Some m' else None, flg, cstate.Fail, None, as))"
prime_pr: fixes f :: "recf"   and n :: "nat"   and g :: "recf"   and m :: "nat" assumes "primerec f n"   and "primerec g (Suc (Suc n))"   and "m = Suc n" shows "primerec (Pr n f g) m"
exact_aux_2: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs" shows "standard_decomp k (exact_aux k m qs)"
span_add1: fixes U :: "'a vec set"   and v :: "'a vec"   and w :: "'a vec" assumes "U \<subseteq> carrier KA.NC.V"   and "v \<in> MN.span U"   and "w \<in> MN.span U" shows "v \<oplus>\<^bsub>KA.NC.V\<^esub> w \<in> MN.span U"
mult_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" shows "(\<otimes>\<^bsub>subgroup_generated G S\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
C: shows "w \<in> sinks_aux I D U (xs @ [x])"
dBox_star_induct: fixes \<Phi> :: "'b upred"   and P :: "'b hrel" shows "`\<Phi> \<and> \<^bold>[P\<^sup>\<star>\<^bold>](\<Phi> \<Rightarrow> \<^bold>[P\<^bold>]\<Phi>) \<Rightarrow> \<^bold>[P\<^sup>\<star>\<^bold>]\<Phi>`"
split_pi_off: shows "x + real (Suc n) * pi = x + real n * pi + pi"
min_maxsimpchain_betw_uniform_length: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set"   and ys :: "'a set list" assumes "min_gallery (x # xs @ [y])"   and "min_gallery (x # ys @ [y])" shows "length xs = length ys"
sorted_snoc: fixes xs :: "'b list"   and y :: "'b" assumes "Sorted_Less.sorted (xs @ [y])" shows "Sorted_Less.sorted xs"
test_num': fixes n :: "int"   and a :: "'a" shows "test_num' n a = (n \<in> \<gamma> a)"
simps_120: fixes x2 :: "'b" shows "set_ptrm (PVar x2) = {x2}"
invertibleD_1: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "invertible A" shows "A\<^sup>-\<^sup>1 ** A = mat (1::'d)"
mult_Mp_1: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.Mp m (poly_mod.Mp m f * g) = poly_mod.Mp m (f * g)"
tagged_division_ofD_1: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "s tagged_division_of i" shows "finite s"
poly_div_diff_left: fixes x :: "'a poly"   and y :: "'a poly"   and z :: "'a poly" shows "(x - y) div z = x div z - y div z"
primitive_dchar_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" shows "primitive_dchar n \<chi> \<equiv> dcharacter n \<chi> \<and> primitive_dchar_axioms n \<chi>"
sagt: shows "(sign ! a = 1) = (0 < poly (qs ! a) x)"
group_closurep_group_closure_eq: fixes S :: "'a set" shows "group_closurep S = (\<lambda>x. x \<in> group_closure S)"
image_image_id_if: fixes f :: "'a \<Rightarrow> 'a"   and M :: "'a set" assumes "\<And>x. f (f x) = x" shows "f ` f ` M = M"
can_step_def: fixes l :: "nat"   and s :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)" shows "C.can_step l s \<equiv> \<exists>s'. clstep l s s'"
cases: fixes a :: "'node"   and n :: "'node"   and S :: "'node set" assumes "a \<in> obs n S" obtains as :: "'edge list" and n' :: "'node" where "a = n'" and "n -as\<rightarrow>* n'" and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S" and "n' \<in> S"
absorb_iff2: fixes a :: "'b word"   and b :: "'b word" shows "(a \<le>s b) = (signed.max a b = b)"
distinct_1: fixes x1 :: "name"   and x2 :: "name" shows "InputS_Rep x1 \<noteq> BoundOutputS_Rep x2"
kat_2_equiv_opp: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> ar z \<le> ar x \<cdot> y) = (r x \<cdot> (y \<cdot> ar z) = (zero_class.zero::'a))"
ok_rctxt_swap: fixes \<phi>s :: "Formula.formula list"   and \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and mr :: "mregex" assumes "ok_rctxt \<phi>s \<kappa> \<kappa>'" shows "from_mregex (\<kappa> mr) \<phi>s = \<kappa>' (from_mregex mr \<phi>s)"
logic_actual_nec_3: fixes \<phi> :: "'d \<Rightarrow> \<o>" shows "[[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<A>\<phi> \<alpha>)]]"
finite_imp_compact: fixes s :: "'c set" assumes "finite s" shows "compact s"
f'f: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> S" shows "f' x = f x"
tends: fixes x :: "'a" assumes "x \<notin> limsup E" shows "(\<lambda>n. g n x) \<longlonglongrightarrow> indicat_real S x"
partition_on_split1: fixes A :: "'a set"   and P :: "'a set set"   and Q :: "'a set set" assumes "partition_on A (P \<union> Q)" shows "partition_on (\<Union> P) P"
blinfun_apply_id_blinfun: shows "blinfun_apply id_blinfun = (\<lambda>x. x)"
simps: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and aa :: "'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f" shows "(aa \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a) = (aa = a \<or> (\<exists>x. aa = f x \<and> x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a) \<or> (\<exists>M. aa = lub M \<and> M \<in> Chains leq \<and> M \<noteq> {} \<and> (\<forall>x. x \<in> M \<longrightarrow> x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a)))"
rtrancl_Image_step: fixes a :: "'a"   and r :: "('a \<times> 'a) set"   and A :: "'a set"   and b :: "'a" assumes "a \<in> r\<^sup>* `` A"   and "(a, b) \<in> r\<^sup>*" shows "b \<in> r\<^sup>* `` A"
vars_of_conj: fixes e1 :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and e2 :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "Bexp.vars (\<lambda>\<sigma>. e1 \<sigma> \<and> e2 \<sigma>) \<subseteq> Bexp.vars e1 \<union> Bexp.vars e2"
hom_dvd_hom: fixes x :: "'a"   and y :: "'a" shows "(hom x dvd hom y) = (x dvd y)"
positive_close_under_left_right_mult_adjoint: fixes M :: "complex mat"   and n :: "nat"   and A :: "complex mat" assumes "M \<in> carrier_mat n n"   and "A \<in> carrier_mat n n"   and "positive A" shows "positive (M * A * adjoint M)"
DefaultPUpdate_def: shows "DefaultPUpdate = Abs_pupdate (\<lambda>D. DefaultPData (Data.DataSpace D))"
sets_Ball: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'c \<Rightarrow> 'd measure"   and i :: "'c" assumes "\<forall>i\<in>I. A i \<in> sets (M i)"   and "i \<in> I" shows "A i \<in> sets (M i)"
bot_less: fixes a :: "'a" shows "(a \<noteq> 0) = (0 < a)"
mtx_row_length_R: fixes signs :: "rat list list"   and ls :: "(nat list \<times> nat list) list" shows "list_all (\<lambda>r. length r = length signs) (map (mtx_row_R signs) ls)"
sum_list_prod_map2: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and as :: "'b list"   and f :: "'d \<Rightarrow> 'c"   and bs :: "'d list" shows "sum_list (map2 g as (map f bs)) = sum_list (map2 (\<lambda>a b. g a (f b)) as bs)"
simps_1: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c" shows "s1 []\<Longrightarrow>\<^bsub>SES\<^esub> s1"
encode_all_frame_axioms_valid: fixes sv :: "sat_plan_variable"   and t :: "nat" assumes "sv \<in> atoms (encode_all_frame_axioms (\<phi> prob_with_noop abs_prob ) t)"   and "t < Suc h" shows "valid_state_var sv"
integral_sum: fixes I :: "'b set"   and M :: "'c measure"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "LINT x|M. (\<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
fresh_append: fixes a :: "atom"   and xs :: "'a list"   and ys :: "'a list" shows "a \<sharp> xs @ ys = (a \<sharp> xs \<and> a \<sharp> ys)"
angle_actrans_sem_eq: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<langle>F\<rangle>_v = (F \<and> v$ \<noteq> $v)"
test_mult: fixes x :: "'a"   and y :: "'a" shows "\<tt>\<tt> (!x \<cdot> !y) = !x \<cdot> !y"
eq_zero_or_degree_less: fixes p :: "'d poly"   and n :: "nat" assumes "degree p \<le> n"   and "coeff p n = (0::'d)" shows "p = 0 \<or> degree p < n"
simps_1: fixes init :: "'a set"   and trans :: "('a \<times> 'a) set"   and more :: "'b" shows "init \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = init"
subst_agreement: fixes t :: "('c, 'e) Term.term"   and r :: "'e \<Rightarrow> ('c, 'd) Term.term"   and s :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "(t \<cdot> r = t \<cdot> s) = (\<forall>v\<in>fv t. Var v \<cdot> r = Var v \<cdot> s)"
subpath_refl: fixes a :: "real"   and g :: "real \<Rightarrow> 'b" shows "subpath a a g = linepath (g a) (g a)"
fps_one_nth: fixes n :: "nat" shows "1 $ n = (if n = 0 then 1::'b else (0::'b))"
facesAt_distinct_def: fixes g :: "graph" shows "facesAt_distinct g \<equiv> \<forall>v\<in>\<V> g. distinct (normFaces (facesAt g v))"
ennreal_pmf_map: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c pmf"   and x :: "'b" shows "ennreal (pmf (map_pmf f p) x) = integral\<^sup>N (measure_pmf p) (indicator (f -` {x}))"
elims: fixes x :: "(real, 'a) vec \<times> (real, 'a) vec"   and xa :: "'a"   and xb :: "real"   and y :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "repv x xa xb = y" obtains v :: "(real, 'a) vec \<times> (real, 'a) vec" and xc :: "'a" and r :: "real" where "x = v" and "xa = xc" and "xb = r" and "y = (\<chi>y. if xc = y then r else fst v $ y, snd v)"
fw_upd_mono: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat" shows "fw_upd m k i j i' j' \<le> m i' j'"
nominal_ts_axioms: shows "nominal_ts (\<turnstile>\<^sub>S) (\<rightarrow>\<^sub>S)"
nn_integral_PairVal: fixes f :: "val \<Rightarrow> ennreal"   and t1 :: "pdf_type"   and t2 :: "pdf_type" assumes "f \<in> borel_measurable (stock_measure (PRODUCT t1 t2))" shows "integral\<^sup>N (stock_measure (PRODUCT t1 t2)) f = \<integral>\<^sup>+ x. f <|fst x, snd x|> \<partial>stock_measure t1 \<Otimes>\<^sub>M stock_measure t2"
l2polR_eq: fixes N :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "N \<noteq> []" shows "Cp (list2policyR N) = list2policy (map Cp N)"
co_2: shows "set (pawalk_verts u (co_path e w p)) = set (pawalk_verts u p) - {w}"
\<tau>J: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {inp}" shows "\<tau>J \<psi> \<in> ptrm 0"
simps_2: fixes n :: "nat"   and us :: "'b vec list"   and w :: "'b vec"   and ws :: "'b vec list" shows "gram_schmidt_sub n us (w # ws) = gram_schmidt_sub n ((adjuster n w us + w) # us) ws"
Cinfinite_Cnotzero: fixes r :: "('c \<times> 'c) set" assumes "Cinfinite r" shows "(r, czero) \<notin> ordIso \<and> Card_order r"
g'_nonneg: fixes x :: "real" assumes "a \<le> ereal x"   and "ereal x \<le> b" shows "0 \<le> g' x"
simps_2: fixes r :: "'a pfm"   and x :: "char list" shows "reduce' r (Pro\<^sub>! x) = (r \<^bold>\<longrightarrow>\<^sub>! Pro\<^sub>! x)"
distance_attains_inf: fixes s :: "'a set"   and a :: "'a" assumes "closed s"   and "s \<noteq> {}" obtains x :: "'a" where "x \<in> s" and "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y"
ln: assumes "w1 \<noteq> NEG_INF \<and> w1 <s 0 \<and> w2 = NEG_INF" shows "P w1 w2"
f_def: shows "f \<equiv> \<lambda>u. u *\<^sub>R a + (1 - u) *\<^sub>R b"
perzeta_conv_hurwitz_zeta_multiplication: assumes "0 < k"   and "s \<noteq> 1" shows "of_nat k powr s * perzeta (real_of_int a / real k) s = (\<Sum>n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int a / real k) * \<i>) * hurwitz_zeta (real n / real k) s)"
gcd_le2_int: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "gcd a b \<le> b"
power_one: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times one n = one"
lens_indep_left_ext: fixes y :: "'e \<Longrightarrow> 'f"   and z :: "'g \<Longrightarrow> 'f"   and x :: "'h \<Longrightarrow> 'e" assumes "y \<bowtie> z" shows "x ;\<^sub>L y \<bowtie> z"
alphaAbs_qAbs_imp_alphaAbs_all_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_qFresh xs x X xs' x' X'"
lin_dep_crit: fixes A :: "'a vec set"   and S :: "'a vec set"   and v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> S"   and "True"   and "v \<in> A"   and "a v \<noteq> (0::'a)"   and "lincomb a A = 0\<^sub>v n" shows "lin_dep S"
CR_onD: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "CR_on r A"   and "a \<in> A"   and "(a, b) \<in> r\<^sup>*"   and "(a, c) \<in> r\<^sup>*" shows "(b, c) \<in> r\<^sup>\<down>"
orderE: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" obtains "a = a \<sqinter> b"
rtrancl_refl: fixes a :: "'ap"   and r :: "('ap \<times> 'ap) set" shows "(a, a) \<in> r\<^sup>*"
exQ_clearQuantifiers: fixes \<phi> :: "atom fm"   and xs :: "real list" assumes "\<And>xs. eval (clearQuantifiers \<phi>) xs = eval \<phi> xs" shows "eval (clearQuantifiers (ExQ \<phi>)) xs = eval (ExQ \<phi>) xs"
simps: fixes \<delta>\<^sub>m :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c" shows "\<Delta>\<^sub>\<times> \<delta>\<^sub>m = (\<lambda>q \<nu> x. case q x of None \<Rightarrow> None | Some y \<Rightarrow> Some (\<delta>\<^sub>m x y \<nu>))"
tar: shows "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>) \<union> ik\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d \<union> (\<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>) \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d))"
reduce_le_plus_consts: fixes x :: "nat"   and n1 :: "nat"   and y :: "nat"   and n2 :: "nat" assumes "x + n1 \<le> y + n2" shows "x \<le> y + (n2 - n1)"
lo_lemma': fixes x :: "nat"   and y :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y" shows "rec_exec rec_lo [x, y] = lo x y"
map_simp: fixes f :: "'j" assumes "J.arr f" shows "U.map f = S.unity"
subformula_atoms: fixes G :: "'a formula"   and F :: "'a formula" assumes "G \<in> set (subformulae F)" shows "atoms G \<subseteq> atoms F"
Scheffe_lemma2: fixes F :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>n. F n \<in> borel_measurable M"   and "integrable M f"   and "AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x"   and "\<And>n. \<integral>\<^sup>+ x. ennreal (norm (F n x)) \<partial>M \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M" shows "(\<lambda>n. \<integral>\<^sup>+ x. ennreal (norm (F n x - f x)) \<partial>M) \<longlonglongrightarrow> 0"
jointfun_inj: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and B :: "'e set"   and b :: "'e" assumes "f \<in> {j. j \<le> n} \<rightarrow> B"   and "inj_on f {j. j \<le> n}"   and "b \<notin> f ` {j. j \<le> n}" shows "inj_on (jointfun n f 0 (\<lambda>k\<in>{0}. b)) {j. j \<le> Suc n}"
safe_distance_2r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance_2r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / 2 / a\<^sub>e + v\<^sub>o\<^sup>2 / 2 / a\<^sub>o"
b_least2_aux2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "b_least2 f x y < y" shows "b_least2 f x (Suc y) = b_least2 f x y"
fpxs_compose_power_distrib: fixes r1 :: "rat"   and r2 :: "rat"   and f :: "'b fpxs" assumes "0 < r1 \<or> 0 < r2" shows "fpxs_compose_power (fpxs_compose_power f r1) r2 = fpxs_compose_power f (r1 * r2)"
pickE: fixes xl :: "Preliminaries.var list"   and al :: "univ list"   and i :: "nat" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> eintT (TE.tpOfV (xl ! i)) (al ! i)"   and "i < length xl" shows "pickE xl al (xl ! i) = al ! i"
ordIso_equivalence_3: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r =o r'" shows "r' =o r"
assms_2: shows "diff_invariant I f U S t\<^sub>0 G"
ss: shows "{a. \<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r} \<subseteq> vfst ` elts r"
antipar_2: shows "src\<^sub>D (F g) = trg\<^sub>D (F f)"
setdist_pos_le: fixes s :: "'b set"   and t :: "'b set" shows "0 \<le> setdist s t"
fofu_II_III: assumes "\<not> Ex isAugmentingPath" shows "\<exists>k'. NCut c s t k' \<and> val = NCut.cap c k'"
rev_append: fixes xs :: "'a list"   and ys :: "'a list" shows "rev (xs @ ys) = rev ys @ rev xs"
contf': shows "continuous_on (cball z r) (\<lambda>u. f u - y)"
sigma_finite_iff_density_finite': fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M" shows "sigma_finite_measure (density M f) = (AE x in M. f x \<noteq> \<infinity>)"
set_discriminating_if: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "set_discriminating n f borel"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "ring_hom hom" shows "comm_ring_hom hom"
rdvd_mult: fixes k :: "int"   and n :: "int"   and x :: "real" assumes "k \<noteq> 0" shows "(real_of_int n * real_of_int k rdvd x * real_of_int k) = (real_of_int n rdvd x)"