hcju/isabelleps huggingface dataset:
consensus_def: fixes vals :: "'a \<Rightarrow> 'b"   and dec :: "'c \<Rightarrow> 'b option"   and rho :: "nat \<Rightarrow> 'a \<Rightarrow> 'c" shows "consensus vals dec rho \<equiv> (\<forall>n p v. dec (rho n p) = Some v \<longrightarrow> v \<in> range vals) \<and> (\<forall>m n p q v w. dec (rho m p) = Some v \<and> dec (rho n q) = Some w \<longrightarrow> v = w) \<and> (\<forall>p. \<exists>n. dec (rho n p) \<noteq> None)"
all_termi: shows "\<forall>j<i. terminate (gs ! j) args"
assms_3: assumes "\<not> c"   and "\<not> c'" shows "(e, e') \<in> R"
is_Tensor_def: fixes "term" :: "'a local.term" shows "is_Tensor term = (\<exists>x31 x32. term = x31 \<^bold>\<otimes> x32)"
LIMSEQ_iff_nz: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
atLeast_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and l :: "'e" shows "ord.atLeast less_eq l \<equiv> {x. less_eq l x}"
rev_take: fixes i :: "nat"   and xs :: "'b list" shows "rev (take i xs) = drop (\<^bold>|xs\<^bold>| - i) (rev xs)"
ffb_prop_var: fixes f :: "'d \<Rightarrow> 'c set" shows "fb\<^sub>\<F> f = uminus \<circ> bd\<^sub>\<F> (op\<^sub>K f) \<circ> uminus"
that: fixes u :: "real" assumes "1 / 2 \<le> u"   and "u \<le> 1"   and "X = u *\<^sub>R A + (1 - u) *\<^sub>R B" shows "thesis"
inv_in_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.iso f"   and "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>D.inv f : b \<rightarrow>\<^sub>D a\<guillemotright>"
partition_on_msetD2: fixes A :: "'a multiset"   and P :: "'a multiset multiset" assumes "partition_on_mset A P" shows "{#} \<notin># P"
simps: fixes p :: "nat"   and n :: "nat" shows "showsp_nat p n = (if n < 10 then shows_string (string_of_digit n) else showsp_nat p (n div 10) \<circ> shows_string (string_of_digit (n mod 10)))"
SeqConseq: fixes P :: "'a set"   and P' :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set"   and a :: "('a, 'b, 'c) ann"   and c :: "('a, 'b, 'c) com"   and Q' :: "'a set"   and A' :: "'a set"   and Q :: "'a set"   and A :: "'a set" assumes "P \<subseteq> P'"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P' a c Q', A'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P a c Q, A"
compl_unique: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "inf x y = bot"   and "sup x y = top" shows "uminus x = y"
simps_1: fixes y :: "'c arctic_delta" shows "MinInfty_delta + y = y"
callcc_cont_def: fixes f :: "('d \<Rightarrow> ('d, 'e) contT) \<Rightarrow> ('d, 'e) contT" shows "callcc_cont f = ContT (\<lambda>k. run_cont (f (\<lambda>x. ContT (\<lambda>_. k x))) k)"
tvs_def_3: shows "\<S>\<bullet>\<C> \<turnstile> ves : [] _> tvs"
intros_5: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<equiv>\<^sub>s P \<oplus> Q \<oplus> R"
\<alpha>nodes_aux_def: fixes g :: "'m" shows "\<alpha>nodes_aux g \<equiv> dom (M_lookup g)"
Lindelof_space_continuous_map_image: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "Lindelof_space X"   and "continuous_map X Y f"   and "f ` topspace X = topspace Y" shows "Lindelof_space Y"
xeq4: fixes x :: "Value"   and s :: "Store"   and t :: "New" assumes "x \<noteq> new s t" shows "s \<equiv>[x] s\<langle>t\<rangle>"
leq_antisym: fixes x :: "('c, 'b) tllist"   and y :: "('c, 'b) tllist" assumes "local.tllist_ord x y"   and "local.tllist_ord y x" shows "x = y"
weakBisimStructCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "P \<approx> Q"
d: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> S"   and "x' \<in> S"   and "dist x' x < d" shows "dist (f x') (f x) < e / 3"
extend_map_UNIV_to_sphere_cofinite: fixes r :: "real"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and L :: "'a set" assumes "DIM('a) \<le> DIM('b)"   and "0 \<le> r"   and "compact S"   and "continuous_on S f"   and "f ` S \<subseteq> sphere a r"   and "\<And>C. \<lbrakk>C \<in> components (- S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}" obtains K :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite K" and "K \<subseteq> L" and "disjnt K S" and "continuous_on (- K) g" and "g ` (- K) \<subseteq> sphere a r" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
is_unit_mult_iff: fixes a :: "'b"   and b :: "'b" shows "is_unit (a * b) = (is_unit a \<and> is_unit b)"
sort_conv_fold: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = fold (linorder.insort_key less_eq (\<lambda>x. x)) xs []"
undef_vec: fixes i :: "nat"   and x :: "'a list" assumes "\<not> i < length x" shows "undef_vec (i - length x) = x ! i"
image_prod: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'f set" shows "(\<lambda>(x, y). (f x, g y)) ` (A \<times> B) = f ` A \<times> g ` B"
healthy_intros_16: fixes f :: "'a \<Rightarrow> 'a" shows "nearly_healthy (wlp (Apply f))"
perm_fresh_fresh: fixes a :: "NominalSequents.var"   and x :: "'a"   and b :: "NominalSequents.var" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
B: shows "index (mtf2 n q xs) q = index xs q - n"
IH: fixes s :: "'a \<Rightarrow> bool option" assumes "are_all_operators_applicable s (op # \<pi>)"   and "are_all_operators_non_interfering (op # \<pi>)" shows "are_all_operators_applicable (s \<then> op) \<pi>"
pubSK_symK: fixes X :: "agent"   and n :: "nat" shows "pubSK X \<noteq> symK n"
sl4: shows "(abscissa E - abscissa F)\<^sup>2 \<in> radical_sqrt"
rw: fixes I :: "nat set" assumes "I \<in> {I. I \<subseteq> {0..<m} \<and> card I = n}" shows "(\<lambda>i. if i < n then sorted_list_of_set I ! i else i) ` {0..<n} = I"
fmdrop_idle': fixes x :: "'a"   and B :: "('a, 'b) fmap" assumes "x \<notin> fmdom' B" shows "fmdrop x B = B"
X5_UNIV: shows "UNIV = set [X5.Xd1, X5.Xd1', X5.Xd2, X5.Xd2', Xd2'']"
hyps_2: assumes "ws' t' = \<lfloor>InWS w\<rfloor>"   and "ws t' \<noteq> \<lfloor>InWS w\<rfloor>" shows "t = t' \<and> Suspend w \<in> set was"
assms: shows "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"
lt_eq_TT_iff: fixes compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b"   and y :: "'b" shows "(Ord.lt compare\<cdot>x\<cdot>y = TT) = (compare\<cdot>x\<cdot>y = LT)"
content_field_poly: fixes f :: "'b poly" shows "content f = (if f = 0 then 0::'b else (1::'b))"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "C'.arr f"   and "C'.cod f = b" shows "C' b f = f"
HNatInfinite_FreeUltrafilterNat: fixes X :: "nat \<Rightarrow> nat" assumes "star_n X \<in> HNatInfinite" shows "\<forall>u. \<forall>\<^sub>F n in \<U>. u < X n"
newton_seq_fact8: fixes l :: "nat" assumes "f \<bullet> ns l \<noteq> \<zero>" shows "local.divide (f \<bullet> ns l) (f' \<bullet> ns l) \<in> nonzero Zp"
And\<^sub>n_semantics: fixes \<Phi> :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set" assumes "finite \<Phi>" shows "w \<Turnstile>\<^sub>n And\<^sub>n \<Phi> = (\<forall>\<phi>\<in>\<Phi>. w \<Turnstile>\<^sub>n \<phi>)"
mono_step'_top: fixes C1 :: "'av st option acom"   and C2 :: "'av st option acom" assumes "C1 \<le> C2" shows "local.step' \<top> C1 \<le> local.step' \<top> C2"
int_tA: shows "invertible (Finite_Cartesian_Product.transpose A ** A)"
C: shows "s setinterleaves ((t, u), insert tick (ev ` A))"
HighSame: shows "high_proc_no_low_output_change s\<^sub>H"
maxsimpchain_append_reduce2: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "gallery (xs @ ys)" shows "gallery ys"
char_poly_matrix: fixes A :: "'b mat"   and n :: "nat"   and k :: "'b" assumes "A \<in> carrier_mat n n" shows "poly (char_poly A) k = det (- char_matrix A k)"
less_eq_mask: fixes n :: "nat" shows "n \<le> mask n"
lookup_def: fixes t :: "('c, 'd) trie" shows "lookup t = lookup_trie (impl_of t)"
ordinal_cases: fixes a :: "ordinal" obtains "a = 0"   | x :: "ordinal" where "a = oSuc x"   | f :: "nat \<Rightarrow> ordinal" where "OrdinalInduct.strict_mono f" and "a = oLimit f"
prefixeq_butlast: fixes xs :: "'b list" shows "prefix (butlast xs) xs"
iso_tuple_UNIV_I: fixes x :: "'b" shows "x \<in> UNIV \<equiv> True"
assignment_rhs\<^sub>e\<^sub>s\<^sub>t_append: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "assignment_rhs\<^sub>e\<^sub>s\<^sub>t (A @ B) = assignment_rhs\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t B"
NONCE_neqv_MPAIR: fixes m :: "nat"   and X :: "freemsg"   and Y :: "freemsg" assumes "NONCE m \<sim> MPAIR X Y" shows "False"
SkipRule: fixes p :: "'a set"   and q :: "'a set" assumes "p \<subseteq> q" shows "\<parallel>- p Basic id q"
snoc_conv_cons: fixes ys :: "'b list"   and y :: "'b" shows "\<exists>x xs. ys @ [y] = x # xs"
revis_extensionality: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and revision :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b"   and \<psi> :: "'b" assumes "AGM_Revision Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL revision"   and "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "revision K \<phi> = revision K \<psi>"
arr_iff_in_hom: fixes f :: "'a" shows "S.arr f = \<guillemotleft>f : S.dom f \<Rightarrow>\<^sub>S S.cod f\<guillemotright>"
init_segment_empty2: fixes S :: "nat set" shows "init_segment S {} = (S = {})"
under_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "Order_Relation.under r a \<equiv> {b. (b, a) \<in> r}"
repeat_Cons_left: fixes n :: "nat"   and xs :: "'a list" shows "repeat (Suc n) xs = xs @ repeat n xs"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G G G\<epsilon>o\<eta>G.map"
small_1_mult: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "h \<in> O[F](\<lambda>_. 1::'b)" shows "(\<lambda>x. f x * h x) \<in> o[F](g)"
weakCongAction: fixes \<alpha> :: "act"   and P :: "ccs" shows "\<alpha>.P \<Longrightarrow>\<alpha> \<prec> P"
AK: fixes q :: "('a, 'b) fm"   and p :: "('a, 'b) fm" shows "\<turnstile> q \<^bold>\<longrightarrow> p \<^bold>\<longrightarrow> q"
insert_var_0: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" shows "insert_var 0 f x = case_nat x f"
sup_upper: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and x :: "'a" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L"   and "x \<in> A" shows "x \<sqsubseteq>\<^bsub>L\<^esub> \<Squnion>\<^bsub>L\<^esub>A"
inverse_arrows_compose: fixes g :: "'a"   and f :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "seq g f"   and "inverse_arrows f f'"   and "inverse_arrows g g'" shows "inverse_arrows (g \<cdot> f) (f' \<cdot> g')"
pell_valuation_nonneg: fixes z :: "int \<times> int" assumes "0 \<le> fst z"   and "0 \<le> snd z" shows "0 \<le> pell_valuation z"
rep_eq: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and xa :: "'a" shows "mapping_of (MPoly_Type.monom x xa) = monomial xa x"
simps: fixes prec :: "nat"   and I :: "float interval list"   and f :: "floatarith"   and i :: "nat" shows "tmf_c prec I f i = compute_bound_fa prec (Mult (deriv_rec f i) (Inverse (Num (fact i)))) I"
extend_transient: fixes F :: "'a program"   and A :: "'a set" shows "(extend h F \<in> transient (extend_set h A)) = (F \<in> transient A)"
assms: shows "run_one_step d i (s, vs, ves, $Relop_f x301 x302) = (s', vs', res)"
single_leadsETo_I: fixes A :: "'d set"   and F :: "'d program"   and CC :: "'d set set"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> F \<in> {x} leadsTo[CC] B" shows "F \<in> A leadsTo[CC] B"
left_neutral: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and z :: "'b::type"   and a :: "'b::type" assumes "monoid (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (z::'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (z::'b::type) (a::'b::type) = a"
fermat_theorem: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "\<not> p dvd a" shows "[a ^ (p - 1) = 1] (mod p)"
relcomp_mono: fixes r' :: "('e \<times> 'f) set"   and r :: "('e \<times> 'f) set"   and s' :: "('f \<times> 'g) set"   and s :: "('f \<times> 'g) set" assumes "r' \<subseteq> r"   and "s' \<subseteq> s" shows "r' O s' \<subseteq> r O s"
i_Exec_Stream_Acc_Output_nth_f_nth: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and n :: "nat" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c n = f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (input \<Down> Suc n) c ! n"
lt1_neq0: fixes x :: "'b word" shows "(1 \<le> x) = (x \<noteq> 0)"
intList_equi: fixes ts :: "'b set list"   and s :: "'b set" assumes "disjoint_list_rec ts" shows "intersection_list s ts = intersection_list_opt s ts"
WHILEI_def: shows "aWHILEI \<equiv> \<lambda>I b f. REC (WHILEI_body abind areturn I b f)"
a: shows "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>\<^esup> i' @ [\<pi> i]"
set_mset_diff: fixes M :: "'a multiset"   and N :: "'a multiset" shows "set_mset (M - N) = {a. count N a < count M a}"
comp_ipresIGWlsAbs: fixes hA :: "'x \<Rightarrow> 'y"   and MOD :: "('z, 'aa, 'ab, 'ac, 'ad, 'ae, 'af, 'x, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'ab, 'ac, 'aj, 'ak, 'al, 'y, 'am) model_scheme"   and hA' :: "'y \<Rightarrow> 'an"   and MOD'' :: "('ao, 'ap, 'ab, 'ac, 'aq, 'ar, 'as, 'an, 'at) model_scheme" assumes "ipresIGWlsAbs hA MOD MOD'"   and "ipresIGWlsAbs hA' MOD' MOD''" shows "ipresIGWlsAbs (hA' \<circ> hA) MOD MOD''"
simps: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and C :: "'b literal set" shows "validate_clause I C = (\<forall>s. ground_clause (subst_cl C s) \<longrightarrow> validate_ground_clause I (subst_cl C s))"
crel_vs1_state_of: fixes P :: "heap \<Rightarrow> bool"   and update :: "'g \<Rightarrow> 'h \<Rightarrow> unit Heap"   and lookup :: "'g \<Rightarrow> 'h option Heap"   and dp :: "'g \<Rightarrow> 'h"   and R :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and a :: "'i"   and b :: "'j Heap" assumes "dp_consistency_heap P update lookup"   and "dp_consistency_heap.crel_vs1 P lookup dp R a b" shows "dp_consistency.crel_vs (heap_mem_defs.lookup' lookup) P dp R a (state_of b)"
B_authenticates_A: fixes K :: "nat"   and A :: "agent"   and Ta :: "nat"   and evs :: "event list"   and B :: "agent"   and Tk :: "nat" assumes "Crypt K \<lbrace>Agent A, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Says A B \<lbrace>Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"
prime_ge_Suc_0_nat: fixes p :: "nat" assumes "prime p" shows "Suc 0 \<le> p"
dist2: shows "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
on_graph_comp: fixes G :: "('d, 'b) labeled_graph"   and f :: "'e \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'e" shows "on_graph G (f \<circ> g) = on_graph G g O on_graph (map_graph_fn G g) f"
sees_wf_mdecl: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a option"   and D :: "String.literal" assumes "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "wf_mdecl wf_md P D (M, Ts, T, m)"
wp2_Assign: fixes x :: "char list"   and e :: "aexp"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>2 (x ::= e) Q = (\<lambda>s. eSuc (Q (s[e/x])))"
11: shows "rawpsubst (subst \<phi> (Var v) x) (zip (map Var vs) xs) = subst (rawpsubst \<phi> (zip (map Var vs) xs)) (Var v) x"
singleton_undefinedI: fixes A :: "'a pred"   and default :: "unit \<Rightarrow> 'a" assumes "\<nexists>!x. eval A x" shows "singleton default A = default ()"
rtc_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z + y ; x \<le> y \<longrightarrow> z ; x\<^sup>\<star> \<le> y"
plus_down_def: fixes prec :: "nat"   and x :: "real"   and y :: "real" shows "plus_down prec x y = truncate_down prec (x + y)"
assms_2: shows "x \<in>\<^sub>\<circ> A"
eq_imp_rtranclp: fixes x :: "'b"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = y" shows "r\<^sup>*\<^sup>* x y"
SUP_nat_Infty: shows "(SUP i. ereal (real i)) = \<infinity>"
filter_True: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x" shows "filter P xs = xs"
arsinh_def: fixes x :: "'a" shows "arsinh x = ln (x + (x\<^sup>2 + (1::'a)) powr of_real (1 / 2))"
inL1': fixes G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "G, H, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "H, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
composite_aux: assumes "\<not> prime n" shows "measure_pmf.expectation (primality_test P n) of_bool < q"
m_closed: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
likelihood_init: fixes s :: "'a"   and os :: "'b list" shows "likelihood s os = emeasure (T (s, obs)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = os}"
sylvester_sum_mat_lower: fixes q :: "'b poly"   and i :: "nat"   and p :: "'b poly" assumes "degree q \<le> i"   and "i < degree p + degree q" shows "(\<Sum>j<degree p + degree q. monom (sylvester_mat p q $$ (i, j)) (degree p + degree q - Suc j)) = monom (1::'b) (degree p + degree q - Suc i) * q"
LLL_invD_11: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "upw \<or> \<mu>_small fs i"
measure_le_0_iff: fixes M :: "'a measure"   and X :: "'a set" shows "(Sigma_Algebra.measure M X \<le> 0) = (Sigma_Algebra.measure M X = 0)"
that: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> vcard xs"   and "xs\<lparr>n\<rparr> = a" shows "thesis"
preserves_dom: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "local.dom (fst f \<star> snd f) = fst (VV.dom f) \<star> snd (VV.dom f)"
subst_lift_cond: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and s :: "('a, 'b) uexpr" shows "\<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> \<lceil>s\<rceil>\<^sub>\<leftarrow> = \<lceil>\<sigma> \<dagger> s\<rceil>\<^sub>\<leftarrow>"
closed_map_paired_continuous_map_right: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "Hausdorff_space Y" shows "closed_map X (prod_topology X Y) (\<lambda>x. (x, f x))"
ACI_norm_rderiv: fixes as :: "'a"   and r :: "'b rexp" shows "\<guillemotleft>rderiv as \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>rderiv as r\<guillemotright>"
q64_def: shows "q64 = {0, 1, 4, 9, 16, 17, 25, 36, 33, 49, 41, 57}"
Plus_def: fixes A :: "'b set"   and B :: "'c set" shows "A <+> B = Sum_Type.Inl ` A \<union> Sum_Type.Inr ` B"
disjuncts: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list" shows "(\<exists>q\<in>set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
normal_bicategory_axioms: shows "normal_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg"
limit_ordinal_not_0: fixes x :: "ordinal" assumes "limit_ordinal x" shows "x \<noteq> 0"
a_comm_monoid: shows "Group.comm_monoid (add_monoid G)"
steps_image: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> nat"   and l :: "nat"   and u :: "nat"   and v :: "nat" assumes "finite A"   and "mono_on g A"   and "inj_on g A" shows "steps A (f \<circ> g) l u v = steps (g ` A) f l u v"
prems_1: shows "tyenv_wellformed mds \<Gamma> \<S> P"
partitions_remaining_Max_part: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "p partitions n"   and "0 < p k" shows "\<forall>i. n - k < i \<and> i \<noteq> k \<longrightarrow> p i = 0"
HInfinite_HFinite_iff: fixes x :: "'b star" shows "(x \<in> HInfinite) = (x \<notin> HFinite)"
stable_pair_onE: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" obtains "- Rf CH (snd XD_XH) = fst XD_XH" and "- Rf (CD_on ds) (fst XD_XH) = snd XD_XH"
\<epsilon>_is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) F'G'.FG.map B.map F'G'.\<epsilon>"
countable_web_def: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "countable_web \<Gamma> \<equiv> web \<Gamma> \<and> countable_web_axioms \<Gamma>"
permutation_lift_signed_lconjperm_eq_signed_list_lconjaction: fixes x :: "'w freeword" assumes "x \<in> FreeGroup S" shows "(\<rightarrow>) (freeword_funlift signed_lconjpermutation x) = signed_list_lconjaction (map fst (freeword x))"
set: fixes ls :: "'a list"   and h :: "heap"   and new_ls :: "'a list" shows "Array.set (fst (Array.alloc ls h)) new_ls (snd (Array.alloc ls h)) = snd (Array.alloc new_ls h)"
ws_dynmethd: fixes G :: "prog"   and statC :: "qtname"   and sig :: "sig"   and statM :: "qtname \<times> methd"   and dynC :: "qtname" assumes "methd G statC sig = Some statM"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC"   and "is_class G statC"   and "ws_prog G" shows "\<exists>dynM. dynmethd G statC dynC sig = Some dynM \<and> is_static dynM = is_static statM \<and> G\<turnstile>resTy dynM\<preceq>resTy statM"
scaleR_nonpos_nonpos: fixes a :: "real"   and b :: "'b" assumes "a \<le> 0"   and "b \<le> (0::'b)" shows "(0::'b) \<le> a *\<^sub>R b"
exec1_appendL: fixes P :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and i' :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list"   and P' :: "instr list" assumes "P \<turnstile> (i, s, stk) \<rightarrow> (i', s', stk')" shows "P' @ P \<turnstile> (size P' + i, s, stk) \<rightarrow> (size P' + i', s', stk')"
tendsto_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "F \<noteq> bot"   and "Limsup F f = Liminf F f" shows "(f \<longlongrightarrow> Limsup F f) F"
cong_add: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c"   and d :: "'c"   and e :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (plus b d) (plus c e) a"
finite_FV0: fixes k :: "'k"   and a :: "'a" shows "finite (FV0 k a)"