hcju/isabelleps huggingface dataset:
functor_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c" shows "functor A B F \<equiv> category A \<and> category B \<and> functor_axioms A B F"
folding_ex: fixes C :: "'a set" assumes "chamber C"   and "C \<in> f \<turnstile> X" shows "\<exists>x. chamber x \<and> x \<notin> f \<turnstile> X \<and> f ` x = C"
least: shows "(LEAST k. \<not> b ((c ^^ k) s)) = 0"
starlike_compact_projective_special: fixes S :: "'c set" assumes "compact S"   and "cball (0::'c) 1 \<subseteq> S"   and "\<And>x u. \<lbrakk>x \<in> S; 0 \<le> u; u < 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x \<in> S - frontier S" shows "S homeomorphic cball (0::'c) 1"
y'_def: shows "y' = (n div y ^ pm + y * int pm) div int p"
f_not_0: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F" shows "f \<noteq> 0"
generating_ring: shows "ring_of_sets \<Omega> G.generated_ring"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G G \<eta>'GoG\<epsilon>'.map"
replicate_count_mset_eq_filter_eq: fixes xs :: "'k list"   and k :: "'k" shows "replicate (count (mset xs) k) k = filter ((=) k) xs"
sound_reduction': fixes \<S>\<^sub>1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and \<S>\<^sub>2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" assumes "\<S>\<^sub>1 \<hookrightarrow> \<S>\<^sub>2" shows "\<lbrakk> \<S>\<^sub>2 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<subseteq> \<lbrakk> \<S>\<^sub>1 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
AB: shows "(\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` A \<union> (@\<^sub>v) (0\<^sub>v n) ` B \<subseteq> carrier_vec (n + m)"
I_nnf: fixes \<phi> :: "atom fm"   and xs :: "real list" shows "R.I (R.nnf \<phi>) xs = R.I \<phi> xs"
simps_1: fixes P :: "'b" shows "strip (SKIP {P}) = SKIP"
Limsup_bounded: fixes X :: "'a \<Rightarrow> 'b"   and C :: "'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F n in F. X n \<le> C" shows "Limsup F X \<le> C"
genby_eq_sum_lists: fixes S :: "'a set" shows "\<langle>S\<rangle> = (\<Union>ss\<in>lists (S \<union> uminus ` S). {sum_list ss})"
is_unit_div_mult_cancel_left: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "is_unit b" shows "a div (a * b) = (1::'a) div b"
adjunct_sup: fixes P :: "'a pred"   and xq :: "'a seq" shows "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
partizan_g: shows "Rep_game g \<in> games_lfp"
Y_arr_ide_1: fixes b :: "'c"   and f :: "'c"   and a :: "'c"   and a' :: "'c" assumes "C.ide b"   and "\<guillemotleft>f : a \<rightarrow> a'\<guillemotright>" shows "\<guillemotleft>Y f b : Hom.map (b, a) \<rightarrow>\<^sub>S Hom.map (b, a')\<guillemotright>"
list_of_oalist_shift_keys: fixes t :: "('a, 'b) pp"   and f :: "'c \<Rightarrow> 'c"   and xs :: "(('a, 'b) pp \<times> nat, 'c) oalist_ntm" shows "list_of_oalist_ntm (shift_map_keys_pprod t f xs) = map_raw (\<lambda>kv. (splus_pprod t (fst kv), f (snd kv))) (list_of_oalist_ntm xs)"
weakenStatImpWeakStatImp: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q"   and "\<And>\<Psi>' R S \<Psi>''. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', R, S) \<in> Rel" shows "\<Psi> \<rhd> P \<lessapprox><Rel> Q"
simps: fixes z :: "'d list"   and W :: "'d list set"   and p :: "'d list" shows "mcp z W p = (prefix p z \<and> p \<in> W \<and> (\<forall>p'. prefix p' z \<and> p' \<in> W \<longrightarrow> length p' \<le> length p))"
SPEC_trans: fixes x :: "'b nres"   and y :: "'b nres"   and Postcond :: "'b \<Rightarrow> bool" assumes "x \<le> y"   and "y \<le> SPEC Postcond" shows "x \<le> SPEC Postcond"
PQ: shows "(P, Q) = diagonal_step_PQ A i k bezout"
residuated_sup: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "f (x + y) = f x + f y"
simps_4: fixes x :: "char"   and y :: "char" shows "proper_interval (Some x) (Some y) = (1 < of_char y - of_char x)"
sum_constant_scaleR: fixes y :: "'c"   and A :: "'d set" shows "(\<Sum>x\<in>A. y) = real (card A) *\<^sub>R y"
valid_dbm_cases: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" assumes "valid_dbm M" obtains "[M]\<^bsub>v,n\<^esub> \<subseteq> V" and "dbm_int M n"
ext_inject: fixes newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a"   and newOID' :: "CTypeId \<Rightarrow> nat"   and newAID' :: "Arraytype \<Rightarrow> nat"   and vals' :: "Location \<Rightarrow> Value"   and more' :: "'a" shows "(\<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = \<lparr>newOID = newOID', newAID = newAID', vals = vals', \<dots> = more'\<rparr>) = (newOID = newOID' \<and> newAID = newAID' \<and> vals = vals' \<and> more = more')"
fls_inverse_zero': assumes "inverse (0::'a) = (0::'a)" shows "inverse 0 = 0"
finite_primes_le: shows "finite {p. prime p \<and> real p \<le> x}"
lnull_ltake: fixes n :: "enat"   and xs :: "'c llist" shows "lnull (ltake n xs) = (lnull xs \<or> n = 0)"
take_bit_xor: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "take_bit n (xor a b) = xor (take_bit n a) (take_bit n b)"
lunit_eqI: fixes f :: "'a"   and \<mu> :: "'a" assumes "B.ide f"   and "\<guillemotleft>\<mu> : f \<star> src f \<rightarrow>\<^sub>B f\<guillemotright>"   and "\<mu> \<star> src f = (f \<star> \<i>[src f]) \<cdot> B.\<a>' (src f) (src f) f" shows "\<mu> = B.lunit f"
Collect_cong: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "_Collect x (P x) = _Collect x (Q x)"
n_context_tables_def: shows "n_context_tables \<equiv> 3"
valid_modif: shows "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"
split_option_all: fixes P :: "'d option \<Rightarrow> bool" shows "(\<forall>x. P x) = (P \<bottom> \<and> (\<forall>x. P (\<lfloor>x\<rfloor>)))"
matchT_TMT_def: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchT_TMT c d \<equiv> \<forall>s t s'. mustT c s \<and> mustT d t \<and> s \<approx> t \<and> (c, s) \<rightarrow>t s' \<longrightarrow> (\<exists>t'. (d, t) \<rightarrow>*t t' \<and> s' \<approx> t')"
eq: assumes "p - 1 = int n" shows "reduced_homology_group (int n) (nsphere n) \<cong> integer_group"
d_OUT_h_minus: fixes i :: "nat"   and x :: "'v" assumes "cycles \<Delta> \<noteq> {}" shows "KIR (h_minus i) x"
trans_m1a: shows "TS.trans m1a = m1a_trans"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_irrefl_before_trans ext" shows "ext_irrefl ext"
strongSimWeakSim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>[Rel] Q" shows "P \<leadsto><Rel> Q"
has_derivative_series: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'a"   and l :: "'b" assumes "convex S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_derivative f' n x) (at x within S)"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x h) - g' x h) \<le> e * norm h"   and "x \<in> S"   and "(\<lambda>n. f n x) sums l" shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_derivative g' x) (at x within S)"
step_3: fixes C' :: "'a literal multiset" assumes "C' \<in># CC'" shows "\<exists>DD\<subseteq>CC. finite DD \<and> C' \<in> saturate DD"
adjacent_nodes_ss_V: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "Graph.adjacent_nodes c u \<subseteq> Graph.V c"
distinct_49: fixes x8 :: "'a" shows "Real \<noteq> \<langle>x8\<rangle>\<^sub>\<T>"
c_ubound: shows "\<forall>x. \<exists>n. x < c n"
order_iff_strict: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># b) = (a \<subset># b \<or> a = b)"
rSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<leadsto>[{uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}] Q"
above_Above_trans: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and C :: "'b set" assumes "trans r"   and "a \<in> above r b"   and "b \<in> Above r C" shows "a \<in> Above r C"
events_before_exist: fixes x :: "('a \<times> ('a, 'b) operation) event"   and i :: "nat" assumes "x \<in> set (history i)" shows "\<exists>pre. pre @ [x] prefix of i"
surjective: fixes r :: "('a, 'b) tagged_packet_scheme" shows "r = \<lparr>p_iiface = p_iiface r, p_oiface = p_oiface r, p_src = p_src r, p_dst = p_dst r, p_proto = p_proto r, p_sport = p_sport r, p_dport = p_dport r, p_tcp_flags = p_tcp_flags r, p_payload = p_payload r, p_tag_ctstate = p_tag_ctstate r, \<dots> = tagged_packet.more r\<rparr>"
cube_form_mult: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int"   and e :: "int" assumes "is_cube_form a b"   and "is_cube_form c d"   and "\<bar>e\<bar> = 1" shows "is_cube_form (a * c + e * 3 * b * d) (a * d - e * b * c)"
nodes_of_def: fixes g :: "(node \<times> node) list" shows "nodes_of g = set (map fst g @ map snd g)"
weak_meet_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter> y \<sqinter> z .= x \<sqinter> (y \<sqinter> z)"
meets_irrefl: fixes p :: "'b" assumes "\<I> p" shows "\<not> p \<parallel> p"
abc_only_cba_3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "\<not> [[b c a]]"
abs_summable_on_cong_neutral: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = (0::'e)"   and "\<And>x. x \<in> B - A \<Longrightarrow> g x = (0::'e)"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> f x = g x" shows "(f abs_summable_on A) = (g abs_summable_on B)"
path_component_linepath: fixes a :: "'a"   and b :: "'a"   and S :: "'a set" assumes "closed_segment a b \<subseteq> S" shows "path_component S a b"
cnf_cnf: fixes F :: "'a formula" assumes "is_nnf F" shows "cnf (cnf_form_of F) = cnf F"
inject: fixes x1 :: "('a, 'b) form multiset"   and x2 :: "('a, 'b) form multiset"   and y1 :: "('a, 'b) form multiset"   and y2 :: "('a, 'b) form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
conv_abscissa_def: fixes f :: "'b fds" shows "conv_abscissa f = (INF s\<in>{s. fds_converges f s}. ereal (s \<bullet> (1::'b)))"
"1_7": fixes k :: "nat"   and xs :: "event_data option list set list"   and t :: "nat"   and z :: "'a" assumes "i \<le> k"   and "Suc k \<le> Mini j js"   and "list_all2 (\<lambda>P. P k) Ps xs"   and "R k t"   and "U k z" shows "U (Suc k) (f xs t z)"
kuratowski: shows "K\<^bsub>3,3\<^esub> (with_proj (gen_contr_graph (with_proj K) (verts3 (with_proj K)))) \<or> K\<^bsub>5\<^esub> (with_proj (gen_contr_graph (with_proj K) (verts3 (with_proj K))))"
flow0_defined: fixes xa :: "real"   and x0 :: "'a" assumes "xa \<in> existence_ivl0 x0" shows "flow0 x0 xa \<in> X"
takefill_alt: fixes fill :: "'a"   and n :: "nat"   and l :: "'a list" shows "takefill fill n l = take n l @ replicate (n - length l) fill"
f_Exec_Stream_expand_aggregate_nth_eq_i_nth: fixes k :: "nat"   and n :: "nat"   and n' :: "nat"   and trans_fun :: "'e message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e message_af"   and c :: "'d"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k"   and "n < n'" shows "f_aggregate (f_Exec_Comp_Stream trans_fun (input \<Down> n' \<odot> k) c) k ag ! n = i_aggregate (i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag n"
main: assumes "1 + length (coeffs f) - length (coeffs g) = 1 + degree f - degree g \<or> degree f = 0 \<and> 1 + length (coeffs f) - length (coeffs g) = 0 \<and> f = 0" shows "(r = 0 \<or> degree r < degree g) \<and> smult (lead_coeff g ^ (1 + length (coeffs f) - length (coeffs g))) (g * 0 + f) = g * q + r"
zero_less_mult_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero (times a b)"   and "less zero a" shows "less zero b"
runit_simps_1: fixes f :: "'a" assumes "B.ide f" shows "B.arr (B.runit f)"
Prefix_is_CSP: fixes c :: "'d \<Rightarrow> 'c" shows "is_CSP_process (R (true \<turnstile> do c \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A)))"
Collect_empty_eq_bot: fixes P :: "'e \<Rightarrow> bool" shows "(Collect P = {}) = (P = bot)"
assms_1: shows "edges G\<^sub>1 \<subseteq> edges G\<^sub>2"
less: fixes n :: "nat" shows "fst (ivl n) < snd (ivl n)"
lset_lconst: fixes x :: "'c"   and b :: "'c" assumes "x \<in> lset (lconst b)" shows "x = b"
step_preserves_termination: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>c' \<down> s'"
ltl_prop_equiv_subst_S: fixes S :: "'b ltl set"   and \<phi> :: "'b ltl"   and m :: "'b ltl \<Rightarrow> 'b ltl option" shows "S \<Turnstile>\<^sub>P subst \<phi> m = (S - dom m \<union> {uu_. \<exists>\<chi> \<chi>'. uu_ = \<chi> \<and> \<chi> \<in> dom m \<and> m \<chi> = Some \<chi>' \<and> S \<Turnstile>\<^sub>P \<chi>'}) \<Turnstile>\<^sub>P \<phi>"
Suc_diff_1: fixes n :: "nat" assumes "0 < n" shows "Suc (n - 1) = n"
tiTable_ack_ok: fixes d :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat"   and stop :: "nat \<Rightarrow> stopType list"   and a1 :: "nat \<Rightarrow> aType list"   and req :: "nat \<Rightarrow> reqType list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and st :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" assumes "\<forall>j\<le>d + 2. lose (t + j) = [False]"   and "ts lose"   and "msg (Suc 0) stop"   and "msg (Suc 0) a1"   and "req (Suc t) \<noteq> [send]"   and "ack t = [connection_ok]"   and "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) b ack i1 vc st" shows "ack (Suc t) = [connection_ok]"
verit_prod_simplify_2: fixes a :: "'i" shows "a * (1::'i) = a"
compl_sup: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) = - x \<sqinter> - y"
simps_3: fixes \<phi> :: "('e, 'f + 'h, 'g) formula"   and \<sigma> :: "'h \<Rightarrow> ('e, 'f, 'g) formula" shows "PFsubst (! \<phi>) \<sigma> = ! (PFsubst \<phi> \<sigma>)"
absorb1: fixes a :: "'d upred"   and b :: "'d upred" assumes "b \<sqsubseteq> a" shows "(a \<and> b) = a"
tuple_in'_def: shows "tuple_in' \<equiv> Mapping.filter (\<lambda>as _. proj_tuple_in_join pos maskL as X) tuple_in"
SymSystem_def: fixes init :: "(nat \<Rightarrow> 'c) set"   and p :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" shows "SymSystem init p r = [: z \<leadsto> (u, x) . u \<in> init \<and> z = x :] \<circ> {.u, x. prec_st p r u x.} \<circ> [: id (\<lambda>(u, x). id ((\<box> lft_rel_st r) u x)) :]"
add_1: fixes x :: "'a" assumes "x \<in> r \<down>m P" shows "\<exists>y\<in>#P -s r \<down>m P. (x, y) \<in> r"
eventually_laplace_integrand_le: fixes M :: "real"   and c :: "real"   and s :: "complex" assumes "exponential_order M c f" shows "\<forall>\<^sub>F t in at_top. cmod (local.laplace_integrand s t) \<le> M * exp (- (Re s - c) * t)"
fds_abs_summable_zeta_real: fixes s :: "real" assumes "1 < s" shows "fds_abs_converges fds_zeta s"
hom_sum: fixes f :: "'d \<Rightarrow> 'c poly poly"   and X :: "'d set" shows "poly_y_x (sum f X) = (\<Sum>x\<in>X. poly_y_x (f x))"
1: shows "B.VVV.arr \<mu>\<nu>\<tau>"
continuous_at_within_divide: fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous (at a within s) f"   and "continuous (at a within s) g"   and "g a \<noteq> (0::'b)" shows "continuous (at a within s) (\<lambda>x. f x / g x)"
order_zmset_exists_foundation': fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c zmultiset"   and t :: "'c" assumes "class.order less_eq less"   and "0 < zcount M t" shows "\<exists>s. less_eq s t \<and> 0 < zcount M s \<and> (\<forall>u. less u s \<longrightarrow> zcount M u \<le> 0)"
cycle_free_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.cycle_free plus less_eq neutral m n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (\<forall>j\<le>n. less_eq (linordered_ab_monoid_add.len plus m i j (rem_cycles i j xs)) (linordered_ab_monoid_add.len plus m i j xs)) \<and> less_eq neutral (linordered_ab_monoid_add.len plus m i i xs)"
rep_nat_pp_pp_inverse: fixes x :: "('b, 'c) pp" shows "abs_nat_pp (rep_nat_pp x) = x"
Bloch_unit: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex" assumes "f holomorphic_on ball a 1"   and "deriv f a = 1" obtains b :: "complex" and r :: "real" where "1 / 12 < r" and "ball b r \<subseteq> f ` ball a 1"
inf_sup_aci_7: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup" shows "sup x (sup y z) = sup y (sup x z)"
extremum_unique: fixes a :: "'c list" shows "prefix a [] = (a = [])"
continuous_map_snd_of: fixes Z :: "'d topology"   and X :: "'e topology"   and Y :: "'f topology"   and f :: "'d \<Rightarrow> 'e \<times> 'f" assumes "continuous_map Z (prod_topology X Y) f" shows "continuous_map Z Y (snd \<circ> f)"
\<psi>_def: shows "\<psi> \<equiv> \<lambda>n x. \<Sum>k = Suc 0..\<sigma> n. if real k / real (\<sigma> n) \<le> g x then inverse (real (\<sigma> n)) else 0"
le_funE: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f \<le> g" obtains "f x \<le> g x"
emeasure_pair_measure_alt: fixes X :: "('e \<times> 'd) set"   and N :: "'e measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M distr N T g)" shows "emeasure (N \<Otimes>\<^sub>M distr N T g) X = \<integral>\<^sup>+ x. emeasure (distr N T g) (Pair x -` X) \<partial>N"
in_lists_conv_set: fixes xs :: "'b list"   and A :: "'b set" shows "(xs \<in> lists A) = (\<forall>x\<in>set xs. x \<in> A)"
vimageE: fixes a :: "'d"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "a \<in> f -` B" obtains x :: "'e" where "f a = x" and "x \<in> B"
Float_num_1: shows "real_of_float (Float 1 0) = 1"
pth_c_3: fixes c :: "real"   and x :: "'a"   and d :: "real"   and y :: "'a"   and z :: "'a" shows "c *\<^sub>R x + (d *\<^sub>R y + z) \<equiv> d *\<^sub>R y + (c *\<^sub>R x + z)"
that: fixes G :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "G \<subseteq> dgrad_p_set d m"   and "finite G"   and "is_Groebner_basis G"   and "pmdl G = pmdl F" shows "thesis"
preserves_reflects_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.arr (B.\<alpha>' f) = B.VVV.arr f"
left_unique_vimage2p: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd" assumes "left_unique P"   and "inj f" shows "left_unique (BNF_Def.vimage2p f g P)"
update_convs_1: fixes mbal' :: "nat \<Rightarrow> nat"   and mbal :: "nat"   and bal :: "nat"   and inp :: "InputsOrNi"   and more :: "'a" shows "mbal_update mbal' \<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr> = \<lparr>mbal = mbal' mbal, bal = bal, inp = inp, \<dots> = more\<rparr>"
cut2: fixes A :: "fmla"   and B :: "fmla"   and C :: "fmla"   and H :: "fmla set" assumes "{A, B} \<turnstile> C"   and "H \<turnstile> A"   and "H \<turnstile> B" shows "H \<turnstile> C"
infsetsum_nonneg_is_SUPREMUM_ennreal: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "f abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "ennreal (infsetsum f A) = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. ennreal (sum f F))"
union_lcomm: fixes M :: "'c multiset"   and N :: "'c multiset"   and K :: "'c multiset" shows "M + (N + K) = N + (M + K)"
simps_1: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" shows "C M2 M1 \<Omega> V m 0 = {}"
GF_Inf_many: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<phi>) = (\<exists>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n \<phi>)"
seqI: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g"   and "\<F>C.dom g = \<F>C.cod f" shows "\<F>C.seq g f"
LNil_eq_ltake_iff: fixes n :: "enat"   and xs :: "'b llist" shows "(LNil = ltake n xs) = (xs = LNil \<or> n = 0)"
graph_union_iff: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" shows "(graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2) = (edges G\<^sub>1 \<subseteq> edges G\<^sub>2 \<and> vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2)"
valid_without_offending_flows: fixes F :: "('b \<times> 'b) set"   and sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" assumes "F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP" shows "sinvar (delete_edges G F) nP"
as0': assumes "S_M S M (D + negs (mset As)) \<noteq> {#}" shows "negs (mset As0') = S DA0'"