hcju/isabelleps huggingface dataset:
normal_complex_mat_has_spectral_decomposition: fixes A :: "complex mat"   and n :: "nat"   and es :: "complex list"   and B :: "complex mat"   and P :: "complex mat"   and Q :: "complex mat" assumes "A \<in> carrier_mat n n"   and "A * adjoint A = adjoint A * A"   and "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:])"   and "unitary_schur_decomposition A es = (B, P, Q)" shows "similar_mat_wit A B P (adjoint P) \<and> diagonal_mat B \<and> diag_mat B = es \<and> unitary P"
if_split_asm: fixes P :: "'b \<Rightarrow> bool"   and Q :: "bool"   and x :: "'b"   and y :: "'b" shows "P (if Q then x else y) = (\<not> (Q \<and> \<not> P x \<or> \<not> Q \<and> \<not> P y))"
not_eventually_impI: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter"   and Q :: "'c \<Rightarrow> bool" assumes "eventually P F"   and "\<not> eventually Q F" shows "\<not> (\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x)"
preference_def: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" shows "preference carrier relation \<equiv> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> y \<in> carrier) \<and> preorder_on carrier relation"
yoneda_functor_axioms: shows "yoneda_functor (\<cdot>) replete_setcat.comp \<chi>.\<phi>"
set_take_auxlist': fixes x :: "nat \<times> event_data option list set \<times> event_data option list set" assumes "x \<in> set (take (length done) (update_until args rel1 rel2 nt auxlist))" shows "check_before I nt x"
power_poly_f_mod_i: fixes ff_ops :: "'a arith_ops_record"   and R :: "'a \<Rightarrow> 'b mod_ring \<Rightarrow> bool"   and p :: "int"   and m :: "'a list \<Rightarrow> 'a list"   and m' :: "'b mod_ring poly"   and f :: "'a list"   and f' :: "'b mod_ring poly"   and n :: "nat" assumes "prime_field_gen ff_ops R p"   and "(ring_ops.poly_rel R ===> ring_ops.poly_rel R) m (\<lambda>x'. x' mod m')"   and "ring_ops.poly_rel R f f'" shows "ring_ops.poly_rel R (power_poly_f_mod_i ff_ops m f n) (power_poly_f_mod m' f' n)"
ginjec_def: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" shows "ginj\<^bsub>F,G\<^esub> f = (f \<in> gHom F G \<and> inj_on f (carrier F))"
DirProds_finite_imp_finite_groups: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and i :: "'e" assumes "finite_group (DirProds G I)"   and "finite I"   and "i \<in> I" shows "finite_group (G i)"
path_component_of_set: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "path_component_of X x y = (\<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y)"
strictly_preserves_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "C.arr a"   and "C.arr b"   and "C.arr c" shows "F \<a>\<^sub>C[a, b, c] = \<a>\<^sub>D[F a, F b, F c]"
ide_is_iso: fixes a :: "'c" assumes "Cop.ide a" shows "Cop.iso a"
inj_on_image_eqpoll_1: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'c set" assumes "inj_on f A" shows "(f ` A \<approx> B) = (A \<approx> B)"
assms: fixes x :: "'a" shows "A *\<^sub>V ket x = B *\<^sub>V ket x"
finite_distinct_list: fixes A :: "'c set" assumes "finite A" shows "\<exists>xs. set xs = A \<and> distinct xs"
mult2_altE: fixes X :: "'a multiset"   and Y :: "'a multiset"   and b :: "bool"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> mult2_alt b ns s" obtains X1 :: "'a multiset" and X2 :: "'a multiset" and Y1 :: "'a multiset" and Y2 :: "'a multiset" where "X = X1 + X2" and "Y = Y1 + Y2" and "(X1, Y1) \<in> multpw ns" and "b \<or> Y2 \<noteq> {#}" and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)"
atMost_iff: fixes i :: "'c list"   and k :: "'c list" shows "(i \<in> sublist_order.atMost k) = sublist i k"
isomorphic_unit_left: fixes f :: "'b" assumes "D.ide f" shows "D.isomorphic (trg\<^sub>D f \<star>\<^sub>D f) f"
B_def: shows "B \<equiv> (\<Sum>n. norm (fds_nth f (n + 2)) / real (n + 2) powr c) * 2 powr c"
SUP_bot_conv_2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 B :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
normalize_positive_dst_ports: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "\<not> has_disc_negated is_Dst_Ports False m" shows "match_list (\<beta>, \<alpha>) (normalize_positive_dst_ports m) a p = matches (\<beta>, \<alpha>) m a p"
list_update_id: fixes xs :: "'a list"   and i :: "nat"   and z :: "'a" assumes "xs ! i = z" shows "xs[i := z] = xs"
imp_cong: fixes H :: "fmla set"   and A :: "fmla"   and A' :: "fmla"   and B :: "fmla"   and B' :: "fmla" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> (A IMP B) IFF A' IMP B'"
rel_witness_resource2: fixes A :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and res :: "('c, 'e) resource"   and res' :: "('d, 'f) resource" assumes "rel_resource (A OO A') B res res'"   and "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'" shows "rel_resource A' (\<lambda>(b, c') c. c = c' \<and> B b c') (rel_witness_resource A A' B (res, res')) res'"
unique_idx_def: fixes bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" shows "unique_idx bs data = ((\<forall>f\<in>set bs. \<forall>g\<in>set bs. fst (snd f) = fst (snd g) \<longrightarrow> f = g) \<and> (\<forall>f\<in>set bs. fst (snd f) < fst data))"
fresh_some: fixes a :: "'a"   and x :: "'b" shows "a \<sharp> Some x = a \<sharp> x"
All_mapping_mono: fixes m :: "('b, 'c) mapping"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>k v. \<lbrakk>k \<in> keys m; P k v\<rbrakk> \<Longrightarrow> Q k v"   and "All_mapping m P" shows "All_mapping m Q"
uminus_QDelta_def: fixes qd :: "QDelta" shows "- qd = QDelta (- qdfst qd) (- qdsnd qd)"
inv_mkarr: fixes t :: "'c local.term" assumes "Can t" shows "local.inv (mkarr t) = mkarr (Inv t)"
language_coinduct: fixes R :: "'a list set \<Rightarrow> 'a list set \<Rightarrow> bool"   and K :: "'a list set"   and L :: "'a list set" assumes "R K L"   and "\<And>K L. R K L \<Longrightarrow> ([] \<in> K) = ([] \<in> L)"   and "\<And>K L x. R K L \<Longrightarrow> R (Deriv x K) (Deriv x L)" shows "K = L"
to_nat_from_nat_id: fixes x :: "nat" assumes "x < CARD('b)" shows "mod_type_class.to_nat (mod_type_class.from_nat x) = x"
env_restr_mono2: fixes S2 :: "'d set"   and S1 :: "'d set"   and m :: "'d \<Rightarrow> 'e" assumes "S2 \<subseteq> S1" shows "m f|` S2 \<sqsubseteq> m f|` S1"
LtA_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "A B C LtA D E F \<equiv> A B C LeA D E F \<and> \<not> A B C CongA D E F"
univ: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "type_definition Rep Abs A" shows "UNIV = Abs ` A"
newton_seq_fact9: fixes n :: "nat" assumes "f \<bullet> ns n \<noteq> \<zero>" shows "val_Zp (ns (Suc n) \<ominus> ns n) = val_Zp (f \<bullet> ns n) - val_Zp (f' \<bullet> ns n)"
assms: shows "r \<in>\<^sub>\<circ> Vset \<alpha>"
pcompose_pCons: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "pCons a p \<circ>\<^sub>p q = [:a:] + q * p \<circ>\<^sub>p q"
hom_inv: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (inv\<^bsub>G\<^esub> x) = inv\<^bsub>H\<^esub> h x"
work_alt_def: fixes return :: "int \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> (int \<Rightarrow> 'a) \<Rightarrow> 'a"   and fail :: "'a"   and catch :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "int list" assumes "fast_product return bind fail catch" shows "fast_product_base.work return bind fail xs = (if 0 \<in> set xs then fail else return (prod_list xs))"
word_leq_le_minus_one: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> y"   and "x \<noteq> 0" shows "x - 1 < y"
oracle_decrypt0'_def: shows "oracle_decrypt0' \<equiv> \<lambda>key (bad, L) (x', c', t'). return_spmf (if (x', c', t') \<in> L \<or> length x' \<noteq> prf_dlen then (None, bad, L) else (decrypt key (x', c', t'), bad \<or> upf_fun (snd key) (x' @ c') = t', L))"
simps_2: fixes k :: "nat"   and n :: "nat"   and ns :: "nat list" shows "into_WW k (n # ns) = (k + n) # into_WW (Suc (k + n)) ns"
Lim_transform: fixes g :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "(g \<longlongrightarrow> a) F"   and "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'c)) F" shows "(f \<longlongrightarrow> a) F"
output_def: fixes resumption :: "('d, 'e, 'f) resumption" shows "output resumption = (case resumption of Pause x21 x22 \<Rightarrow> x21)"
finite'_intersection_of_inc: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "P S" shows "(finite' intersection_of P) S"
codomains_char: fixes f :: "'c" shows "C.codomains f = (if C.arr f then {C.cod f} else {})"
trancl_rtrancl_trancl: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
ide_is_iso: fixes a :: "'a" assumes "A.ide a" shows "A.iso a"
of_nat_0_eq_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(zero = semiring_1.of_nat one plus zero n) = (0 = n)"
lea__nlta: 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 D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "\<not> Tarski_neutral_dimensionless.LtA Bet Cong D E F A B C"
unique_connecting_path_connects: fixes v :: "'c"   and w :: "'c" assumes "v \<in> V\<^bsub>T\<^esub>"   and "w \<in> V\<^bsub>T\<^esub>" shows "v \<leadsto>v \<leadsto>\<^bsub>T\<^esub> w\<leadsto>\<^bsub>T\<^esub> w"
assms: shows "moebius_pt M1 = moebius_pt M2"
simps: fixes lc :: "bool"   and base :: "'a word"   and ml :: "nat"   and w :: "'a word" shows "string_of_word lc base ml w = (if base < 2 \<or> LENGTH('a) < 2 then undefined else if w < base \<and> ml = 0 then string_of_word_single lc w else string_of_word lc base (ml - 1) (w div base) @ string_of_word_single lc (w mod base))"
Ri_least: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'c"   and z :: "'c" assumes "mono f"   and "x \<le> f y" shows "Ri x y z \<le> f z"
holomorphic_on_cos: fixes S :: "complex set" shows "cos holomorphic_on S"
irrefl_nest: fixes a :: "'a set" shows "\<not> a \<lless> a"
is_inf_prod: fixes p :: "'c \<times> 'd"   and q :: "'c \<times> 'd" shows "is_inf p q (fst p \<sqinter> fst q, snd p \<sqinter> snd q)"
le_cSup_finite: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set"   and x :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "finite X"   and "x \<in> X" shows "less_eq x (Sup X)"
hd_or_tl_gt: fixes ys :: "'a list"   and xs :: "'a list"   and y :: "'a"   and x :: "'a" assumes "length ys = length xs"   and "ext gt (y # ys) (x # xs)" shows "gt y x \<or> ext gt ys xs"
non_volatile_writes_unshared_no_outstanding_non_volatile_Write\<^sub>s\<^sub>b: fixes \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "non_volatile_writes_unshared \<S> sb"   and "\<forall>a\<in>dom \<S>' - dom \<S>. a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb" shows "non_volatile_writes_unshared \<S>' sb"
sorted_wrt_dist_nearest_neighbors: fixes p :: "(real, 'b) vec"   and n :: "nat"   and kdt :: "'b kdt" shows "sorted_wrt_dist p (nearest_neighbors n p kdt)"
prj_in_hom_1: shows "Maps.in_hom SPN_fgh.\<nu>\<pi>.prj\<^sub>1 (Maps.pbdom SPN_fgh.\<nu>.leg0 SPN_fgh.\<pi>.leg1) SPN_fgh.\<nu>.apex"
member_rec_2: fixes y :: "'e" shows "List.member [] y = False"
facet2_subset: fixes z :: "'b set"   and x :: "'b set"   and y :: "'b set" assumes "z \<lhd> x"   and "z \<lhd> y"   and "x \<inter> y - z \<noteq> {}" shows "order.greater_eq y x"
disc_2: fixes x21 :: "variable"   and x22 :: "String.literal set" shows "\<not> is_Ty (Tv x21 x22)"
f_def: shows "f \<equiv> \<lambda>M. {t \<cdot> \<delta> |t \<delta>. t \<in> M \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}}"
deriv_to_deriv: fixes C :: "'b sequent"   and n :: "nat"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "(C, n) \<in> derivable R" shows "C \<in> derivable' R"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<phi>"
Int_insert_left_if1: fixes a :: "'c"   and C :: "'c set"   and B :: "'c set" assumes "a \<in> C" shows "insert a B \<inter> C = insert a (B \<inter> C)"
rbd_irbd_galois: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and R :: "('c \<times> 'd) set" assumes "Sup_pres \<phi>" shows "(\<phi> = bd\<^sub>\<R> R) = (R = bd\<^sup>-\<^sub>\<R> \<phi>)"
max_mono: fixes R :: "('c, 'd) ordered_semiring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ordered_semiring R"   and "x \<succeq>\<^bsub>R\<^esub> y"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "Max\<^bsub>R\<^esub> z x \<succeq>\<^bsub>R\<^esub> Max\<^bsub>R\<^esub> z y"
lookup_except_singleton: fixes p :: "'b \<Rightarrow>\<^sub>0 'a"   and t :: "'b" shows "lookup (except p {t}) t = (0::'a)"
deriv_body: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) the (\<Gamma> p) (Q Z),(A Z)"
collapse_1: fixes literal :: "'b literal" assumes "is_pos literal" shows "Pos (atm_of literal) = literal"
vrat_minus_def: shows "vrat_minus = (\<lambda>x\<in>\<^sub>\<circ>\<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (rat_of_vrat (x\<lparr>0\<^sub>\<nat>\<rparr>) - rat_of_vrat (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<rat>)"
causal': shows "causal_program_history is\<^sub>s\<^sub>b (sb @ sb')"
inj_add_left: fixes a :: "'a" shows "inj ((+) a)"
edge_preserving_atomic: fixes h1 :: "('a \<times> 'b) set"   and e1 :: "('c \<times> 'a \<times> 'a) set"   and e2 :: "('c \<times> 'b \<times> 'b) set"   and v1 :: "'a"   and v1' :: "'b"   and v2 :: "'a"   and v2' :: "'b"   and k :: "'c" assumes "edge_preserving h1 e1 e2"   and "(v1, v1') \<in> h1"   and "(v2, v2') \<in> h1"   and "(k, v1, v2) \<in> e1" shows "(k, v1', v2') \<in> e2"
entailsI: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "\<And>s. P s \<le> Q s" shows "P \<tturnstile> Q"
zero_le_odd_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "odd n" shows "less_eq zero (power.power one times a n) = less_eq zero a"
sum_sup: fixes A :: "'b set"   and z :: "'b" assumes "finite A" shows "(\<Sum> A \<le> z) = (\<forall>a\<in>A. a \<le> z)"
disjoint_union: fixes A :: "'d set"   and B :: "'d set set"   and b1 :: "'d set"   and b2 :: "'d set" assumes "Congruence.partition A B"   and "b1 \<in> B"   and "b2 \<in> B"   and "b1 \<inter> b2 \<noteq> {}" shows "b1 = b2"
rel_FGf_def: fixes L1 :: "'q \<Rightarrow> 'aa \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 'ab \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2 :: "'u \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra2 :: "'x \<Rightarrow> 'af \<Rightarrow> bool" shows "rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 = rel_F L1 L2 (=) Co1 Co2 (=) Contra1 Contra2 (=)"
homotopic_loops_imp_path: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_loops s p q" shows "path p \<and> path q"
ENC_FAnd: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" assumes "wf_formula n (FAnd \<phi> \<psi>)" shows "lang n (ENC n (FOV (FAnd \<phi> \<psi>))) \<subseteq> lang n (ENC n (FOV \<phi>)) \<inter> lang n (ENC n (FOV \<psi>))"
intros_14: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('a, 'b) mformula"   and \<phi>' :: "Formula.formula"   and first :: "bool"   and nts :: "nat list"   and I :: "\<I>" assumes "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"   and "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R \<phi> \<phi>'"   and "first = (Monitor.progress \<sigma> P \<phi>' j = 0)"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [Monitor.progress \<sigma> P \<phi>' j - 1..<j] nts" shows "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R (MNext I \<phi> first nts) (formula.Next I \<phi>')"
process_mru_is_Some: fixes v_f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and a :: "'e"   and rv :: "'c \<times> 'd" shows "(process_mru v_f a = Some rv) = (vote_set v_f {a} \<noteq> {} \<and> rv = Max_by fst (vote_set v_f {a}))"
inner_prod_csqrt: fixes v :: "complex vec" shows "csqrt \<langle>v|v\<rangle> = complex_of_real \<parallel>v\<parallel>"
expansion_gamma_onE: fixes A :: "'a set"   and As :: "'a set set"   and f :: "'a set \<Rightarrow> 'a set"   and c :: "'a" assumes "expansion_gamma_on A As f"   and "\<Union> As \<subseteq> A"   and "As \<noteq> {}"   and "\<forall>x\<in>As. c \<in> f x" shows "c \<in> f (\<Union> As)"
multiplicity_eq_zero_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "(factorial_semiring.multiplicity times one p x = 0) = (\<not> dvd.dvd times p x)"
to_struct_map_closed: fixes f :: "'a \<Rightarrow> 'b"   and T :: "('a, 'c) partial_object_scheme"   and S :: "('b, 'd) partial_object_scheme" assumes "f \<in> carrier T \<rightarrow> carrier S" shows "to_struct_map T f \<in> struct_maps T S"
simps: fixes prec :: "nat"   and a :: "floatarith"   and bs :: "float interval option list" shows "approx' prec a bs = (case approx prec a bs of None \<Rightarrow> None | Some ivl \<Rightarrow> Some (round_interval prec ivl))"
simps_19: fixes vb :: "'b"   and vc :: "value list"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (In vb vc) (Eq v va) = False"
exhaust_sel: fixes stream :: "'c stream" obtains "stream = shd stream ## stl stream"
knows_subset_knows_Notes: fixes A :: "agent"   and evs :: "event list"   and A' :: "agent"   and X :: "msg" shows "knows A evs \<subseteq> knows A (Notes A' X # evs)"
elims_2: fixes x :: "bool"   and xa :: "'b \<times> 'c set" assumes "pure_assn_raw x xa" obtains b :: "bool" and h :: "'b" and as :: "'c set" where "x = b" and "xa = (h, as)" and "as = {} \<and> b"
not_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
axioms_1: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex X"
fps_lcm_def: shows "lcm = normalization_euclidean_semiring_class.lcm"
fls_base_factor_delta: fixes a :: "int"   and c :: "'a" shows "fls_base_factor (Abs_fls (\<lambda>n. if n = a then c else (0::'a))) = fls_const c"
negate_def: fixes P :: "'c \<Rightarrow> bool" shows "\<N> P = (\<lambda>s. \<not> P s)"
acyclic_4b_def: fixes x :: "'a" shows "acyclic_4b x \<equiv> \<forall>y\<le>x. x \<sqinter> y\<^sup>\<star> = y"
connected_component_of_equiv: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "connected_component_of X x y = (x \<in> topspace X \<and> y \<in> topspace X \<and> connected_component_of X x = connected_component_of X y)"
assms: shows "\<forall>t. 0 < t \<and> t < 1 \<longrightarrow> g t = c"
x_in_dom_allow: shows "x \<in> dom (C (AllowPortFromTo c d po))"
g2: shows "R_Chop(Abs_real_int (left r4, right r2),r4,r2)"
le_Pinf: fixes x :: "'a extended" shows "x \<le> \<infinity>"
assms: shows " b \<in>B \<langle>G\<rangle> "
broadcast_only_valid_msgs: fixes pre :: "('a \<times> 'b) event list"   and m :: "'a \<times> 'b"   and i :: "nat" assumes "pre @ [Broadcast m] prefix of i" shows "\<exists>state. apply_operations pre = Some state \<and> valid_msg state m"
birkhoff_sum_1_3: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "A.birkhoff_sum f (Suc 0) x = f x"
vec_mat_Tensor_vector_id: fixes v :: "int poly fract list list" shows "rat_poly.vec_mat_Tensor [1] v = v"
ratfps_cutoff_def: fixes n :: "nat"   and x :: "'b ratfps" shows "ratfps_cutoff n x = poly_of_list (map (ratfps_nth x) [0..<n])"