hcju/isabelleps huggingface dataset:
concurrent_add_remove_independent: fixes i :: "'a"   and e1 :: "'b"   and ir :: "'a"   and "is" :: "'a set"   and e2 :: "'b"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Add i e1) (ir, Rem is e2)"   and "\<not> hb (ir, Rem is e2) (i, Add i e1)"   and "xs prefix of j"   and "(i, Add i e1) \<in> set (node_deliver_messages xs)"   and "(ir, Rem is e2) \<in> set (node_deliver_messages xs)" shows "i \<notin> is"
InlI: fixes a :: "'d"   and A :: "'d set"   and B :: "'e set" assumes "a \<in> A" shows "Inl a \<in> A <+> B"
bind_spmf_spmf_of_set: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "(\<bind>) (spmf_of_set A) = (\<bind>) (pmf_of_set A)"
rev_ord0_le_less_trans: fixes y :: "ord0"   and z :: "ord0"   and x :: "ord0" assumes "y < z"   and "x \<le> y" shows "x < z"
nn_integral_null_delta: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "A \<in> sets M"   and "B \<in> sets M"   and "A \<Delta> B \<in> null_sets M" shows "set_nn_integral M A f = set_nn_integral M B f"
temp0: assumes "e = Cvtop t1 Convert t sx" shows "type_update tn' [TSome t] (Type [arity_1_result e]) = tm'"
not_in_Rep_preal_ub: fixes x :: "rat"   and xa :: "preal"   and y :: "rat" assumes "x \<notin> Rep_preal xa"   and "y \<in> Rep_preal xa"   and "0 < x" shows "y < x"
TOTAL_neq_CONS: shows "TOTAL \<noteq> CONS"
similar_termTr: fixes A :: "'b set"   and a :: "'b"   and n :: "nat"   and m :: "'a" assumes "ideal R A"   and "a \<in> A" shows "\<forall>s f. s \<in> {j. j \<le> n} \<rightarrow> A \<and> f \<in> {j. j \<le> n} \<rightarrow> carrier M \<and> m \<in> f ` {j. j \<le> n} \<longrightarrow> (\<exists>t\<in>{j. j \<le> n} \<rightarrow> A. \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n \<plusminus> a \<cdot>\<^sub>s m = \<Sigma>\<^sub>e M (\<lambda>j. t j \<cdot>\<^sub>s f j) n)"
finite_closed_segment: fixes a :: "'b"   and b :: "'b" shows "finite {a--b} = (a = b)"
map_of_asetI2: fixes a :: "'c"   and S :: "('c \<times> 'b) set" assumes "\<forall>b. (a, b) \<notin> S" shows "map_of_aset S\<langle>a\<rangle> = None"
HFinite_HComplex: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "HComplex x y \<in> HFinite"
prev: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S" shows "\<exists>h. \<forall>z\<in>S. (h has_field_derivative f z) (at z)"
take_rev: fixes n :: "nat"   and xs :: "'g list" shows "take n (rev xs) = rev (drop (length xs - n) xs)"
add1: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ccw_vector_space ccw S"   and "ccw (0::'b) a b"   and "ccw (0::'b) c b" shows "ccw (0::'b) (a + c) b"
subspace_inter: fixes A :: "('a \<Rightarrow> real) set"   and B :: "('a \<Rightarrow> real) set" assumes "A \<subseteq> diff_fun_space"   and "B \<subseteq> diff_fun_space"   and "diff_fun_space.m1.subspace A"   and "diff_fun_space.m1.subspace B" shows "diff_fun_space.m1.subspace (A \<inter> B)"
distinct_finfun_to_list: fixes f :: "('a, 'b) finfun" shows "distinct (finfun_to_list f)"
the_left_decr: fixes xs :: "int list"   and i :: "nat" assumes "has_left xs i" shows "the_left xs i < i"
ptrm_alpha_equiv_prm: fixes X :: "'b ptrm"   and Y :: "'b ptrm"   and \<pi> :: "'b prm" assumes "X \<approx> Y" shows "\<pi> \<bullet> X \<approx> \<pi> \<bullet> Y"
i: shows "finite {(f (a, b), b) |a b. (a, b) \<in> p}"
normal_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" shows "H \<lhd> G \<equiv> subgroup H G \<and> Group.group G \<and> normal_axioms H G"
rep_eq: fixes x :: "'a mpoly" shows "MPoly_Type.degree x = (\<lambda>v. Max (insert 0 ((\<lambda>m. lookup m v) ` keys (mapping_of x))))"
tt_tail: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.tail term_of_pair ord_term p \<noteq> 0" shows "ordered_term.tt term_of_pair ord_term (ordered_term.tail term_of_pair ord_term p) = ordered_term.tt term_of_pair ord_term p"
assign_axiom_def: shows "assign_axiom \<equiv> [[vid1 := $f fid1 local.empty]]$\<phi> vid1 (singleton (Var vid1)) \<leftrightarrow> $\<phi> vid1 (singleton ($f fid1 local.empty))"
zcarr: shows "z \<in> carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))"
l11_25: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and C' :: "'a"   and P' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.InAngle Bet P A B C"   and "Tarski_neutral_dimensionless.Out Bet B A' A"   and "Tarski_neutral_dimensionless.Out Bet B C' C"   and "Tarski_neutral_dimensionless.Out Bet B P' P" shows "Tarski_neutral_dimensionless.InAngle Bet P' A' B C'"
f: shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> equivalence_map f"
Base: fixes B :: "('d, 'e, 'f) boundOutput" shows "\<lparr>\<nu>*[]\<rparr>B = B"
list_all_iff_fset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_all P xs = fBall (fset_of_list xs) P"
preserves_right_adjoint: fixes g :: "'a" assumes "C.is_right_adjoint g" shows "D.is_right_adjoint (F g)"
measurable: shows "(\<lambda>x1. emeasure (lborel_f n) {x \<in> space (lborel_f n). x(n := x1) \<in> A}) \<in> borel_measurable lborel"
prod_list_distinct_conv_prod_set: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "distinct xs" shows "prod_list (map f xs) = prod f (set xs)"
max_in_chainI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "nat"   and Y :: "nat \<Rightarrow> 'a" assumes "class.po below"   and "\<And>j. i \<le> j \<Longrightarrow> Y i = Y j" shows "po.max_in_chain i Y"
nonInterference_path_to_Low: fixes cf :: "'var \<Rightarrow> 'val option"   and cf' :: "'var \<Rightarrow> 'val option"   and S :: "'node SDG_node set"   and as :: "'edge list"   and as' :: "'edge list" assumes "[cf] \<approx>\<^sub>L [cf']"   and "(_High_) \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "CFG_node (_Low_) \<in> S"   and "(_Entry_) -as\<rightarrow>\<^sub>\<surd>* (_Low_)"   and "preds (kinds as) [(cf, undefined)]"   and "(_Entry_) -as'\<rightarrow>\<^sub>\<surd>* (_Low_)"   and "preds (kinds as') [(cf', undefined)]" shows "map fst (transfers (kinds as) [(cf, undefined)]) \<approx>\<^sub>L map fst (transfers (kinds as') [(cf', undefined)])"
Always_LeadsTo_pre: fixes F :: "'c program"   and INV :: "'c set"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> Always INV" shows "(F \<in> INV \<inter> A \<longmapsto>w A') = (F \<in> A \<longmapsto>w A')"
update_by_fun_pair_ko_ntm_def: fixes ko :: "'a nat_term_order" shows "update_by_fun_pair_ko_ntm ko \<equiv> comparator.update_by_fun_pair (nat_term_compare_inv ko)"
exec_gpv_obsf_resource: fixes gpv :: "('a, 'b, 'c) gpv"   and res :: "('b, 'c) resource" shows "exec_gpv run_resource (map_gpv' id id option_of_exception (gpv_stop gpv)) (obsf_resource res) \<upharpoonleft> {(Some x, y) |x y. True} = map_spmf (map_prod Some obsf_resource) (exec_gpv run_resource gpv res)"
facetrel_psubset: fixes y :: "'b set"   and x :: "'b set" assumes "y \<lhd> x" shows "order.greater x y"
reindex_bij_betw: fixes h :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set"   and g :: "'e \<Rightarrow> 'a" assumes "bij_betw h S T" shows "F (\<lambda>x. g (h x)) S = F g T"
Nmlize_Vcomp_Arr_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t \<^bold>\<cdot> Dom t\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
inj_enumerate: fixes S :: "'a set" assumes "infinite S" shows "inj (wellorder_class.enumerate S)"
DBPass_visible: shows "\<turnstile> <DBPass>_(inp, mid, out, q1, q2) = DBPass"
hyps: assumes "x \<in> set ys" shows "listgcd ys dvd x"
inter_union_distrib_right: fixes C :: "'a multiset"   and A :: "'a multiset"   and B :: "'a multiset" shows "C + A \<inter># B = (C + A) \<inter># (C + B)"
State_Idle_imp_exists_state_change: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and n :: "nat" assumes "\<not> State_Idle localState output_fun trans_fun (localState c)"   and "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c))" shows "\<exists>i<n. \<not> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>i\<^esup> c)) \<and> (\<forall>j\<le>n. i < j \<longrightarrow> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>j\<^esup> c)))"
assoc: fixes a :: "'k"   and b :: "'k"   and c :: "'k" shows "a + b + c = a + (b + c)"
get_itvl_3: fixes x :: "real_alg_3" shows "real_of_3 x \<in>\<^sub>i get_itvl_3 x"
fvc: fixes v :: "variable \<Rightarrow> real"   and w :: "variable \<Rightarrow> real" assumes "Vagree v w (- {z})" shows "term_sem I t v = term_sem I t w"
y_def: shows "y \<equiv> real_of_1 (p2, l2, r2)"
tr_s_ifI: fixes x :: "x"   and T :: "x \<Rightarrow> x option"   and x' :: "x"   and y :: "x"   and y' :: "x"   and s1 :: "s"   and s1' :: "s"   and s2 :: "s"   and s2' :: "s" assumes "(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'"   and "(case T y of None \<Rightarrow> y | Some x' \<Rightarrow> x') = y'"   and "tr_s T s1 s1'"   and "tr_s T s2 s2'" shows "tr_s T (s_if x y s1 s2) (s_if x' y' s1' s2')"
term_rest: shows "\<forall>t. \<Gamma>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> \<Rightarrow> t \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> t"
plane_final_facesAt: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "Invariants.inv g"   and "final g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "final f"
sets_measure_pmf: fixes p :: "'a pmf" shows "measure_pmf.events p = UNIV"
THROW_refine: fixes ei :: "'c"   and e :: "'d"   and E :: "('c \<times> 'd) set"   and R :: "('e \<times> 'f) set" assumes "(ei, e) \<in> E" shows "THROW ei \<le> \<Down>\<^sub>E E R (THROW e)"
cod_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "ide a"   and "ide b"   and "ide c" shows "cod \<a>[a, b, c] = a \<otimes> b \<otimes> c"
goa: shows "grouped_out_arcs G_list = map (\<lambda>u. filter (\<lambda>x. fst x = u) (snd G_list)) vs"
assms: shows "length (cs\<^bsup>\<pi>\<^esup> i) \<noteq> 1"
pseudo_inverse_finite_set: fixes u :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "filterlim u sequentially sequentially" shows "finite {N. u N \<le> n}"
map_upd_Some_unfold: fixes m :: "'g \<Rightarrow> 'f option"   and a :: "'g"   and b :: "'f"   and x :: "'g"   and y :: "'f" shows "((m(a \<mapsto> b)) x = \<lfloor>y\<rfloor>) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = \<lfloor>y\<rfloor>)"
eq_access: shows "\<forall>l. s1@@l = s2@@l"
sum_left: fixes g :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'a"   and S :: "'c set"   and x :: "'b" shows "blinfun_apply (sum g S) x = (\<Sum>i\<in>S. blinfun_apply (g i) x)"
coeff_Q': fixes f :: "nat \<Rightarrow> complex"   and n :: "nat"   and k :: "nat" shows "poly.coeff (power_sum_puzzle_existence.Q' f n) k = (if n < k then 0 else (- 1) ^ (n - k) * power_sum_puzzle_existence.e' f n (n - k))"
fgmodule_condition_def: fixes R :: "('e, 'f) Ring_scheme"   and f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and i :: "'g \<Rightarrow> 'g"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and A :: "'g set"   and z :: "'g" shows "fgmodule_condition R f i s A z = (commute_bpp f (aug_pm_set z i (s_set R s A)) \<and> assoc_bpp (aug_pm_set z i (s_set R s A)) f \<and> ipp_cond1 (s_set R s A) i \<and> ipp_cond2 z (s_set R s A) i f \<and> ipp_cond3 z i \<and> inv_ipp z i f (s_set R s A) \<and> zeroA z i f (s_set R s A) z \<and> sop_distr1 R s f i (s_set R s A) z \<and> sop_distr2 R s f i (s_set R s A) z \<and> sop_assoc R s (aug_pm_set z i (s_set R s A)) \<and> sop_inv R s i (s_set R s A) \<and> sop_one R s (aug_pm_set z i (s_set R s A)) \<and> sop_z R s z)"
lnull_ldropn: fixes n :: "nat"   and xs :: "'c llist" shows "lnull (ldropn n xs) = (llength xs \<le> enat n)"
cod_trg_eq_dom_trg: shows "\<mu>'.cod.trg = \<mu>'.dtrg"
direct_arc_swapped: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b" assumes "pair_sym_digraph G" shows "pair_pre_digraph.direct_arc (u, v) = pair_pre_digraph.direct_arc (v, u)"
rComm2: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C"   and "x\<alpha> = \<tau>"   and "xT = \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'" shows "Prop"
inject: fixes x21 :: "('b btree \<times> 'b) list"   and x22 :: "'b btree"   and y21 :: "('b btree \<times> 'b) list"   and y22 :: "'b btree" shows "(Node x21 x22 = Node y21 y22) = (x21 = y21 \<and> x22 = y22)"
fres_bot_bot: shows "prod.fres bot_prod bot_prod = top_prod"
reset_set11: fixes cs :: "'a list"   and d :: "'b"   and u :: "'a \<Rightarrow> 'b" shows "\<forall>c. c \<notin> set cs \<longrightarrow> ([cs\<rightarrow>d]u) c = u c"
intersection_number_empty_iff: fixes b1 :: "'a set"   and b2 :: "'a set" assumes "finite b1" shows "(b1 \<inter> b2 = {}) = (b1 |\<inter>| b2 = 0)"
dynProcModifyReturnSameFaults_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
c': shows "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
saturated_Extend: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and f :: "nat \<Rightarrow> ('c, 'b) fm list \<times> 'b" assumes "infinite UNIV"   and "finite A"   and "consistent A S"   and "finite (\<Union> (block_nominals ` S))"   and "surj f" shows "saturated (Extend A S f)"
respond_imp_not_used: fixes PA :: "msg"   and RB :: "msg"   and KAB :: "nat"   and evs :: "event list" assumes "(PA, RB, KAB) \<in> respond evs" shows "Key KAB \<notin> used evs"
simps_14: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Package of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f2"
condensation_inequality_2: fixes n :: "nat" assumes "\<And>m n. \<lbrakk>0 < m; m \<le> n\<rbrakk> \<Longrightarrow> f n \<le> f m" shows "sum f {1..<n} \<le> (\<Sum>k = 1..<n. f (2 ^ Discrete.log k))"
minus_minus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "uminus (uminus a) = a"
count_roots_between_correct: fixes p :: "real poly"   and a :: "real"   and b :: "real" shows "count_roots_between p a b = card {x. a < x \<and> x \<le> b \<and> poly p x = 0}"
FP_Orig_def: fixes F :: "'b program" shows "FP_Orig F \<equiv> \<Union> {A. \<forall>B. F \<in> stable (A \<inter> B)}"
sees_method_decl_above: fixes 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 "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
mod_2_Suc_iff: fixes x :: "nat" shows "(x mod 2 = Suc 0) = (x = Suc (2 * (x div 2)))"
hyps: fixes b :: "int"   and a :: "int" assumes "nat \<bar>b\<bar> < nat \<bar>ba__\<bar>" shows "gcd (of_int a) (of_int b) = of_int (gcd a b)"
approx_un_def: fixes p :: "nat"   and f :: "float interval \<Rightarrow> float interval option"   and a :: "((real \<times> real pdevs) \<times> real) option" shows "approx_un p f a = a \<bind> (\<lambda>rd. f (ivl_of_aform_err p rd) \<bind> (\<lambda>ivl. Some (ivl_err (real_interval ivl))))"
ipresIGWlsAbs_def: fixes hA :: "'u \<Rightarrow> 'v"   and MOD :: "('w, 'x, 'y, 'z, 'aa, 'ab, 'ac, 'u, 'ad) model_scheme"   and MOD' :: "('ae, 'af, 'y, 'z, 'ag, 'ah, 'ai, 'v, 'aj) model_scheme" shows "ipresIGWlsAbs hA MOD MOD' \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD' (us, s) (hA A)"
last_index_le_size: fixes xs :: "'a list"   and x :: "'a" shows "last_index xs x \<le> length xs"
take_bit_not_take_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (not (take_bit n a)) = take_bit n (not a)"
bij_betw_chambers: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C" shows "bij_betw f C (f ` C)"
adopt_node_locs_def: fixes parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr" shows "adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent owner_document document_ptr = (if parent = None then {} else remove_child_locs (the parent) owner_document) \<union> set_disconnected_nodes_locs document_ptr \<union> set_disconnected_nodes_locs owner_document"
unhindered_linkable_bounded: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "bounded_countable_bipartite_web \<Gamma>"   and "\<not> hindered \<Gamma>" shows "linkable \<Gamma>"
diff_diff_add: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<ominus>\<^sub>a b \<ominus>\<^sub>a c = a \<ominus>\<^sub>a (b \<oplus>\<^sub>a c)"
bounded_countable_network_axioms_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "bounded_countable_network_axioms \<Delta> \<equiv> \<forall>x. x \<in> \<^bold>V\<^bsub>\<Delta>\<^esub> \<longrightarrow> x \<noteq> source \<Delta> \<longrightarrow> d_OUT (capacity \<Delta>) x < \<top>"
suntil_lfp: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool" shows "\<phi> suntil \<psi> = lfp (\<lambda>P s. \<psi> s \<or> \<phi> s \<and> P (stl s))"
prefix_code_1: fixes xs :: "'d list" shows "prefix [] xs = True"
eclose_minimal: fixes Y :: "hf"   and X :: "hf" assumes "Transset Y"   and "X \<le> Y" shows "eclose X \<le> Y"
quat_divide_numeral: fixes x :: "quat"   and w :: "num" shows "x / numeral w = x /\<^sub>R numeral w"
vD_invalidate_vD_not_dests: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> vD (invalidate rt dests)" shows "dip \<in> vD rt \<and> dests dip = None"
Int_space_eq2: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> fmeasurable M" shows "x \<inter> space M = x"
Bindv_def: shows "Bindv = Conv (Some (''Bind'', TypeExn (Short ''Bind''))) []"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'c \<Rightarrow> 'a"   and x :: "'c pmf"   and y :: "'b pmf" shows "rel_pmf Sb (map_pmf i x) y = rel_pmf (\<lambda>x. Sb (i x)) x y"
cong_pow: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a"   and n :: "nat" 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" shows "unique_euclidean_semiring.cong modulo (power.power one times b n) (power.power one times c n) a"
authKeys_insert: fixes A :: "agent"   and K :: "nat"   and Peer :: "agent"   and Ta :: "nat"   and evs :: "event list" shows "authKeys (Says Kas A (Crypt (shrK A) \<lbrace>Key K, Agent Peer, Number Ta, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K, Number Ta\<rbrace>\<rbrace>) # evs) = insert K (authKeys evs)"
zip_update: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b"   and ys :: "'c list"   and y :: "'c" shows "zip (xs[i := x]) (ys[i := y]) = (zip xs ys)[i := (x, y)]"
nb\<^sub>1\<^sub>8: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> dom ((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op'))" shows "((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op')) (v, a) = (\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) (v, a)"
finiteI_independent: fixes B :: "'a set" assumes "independent B" shows "finite B"