hcju/isabelleps huggingface dataset:
SUFFIX_def: fixes r :: "'b rexp" shows "SUFFIX r = PLUS (sorted_list_of_set {\<guillemotleft>lderivs w r\<guillemotright> |w. wf_word n w})"
surjective_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "surjective x" shows "surjective (x \<squnion> y)"
minSetOfComponents_def: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "minSetOfComponents i chSet \<equiv> outSetOfComponents i chSet \<union> \<Union> (Sources i ` outSetOfComponents i chSet)"
normal_subgroup_axioms: shows "normal_subgroup Normal G (\<cdot>) \<one>"
rec_pdec1_def: shows "rec_pdec1 = CN rec_minus [recf.Id 1 0, CN rec_triangle [CN rec_max_triangle [recf.Id 1 0]]]"
CatchSwap: 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 R :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 Q,R" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Catch c\<^sub>1 c\<^sub>2 Q,A"
ccFromList_below_ccApprox: fixes xs :: "Vars.var list"   and t :: "Vars.var ttree" assumes "xs \<in> paths t" shows "ccFromList xs \<sqsubseteq> ccApprox t"
surjec_to_mimg: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "surjec\<^bsub>M,mimg\<^bsub>R M,N\<^esub> f\<^esub> f"
disc_eq_case: fixes prod :: "'c \<times> 'd" shows "(prod = prod) = (case prod of (uu_, uua_) \<Rightarrow> True)"
bst_wrt_mono: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b tree" assumes "\<And>x y. P x y \<Longrightarrow> Q x y"   and "bst_wrt P t" shows "bst_wrt Q t"
\<psi>: shows "\<psi> (map (sdrop (Suc i)) \<pi>l)"
jkbpCn_jkbpC_inc: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> jkbpC\<^bsub>n\<^esub>" shows "t \<in> jkbpC"
Aenc_def: shows "Aenc \<equiv> map_fun Re (map_fun Re Ab) cAenc"
bisimClosed: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<sim> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<sim> p \<bullet> Q"
simps: fixes inPorts :: "'a \<Rightarrow> 'b fset"   and vertices :: "'c fset"   and nodeOf :: "'c \<Rightarrow> 'a"   and v :: "'c"   and p :: "'b" shows "Vertex_Graph.valid_in_port inPorts vertices nodeOf (v, p) = (v |\<in>| vertices \<and> p |\<in>| inPorts (nodeOf v))"
bin_nth_zero: fixes n :: "nat" shows "\<not> bit 0 n"
simps_6: fixes x :: "nat"   and e :: "cexpr"   and e' :: "cexpr"   and t :: "pdf_type" shows "cexpr_comp_aux x e (\<integral>\<^sub>c e' \<partial>t) = \<integral>\<^sub>c cexpr_comp_aux (Suc x) (map_vars Suc e) e' \<partial>t"
Pi_empty: fixes B :: "'a \<Rightarrow> 'b set" shows "Pi {} B = UNIV"
axioms_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "set_valued_functor C S F" shows "replete_set_category S"
scalar_product_append: fixes ns :: "nat list"   and xs :: "int list"   and ys :: "int list" shows "scalar_product ns (xs @ ys) = scalar_product (take (length xs) ns) xs + scalar_product (drop (length xs) ns) ys"
extend_distance_fp: shows "extend_distance_fp = extend_distance extend_distance_fp"
gr_zeroI: fixes n :: "'a multiset" assumes "n = {#} \<Longrightarrow> False" shows "{#} \<subset># n"
linkage: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> A \<Gamma> \<Longrightarrow> d_OUT f x = weight \<Gamma> x" shows "linkage \<Gamma> f"
conv_invol: fixes x :: "'a" shows "(x\<^sup>T)\<^sup>T = x"
real_cond_exp_ge_c: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable M f"   and "AE x in M. c \<le> f x" shows "AE x in M. c \<le> real_cond_exp M Invariants f x"
det_linear_row_sum: fixes S :: "'a set"   and k :: "'c"   and a :: "'c \<Rightarrow> 'a \<Rightarrow> ('b, 'c) vec"   and c :: "'c \<Rightarrow> ('b, 'c) vec" assumes "finite S" shows "det (\<chi>i. if i = k then sum (a i) S else c i) = (\<Sum>j\<in>S. det (\<chi>i. if i = k then a i j else c i))"
wfE_min': fixes R :: "('a \<times> 'a) set"   and Q :: "'a set" assumes "wf R"   and "Q \<noteq> {}" obtains z :: "'a" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
lower_upper_asymptotic_density_complement: fixes A :: "nat set" shows "lower_asymptotic_density A = 1 - upper_asymptotic_density (UNIV - A)"
run_invar: fixes I :: "nat \<times> nat \<times> (nat \<Rightarrow> thread) \<Rightarrow> bool"   and i :: "nat" assumes "A.is_invar I" shows "I (s i)"
in_bigomega_zero: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "f \<in> \<Omega>[F](\<lambda>x. 0::'c)"
aug_inf_def: shows "Z\<^sub>\<infinity> = {z. z \<noteq> - \<infinity>}"
sorted_sublist_wrt_snoc': fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat" assumes "sorted_sublist_wrt R xs lo (hi - 1)"   and "lo \<le> hi"   and "hi < length xs"   and "\<forall>j. lo \<le> j \<and> j < hi \<longrightarrow> R (xs ! j) (xs ! hi)" shows "sorted_sublist_wrt R xs lo hi"
assms_8: shows "\<not> is_sig_red (\<prec>\<^sub>t) (=) G q"
rcosets_closed: fixes H :: "'a set"   and g :: "'a"   and M :: "'a set" assumes "subgroup H G"   and "g \<in> carrier G"   and "M \<in> rcosets H" shows "M #> g \<in> rcosets H"
inj_iff: fixes f :: "'a \<Rightarrow> 'b" shows "inj f = (inv f \<circ> f = id)"
negligible_Union: fixes \<T> :: "'c set set" assumes "finite \<T>"   and "\<And>t. t \<in> \<T> \<Longrightarrow> negligible t" shows "negligible (\<Union> \<T>)"
Apollonius: fixes B :: "'a"   and C :: "'a"   and b :: "real"   and A :: "'a"   and c :: "real"   and d :: "real"   and m :: "real" assumes "B \<noteq> C"   and "b = dist A C"   and "c = dist B A"   and "d = dist A (midpoint B C)"   and "m = dist B (midpoint B C)" shows "b\<^sup>2 + c\<^sup>2 = 2 * (m\<^sup>2 + d\<^sup>2)"
sep_dom2: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp (list2FWpolicy (separate p)))" shows "x \<in> dom (Cp (list2FWpolicy p))"
n_in_restr: shows "n \<^bold>\<in> restrict v (clm ts) c"
SA: shows "nths (xs @ [x]) A = nths xs A @ nths [x] {j. j + length xs \<in> A}"
bound1a: shows "real_of_int (sint w1) < real_of_int (sint POS_INF)"
ord_resolve_12: shows "\<forall>i<n. strictly_maximal_wrt (As ! i) (Cs ! i)"
subgroup_intersect: fixes H :: "'a set"   and H' :: "'a set" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)"   and "subgroup H' (G\<lparr>carrier := H__\<rparr>)" shows "subgroup (H \<inter> H') (G\<lparr>carrier := H__\<rparr>)"
case_prodI2: fixes p :: "'b \<times> 'd"   and c :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
vbrelation_vconverse: shows "vbrelation (vreal_inverse\<inverse>\<^sub>\<circ>)"
tree2_cases: fixes y :: "('c \<times> 'd) tree" obtains "y = \<langle>\<rangle>"   | x21 :: "('c \<times> 'd) tree" and a :: "'c" and b :: "'d" and x23 :: "('c \<times> 'd) tree" where "y = \<langle>x21, (a, b), x23\<rangle>"
map_eq_conv: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and g :: "'f \<Rightarrow> 'e" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
r_contains: fixes xs :: "nat list"   and x :: "nat" shows "eval (r_contains xs) [x] \<down>= (if x \<in> set xs then 1 else 0)"
normalize_positive_src_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_Src_Ports False m" shows "match_list (\<beta>, \<alpha>) (normalize_positive_src_ports m) a p = matches (\<beta>, \<alpha>) m a p"
strip2_empty_mrexp2: fixes r :: "'b rexp" shows "strip2 (empty_mrexp2 r) = empty_mrexp r"
inject: fixes x1 :: "t list"   and x2 :: "t list"   and y1 :: "t list"   and y2 :: "t list" shows "(x1 _> x2 = y1 _> y2) = (x1 = y1 \<and> x2 = y2)"
phl_test_cond: fixes p :: "'a"   and b :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "test p"   and "test b"   and "\<lbrace>p \<cdot> b\<rbrace>x\<lbrace>q\<rbrace>"   and "\<lbrace>p \<cdot> !b\<rbrace>y\<lbrace>q\<rbrace>" shows "\<lbrace>p\<rbrace>if b then x else y fi\<lbrace>q\<rbrace>"
span_Int_subset: fixes A :: "'c set"   and B :: "'c set" shows "ideal (A \<inter> B) \<subseteq> ideal A \<inter> ideal B"
arccos_arcsin_eq: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "arccos x = pi / 2 - arcsin x"
sfoldl_op2_root2_rep2_split: fixes pat :: "[:'r:]"   and xs :: "[:'r:]" assumes "pat \<noteq> \<bottom>" shows "sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>xs = rep2\<cdot>pat\<cdot>(split\<cdot>pat\<cdot>xs)"
f_shrink_last_Cons: fixes k :: "nat"   and xs :: "'d list"   and ys :: "'d list" assumes "0 < k"   and "length xs = k" shows "(xs @ ys) \<div>\<^sub>l k = last xs # ys \<div>\<^sub>l k"
take_all: fixes xs :: "'f list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
power_minus1_even: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus one) (2 * n) = one"
I_in_S: fixes s :: "'s" assumes "pmf I s \<noteq> 0" shows "s \<in> S"
igSwapIGOpSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapIGOpSTR MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
dghm_comp_ObjMap_vrange: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
prems_3: fixes x :: "real \<times> real" assumes "x \<in> set (x__ # xs)" shows "coll 0 x (sum_list (x__ # xs))"
bounded_iff: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" shows "(a \<le>s signed.min b c) = (a \<le>s b \<and> a \<le>s c)"
someI2_ex: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
Confidentiality_S_temporal: fixes A :: "agent"   and K' :: "nat"   and T :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>Number T, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "\<not> expiredK T evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Key K \<notin> analz (knows Spy evs)"
linorder_injI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. x < y \<Longrightarrow> f x \<noteq> f y" shows "inj f"
bisim: shows "abs.strong_low_bisim_mm \<R>\<^sub>A"
bit_word_ucast_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (UCAST('b \<rightarrow> 'a) w) n = (n < LENGTH('b) \<and> n < LENGTH('a) \<and> bit w n)"
rhigha_rhigh: shows "(rep \<propto> higha) q = (rep \<propto> high) q"
of_rat_eq_iff: fixes a :: "rat"   and b :: "rat" shows "(of_rat a = of_rat b) = (a = b)"
butlast_prefix_imp_length_not_gt: fixes s :: "'b list"   and t :: "'b list" assumes "0 < length s"   and "strict_prefix (butlast s) t" shows "\<not> length t < length s"
domains_char: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.domains f = (if A1xA2.arr f then {A1xA2.dom f} else {})"
prim_spanning_invariant_def: fixes t :: "'a"   and v :: "'a"   and g :: "'a"   and r :: "'a" shows "prim_spanning_invariant t v g r \<equiv> prim_precondition g r \<and> v\<^sup>T = r\<^sup>T * t\<^sup>\<star> \<and> spanning_tree t (v * v\<^sup>T \<sqinter> g) r"
in_rel_Fab: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'Fa"   and y :: "'Fb" shows "rel_Fab A x y = (\<exists>z. z \<in> {x. set_Fab x \<subseteq> {(x, y). A x y}} \<and> \<pi>_Faba z = x \<and> \<pi>_Fabb z = y)"
le_some_optE: fixes m :: "'a"   and x :: "'a option" assumes "Some m \<le> x" obtains m' :: "'a" where "x = Some m'" and "m \<le> m'"
absorb_iff1: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (max a b = a)"
poly_deg_leI: fixes p :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'd"   and d :: "'c" assumes "\<And>t. t \<in> keys p \<Longrightarrow> deg_pm t \<le> d" shows "poly_deg p \<le> d"
in_rtrancl_on_in_F_2: fixes a :: "'a"   and b :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> rtrancl_on F r" shows "b \<in> F"
binomial_Vandermonde: fixes a :: "nat"   and b :: "nat"   and n :: "nat" shows "(\<Sum>k = 0..n. (a choose k) * (b choose (n - k))) = a + b choose n"
split_at_take_drop_conv: fixes n :: "nat"   and xs :: "'a list" shows "split_at n xs = (take n xs, drop n xs)"
fract_content_def: fixes p :: "'b fract poly" shows "fract_content p = (let d = Lcm (snd ` quot_of_fract ` set (coeffs p)) in Fraction_Field.Fract (content (map_poly (fst \<circ> quot_of_fract) (smult (to_fract d) p))) d)"
card_of_Card_order: fixes A :: "'a set" shows "Card_order |A|"
nn_integral_fst_count_space: fixes f :: "'b \<times> 'c \<Rightarrow> ennreal" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV = integral\<^sup>N (count_space UNIV) f"
errMOD_igSubstIGOpSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igWlsAbsIsInBar Delta MOD"   and "FixSyn.igVarIPresIGWls Delta MOD"   and "FixSyn.igOpIPresIGWls Delta MOD"   and "FixSyn.igSubstIPresIGWls Delta MOD"   and "FixSyn.igSubstAbsIPresIGWlsAbs Delta MOD"   and "igWlsDisj MOD"   and "FixSyn.igWlsAbsDisj Delta MOD"   and "FixSyn.igSubstIGOp Delta MOD" shows "FixSyn.igSubstIGOpSTR Delta (FixSyn.errMOD Delta MOD)"
state_set_card_ii: fixes a :: "'a"   and X :: "'a list" shows "card ((#) a ` state_set X) = card (state_set X)"
cont2cont_lambda: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'i" assumes "\<And>y. cont (\<lambda>x. f x y)" shows "cont f"
alphaAbs_all_equal_or_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "alphaAbs_all_equal_or_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
subterms_refl: fixes p :: "('e, 'f, 'g, 'h) seqp" shows "p \<in> subterms p"
smult_monom: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "Polynomial.smult a (monom b n) = monom (a * b) n"
focus_times: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "focus X (p * q) = focus X p * focus X q"
rel_Grp: fixes A1 :: "'e set"   and f1 :: "'e \<Rightarrow> 'h"   and A2 :: "'f set"   and f2 :: "'f \<Rightarrow> 'i" shows "rel_gpv (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. results'_gpv x \<subseteq> A1 \<and> outs'_gpv x \<subseteq> A2} (map_gpv f1 f2)"
mn_4: shows "fds_nth g n \<noteq> (0::'a)"
Cod_TensorDiag: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u" shows "Cod (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u"
leadsto_false: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> #False) = (\<box>\<not> F)"
conjugate_r_def: fixes x :: "'a"   and y :: "'a" shows "x \<rhd> y \<equiv> - (x \<rightarrow> - y)"
P: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'd measure"   and M :: "nat \<Rightarrow> 'd measure"   and i :: "nat" assumes "Ionescu_Tulcea P M" shows "P i \<in> Pi\<^sub>M {0..<i} M \<rightarrow>\<^sub>M subprob_algebra (M i)"
le: fixes X :: "'v set"   and n :: "nat" assumes "X \<subseteq> A_n n" shows "integral\<^sup>N (count_space X) (weight \<Gamma>) \<le> integral\<^sup>N (count_space (Y2 n X)) (weight \<Gamma>)"
norm_f: shows "\<exists>K. \<forall>n x. norm (f n *\<^sub>v x) \<le> norm x * K"
Gcd_ratfps_def: shows "Gcd = normalization_euclidean_semiring_class.Gcd"
char_poly_defs_1: fixes A :: "'b mat" shows "char_poly A = det (char_poly_matrix A)"
gnodes_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "gnodes A w V \<equiv> transition_system_initial.nodes (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V) (\<lambda>v. v \<in> ginitial A \<inter> V)"
simps_1: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat" shows "trace d i [] = []"
cos_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X0 :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X0"   and "cos_aform_err p X0 = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "cos x \<in> aform_err e Y"
wens_single_finite_mono: fixes m :: "nat"   and n :: "nat"   and act :: "('b \<times> 'b) set"   and B :: "'b set" assumes "m \<le> n" shows "wens_single_finite act B m \<subseteq> wens_single_finite act B n"
h_w1: fixes x :: "'b \<times> 'c"   and x' :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x x'"   and "less_eq_prod (times_prod' x' y) z" shows "less_eq_prod (times_prod' x y) z"
times_ennreal_def: shows "(*) \<equiv> map_fun enn2ereal (map_fun enn2ereal e2ennreal) (*)"
put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok: fixes h :: "(_) heap"   and ptr :: "(_) character_data_ptr"   and setter :: "('l \<Rightarrow> 'l) \<Rightarrow> 'h option RCharacterData_scheme \<Rightarrow> 'h option RCharacterData_scheme"   and v :: "'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr setter v"
convex_halfspace_le: fixes a :: "'c"   and b :: "real" shows "convex {x. a \<bullet> x \<le> b}"
inv_init_imp_prop_safe_def: fixes c :: "('b, 't, 'loc, 'c) Combined.configuration_scheme" shows "inv_init_imp_prop_safe c = (\<forall>p. init c p \<longrightarrow> prop_safe (prop_config c p))"
sorted_hyp_var: shows "\<forall>q1<length l. \<forall>q2<length l. q1 < q2 \<longrightarrow> l ! q1 < l ! q2"
length_finite_permutations_of_set: fixes xs :: "'a list"   and A :: "'a set" assumes "xs \<in> permutations_of_set A" shows "length xs = card A"
simps_2: fixes x2 :: "nat" shows "0 \<noteq> Suc x2"
inv_chart_smooth_on: fixes c :: "('c, 'c) chart"   and k :: "enat" assumes "c \<in> manifold_eucl.atlas k" shows "k-smooth_on (codomain c) (inv_chart c)"
monofunI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" shows "monofun f"
pcr_cr_eq: shows "pcr_mapping_rbt (=) (=) = cr_mapping_rbt"
intensional_simps_4: fixes y :: "'b \<Rightarrow> bool" shows "\<not> \<not> y \<equiv> y"
lo_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_lo [x, y] = lo x y"
dy_fake_chan_extr_insert: fixes M :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" assumes "M \<in> dy_fake_chan bad IK CH" shows "extr bad IK (insert M CH) \<subseteq> dy_fake_msg bad IK CH"
s01: shows "(\<lambda>x. (x - a) /\<^sub>R norm (x - a)) ` S \<subseteq> sphere (0::'a) 1"
distinct_1: fixes x1 :: "nat"   and x21 :: "'a"   and x22 :: "'a convnet" shows "Input x1 \<noteq> Conv x21 x22"
tail_invar_cons_down: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "tail_invar (t # bq)" shows "tail_invar bq"
simps_1: fixes \<phi> :: "fml" shows "Noto (Afml \<phi>) = Afml (! \<phi>)"
n_distrib_alt: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "!x \<cdot> !z = !y \<cdot> !z"   and "!x \<oplus> !z = !y \<oplus> !z" shows "!x = !y"
runit'_naturality: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "(\<mu> \<star> src \<mu>) \<cdot> \<r>\<^sup>-\<^sup>1[local.dom \<mu>] = \<r>\<^sup>-\<^sup>1[cod \<mu>] \<cdot> \<mu>"
transpose_singleton: fixes j :: "nat"   and i :: "nat"   and a :: "'b" shows "transpose_matrix (singleton_matrix j i a) = singleton_matrix i j a"
plossless_inline: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('e, 'f) \<I>"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'd, 'e, 'f) gpv"   and s :: "'d" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> plossless_gpv \<I>' (callee s x)"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>" shows "plossless_gpv \<I>' (inline callee gpv s)"
rel_interior_closure_convex_segment: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "convex S"   and "a \<in> rel_interior S"   and "b \<in> closure S" shows "open_segment a b \<subseteq> rel_interior S"
assms_1: shows "\<Psi> \<rhd> P \<longmapsto> Rs"
assms_2: shows "(f has_field_derivative D) (at x within B)"
permutes_bij: fixes p :: "'b \<Rightarrow> 'b"   and s :: "'b set" assumes "p permutes s" shows "bij p"
Levellist_ext_to_all: fixes hds :: "ref list"   and "next" :: "ref \<Rightarrow> ref"   and P :: "ref list list \<Rightarrow> bool"   and Q :: "bool" shows "((\<exists>ll. Levellist hds next ll \<and> P ll) \<longrightarrow> Q) = (\<forall>ll. Levellist hds next ll \<and> P ll \<longrightarrow> Q)"
set_homE_5: fixes f :: "'b set_arrow"   and U :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "f \<in> Hom\<^bsub>set_cat U\<^esub> A B" shows "set_cod f = B"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) (tab\<^sub>0 f)"
map_of_Cons_code_1: fixes k :: "'d" shows "map_of [] k = None"
assms: shows "f $ subdegree f * g $ subdegree g \<noteq> (0::'a)"
dom_m_fmempty: shows "dom_m fmempty = {#}"
simps: fixes v :: "'d" shows "terminal_vertex v = (v |\<in>| vertices \<and> terminal_node (nodeOf v))"
RPDs_refl: fixes r :: "mregex" shows "r \<in> RPDs r"
assms_10: shows "callee_invariant_on oracle1 I1 \<I>"
iT_iff_1: fixes x :: "nat"   and n :: "nat" shows "(x \<in> [n\<dots>]) = (n \<le> x)"
absn_dist_eta: fixes n :: "nat"   and t :: "dB" shows "(Abs ^^ n) (var_dist (rev [0..<n]) (liftn n t 0)) \<leftrightarrow> t"
del: fixes p :: "int mpoly"   and A :: "int mpoly multiset"   and \<V> :: "nat set" assumes "p \<in># A" shows "PAC_Format (\<V>, A) (\<V>, remove1_mset p A)"
bij_h: shows "bij_betw (\<lambda>j. j + i - degree g) {0..<Suc (degree g)} {i - degree g..<Suc i}"
tab_in_vhom': shows "\<guillemotleft>\<tau> : t\<^sub>1 \<Rightarrow> t \<star> t\<^sub>0\<guillemotright>"