hcju/isabelleps huggingface dataset:
compute_\<ff>s_inner_bounds_2: fixes s :: "'a list"   and \<ff>s :: "nat list"   and ix :: "nat"   and j :: "nat"   and i :: "nat" assumes "I_out_cb s (\<ff>s, ix, j)"   and "j < length \<ff>s"   and "I_in_cb s j i" shows "j - 1 < length s"
ABinds_restrict_edom: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "ABinds (restrictA (edom ae) \<Gamma>)\<cdot>ae = ABinds \<Gamma>\<cdot>ae"
A: shows "\<forall>a\<in>A. \<forall>b\<in>A. a \<sqsubseteq> b"
card_of_Times_ordLess_infinite: fixes C :: "'c set"   and A :: "'d set"   and B :: "'e set" assumes "infinite C"   and "|A| <o |C|"   and "|B| <o |C|" shows "|A \<times> B| <o |C|"
pred_transfer: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "((R ===> (=)) ===> list_all2 R ===> (=)) list_all list_all"
lprefix_llength_eq_imp_eq: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lprefix xs ys"   and "llength xs = llength ys" shows "xs = ys"
sel: fixes x :: "pheap \<Rightarrow> bool" shows "assn_fn (Assn x) = x"
wp1_terminates: fixes c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "wp\<^sub>1 c Q l s" shows "\<down> (c, s)"
is_aligned_iff_udvd: fixes w :: "'b word"   and n :: "nat" shows "is_aligned w n = (2 ^ n udvd w)"
hom_deg_set_subset: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and z :: "nat" assumes "homogeneous_set A" shows "hom_deg_set z A \<subseteq> A"
boundedE: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "b \<squnion> c \<le> a" obtains "b \<le> a" and "c \<le> a"
C: shows "w = \<lbrakk>\<guillemotleft>C\<guillemotright>\<rbrakk>e"
eval_wp_Abort: fixes P :: "'a \<Rightarrow> real" shows "wp Abort P = (\<lambda>s. 0)"
regular_bot_top_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<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 bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linear_aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "(x = uminus (uminus x)) = (x = bot \<or> x = top)"
Quotient3_rep_abs: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and r :: "'a" assumes "Quotient3 R Abs Rep"   and "R r r" shows "R (Rep (Abs r)) r"
merge_token_succeeds: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and y :: "nat"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "(x, y) \<in> mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i" shows "mojmir_def.token_succeeds \<delta> q\<^sub>0 w F x = mojmir_def.token_succeeds \<delta> q\<^sub>0 w F y"
"2_4": shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"
i_expand_assoc: fixes f :: "nat \<Rightarrow> 'b message_af"   and a :: "nat"   and b :: "nat" shows "f \<odot> a \<odot> b = f \<odot> (a * b)"
t_p2r: fixes P :: "'a \<Rightarrow> bool" shows "rel_dioid_tests.t \<lceil>P\<rceil> = \<lceil>P\<rceil>"
perm_rem_def: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" shows "perm_rem x f \<equiv> if f x \<noteq> x then x \<rightleftharpoons>\<^sub>F f x \<circ> f else f"
chamber_card: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D" shows "card C = card D"
get_ACS_union_get_IFS: fixes M :: "'b SecurityInvariant_configured list" shows "set (get_ACS M) \<union> set (get_IFS M) = set M"
ipurge_ref_subset: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and X :: "'c set" shows "ipurge_ref I D u xs X \<subseteq> X"
cong_pow: fixes b :: "'b"   and c :: "'b"   and a :: "'b"   and n :: "nat" assumes "[b = c] (mod a)" shows "[b ^ n = c ^ n] (mod a)"
perm_simps_5: fixes p :: "perm"   and tvar :: "Syntax.tvar"   and ty :: "ty" shows "p \<bullet> Mu tvar ty = Mu (p \<bullet> tvar) (p \<bullet> ty)"
se_sat_imp_sat: fixes c :: "('d, 'e) conf"   and l :: "('d, 'e) label"   and c' :: "('d, 'e) conf" assumes "se c l c'"   and "Conf.sat c'" shows "Conf.sat c"
finite_bound: fixes F :: "'a set" assumes "finite F" shows "(\<Sum>i\<in>F. (cmod (if a = i then 1 else 0))\<^sup>2) \<le> 1"
mult_left_less_imp_less: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times c a) (times c b)"   and "less_eq zero c" shows "less a b"
elims: fixes x :: "'c tree"   and y :: "'c list" assumes "postorder x = y" obtains "x = \<langle>\<rangle>" and "y = []"   | l :: "'c tree" and xa :: "'c" and r :: "'c tree" where "x = \<langle>l, xa, r\<rangle>" and "y = postorder l @ postorder r @ [xa]"
reachable_oseq_seqp_sos: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and OA :: "((nat \<Rightarrow> 'e) \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and I :: "'f seq_action \<Rightarrow> bool"   and i :: "nat"   and A :: "('e \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "(\<sigma>, p) \<in> reachable OA I"   and "initiali i (init OA) (init A)"   and "automaton.trans OA = oseqp_sos \<Gamma> i"   and "automaton.trans A = seqp_sos \<Gamma>" shows "\<exists>\<xi>. \<sigma> i = \<xi> \<and> (\<xi>, p) \<in> reachable A I"
algebraic_int_nth_root_real: fixes x :: "real"   and n :: "nat" assumes "algebraic_int x" shows "algebraic_int (root n x)"
Pow_fold: fixes A :: "'b set" assumes "finite A" shows "Pow A = Finite_Set.fold (\<lambda>x A. A \<union> insert x ` A) {{}} A"
polynomial_dense_repr: fixes K :: "'a set"   and p :: "'a list" assumes "polynomial K p"   and "p \<noteq> []" shows "dense_repr p = (lead_coeff p, degree p) # dense_repr (local.normalize (tl p))"
inducts: fixes p :: "'b \<times> 'c"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "fstsp p x"   and "P (fst p)" shows "P x"
c2ka_axioms: shows "c2ka (\<circ>) \<lambda>"
brec_invar_add_def: fixes Qi :: "'a set" shows "brec_invar_add Qi \<equiv> {(Q, W, rcm, qwit). hm.invar Q \<and> distinct (fifo.\<alpha> W) \<and> hm.invar rcm \<and> (case qwit of None \<Rightarrow> Qi \<inter> dom (hm.\<alpha> Q) = {} | Some q \<Rightarrow> q \<in> Qi \<inter> dom (hm.\<alpha> Q))}"
geodesic_segment_topology_6: fixes G :: "'b set" assumes "geodesic_segment G" shows "G \<noteq> {}"
true_conj_zero_1: fixes P :: "'a upred" shows "(P \<and> false) = false"
circline_set_y_axis_I: fixes z :: "complex" assumes "is_imag z" shows "of_complex z \<in> circline_set y_axis"
simps_1: fixes s :: "'b stream" shows "[] @- s = s"
f': fixes t :: "real" assumes "0 < t" shows "(f has_vector_derivative f' t) (at t)"
rga_ops_ref_exists: fixes pre :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and suf :: "('a \<times> 'a option) list" assumes "rga_ops (pre @ (oid, Some ref) # suf)" shows "ref \<in> fst ` set pre"
rel_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "list_all2 R\<inverse>\<inverse> = (list_all2 R)\<inverse>\<inverse>"
minus_eq: fixes x :: "real"   and y :: "real" shows "x + - y = x - y"
deg_mult_pols: fixes p :: "'a"   and q :: "'a" assumes "Idomain S"   and "p \<in> carrier R"   and "p \<noteq> \<zero>"   and "q \<in> carrier R"   and "q \<noteq> \<zero>" shows "p \<cdot>\<^sub>r q \<noteq> \<zero> \<and> deg_n R S X (p \<cdot>\<^sub>r q) = deg_n R S X p + deg_n R S X q"
SN_trancl_SN_conv: fixes R :: "('a \<times> 'a) set" shows "SN (R\<^sup>+) = SN R"
absorb2: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "a \<squnion> b = b"
foldr_apply_guards: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "(gval (foldr gAnd G (Bc True)) s = true) = apply_guards G s"
q_def: shows "q = embed_pmf (\<lambda>x. pmf p x + \<epsilon> * (pmf q' x - pmf p' x))"
abs_finite_2_def: shows "abs = (\<lambda>x. x)"
refl: fixes a :: "'d multiset" shows "a \<subseteq># a"
res_assoc_iff2: shows "(\<forall>x y z. x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z) = (\<forall>x y z. x \<lhd> y \<cdot> z = x \<lhd> z \<lhd> y)"
affine_parallel_reflex: fixes S :: "'c set" shows "affine_parallel S S"
map_def: fixes f :: "'a" shows "\<rr>.map f = (if S.arr f then \<r>[S.cod f] \<cdot>\<^sub>S R f else S.null)"
Sup_upper2: fixes u :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set"   and v :: "'c \<times> 'd" assumes "s2p A u"   and "less_eq_prod v u" shows "less_eq_prod v (Sup_prod A)"
degen_def: fixes cs :: "('b \<Rightarrow> bool) list" shows "degen cs \<equiv> \<lambda>(a, k). length cs \<le> k \<or> (cs ! k) a"
inner_gt_zero_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(0 < inner x x) = (x \<noteq> zero)"
T_A_A'_leq': fixes n :: "nat" assumes "n \<le> length qs" shows "T_A n \<le> T_A' n"
choose_mult_lemma: fixes m :: "nat"   and r :: "nat"   and k :: "nat" shows "(m + r + k choose (m + k)) * (m + k choose k) = (m + r + k choose k) * (m + r choose m)"
alpha_Tree_tAct: fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree"   and f :: "'d"   and \<alpha> :: "'c" assumes "t1 =\<^sub>\<alpha> t2" shows "tAct f \<alpha> t1 =\<^sub>\<alpha> tAct f \<alpha> t2"
WTrtTry: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and V :: "char list"   and C :: "char list"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E(V \<mapsto> Class C),h \<turnstile> e\<^sub>2 : T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2" shows "P,E,h \<turnstile> try e\<^sub>1 catch(C V) e\<^sub>2 : T\<^sub>2"
pt_prm: shows "pt TYPE(('x \<times> 'x) list) TYPE('x)"
inorder_paint: fixes c :: "tcolor"   and t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "Tree2.inorder (paint c t) = Tree2.inorder t"
equivalent_eq_exists_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and s :: "'b trm"   and C :: "'b literal set"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and \<sigma> :: "('b \<times> 'b trm) list"   and L1 :: "'b literal" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.equivalent_eq_exists trm_ord t s C I \<sigma> L1 = (\<exists>L\<in>C - {L1}. \<exists>u v. basic_superposition.orient_lit_inst trm_ord L u v pos \<sigma> \<and> t \<lhd> \<sigma> = u \<lhd> \<sigma> \<and> I (s \<lhd> \<sigma>) (v \<lhd> \<sigma>))"
f: shows "steps0 (1, l, r) (tp @ shift (mopup n) off) stp = (s'', l'', r'') \<and> is_final (s'', l'', r'')"
simps_9: fixes M :: "'d"   and P :: "('d, 'e, 'f) psi" shows "guarded' (\<rparr>M.P) = False"
simps_1: fixes i :: "nat" shows "gso_int i 0 = fs ! i"
ind: shows "indecomposable (\<omega> \<up> (1 + \<nu> * ord_of_nat n))"
ordLeq_ordIso_trans: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set"   and r'' :: "('h \<times> 'h) set" assumes "r \<le>o r'"   and "r' =o r''" shows "r \<le>o r''"
assms_3: fixes theta :: "('a \<times> 'a) set" shows "Retr1 theta \<subseteq> Retr2 theta"
ground_gt_unary: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "ground t" shows "\<not> gt_unary t s"
continuous_map_iff_continuous2: fixes g :: "'d \<Rightarrow> 'e" shows "continuous_map euclidean euclidean g = continuous_on UNIV g"
pr_conv_2_to_3_def: shows "pr_conv_2_to_3 = (\<lambda>f. pr_conv_1_to_3 (pr_conv_2_to_1 f))"
B_1: fixes u :: "real" assumes "u \<in> A - {b}" shows "u + d \<le> next_in A u"
fps_mult_fps_X_power_nonzero_2: fixes f :: "'a fps"   and n :: "nat" assumes "f \<noteq> 0" shows "f * fps_X ^ n \<noteq> 0"
I: fixes c :: "real" assumes "k \<le> c" shows "(f has_integral I c) {a..c}"
assms_4: shows "\<not> (x \<notin> ins P \<and> x \<notin> ins Q)"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> es : t1s _> t2s"
bind_advantage: fixes \<A> :: "'a \<times> 'a \<Rightarrow> (('a \<times> 'a) \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "chaum_ped_\<Sigma>_commit.abstract_com.bind_advantage \<A> \<le> chaum_ped_\<Sigma>_commit.rel_advantage (chaum_ped_\<Sigma>_commit.adversary \<A>)"
ListUpdate_updates: fixes x :: "'a"   and xs :: "('a \<times> 'b \<times> 'c) list"   and l'' :: "'a \<Rightarrow> 'd"   and l :: "'a \<Rightarrow> 'd" assumes "x \<in> lesvars xs" shows "ListUpdate l'' xs (\<lambda>e. l) x = l x"
incl_L: shows "fst ` \<Union> ((edges \<circ> snd) ` Rules) \<subseteq> L"
CANDs_preserved_1: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and M' :: "trm" assumes "<a>:M \<in> \<parallel><B>\<parallel>"   and "M \<longrightarrow>\<^sub>a* M'" shows "<a>:M' \<in> \<parallel><B>\<parallel>"
continuous_on_cos: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. cos (f x))"
enn2ereal_Ioi: fixes a :: "ereal" shows "enn2ereal -` {a<..} = (if 0 \<le> a then {e2ennreal a<..} else UNIV)"
gext_def: fixes s :: "st"   and s' :: "st" shows "s\<le>|s' \<equiv> \<forall>r. ! obj:globs s r: ? obj':globs s' r: tag obj' = tag obj"
r_sign_def: shows "r_sign \<equiv> r_shrink (Pr 1 Z (r_constn 2 1))"
lunit_char_2: fixes a :: "'a" assumes "ide a" shows "\<I>\<^sub>1 \<otimes>\<^sub>1 \<l>\<^sub>1[a] = (\<iota>\<^sub>1 \<otimes>\<^sub>1 a) \<cdot> \<a>\<^sub>1\<^sup>-\<^sup>1[\<I>\<^sub>1, \<I>\<^sub>1, a]"
shadow_root_ptr_casts_commute: fixes ptr :: "(_) object_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" shows "(cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some shadow_root_ptr) = (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = ptr)"
matrix_works: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and x :: "('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "matrix f *v x = f x"
decompose_equation_def: fixes e :: "'b equation"   and t :: "'b trm"   and s :: "'b trm" shows "decompose_equation e t s = (e = Eq t s \<or> e = Eq s t)"
hfset_HF: fixes A :: "hf set" assumes "finite A" shows "hfset (HF A) = A"
Q'_def: shows "Q' = (\<Prod>X\<in>Roots'. [:- root' X, 1:])"
igSubstIPresIGWlsSTR_imp_igSubstIPresIGWls: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" assumes "igSubstIPresIGWlsSTR MOD" shows "igSubstIPresIGWls MOD"
hoare_r_weaken_pre_2: fixes q :: "'a upred"   and Q :: "'a hrel"   and r :: "'a upred"   and p :: "'a upred" assumes "\<lbrace>q\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u" shows "\<lbrace>p \<and> q\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u"
aezdeltafun_decomp': fixes b :: "'a"   and a :: "'b" shows "b \<delta>\<delta> a = (1::'a) \<delta>\<delta> a * (b \<delta>\<delta> (0::'b))"
simps_2: fixes \<A> :: "'b \<Rightarrow> bool"   and k :: "'b" shows "lit_semantics \<A> (k\<inverse>) = (\<not> \<A> k)"
not_le_imp_less: fixes y :: "'a Word.word"   and x :: "'a Word.word" assumes "\<not> y \<le>s x" shows "x <s y"
subtop: fixes s :: "'a set" shows "(s open) = (\<exists>t. t open\<^bsub>T\<^esub> \<and> s = t \<inter> carrier)"
homogeneous_set_def: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "homogeneous_set A = (\<forall>a\<in>A. \<forall>n. hom_component a n \<in> A)"
card_Suc_eq: fixes A :: "'c set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
normalize_zero: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(vec_normalize v = 0\<^sub>v n) = (v = 0\<^sub>v n)"
binomial_def: fixes n :: "nat"   and k :: "nat" shows "n choose k = card {K \<in> Pow {0..<n}. card K = k}"
conjugate_exponent_ennrealI_1: fixes p :: "ennreal"   and q :: "ennreal" assumes "1 / p + 1 / q = 1" shows "1 \<le> p"
fds_nth_0: fixes f :: "'b fds" shows "f $ 0 = (0::'b)"
pcompose_pCons: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "pCons a p \<circ>\<^sub>p q = [:a:] + q * p \<circ>\<^sub>p q"
a_get_ancestors_locs_def: shows "Shadow_DOM.get_ancestors_locs = Shadow_DOM.get_parent_locs"
\<phi>: shows "local.iso \<phi> \<and> \<guillemotleft>\<phi> : b \<star> b \<Rightarrow> b\<guillemotright>"
return_edge_determines_call_and_sum_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f" obtains a' :: "'a" and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool" and r' :: "'e" and fs' :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" where "a \<in> get_return_edges a'" and "valid_edge a'" and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p\<^esub>fs'" and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses (CFG_node (sourcenode a')) None (Some (p, True)) True (CFG_node (targetnode a))"
nth_restrict: fixes i :: "nat"   and A :: "nat set"   and v :: "'c option list" assumes "i \<in> A"   and "i < length v" shows "restrict A v ! i = v ! i"
f'_def: shows "f' = (\<lambda>x. poly q' x / poly p' x)"
oracle_encrypt3_def: shows "oracle_encrypt3 = (\<lambda>(k_prf, k_upf) b (bad, D) (msg1, msg0). case length msg1 = prf_clen \<and> length msg0 = prf_clen of True \<Rightarrow> spmf_of_set prf_domain \<bind> (\<lambda>x. spmf_of_set (nlists UNIV prf_clen) \<bind> (\<lambda>P. let (p, F) = case D x of None \<Rightarrow> (P, False) | Some r \<Rightarrow> (P, True); c = p [\<oplus>] (if b then msg1 else msg0); t = upf_fun k_upf (x @ c) in return_spmf (Some (x, c, t), bad \<or> F, D(x \<mapsto> p)))) | False \<Rightarrow> return_spmf (None, bad, D))"
prems_3: shows "P,shp s1 \<turnstile>\<^sub>b (es1,b1) \<surd>"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "G.awalk u (e # es) w = (e \<in> parcs G \<and> u = fst e \<and> G.awalk (snd e) es w)"
DERIV_minus: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. - f x) has_field_derivative - D) (at x within s)"
hchop_weaken2: fixes \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. \<phi> ts v \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> True \<and> \<phi> ts u)"
smultfunset_def: fixes G :: "'g set"   and FH :: "('f, 'g) aezfun set" shows "BaseFGMod.smultfunset G FH \<equiv> {f. \<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = a \<delta>\<delta> g \<cdot> f x}"
lm077: fixes f :: "'c \<times> 'b \<Rightarrow> 'a"   and x :: "'c"   and Y :: "'b set" shows "(\<Sum>y\<in>Y. f (x, y)) = sum f ({x} \<times> Y)"
assms: shows "\<^bold>|us\<^bold>| = 1"
m_nth': fixes j :: "nat"   and ia :: "nat" assumes "j < DIM('a)"   and "ia < max_Var_floatarith (fas ! j)" shows "ia < DIM('a) + length ramsch"
mgu_unifier: fixes As :: "'a list"   and n :: "nat"   and AAs :: "'a multiset list"   and \<sigma> :: "'s"   and i :: "nat"   and A :: "'a" assumes "length As = n"   and "length AAs = n"   and "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))"   and "i < n"   and "A \<in># AAs ! i" shows "A \<cdot>a \<sigma> = As ! i \<cdot>a \<sigma>"
option2set_def: fixes n :: "'b option" shows "option2set n \<equiv> case n of None \<Rightarrow> {} | Some s \<Rightarrow> {s}"
is_singletonI': fixes A :: "'c set" assumes "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "is_singleton A"
ereal_minus_le: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>" shows "(x - y \<le> z) = (x \<le> z + y)"
quot_of_fract_0: shows "quot_of_fract 0 = (0::'b, 1::'b)"
cancel_hyp: shows "matr_option (dim_row (M_mat signs subsets)) (mat_inverse (M_mat signs subsets)) * M_mat signs subsets *\<^sub>v w_vec p qs signs = w_vec p qs signs"
div_sum: fixes a :: "nat"   and A :: "nat \<Rightarrow> nat"   and D :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "\<forall>x. a dvd A x" shows "a dvd (\<Sum>x<n. A x * D x)"
min_height_size_if_incomplete: fixes kdt :: "'b kdt" assumes "\<not> KD_Tree.complete kdt" shows "2 ^ min_height kdt < KD_Tree.size_kdt kdt"
code: fixes x :: "complex" shows "- x = Complex (- Re x) (- Im x)"
ide_hcomp: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.ide \<nu>"   and "S.ide \<mu>"   and "S.src \<nu> = S.trg \<mu>" shows "S.ide (\<nu> \<star>\<^sub>S \<mu>)"
inf_last_ti_nonempty: fixes s :: "nat \<Rightarrow> 'd list"   and t :: "nat"   and k :: "nat" assumes "s t \<noteq> []" shows "inf_last_ti s (t + k) \<noteq> []"
add_nonneg_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (plus a b)"
suc\<rho>': shows "\<rho>' (Suc k') = \<pi>' (Suc k')"
nodes_target_elim: fixes q :: "'state" assumes "q \<in> nodes" obtains r :: "'action list" and p :: "'state" where "init p" and "path r p" and "q = target r p"
mod_mult_right_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b mod c) mod c = a * b mod c"