hcju/isabelleps huggingface dataset:
SLC_join\<^sub>\<epsilon>: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r'' :: "'r"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r''))])"   and "s\<^sub>2 = \<epsilon>"   and "s\<^sub>1 r'' = None"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
is_projection_on_bounded_clinear: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "bounded_clinear \<pi>"
le_sup_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y \<le> z) = (x \<le> z \<and> y \<le> z)"
line_integral_exists_joinD1: fixes f :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "line_integral_exists f basis (g1 +++ g2)"   and "valid_path g1" shows "line_integral_exists f basis g1"
image_subsetI: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and B :: "'f set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
runit_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (monoidal_category.runit C T \<alpha> \<iota> a) (T (a, monoidal_category.unity C \<iota>)) a"
simps_1: shows "remdups_adj \<epsilon> = \<epsilon>"
r_3: fixes x :: "nat" shows "eval r [x] = f x"
prems_2: shows "y \<in> set (remdups_wrt f (a # xs))"
gr_zeroI: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
RedT_def: shows "RedT \<equiv> rtrancl3p redT"
count_wc_queries_def: fixes Rs :: "('a \<times> 'a) set set"   and sorter :: "'a sorter" shows "count_wc_queries Rs sorter = (if Rs = {} then 0 else MAX R\<in>Rs. count_queries R sorter)"
dither_halts: shows "{\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1>)} dither {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1>)}"
thetaIfTT_def: shows "thetaIfTT \<equiv> {(if tst then c1 else c2, if tst then d1 else d2) |tst c1 c2 d1 d2. compatTst tst \<and> c1 \<approx>T d1 \<and> c2 \<approx>T d2}"
extensional'_monotone: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "extensional' A \<subseteq> extensional' B"
inner_sum_left_Basis: fixes b :: "'a"   and f :: "'a \<Rightarrow> real" assumes "b \<in> Basis" shows "inner (\<Sum>i\<in>Basis. f i *\<^sub>R i) b = f b"
integrable_Beta': fixes a :: "real"   and b :: "real" assumes "0 < a"   and "0 < b" shows "(\<lambda>t. t powr (a - 1) * (1 - t) powr (b - 1)) integrable_on {0..1}"
IH: shows "(\<forall>x\<in>S. \<forall>\<^sub>\<infinity>a. P x a) = (\<forall>\<^sub>\<infinity>i. \<forall>x\<in>S. P x i)"
stopping_time_le_const: fixes T :: "'a \<Rightarrow> 't"   and s :: "'t"   and t :: "'t" assumes "stopping_time F T"   and "s \<le> t" shows "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> \<le> s)"
2: shows "v * - v\<^sup>T \<sqinter> - - g = bot"
prod_coprime_left: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>i. i \<in> A \<Longrightarrow> algebraic_semidom.coprime times one (f i) a" shows "algebraic_semidom.coprime times one (comm_monoid_mult.prod times one f A) a"
sphere_cball: fixes z :: "'b"   and r :: "real" shows "sphere z r \<subseteq> cball z r"
sq_mtx_diag_simps_1: fixes i :: "'a"   and j :: "'a"   and f :: "'a \<Rightarrow> real" assumes "i = j" shows "sq_mtx_diag f $$ i $ j = f i"
bind_cong: fixes x :: "'b option"   and y :: "'b option"   and f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "Option.bind x f = Option.bind y g"
llast_body_def: shows "llast_body \<equiv> \<mu> llast_body. \<Lambda> f v. Fixrec.run\<cdot>(match_lcons\<cdot>v\<cdot>(\<Lambda> x yys. Fixrec.succeed\<cdot>(case yys of lnil \<Rightarrow> x | y :@ ys \<Rightarrow> f\<cdot>yys)))"
abs_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs one = one"
stable_pair_on_Xd: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set"   and x :: "'a" assumes "stable_pair_on ds XD_XH"   and "x \<in> match XD_XH" shows "Xd x \<in> ds"
is_word_b1X: shows "is_word (b1 @ [X])"
sup_bool_def: fixes P :: "bool"   and Q :: "bool" shows "sup P Q = (P \<or> Q)"
vdiff_vsingleton_in: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {a} -\<^sub>\<circ> A = []\<^sub>\<circ>"
inv_imp: shows "\<forall>s. s \<in> wa_invar (det_wa_wa WA) \<and> s \<notin> wa_cond (det_wa_wa WA) \<longrightarrow> Q s"
eventually_bound_pairE: fixes t0 :: "'b"   and x0 :: "'c"   and f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "isCont f (t0, x0)" obtains B :: "real" where "1 \<le> B" and "\<forall>\<^sub>F e in at_right 0. \<forall>x\<in>cball t0 e \<times> cball x0 e. norm (f x) \<le> B"
fimageI: fixes x :: "'m"   and A :: "'m fset"   and f :: "'m \<Rightarrow> 'n" assumes "x |\<in>| A" shows "f x |\<in>| f |`| A"
lub: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and z :: "'c"   and y :: "'c" assumes "Semilat A r f"   and "x \<sqsubseteq>\<^bsub>r\<^esub> z"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z"
div_conv_mult_inverse: fixes a :: "'a"   and b :: "'a" shows "a / b = a \<cdot> inverse b"
of_int_0_less_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (ring_1.of_int one plus zero minus z) = (0 < z)"
hyps: assumes "t \<in> jkbpC" shows "simAbs (runJP k_mkAlgAuto t a) = simAbs (runJP mkAutoSim t a)"
inverse_distrib_swap: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "group f z inverse" shows "inverse (f a b) = f (inverse b) (inverse a)"
tauChainInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l P'"   and "F P"   and "\<And>P' P''. \<lbrakk>P \<Longrightarrow>\<^sub>\<tau>\<^sub>l P'; P' \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P''; F P'\<rbrakk> \<Longrightarrow> F P''" shows "F P'"
tensor_lookup_eqI: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A = dims B"   and "\<And>is. is \<lhd> dims A \<Longrightarrow> lookup A is = lookup B is" shows "A = B"
wgreater_lemma: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>L r1"   and "w2 \<equiv>\<^sub>U r2"   and "wgreater w1 w2" shows "r2 < r1"
nres_more_simps_3: fixes X :: "'b set"   and x :: "'b" shows "(RES X = RETURN x) = (X = {x})"
r_cons_else: fixes e :: "nat"   and es :: "nat" shows "eval r_cons_else [e, es] \<down>= Suc (prod_encode (e_length es, prod_encode (e, pdec2 (es - 1))))"
exp_ge_add_one_self_aux: fixes x :: "real" assumes "0 \<le> x" shows "1 + x \<le> exp x"
primes_omega_primorial: fixes x :: "real" shows "primes_omega (primorial x) = nat \<lfloor>primes_pi x\<rfloor>"
T_UU: shows "T \<bottom> = {s. front_tickFree s}"
linord_of_list_refl_on: fixes xs :: "'b list" shows "refl_on (set xs) (linord_of_list xs)"
has_integral_bound_spike_finite: fixes B :: "real"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b" assumes "0 \<le> B"   and "finite S"   and "(f has_integral i) (cbox a b)"   and "\<And>x. x \<in> cbox a b - S \<Longrightarrow> norm (f x) \<le> B" shows "norm i \<le> B * content (cbox a b)"
Var: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>" shows "\<Gamma> \<turnstile> Var x : T"
typeVal: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and V :: "val"   and lev :: "secLevel" shows "\<Gamma> \<turnstile> Val V : lev"
index_size_conv: fixes xs :: "'c list"   and x :: "'c" shows "(index xs x = length xs) = (x \<notin> set xs)"
code: fixes xs :: "('a, 'b) tllist" shows "llist_of_tllist xs = (case xs of TNil x \<Rightarrow> LNil | TCons x xs' \<Rightarrow> LCons x (llist_of_tllist xs'))"
qcarr: shows "homologous_rel_set 0 X {} (frag_of f) \<in> carrier (homology_group 0 X)"
ctxt_sub_cong: fixes e1 :: "db"   and e2 :: "db"   and C :: "db" assumes "closed e1"   and "closed e2"   and "evalDdb e1\<cdot>env_empty_db = evalDdb e2\<cdot>env_empty_db" shows "evalDdb (C<e1>)\<cdot>env_empty_db = evalDdb (C<e2>)\<cdot>env_empty_db"
assms_2: shows "LLL_invariant upw i (fs_state state)"
insert_inc1: fixes A :: "'c set"   and a :: "'c" shows "A \<subseteq> insert a A"
P2_def: shows "P2 \<equiv> \<lambda>tp. tp = ([Bk], <(code_tcontra, code_tcontra)>)"
ord_term_canc: fixes t :: "'a"   and v :: "'t"   and w :: "'t" assumes "t \<oplus> v \<preceq>\<^sub>t t \<oplus> w" shows "v \<preceq>\<^sub>t w"
G_scc_of_tail_not_in: shows "G.scc_of (tail G a) \<notin> H.sccs_verts"
sum_dcharacter_block: shows "sum \<chi>\<^sub>0 {..<n} = (if \<chi>\<^sub>0 = \<chi>\<^sub>0 then of_nat (totient n) else 0)"
poly_linear_rem: fixes h :: "'a"   and t :: "'a list"   and a :: "'a" shows "\<exists>q r. h # t = [r] +++ [- a, 1::'a] *** q"
xor_def: fixes x :: "'g"   and y :: "'g" shows "x \<oplus> y = (x \<squnion> y) \<sqinter> - (x \<sqinter> y)"
bool_mask': fixes x :: "'a word" assumes "2 < LENGTH('a)" shows "(0 < x && 1) = (x && 1 = 1)"
elims: fixes x :: "('c, 'd) chain"   and y :: "'c" assumes "pre x = y" obtains P :: "'c" where "x = \<lbrace> Pa__ \<rbrace>" and "y = Pa__"   | P :: "'c" and uu_ :: "'d" and uv_ :: "('c, 'd) chain" where "x = \<lbrace> P \<rbrace> \<cdot> uu_ \<cdot> uv_" and "y = P"
sign_bl_bin': fixes bs :: "bool list"   and w :: "int" shows "bin_sign (bl_to_bin_aux bs w) = bin_sign w"
assms_2: shows "n \<in> set (\<alpha>n g)"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_intrvl.proper_interval_Compl_set_aux less proper_interval ao (x # xs) (y # ys) = (if less x y then \<not> proper_interval ao (Some x) \<and> proper_intrvl.proper_interval_Compl_set_aux less proper_interval (Some x) xs (y # ys) else if less y x then \<not> proper_interval ao (Some y) \<and> proper_intrvl.proper_interval_Compl_set_aux less proper_interval (Some y) (x # xs) ys else \<not> proper_interval ao (Some x) \<and> (ys = [] \<longrightarrow> xs \<noteq> []))"
oalist_inv_rawD2: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs" shows "sorted_wrt lt (map fst xs)"
NCol_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C \<or> \<not> Col A C B \<or> \<not> Col B A C \<or> \<not> Col B C A \<or> \<not> Col C A B \<or> \<not> Col C B A" shows "\<not> Col A B C"
fMax_Diff_0: fixes m :: "nat"   and P :: "nat fset" assumes "Suc m |\<in>| P" shows "fMax (P |-| {|0|}) = fMax P"
qmsg_msgs_not_empty: shows "qmsg \<TTurnstile> onl \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G (\<lambda>(msgs, l). l = ()-:1 \<longrightarrow> msgs \<noteq> [])"
semiring_norm_172: fixes v :: "num"   and w :: "num"   and y :: "'b" shows "- numeral v * (- numeral w * y) = numeral (v * w) * y"
4: shows "c * x + (y * (A - A') + (y * A' - c') + (c' - c)) * x \<le> y * b'"
gs_gs: shows "image_mset pl.Mp (mset gs) = image_mset pl.Mp (mset gs1) + image_mset pl.Mp (mset gs2)"
bb_aa_p: shows "\<forall>\<^sub>F k in sequentially. \<forall>p. bb (k + p) / aa (k + p) \<le> 1 / M ^ p * (bb k / aa k)"
less_ceiling_iff: fixes z :: "int"   and x :: "'b" shows "(z < \<lceil>x\<rceil>) = (of_int z < x)"
smooth_on_divide: fixes k :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<noteq> (0::'b)" shows "k-smooth_on S (\<lambda>x. f x / g x)"
d_approx_main: fixes ii :: "nat" assumes "ii \<le> m"   and "m \<noteq> 0" shows "rat_of_int (d fs ii) \<le> rat_of_nat (N ^ ii)"
cases: fixes x :: "real \<times> real"   and a :: "(real \<times> real) list" assumes "ccw'.sortedP x a" obtains "a = []"   | ys :: "(real \<times> real) list" and xa :: "real \<times> real" where "a = xa # ys" and "\<forall>y\<in>set ys. ccw' x xa y" and "ccw'.sortedP x ys"
hyp2: shows "(\<Sum>s\<leftarrow>remdups (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)). z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s})) = (\<Sum>s\<in>set (remdups (map (consistent_sign_vec_copr qs) (characterize_root_list_p p))). z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))"
spmf_lub_spmf: fixes x :: "'a" assumes "Y \<noteq> {}" shows "spmf local.lub_spmf x = (\<Squnion>p\<in>Y. spmf p x)"
d: fixes r :: "rat" shows "local.initial_upper_bound r = rat_of_int (root_rat_ceiling n r)"
fst_convol: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "fst \<circ> BNF_Def.convol f g = f"
bal_all_subtrees_equal: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node ts t)" shows "\<forall>s1\<in>set (subtrees ts). \<forall>s2\<in>set (subtrees ts). height s1 = height s2"
keys_tail_less_lt: fixes v :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys (i.punit.tail p)" shows "lex_pm_strict v (i.lpp p)"
closed_atLeastAtMost: fixes a :: "'a"   and b :: "'a" shows "closed {a..b}"
disjointI: fixes a :: "'a set"   and b :: "'a set" assumes "\<And>x. \<lbrakk>x \<in> a; x \<in> b\<rbrakk> \<Longrightarrow> False" shows "a \<inter> b = {}"
lens_put_comm: fixes X :: "'c \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and v :: "'e"   and u :: "'c" assumes "X \<bowtie> Y" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"
sys_load_def: fixes p :: "'b process_name"   and ract :: "('a, 'd) mem_load_action" shows "sys_load p ract = do_load_action ract \<circ> processors_view_of_memory p"
vimage_vinsert: fixes r :: "V"   and a :: "V"   and A :: "V" shows "r `\<^sub>\<circ> vinsert a A = r `\<^sub>\<circ> set {a} \<union>\<^sub>\<circ> r `\<^sub>\<circ> A"
thm_noncont_e_e_3: fixes v :: "i" shows "[NonContingent L in v]"
adj_map': fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<sim> D"   and "f ` C \<noteq> f ` D" shows "f ` C \<sim> f ` D"
that: fixes p :: "'a set set" assumes "p division_of \<Union> f" shows "thesis"
fls_base_factor_of_nat: fixes n :: "nat" shows "fls_base_factor (of_nat n) = of_nat n"
poslivenessE: fixes A :: "'b set"   and P :: "'b llist set"   and s :: "'b llist" assumes "posliveness A P" obtains t :: "'b llist" where "t \<in> A\<^sup>\<infinity>" and "s @@ t \<in> P"   | "s \<notin> A\<^sup>\<clubsuit>"
rsd_winners_Cons: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and alts :: "'a set"   and i :: "'b"   and agents :: "'b list" shows "rsd_winners R alts (i # agents) = Max_wrt_among (R i) (rsd_winners R alts agents)"
greaterThanLessThan_borel: fixes a :: "'c"   and b :: "'c" shows "{a<..<b} \<in> sets borel"
add_1: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "int poly \<times> rat \<times> rat" assumes "invariant_1_2 x"   and "invariant_1_2 y" shows "invariant_2 (add_1 x y) \<and> real_of_2 (add_1 x y) = real_of_1 x + real_of_1 y"
distinct_set_mset_eq: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "distinct_mset M"   and "distinct_mset N"   and "set_mset M = set_mset N" shows "M = N"
reachable_0: fixes s :: "'d"   and A :: "('e set \<times> 'e set \<times> 'e set) \<times> 'd set \<times> ('d \<times> 'e \<times> 'd) set" assumes "s \<in> starts_of A" shows "reachable A s"
comp_identity_functor: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "functor A B F" shows "identity_functor.map B \<circ> F = F"
zero_simps_4: shows "IEEE.exponent (- 0) = 0"
inf_sup_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
relabel_increase_u: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Labeling c s t f l"   and "Network.relabel_precond c t f l u" shows "l u < Network.relabel_effect c f l u u"
image_ident: fixes Y :: "'a set" shows "(\<lambda>x. x) ` Y = Y"
SD_is_preorder: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Order_Predicates.preorder_on (lotteries_on {x. R x x}) (SD R)"
mult\<^sub>H\<^sub>O: fixes M :: "'b multiset"   and N :: "'b multiset" shows "((M, N) \<in> mult {(x, y). x < y}) = (M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x)))"
redundancy_criterion_standard_extension: fixes \<Gamma> :: "'b inference set"   and \<Gamma>' :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" assumes "\<Gamma> \<subseteq> \<Gamma>'"   and "redundancy_criterion \<Gamma> Rf Ri" shows "redundancy_criterion \<Gamma>' Rf (\<lambda>N. Ri N \<union> (\<Gamma>' - \<Gamma>))"
coprime_set_def: fixes prime_set :: "nat set"   and N :: "nat" assumes "rsa_base prime_set" shows "coprime_set N \<equiv> {x. coprime x N \<and> 1 < x \<and> x < N}"
weakenTransition_def: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and \<alpha> :: "'d action"   and P' :: "('d, 'e, 'f) psi" assumes "weak substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "weak.weakenTransition substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P \<alpha> P' \<equiv> (\<exists>P''' P''. (P, P''') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>* \<and> env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P''' (\<alpha> \<prec> P'') \<and> (P'', P') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>*) \<or> P = P' \<and> \<alpha> = \<tau>"
exists_ctran_Await_None: fixes x :: "('b com option \<times> 'b) list"   and b :: "'b set"   and c :: "'b com"   and s :: "'b"   and i :: "nat" assumes "x \<in> cptn"   and "x ! 0 = (Some (Await b c), s)"   and "i < length x"   and "fst (x ! i) = None" shows "\<exists>j<i. x ! j -c\<rightarrow> x ! Suc j"
naturality: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "\<rho>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (H \<mu> \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C \<mu>)) = (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D \<rho>\<^sub>1 (C.dom \<mu>)"
abs_eq: fixes y :: "('c \<times> nat) list" assumes "eq_onp (\<lambda>x. x \<in> Collect monom_inv) y y" shows "monom_list (Abs_monom y) = y"
related_match: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "term"   and u :: "'d"   and env :: "(name, 'd) fmap"   and t :: "'c" assumes "term_struct_rel P"   and "match x u = Some env"   and "P t u" obtains env' :: "(name, 'c) fmap" where "match x t = Some env'" and "fmrel P env' env"
bij_betw_sequence_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A" shows "bij_betw (sequence_of A enum) {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} {xs. set xs \<subseteq> B \<and> length xs = card A \<and> distinct xs}"
Rep_resumption_inverse: fixes x :: "('f, 'g, 'h, 'i) resumption" shows "Abs_resumption (Rep_resumption x) = x"
permutes_inj: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "inj p"
simps_1: fixes \<phi> :: "fml" shows "Testo (Afml \<phi>) = Agame (? \<phi>)"
approx_SComplex_mult_cancel_zero: fixes a :: "complex star"   and x :: "complex star" assumes "a \<in> SComplex"   and "a \<noteq> 0"   and "a * x \<approx> 0" shows "x \<approx> 0"
convert_extTA_simps_7: fixes f :: "'i \<Rightarrow> 'e"   and las :: "'c lock_actions"   and tas :: "('d, 'i, 'f) new_thread_action list"   and was :: "'d conditional_action list"   and cas :: "('d, 'g) wait_set_action list"   and "is" :: "'d interrupt_action list"   and obs :: "'h list" shows "convert_extTA f (las, tas, was, cas, is, obs) = (las, map (convert_new_thread_action f) tas, was, cas, is, obs)"
strict_mono_leD: fixes r :: "'a \<Rightarrow> 'b"   and m :: "'a"   and n :: "'a" assumes "strict_mono r"   and "m \<le> n" shows "r m \<le> r n"