hcju/isabelleps huggingface dataset:
is_auto_reducedD: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_auto_reduced B"   and "b \<in> B" shows "\<not> is_red (B - {b}) b"
e'_def: shows "e' \<equiv> \<lambda>i. if i < d then e i else if i < d + length X then e'' i else 0"
mk_and_def: fixes x :: "'b ltln"   and y :: "'b ltln" shows "mk_and x y \<equiv> case x of true\<^sub>n \<Rightarrow> y | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> case y of true\<^sub>n \<Rightarrow> x | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> x and\<^sub>n y"
fut_derivs_def: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and LESS :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> bool"   and assigns :: "nat \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'd"   and nvars :: "'c \<Rightarrow> 'b"   and Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and CONS :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and SNOC :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and Length :: "'c \<Rightarrow> nat"   and extend :: "'a \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> 'e"   and size :: "'e \<Rightarrow> 'b"   and zero :: "'b \<Rightarrow> 'e"   and alphabet :: "'b \<Rightarrow> 'e list"   and eval :: "'d \<Rightarrow> nat \<Rightarrow> bool"   and downshift :: "'d \<Rightarrow> 'd"   and upshift :: "'d \<Rightarrow> 'd"   and add :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and cut :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and len :: "'d \<Rightarrow> nat"   and restrict :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and Restrict :: "'a \<Rightarrow> nat \<Rightarrow> ('f, 'a) aformula"   and lformula0 :: "'f \<Rightarrow> bool"   and FV0 :: "'a \<Rightarrow> 'f \<Rightarrow> nat set"   and find0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> bool"   and wf0 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and decr0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> 'f"   and satisfies0 :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and nullable0 :: "'f \<Rightarrow> bool"   and lderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and rderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and idx :: "'b"   and k :: "nat"   and \<phi> :: "('f, 'a) aformula" assumes "Formula SUC LESS assigns nvars Extend CONS SNOC Length extend size zero alphabet eval downshift upshift add cut len restrict Restrict lformula0 FV0 find0 wf0 decr0 satisfies0 nullable0 lderiv0 rderiv0" shows "Formula.fut_derivs extend zero find0 decr0 rderiv0 idx k \<phi> \<equiv> ((Formula_Operations.norm find0 decr0 \<circ> Formula_Operations.deriv extend rderiv0 (zero idx)) ^^ k) \<phi>"
differentiable_on_ident: fixes S :: "'a set" shows "(\<lambda>x. x) differentiable_on S"
comp_inj_on_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and f' :: "'c \<Rightarrow> 'd" assumes "inj_on f A" shows "inj_on f' (f ` A) = inj_on (f' \<circ> f) A"
opp_chambers_subset_opp_half_apartment: shows "order.greater_eq folding_g.Y (folding_g.\<C> - g \<turnstile> folding_g.\<C>)"
simps: fixes a1 :: "nat"   and a2 :: "intv" shows "valid_intv a1 a2 = ((\<exists>d c. a1 = c \<and> a2 = Const d \<and> 0 \<le> d \<and> d \<le> c) \<or> (\<exists>d c. a1 = c \<and> a2 = Intv d \<and> 0 \<le> d \<and> d < c) \<or> (\<exists>c. a1 = c \<and> a2 = Greater c))"
distinct_map_filter: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "distinct (map f xs)" shows "distinct (map f (filter P xs))"
assms: assumes "P x" shows "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"
ltakeWhile_repeat: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "ltakeWhile P (repeat x) = (if P x then repeat x else LNil)"
reduced_antimono: fixes X :: "'d set"   and Y :: "'d set"   and t :: "'d ifex" assumes "X \<subseteq> Y"   and "reduced t Y" shows "reduced t X"
eq_lesspoll_trans: fixes X :: "'a set"   and Y :: "'b set"   and Z :: "'c set" assumes "X \<approx> Y"   and "Y \<prec> Z" shows "X \<prec> Z"
poly_div_mult_right: fixes x :: "'b poly"   and y :: "'b poly"   and z :: "'b poly" shows "x div (y * z) = x div y div z"
Derives1_Derives1_swap: fixes i :: "nat"   and j :: "nat"   and a :: "symbol list"   and p :: "symbol \<times> symbol list"   and b :: "symbol list"   and q :: "symbol \<times> symbol list"   and c :: "symbol list" assumes "i < j"   and "Derives1 a j p b"   and "Derives1 b i q c" shows "\<exists>b'. Derives1 a i q b' \<and> Derives1 b' (j - 1 + length (snd q)) p c"
comp_ide_self: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "a \<cdot> a = a"
cNode: fixes r :: "'rule"   and s :: "'sequent"   and cts :: "('sequent, 'rule, 'link) ctree fset" assumes "r \<in> R"   and "eff r s (seqOf |`| cts)"   and "\<And>ct'. ct' |\<in>| cts \<Longrightarrow> cwf ct'" shows "cwf (cNode (s, r) cts)"
conjunct2: fixes P :: "bool"   and Q :: "bool" assumes "P \<and> Q" shows "Q"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> parts H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<langle>X, Y\<rangle> \<in> parts H; P \<langle>X, Y\<rangle>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<langle>X, Y\<rangle> \<in> parts H; P \<langle>X, Y\<rangle>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>X Y. \<lbrakk>Enc X Y \<in> parts H; P (Enc X Y)\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>Aenc X Y \<in> parts H; P (Aenc X Y)\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>Sign X Y \<in> parts H; P (Sign X Y)\<rbrakk> \<Longrightarrow> P X" shows "P x"
m2_def: shows "m2 = Min {k. k < length xs \<and> k < j \<and> xs ! k = i}"
interchange_columns_code: fixes A :: "(('c, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and n :: "'b"   and m :: "'b"   and i :: "'d" shows "($h) (interchange_columns_row A n m i) = (\<lambda>j. if j = n then A $h i $h m else if j = m then A $h i $h n else A $h i $h j)"
coprime_0_1: fixes p :: "'a" shows "coprime p (0::'a) = (p dvd (1::'a))"
star_inductr_var_equiv: fixes y :: "'a"   and x :: "'a" shows "\<lparr>y\<rparr>x\<lparr>y\<rparr> = \<lparr>y\<rparr>x\<^sup>\<star>\<lparr>y\<rparr>"
minBasis_def: fixes B :: "'a set" shows "minBasis B \<equiv> basis B \<and> (\<forall>B'. basis B' \<longrightarrow> sum weight B \<le> sum weight B')"
nonempty_agents: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R" shows "agents \<noteq> {}"
sum_sum_distl: fixes X :: "'b set"   and Y :: "'b set" assumes "finite X"   and "finite Y" shows "\<Sum> ((\<lambda>x. x \<cdot> \<Sum> Y) ` X) = \<Sum> {x \<cdot> y |x y. x \<in> X \<and> y \<in> Y}"
lookup_Mapping: fixes xs :: "('b \<times> 'c) list" shows "Mapping.lookup (AList_Mapping.Mapping xs) = map_of xs"
Inl_inject: fixes x :: "'b"   and y :: "'b" assumes "Inl x = Inl y" shows "x = y"
bisim_def: shows "Strong_Early_Bisim.bisim \<equiv> {(xa, x). Strong_Early_Bisim.bisimp xa x}"
complement_character: fixes x :: "i" assumes "\<exists>w. \<not> O w x" shows "\<forall>w. P w (\<midarrow> x) = (\<not> O w x)"
intros: fixes G :: "('a, 'b) graph_scheme"   and v :: "'a"   and p :: "'a list" assumes "path G v p v"   and "p \<noteq> []"   and "distinct p" shows "cycle G p"
point_abscissa_minus: fixes A :: "point" shows "abscissa (- A) = - abscissa A"
r_sub_def: shows "r_sub \<equiv> r_swap (Pr 1 (recf.Id 1 0) (Cn 3 r_dec [recf.Id 3 1]))"
drop_bit_and: 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 n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "drop_bit n (and a b) = and (drop_bit n a) (drop_bit n b)"
has_integral_substitution_general: fixes s :: "real set"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and c :: "real"   and d :: "real"   and f :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> real" assumes "finite s"   and "a \<le> b"   and "g ` {a..b} \<subseteq> {c..d}"   and "continuous_on {c..d} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})" shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral integral {g a..g b} f - integral {g b..g a} f) {a..b}"
assms_1: shows "\<Psi> \<rhd> Q \<longmapsto> Rs"
integral_eq_zero_AE: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "AE x in M. f x = (0::'d)" shows "integral\<^sup>L M f = (0::'d)"
extremum_unique: fixes a :: "'a list" shows "prefix a [] = (a = [])"
sumUnionDisjoint2: fixes X :: "'c set set"   and f :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>X. finite x"   and "is_non_overlapping X" shows "sum f (\<Union> X) = sum (sum f) X"
pr_gr_at_Suc: fixes x :: "nat" shows "pr_gr (Suc x) = g_step (pr_gr x) (c_fst x)"
distinct_primes_upto: fixes n :: "nat" shows "distinct (primes_upto n)"
mono_Span_subset: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us \<subseteq> Span K Vs"   and "set Vs \<subseteq> carrier R" shows "Span K Us \<subseteq> Span K Vs"
mgu_wf_trm: fixes s :: "('fun, 'a) Term.term"   and t :: "('fun, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and v :: "'a" assumes "mgu s t = Some \<sigma>"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<sigma> v)"
of_discrete_inverse: fixes x :: "'b discrete" shows "discrete (of_discrete x) = x"
zs_trans: fixes ys :: "'a list"   and xs :: "'a list" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "lexext gt zs ys"   and "lexext gt ys xs" shows "lexext gt zs xs"
extendf_deriv_bounded: fixes f' :: "('sf + 'sz, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and i :: "'sf"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and x :: "(real, 'sz) vec" assumes "dfree f'"   and "is_interp I" shows "bounded_linear (extendf_deriv I i f' \<nu> x)"
x_axis_integral_measurable: shows "(\<lambda>x. integral UNIV (\<lambda>y. if (x, y) \<in> Dy_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> i) j (x, y) else 0)) \<in> borel_measurable lborel"
coeffM_def: fixes A :: "(('c poly, 'd) vec, 'd) vec"   and n :: "nat" shows "coeffM A n = (\<chi>i j. coeff (A $ i $ j) n)"
psubst_new_free: fixes c :: "'a"   and n :: "'a"   and p :: "('a, 'b) form" assumes "c \<noteq> n" shows "new n (psubst (id(n := c)) p)"
post_rule: fixes P :: "assn"   and f :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and R :: "'a \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<forall>x. Q x \<Longrightarrow>\<^sub>A R x" shows "<P> f <R>"
empty_is_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "({} = f ` A) = (A = {})"
ide_src: fixes \<mu> :: "'c" assumes "C'.arr \<mu>" shows "C'.ide (C'.src \<mu>)"
fresh_10: fixes a :: "tyvrs"   and x2 :: "trm"   and x1 :: "ty" shows "a \<sharp> x2 \<cdot>\<^sub>\<tau> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
swap_swap: fixes p :: "'a \<times> 'b" shows "prod.swap (prod.swap p) = p"
dsafe_Fun_simps: fixes i :: "'a"   and args :: "'b \<Rightarrow> ('a, 'b) trm" shows "dsafe ($f i args) = (\<forall>x. dsafe (args x))"
cases: fixes a1 :: "'var"   and a2 :: "'edge list"   and a3 :: "'edge list"   and n' :: "'node" assumes "(a1, a2, a3) \<in> dependent_live_vars n'" obtains V :: "'var" where "a1 = Va__" and "a2 = []" and "a3 = []" and "Va__ \<in> Use n'"   | V :: "'var" and a :: "'edge" and as' :: "'edge list" and n'' :: "'node" and as'' :: "'edge list" where "a1 = Vb__" and "a2 = []" and "a3 = a__ # as'__ @ as''" and "Vb__ \<in> Use (sourcenode a__)" and "sourcenode a__ -a__ # as'__\<rightarrow>\<^sub>c\<^sub>d n''" and "n'' -as''\<rightarrow>\<^sub>d* n'"   | V :: "'var" and as' :: "'edge list" and as :: "'edge list" and V' :: "'var" and a :: "'edge" where "a1 = V'" and "a2 = []" and "a3 = aa__ # asa__" and "(Vc__, as'a__, asa__) \<in> dependent_live_vars n'" and "V' \<in> Use (sourcenode aa__)" and "n' = last (targetnodes (aa__ # asa__))" and "sourcenode aa__ -{Vc__}aa__ # as'a__\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (aa__ # as'a__))"   | V :: "'var" and as' :: "'edge list" and as :: "'edge list" and a :: "'edge" where "a1 = V" and "a2 = a # as'" and "a3 = a # as" and "(V, as', as) \<in> dependent_live_vars n'" and "n' = last (targetnodes (a # as))" and "\<not> sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (a # as'))"
simps_1: fixes x :: "'b list"   and ya :: "'b list" shows "(IArray x = IArray ya) = (x = ya)"
simps_1: fixes uu :: "'b"   and B :: "'b"   and uv :: "'c" shows "Cod (MkArr uu B uv) = B"
assms_1: shows "\<Union> (subfrmlsr ` A) \<subseteq> B"
eval_binop_type: fixes dt :: "loc \<Rightarrow> ty option"   and binop :: "binop"   and v1 :: "val"   and v2 :: "val" shows "typeof dt (eval_binop binop v1 v2) = Some (PrimT (binop_type binop))"
linear_surjective_isomorphism: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f"   and "surj f"   and "vec.dim UNIV = vec.dim UNIV" shows "\<exists>f'. linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
word_sub_1_le: fixes x :: "'b word" assumes "x \<noteq> 0" shows "x - 1 \<le> x"
cauchy_product: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "bound \<zero>\<^bsub>S\<^esub> n f"   and "bound \<zero>\<^bsub>S\<^esub> m g"   and "f \<in> {..n} \<rightarrow> carrier S"   and "g \<in> {..m} \<rightarrow> carrier S" shows "(\<Oplus>\<^bsub>S\<^esub>k\<in>{..n + m}. \<Oplus>\<^bsub>S\<^esub>i\<in>{..k}. f i \<otimes>\<^bsub>S\<^esub> g (k - i)) = finsum S f {..n} \<otimes>\<^bsub>S\<^esub> finsum S g {..m}"
simps_2: fixes g :: "'d \<Rightarrow> 'd" shows "merge_guards (Basic g) = Basic g"
fmrelI: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "'e \<rightharpoonup>\<^sub>f 'f"   and n :: "'e \<rightharpoonup>\<^sub>f 'g" assumes "\<And>x. rel_option R (fmlookup m x) (fmlookup n x)" shows "fmrel R m n"
mod_mult_self4: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times b c) a) b = modulo a b"
wellformed_1: fixes G :: "('b, 'c) pre_digraph"   and e :: "'c" assumes "wf_digraph G"   and "e \<in> arcs G" shows "tail G e \<in> verts G"
F: shows "offs_none (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) ub xs index key mi ma i \<longrightarrow> fill xs (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) index key ub mi ma ! i = None"
composite_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'c" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
rtr_nomac_e3: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and pi :: "'b Simple_Packet.simple_packet_ext"   and po :: "'b Simple_Packet.simple_packet_ext"   and ifl :: "interface list"   and i :: "interface"   and mlf :: "'b word \<Rightarrow> 48 word option"   and i2 :: "48 word" assumes "simple_linux_router_nol12 rt fw pi = Some po"   and "iface_packet_check ifl pi = Some i"   and "mlf (fromMaybe (p_dst pi) (next_hop (routing_table_semantics rt (p_dst pi)))) = Some i2" shows "\<exists>po'. simple_linux_router rt fw mlf ifl pi = Some po'"
IH: assumes "\<not> y ^ p \<le> n"   and "n \<le> y' ^ p"   and "root_newton_int_main y' n = (x, b)" shows "n < (x + 1) ^ p"
wordinterval_to_set_ipcidr_tuple_to_wordinterval: fixes b :: "'b word"   and m :: "nat" shows "wordinterval_to_set (ipcidr_tuple_to_wordinterval (b, m)) = ipset_from_cidr b m"
set_child_nodes_get_child_nodes_is_l_set_child_nodes_get_child_nodes: shows "l_set_child_nodes_get_child_nodes DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs"
update_def: fixes a :: "'b array"   and i :: "nat"   and x :: "'b"   and h :: "heap" shows "Array.update a i x h = Array.set a ((Array.get h a)[i := x]) h"
NSTD_lemma2: fixes c :: "nat" shows "(rec_exec rec_NSTD [c] = Suc 0) = NSTD c"
fps_const_mult_right: fixes f :: "'a fps"   and c :: "'a" shows "f * fps_const c = Abs_fps (\<lambda>n. fps_nth f n * c)"
simps_9: fixes v :: "poly"   and vc :: "nat"   and vb :: "poly" shows "Reflected_Multivariate_Polynomial.degree (CN v (Suc vc) vb) = 0"
mod_mult_self1_is_0: fixes b :: "'a"   and a :: "'a" shows "b * a mod b = (0::'a)"
assms: shows "\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> en r"
Fault: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and f :: "'c" shows "\<Gamma>\<turnstile>c \<down> Fault f"
rel_chain_Cons: fixes x :: "'b"   and xs :: "'b list"   and r :: "('b \<times> 'b) set" shows "rel_chain (x # xs) r = (if xs = [] then True else (x, hd xs) \<in> r \<and> rel_chain xs r)"
Dup_branch: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and extra :: "(('b, 'c) fm list \<times> 'c) list" assumes "Dup p i branch xs" shows "Dup p i (extra @ branch) xs"
obtain_function_with_partition: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P \<le> card B" shows "\<exists>f\<in>A \<rightarrow>\<^sub>E B. (\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = P"
mr_DA: shows "applied_rule_rev C x p = \<lfloor>DenyAll\<rfloor>"
is_complete_def: fixes x :: "item" shows "is_complete x = (length (item_rhs x) \<le> item_dot x)"
sat_int_to_sat_q: fixes N :: "'f set" shows "empty_ord.saturated N = (\<forall>qi\<in>Q. calculus.saturated Inf_F (Red_I_\<G>_q qi) N)"
intro: fixes U\<^sub>1 :: "'a set"   and inf\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot\<^sub>1 :: "'a"   and U\<^sub>2 :: "'b set"   and inf\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot\<^sub>2 :: "'b" assumes "bounded_lattice_bot_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 bot\<^sub>1"   and "bounded_lattice_bot_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 bot\<^sub>2" shows "bounded_lattice_bot_pair_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 bot\<^sub>1 U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 bot\<^sub>2"
y'_def: shows "y' = snd (normalize_quot (x, y))"
pmult_one: fixes p :: "'a list" shows "[1::'a] *** p = p"
pfp_termination: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y"   and "acc {(x, y). x \<sqsubseteq> y}"   and "x0 \<sqsubseteq> f x0" shows "\<exists>x. pfp f x0 = Some x"
self_fin_delta_pf_trad_strat: fixes der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and q :: "real"   and v0 :: "real" assumes "der \<in> borel_measurable (G matur)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "trading_strategy (self_fin_delta_pf N der matur v0)"
real_is_int_sub: fixes a :: "real"   and b :: "real" assumes "real_is_int a"   and "real_is_int b" shows "real_is_int (a - b)"
ccw_antisym: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "distinct3 p q r"   and "in3 S p q r"   and "ccw p q r" shows "\<not> ccw p r q"
leadsTo_refl: fixes F :: "'b program"   and A :: "'b set" shows "F \<in> A \<longmapsto> A"
not_asym_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Y"   and "y \<in> Y" shows "(\<not> x \<sqsubset> y) = (y \<sqsubseteq> x)"
path_connectedin_PiE: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "path_connectedin (product_topology X I) (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. path_connectedin (X i) (S i)))"
complete_iff_height: fixes t :: "'c tree" shows "complete t = (min_height t = height t)"
sup_bot_left: fixes x :: "'a" shows "top \<sqinter> x = x"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>h. \<lbrakk>P (Suc h); P h\<rbrakk> \<Longrightarrow> P (Suc (Suc h))" shows "P a0"
lActive_least: fixes n :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "\<exists>i\<ge>n. enat i < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "\<exists>i\<ge>n. enat i < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub> \<and> \<not> (\<exists>k\<ge>n. k < i \<and> enat k < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t k\<^esub>)"
step: shows "(if is_final_dstate (N, P, Q, n) then let (uu_, uu_, Q, uu_) = (N, P, Q, n) in Some (map fst Q) else self_call (deterministic_RP_step (N, P, Q, n))) = Some R"
AA6: shows "Esigma (3\<^sup>2 * 7 * 13 * 107) = 157248"
cSUP_const: fixes A :: "'b set"   and c :: "'c multiset" assumes "A \<noteq> {}" shows "(SUP x\<in>A. c) = c"
fps_sub_nth: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "(f - g) $ n = f $ n - g $ n"
subs_sub_rel_of: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb" shows "sub_rel_of (red prb) (subs prb)"
ub_sum: fixes f :: "'d \<Rightarrow> 'c"   and i :: "'d" shows "f i \<le> sup_monoid.sum f {k. True}"
ipresWlsAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'd, 'j) model_scheme" shows "ipresWlsAbs hA MOD \<equiv> \<forall>us s A. wlsAbs (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (hA A)"
last_seg_impl_correct: fixes SBIP :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list" assumes "GS_invar SBIP"   and "GS.p_\<alpha> SBIP \<noteq> []" shows "GS.last_seg_impl SBIP \<le> SPEC (\<lambda>r. r = last (GS.p_\<alpha> SBIP))"
rev_rev_ident: fixes xs :: "'c list" shows "rev (rev xs) = xs"
under_def: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "under r a \<equiv> {b. (b, a) \<in> r}"
bisimCoinductAux: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>[(X \<union> bisim)] Q \<and> (Q, P) \<in> X" shows "P \<sim> Q"
filter_merge: fixes cmp :: "'b comparator"   and xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "Sorting_Algorithms.sorted cmp xs"   and "Sorting_Algorithms.sorted cmp ys" shows "filter P (Sorting_Algorithms.merge cmp xs ys) = Sorting_Algorithms.merge cmp (filter P xs) (filter P ys)"