hcju/isabelleps huggingface dataset:
abs_of_nonpos: fixes abs :: "'d \<Rightarrow> 'd"   and 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 a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a zero" shows "abs a = uminus a"
ilist_Previous_nth_if: fixes f :: "nat \<Rightarrow> 'd"   and init :: "'d"   and n :: "nat" shows "f\<^bsup>\<leftarrow> init\<^esup> n = (if n = 0 then init else f (n - Suc 0))"
cfun_map_ID: shows "cfun_map\<cdot>ID\<cdot>ID = ID"
transpose_of_unitary_is_unitary: fixes U :: "complex Matrix.mat" assumes "unitary U" shows "unitary U\<^sup>t"
that: fixes a :: "V"   and b :: "V" assumes "x = [a, b]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> A"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
MarkGuardsD_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P mark_guards f c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P c Q,A"
bal_list_merge: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and as :: "('c btree \<times> 'c) list"   and a :: "'c btree"   and x :: "'c"   and bs :: "('c btree \<times> 'c) list"   and b :: "'c btree" assumes "split split"   and "bal_up\<^sub>i (Up\<^sub>i (Node as a) x (Node bs b))" shows "bal (Node (as @ (a, x) # bs) b)"
resE_positive: fixes e :: "nat \<times> nat" assumes "e \<in> cf.E" shows "(0::'capacity) < cf e"
wellformed_term_def: fixes t :: "Term_Class.term" shows "pre_strong_term_class.wellformed t = Term.wellformed t"
simps: fixes M :: "hmultiset multiset" shows "Hereditary_Multiset.Rep_hmultiset (HMSet M) = MSet (image_mset Hereditary_Multiset.Rep_hmultiset M)"
c2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma> \<cdot> y = x \<cdot> 1\<^sub>\<pi> \<parallel> y"
Vagree_def: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and V :: "('d + 'd) set" shows "Vagree \<nu> \<nu>' V \<equiv> (\<forall>i. Inl i \<in> V \<longrightarrow> fst \<nu> $ i = fst \<nu>' $ i) \<and> (\<forall>i. Inr i \<in> V \<longrightarrow> snd \<nu> $ i = snd \<nu>' $ i)"
ss_4: shows "rs \<bullet>\<cdot> ms = ss \<bullet>\<cdot> ms"
in_range_if_ex_key: fixes v :: "'a"   and m :: "'b \<Rightarrow> 'a option" shows "(v \<in> ran m) = (\<exists>k. m k = Some v)"
span_with: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "module scale" shows "module.span scale = span_with (+) (0::'d) scale"
homotopy_eqv_cohomotopic_triviality_null_imp: fixes S :: "'b set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and U :: "'d set" assumes "S homotopy_eqv T"   and "continuous_on T f"   and "f ` T \<subseteq> U"   and "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk> \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) S U f (\<lambda>x. c)" obtains c :: "'d" where "homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. c)"
rep_eq: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and x :: "'l word" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus x = ring_1.of_int one plus zero minus (signed_take_bit (LENGTH('l) - Suc 0) (Word.rep x))"
RangePI: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "r a b" shows "Rangep r b"
"3_2": shows "real (count_derangements n) = fact n * (\<Sum>k = 0..n. (- 1) ^ k / fact k)"
freshFreeDerivative_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> \<alpha> \<prec> P'"   and "y \<sharp> P" shows "y \<sharp> P'"
BinOpThrow1: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and bop :: "bop"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Throw r \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
assms_2: shows "(\<Sum>i\<le>n. p i * i) = n"
group_def: fixes G :: "('f, 'g) monoid_scheme" shows "Group.group G \<equiv> Group.monoid G \<and> Group.group_axioms G"
ZObis_finished_discr_L: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(c, d) \<in> PL_Indis.ZObis aval tval cval indis"   and "proper d"   and "finished d" shows "PL_Indis.discr aval tval cval indis c"
step_foldL_def: fixes g :: "'e \<Rightarrow> (String.literal, nat) assoc_list \<times> 'f \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> bool \<Rightarrow> 'd list \<times> edgeIndex \<times> (String.literal, nat) assoc_list"   and stepss :: "'e list list"   and lbls :: "(String.literal, nat) assoc_list"   and pri :: "'f"   and pos :: "nat"   and nxt :: "edgeIndex"   and onxt :: "edgeIndex option" shows "step_foldL g stepss lbls pri pos nxt onxt = foldr (step_foldL_step g pri onxt) stepss (pos, nxt, lbls, [], [])"
Cons: fixes x :: "'b"   and y :: "'c"   and r :: "('b \<times> 'c) set"   and xs :: "'b list"   and ys :: "'c list" assumes "(x, y) \<in> r"   and "(xs, ys) \<in> listrel r" shows "(x # xs, y # ys) \<in> listrel r"
ennreal_sum_eq_top: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I = \<top>) = (\<exists>i\<in>I. f i = \<top>)"
eq: shows "fst (SG ! i) ! j = (p \<leftrightarrow> q)"
t_prod_iff: fixes w :: "'a \<times> ('b \<times> 'a) list"   and X :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "(w \<in> t_prod X Y) = (\<exists>u v. w = t_fusion u v \<and> u \<in> X \<and> v \<in> Y \<and> Dioid_Models.last u = first v)"
simps_5: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and C :: "char list" shows "(P,h \<turnstile> v :\<le> Class C) = (P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Class C\<rfloor> \<or> P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>NT\<rfloor>)"
iter_widen_step'_Com: fixes c :: "ivl Abs_State.st option acom"   and c' :: "ivl Abs_State.st option acom"   and X :: "char list set" assumes "iter_widen (step_ivl \<top>) c = Some c'"   and "vars (strip c) \<subseteq> X"   and "c \<in> Com X" shows "c' \<in> Com X"
unit_factor_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero" shows "dvd.dvd times (unit_factor a) b"
mult_poly_0_1: fixes q :: "'b poly" shows "0 * q = 0"
that: fixes xs :: "'a list" assumes "A = set xs"   and "distinct xs"   and "length xs = card A" shows "thesis"
Cons: fixes x :: "'b"   and xs :: "'b list" shows "sum_list (x # xs) = x + sum_list xs"
interval_measure_def: fixes F :: "real \<Rightarrow> real" shows "interval_measure F = extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a<..b}) (\<lambda>(a, b). ennreal (F b - F a))"
at_ds2: fixes a :: "'a"   and b :: "'a"   and pi :: "('a \<times> 'a) list" assumes "at TYPE('a)" shows " ([(a, b)] @ pi) \<triangleq> (pi @ [(rev pi \<bullet> a, rev pi \<bullet> b)]) "
rel_fun_trans: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'd"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'a"   and f' :: "'f \<Rightarrow> 'b" assumes "rel_fun P Q g g'"   and "rel_fun R P f f'" shows "rel_fun R Q (\<lambda>x. g (f x)) (\<lambda>y. g' (f' y))"
upd_None_map_le: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a" shows "f(x := None) \<subseteq>\<^sub>m f"
assms: shows "\<xi> = (- b + ccsqrt (b\<^sup>2 - 4 * c)) / 2 \<or> \<xi> = (- b - ccsqrt (b\<^sup>2 - 4 * c)) / 2"
mono_sinvar: fixes N :: "'v set"   and E :: "('v \<times> 'v) set"   and E' :: "('v \<times> 'v) set" assumes "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "c_sinvar m \<lparr>nodes = N, edges = E\<rparr>" shows "c_sinvar m \<lparr>nodes = N, edges = E'\<rparr>"
bisim_inv_\<tau>s1_inv: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and s1' :: "'x1 \<times> 'm1" assumes "bisim_inv"   and "t \<turnstile> s1 \<approx> s2"   and "r1.silent_moves t s1 s1'" obtains s2' :: "'x2 \<times> 'm2" where "t \<turnstile> s1' \<approx> s2'"
ord_resolve_1: shows "DA = D_twin + negs (mset As)"
cauchy_def: fixes S :: "nat \<Rightarrow> 'c" shows "Cauchy S = (\<forall>e>0. \<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (S m) (S n) < e)"
prime_sum_upto_eq_0: fixes x :: "real"   and f :: "nat \<Rightarrow> 'a" assumes "x < 2" shows "prime_sum_upto f x = (0::'a)"
coincides_on_stocks_viable: fixes Mkt :: "('a, 'c) discrete_market"   and Mkt2 :: "('a, 'c) discrete_market" assumes "coincides_on Mkt Mkt2 (stocks Mkt)"   and "viable_market Mkt" shows "viable_market Mkt2"
inf_prefix_commute: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<sqinter> ys = ys \<sqinter> xs"
strictly_precedes_alt_def2: fixes K\<^sub>2 :: "clock"   and K\<^sub>1 :: "clock" shows "{\<rho>. \<forall>n. #\<^sub>\<le> \<rho> K\<^sub>2 n \<le> #\<^sub>< \<rho> K\<^sub>1 n} = {\<rho>. \<not> hamlet (Rep_run \<rho> 0 K\<^sub>2) \<and> (\<forall>n. #\<^sub>\<le> \<rho> K\<^sub>2 Suc n \<le> #\<^sub>\<le> \<rho> K\<^sub>1 n)}"
GuaranteeStrip: fixes P :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f"   and g :: "'d set" assumes "P \<subseteq> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P guaranteeStrip f g c Q,A"
pt_swap_bij: fixes a :: "'b"   and b :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
of_nat_eq_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
one_smult: fixes m :: "'v" assumes "m \<in> V" shows "1 \<cdot> m = m"
s\<^sub>0: shows "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None), sh\<^sub>0)"
lookup_abs_nat_pp_pp: fixes t :: "(nat, nat) pp" shows "lookup_pp (abs_nat_pp t) = (\<lambda>n. abs_nat (lookup_pp t (rep_nat n)))"
space_density: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "space (density M f) = space M"
is_Zp_cauchy_def: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p" shows "padic_integers.is_Zp_cauchy p s = (s \<in> carrier padic_int p\<^bsup>\<omega>\<^esup> \<and> (\<forall>n. \<exists>N. \<forall>m k. N < m \<and> N < k \<longrightarrow> eint n < padic_integers.val_Zp_dist p (s m) (s k)))"
next_in_basics_3: fixes u :: "'a" assumes "u \<in> {a..<b}" shows "A \<inter> {u<..<next_in A u} = {}"
possible_steps_0: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "length i = 1" shows "possible_steps drinks 0 r STR ''select'' i = {|(1, select)|}"
isMinim_def: fixes A :: "'a set"   and b :: "'a" shows "isMinim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (b, a) \<in> r)"
assms_1: shows "isotone (\<lambda>x. \<mu> f x)"
coprime_minus_left_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "algebraic_semidom.coprime times one (uminus a) b = algebraic_semidom.coprime times one a b"
ran_m_mapsto_upd_notin: fixes C :: "'a"   and N :: "('a, 'b) fmap"   and C' :: "'b" assumes "C \<notin># dom_m N" shows "ran_m (fmupd C C' N) = add_mset C' (ran_m N)"
mult2_alt_s_def: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "mult2_alt_s ns s = {(X, Y). \<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> Y2 \<noteq> {#} \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s))}"
funs_term_Fun_subterm: fixes f :: "'a"   and t :: "('a, 'b) Term.term" assumes "f \<in> funs_term t" shows "\<exists>T. Fun f T \<sqsubseteq> t"
ac_Zp_factors': fixes x :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp" shows "x = [p] \<cdot> \<one> [^] ord_Zp x \<otimes> ac_Zp x"
group_left_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group f z inverse" shows "f z a = a"
vsv_vfsequence_two: fixes gf :: "V" assumes "vsv gf"   and "\<D>\<^sub>\<circ> gf = 2\<^sub>\<nat>" shows "[gf\<lparr>[]\<^sub>\<circ>\<rparr>, gf\<lparr>1\<^sub>\<nat>\<rparr>]\<^sub>\<circ> = gf"
digitToNlist_inv_1: fixes d :: "('a, 'b) Digit" assumes "FingerTreeStruc.is_measured_digit d" shows "\<forall>x\<in>set (FingerTreeStruc.digitToNlist d). FingerTreeStruc.is_measured_node x"
cobounded1: fixes a :: "'t"   and b :: "'t" shows "a \<preceq>\<^sub>t ord_term_lin.max a b"
mono_bal_i': fixes n :: "nat"   and h :: "nat"   and n' :: "nat"   and h' :: "nat" assumes "ASSUMPTION (bal_i n h)"   and "n \<le> n'"   and "h' \<le> h" shows "bal_i n' h'"
unitary_elim: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat"   and P :: "'c mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "P \<in> carrier_mat n n"   and "unitary P"   and "P * A * adjoint P = P * B * adjoint P" shows "A = B"
Dube_is_GB_cofactor_bound: fixes X :: "'a set"   and F :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite X"   and "finite F"   and "F \<subseteq> P[X]" shows "is_GB_cofactor_bound F (Dube (Suc (card X)) (maxdeg F))"
homeomorphic_to_disc_imp_simply_connected: assumes "S = {} \<or> S homeomorphic ball 0 1" shows "simply_connected S"
de_def: shows "de \<equiv> snd \<circ> quotient_of"
bag_no_drop: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and s :: "'e"   and t :: "'e"   and v :: "'c"   and w :: "'c" assumes "TreeDecomposition G T bag"   and "s \<rightarrow>\<^bsub>T\<^esub> t"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<in> bag s"   and "w \<notin> bag s"   and "w \<in> TreeDecomposition.left_part T bag t s" shows "v \<in> bag t"
hom_uminus: fixes x :: "'a poly"   and i :: "nat" shows "coeff (- x) i = - coeff x i"
Subst': fixes Ax :: "('a, 'b) Axioms"   and B :: "'a"   and d :: "('a, 'b) Expression"   and C :: "'a"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and e2 :: "('a, 'b) Expression" assumes "Axioms Ax"   and "Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : B \<turnstile> d : C"   and "Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Axioms.Theory Ax" shows "Vx : A \<turnstile> sub e1 in d\<equiv>sub e2 in d : C \<in> Axioms.Theory Ax"
gfun_intro: fixes f :: "(val \<times> val) fset"   and x :: "nat"   and \<rho> :: "(nat \<times> bval) list"   and e :: "exp" assumes "\<forall>v1 v2. (v1, v2) \<in> fset f \<longrightarrow> (\<forall>v\<in>good v1. \<exists>v'. (x, v) # \<rho> \<turnstile> e \<Down> v' \<and> v' \<in> good v2)" shows "good_fun f x e \<rho>"
arr_Set_ArrVal_vdomain: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>"
Minimal_Solutions_sym: fixes a :: "nat list"   and b :: "nat list"   and xs :: "nat list"   and ys :: "nat list" assumes "0 \<notin> set a"   and "0 \<notin> set b" shows "(xs, ys) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b \<longrightarrow> (ys, xs) \<in> Linear_Diophantine_Equations.Minimal_Solutions b a"
l2norm_le: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" shows "(l2norm S f \<le> l2norm S g) = (l2product S f f \<le> l2product S g g)"
det_nfa_reach': fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and bd :: "nat option bdd"   and ls :: "bool list list"   and q1 :: "bool list"   and i :: "nat"   and q2 :: "bool list"   and j :: "nat"   and bss :: "bool list list" assumes "subset_DFS A n"   and "subset_dfs A (nfa_startnode A) = (bd, ls)"   and "\<exists>bs. nfa_reach A (nfa_startnode A) bs q1 \<and> list_all (is_alph n) bs"   and "q1 = ls ! i"   and "q2 = ls ! j"   and "i < length ls"   and "j < length ls"   and "list_all (is_alph n) bss" shows "nfa_reach A q1 bss q2 = (dfa_reach (det_nfa A) i bss j \<and> nfa_is_node A q2)"
iso_is_retraction: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.iso f" shows "Maps.retraction f"
assms_3: shows "rk {A, C, \<beta>} = 2"
par_comp_prot_impl_par_comp: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set"   and Sec :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" assumes "par_comp_prot \<P> Sec"   and "\<A> \<in> \<P>" shows "par_comp \<A> Sec"
assms_4: shows "h2 \<in> O[F](h1)"
purgeIdle_purge_ex: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list"   and ucl :: "('b \<times> 'c) list"   and ss' :: "('a, 'b, 'c) state"   and GH :: "'b set" assumes "GM_sec_model.wffp do sl"   and "list_all isState sl"   and "map getGMUserCom (tl sl) = ucl" shows "\<exists>sl'. hd sl' = ss' \<and> GM_sec_model.wffp do sl' \<and> list_all2 (GM_sec_model.f GH) (tl sl) (tl sl') \<and> map getGMUserCom (GM_sec_model.purgeIdle (tl sl')) = GM_sec_model.purge GH ucl"
prems_2: shows "mod_type_class.to_nat i \<le> Suc n"
h8b: assumes "0 < Av"   and "Bv \<le> 0"   and "Av\<^sup>2 \<le> Bv\<^sup>2 * Cv" shows "Av + Bv * sqrt Cv \<le> 0"
that: fixes u :: "'a \<Rightarrow> 'b"   and e :: "real"   and r :: "real" assumes "f (x, u x) = (0::'c)"   and "u x = y"   and "\<And>s. s \<in> cball x e \<Longrightarrow> f (s, u s) = (0::'c)"   and "continuous_on (cball x e) u"   and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> S"   and "0 < e"   and "(u has_derivative blinfun_apply (- T o\<^sub>L f' (x, y) o\<^sub>L embed1_blinfun)) (at x)"   and "0 < r"   and "\<And>U v s. \<lbrakk>v x = y; \<And>s. s \<in> U \<Longrightarrow> f (s, v s) = (0::'c); U \<subseteq> cball x e; continuous_on U v; s \<in> U; (s, v s) \<in> ball (x, y) r\<rbrakk> \<Longrightarrow> u s = v s" shows "thesis"
nrows: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = nrows A"
exists_next_if_can_occur: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.can_occur channel trans send recv ev c" shows "\<exists>c'. distributed_system.next channel trans send recv c ev c'"
set_insert: fixes x :: "'b"   and xs :: "'b list" shows "set (List.insert x xs) = Set.insert x (set xs)"
Q_hit_pre_not_Nil: shows "Q_hit_pre \<noteq> []"
comp_assoc: fixes h :: "'a2"   and g :: "'a2"   and f :: "'a2" shows "(h \<cdot>\<^sub>2 g) \<cdot>\<^sub>2 f = h \<cdot>\<^sub>2 g \<cdot>\<^sub>2 f"
character_data_ptr_kinds_commutes: fixes character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap" shows "(cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr |\<in>| node_ptr_kinds h) = (character_data_ptr |\<in>| character_data_ptr_kinds h)"
univ_preserves_predicate: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "equiv A r"   and "P respects r" shows "{x \<in> A. P x} // r = {X \<in> A // r. univ P X}"
p_def: shows "p = (\<lambda>x. a * x\<^sup>2 + b * x + c)"
prob_compl: fixes A :: "nat set set set" assumes "A \<in> ES.events" shows "ES.prob (space ES.P - A) = 1 - ES.prob A"
less_add_same_cancel2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subset># b + a) = ({#} \<subset># b)"
cmp_in_hom_2: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b"   and "src\<^sub>C a = trg\<^sub>C b" shows "\<guillemotleft>\<Phi> (a, b) : F a \<star>\<^sub>D F b \<Rightarrow>\<^sub>D F (a \<star>\<^sub>C b)\<guillemotright>"
comp: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and h :: "'c \<Rightarrow> 'g"   and i :: "'d \<Rightarrow> 'h" shows "map_sum f g \<circ> map_sum h i = map_sum (f \<circ> h) (g \<circ> i)"
of_int_leI: fixes w :: "int"   and z :: "int" assumes "w \<le> z" shows "of_int w \<le> of_int z"
le_diff_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (minus c b) = less_eq (plus a b) c"
subst_apply_terms_empty: fixes M :: "('a, 'b) Term.term set" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t Var = M"
add_left_imp_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<oplus>\<^sub>a b = a \<oplus>\<^sub>a c" shows "b = c"
assms_2: shows "is_serial_solution_for_problem \<Pi> \<pi>"
P_LCons: shows "ldropn n P = LCons v (ltl (ldropn n P))"
fmrel_on_fsetI: fixes S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "('d, 'e) fmap"   and n :: "('d, 'f) fmap" assumes "\<And>x. x |\<in>| S \<Longrightarrow> rel_option P (fmlookup m x) (fmlookup n x)" shows "fmrel_on_fset S P m n"