hcju/isabelleps huggingface dataset:
pp_dist_inf: fixes 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 x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (uminus (inf x y)) = inf (uminus (uminus x)) (uminus (uminus y))"
mult_vec_minus_mult_left: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_vec pair_of_term term_of_pair (- p) q = - term_powerprod.mult_vec pair_of_term term_of_pair p q"
ffact_Suc_nat: fixes n :: "nat"   and m :: "nat" shows "ffact (Suc n) m = m * ffact n (m - 1)"
pps_to_list_def: fixes S :: "'t set" shows "pps_to_list S = rev (ord_term_lin.sorted_list_of_set S)"
simps_1: fixes k :: "nat"   and \<Gamma> :: "binding list" shows "\<down>\<^sub>e 0 k \<Gamma> = \<Gamma>"
rel: shows "SourceTerm SP \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> SourceTerm SQ"
lambdabinder2_def: shows "\<^bold>\<lambda>\<^sup>2 \<equiv> map_fun (map_fun id (map_fun id eval\<o>)) make\<Pi>\<^sub>2 (\<lambda>\<phi> u v s w. \<exists>x y. \<nu>\<upsilon> x = u \<and> \<nu>\<upsilon> y = v \<and> \<phi> x y s w)"
H: shows "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>"
index_col: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "col A j $ i = A $$ (i, j)"
distincts_distinct: fixes xss :: "'b list list" assumes "distincts xss" shows "distinct xss"
elims_2: fixes x :: "atom set \<times> 'e"   and xa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xb :: "'e \<Rightarrow> atom set"   and xc :: "perm"   and xd :: "atom set \<times> 'e" assumes "x \<approx>set xa xb xc xd" obtains bs :: "atom set" and x :: "'e" and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" and f :: "'e \<Rightarrow> atom set" and p :: "perm" and cs :: "atom set" and y :: "'e" where "xb__ = (bs, x)" and "xa = R" and "xb = f" and "xc = p" and "xd = (cs, y)" and "f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs"
measurable_product_coordinates: fixes i :: "'d" shows "(\<lambda>x. x i) \<in> borel_measurable borel"
ide_in_hom: fixes a :: "'b" shows "C'.ide a = C'.in_hom a a a"
ccompare: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "ccompare = Some comp" shows "comparator comp"
c6: fixes R :: "('a \<times> 'a set) set" shows "R \<cdot> 1\<^sub>\<pi> \<subseteq> 1\<^sub>\<pi>"
tm_comp_next: fixes tp :: "cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and n :: "nat"   and tp' :: "cell list \<times> cell list"   and B :: "(action \<times> nat) list" assumes "steps0 (1, tp) A n = (0, tp')"   and "tm_wf0 A" obtains n' :: "nat" where "steps0 (1, tp) (A |+| B) n' = (Suc (length A div 2), tp')"
comp_surj: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'd" assumes "surj f"   and "surj g" shows "surj (g \<circ> f)"
ws_update_def: fixes E :: "('c \<times> 'b) set"   and u :: "'c"   and p :: "'c list"   and V :: "'b set"   and ws :: "'b \<Rightarrow> 'c list option" shows "ws_update E u p V ws \<equiv> RETURN (V \<union> E `` {u}, ws ++ (\<lambda>v. if (u, v) \<in> E \<and> v \<notin> V then Some (u # p) else None))"
Pi_pmf_map: fixes A :: "'a set"   and f :: "'c \<Rightarrow> 'b"   and dflt :: "'c"   and dflt' :: "'b"   and g :: "'a \<Rightarrow> 'c pmf" assumes "finite A"   and "f dflt = dflt'" shows "Pi_pmf A dflt' (\<lambda>x. map_pmf f (g x)) = map_pmf ((\<circ>) f) (Pi_pmf A dflt g)"
characteriseLast: fixes C :: "'a sequent"   and m :: "nat"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "(C, m + 1) \<in> derivable R" shows "\<exists>Ps. Ps \<noteq> [] \<and> (Ps, C) \<in> R \<and> (\<forall>p\<in>set Ps. \<exists>n\<le>m. (p, n) \<in> derivable R)"
ProcNoRec1: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'d \<Rightarrow> 'f set"   and p :: "'e"   and Q :: "'d \<Rightarrow> 'f set"   and A :: "'d \<Rightarrow> 'f set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) Call p (Q Z),(A Z)"
diagonal_to_Smith_i_PQ': fixes P' :: "(('a, 'rows) vec, 'rows) vec"   and B' :: "(('a, 'cols) vec, 'rows) vec"   and Q' :: "(('a, 'cols) vec, 'cols) vec"   and xs :: "nat list"   and i :: "nat" assumes "(P', B', Q') = diagonal_to_Smith_i_PQ xs i bezout (P, B, Q)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "isDiagonal B"   and "i \<notin> set xs"   and "i < min (nrows A) (ncols A)" shows "B' = P' ** A ** Q' \<and> invertible P' \<and> invertible Q'"
simps_1: fixes x2 :: "'h"   and y2 :: "'h" shows "(Some x2 = Some y2) = (x2 = y2)"
simps: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv" shows "Set2_Join.diff join t1 t2 = (if t1 = \<langle>\<rangle> then \<langle>\<rangle> else if t2 = \<langle>\<rangle> then t1 else case t2 of \<langle>l2, (a, x), r2\<rangle> \<Rightarrow> let (l1, uu_, r1) = Set2_Join.split join t1 a; l' = Set2_Join.diff join l1 l2 in Let (Set2_Join.diff join r1 r2) (Set2_Join.join2 join l'))"
subst_swap_same: fixes x :: "Vars.var"   and e :: "exp"   and y :: "Vars.var" assumes "atom x \<sharp> e" shows "(x \<leftrightarrow> y) \<bullet> e = e[y::=x]"
trans_r: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
fmsubset_restrict_set: fixes S :: "'a set"   and m :: "('a, 'b) fmap" shows "fmrestrict_set S m \<subseteq>\<^sub>f m"
l2_11_b: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A' B' C'"   and "Cong A B A' B'"   and "Cong B C B' C'" shows "Cong A C A' C'"
intro: fixes p :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and P :: "jvm_prog"   and C :: "char list"   and Ts :: "ty list" assumes "wf_prog p P"   and "is_class P C"   and "set Ts \<subseteq> types P" shows "start_context P Ts p C"
aneg_less: fixes x :: "ant"   and y :: "ant" shows "(\<not> x < y) = (y \<le> x)"
o_def: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "f \<circ> g = (\<lambda>x. f (g x))"
adj_path_V: fixes v :: "'v"   and ps :: "'v list" assumes "adj_path v ps" shows "set ps \<subseteq> V"
all_is_branch_def: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set" shows "all_is_branch p i branch xs \<equiv> \<forall>(v, v')\<in>xs. v < length branch \<longrightarrow> is_at p i branch v v'"
inject_1: fixes x1 :: "brick list"   and y1 :: "brick list" shows "(basic x1 = basic y1) = (x1 = y1)"
infinite_set_mset_mset_set: fixes A :: "'a set" assumes "infinite A" shows "set_mset (mset_set A) = {}"
inf_le1: fixes x :: "'k multiset"   and y :: "'k multiset" shows "x \<inter># y \<subseteq># x"
sgn_bcontfun_def: fixes f :: "'a \<Rightarrow>\<^sub>C 'b" shows "sgn f = f /\<^sub>R norm f"
valid_restrict_to_valid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and M :: "'d set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>|\<^bsub>M\<^esub>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
emeasure_eq_measure: fixes p :: "'b spmf"   and A :: "'b set" shows "emeasure (measure_spmf p) A = ennreal (Sigma_Algebra.measure (measure_spmf p) A)"
AllN: fixes xs :: "real list" shows "eval (QE_dnf opt step \<phi>) xs = eval \<phi> xs"
trancl_reflcl: fixes r :: "('a \<times> 'a) set" shows "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
LD_conv_def: fixes L :: "'c set"   and ars :: "('d \<times> 'd) set" shows "LD_conv L ars = (\<exists>r lrs. ars = unlabel lrs \<and> trans r \<and> wf r \<and> (\<forall>p. local_peak lrs p \<longrightarrow> (\<exists>\<gamma>1 \<gamma>2 \<gamma>3 \<delta>1 \<delta>2 \<delta>3. LT lrs r (fst p, snd p, \<gamma>1, \<gamma>2, \<gamma>3, \<delta>1, \<delta>2, \<delta>3))))"
disconnected_nodes_eq2_h2: fixes doc_ptr :: "(_) document_ptr" shows "|h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
safe_ln_def: fixes x :: "real" shows "safe_ln x = (if 0 < x then ln x else 0)"
nextsuffix_def: fixes s :: "nat \<Rightarrow> 'a" shows "nextsuffix s \<equiv> s |\<^sub>s nextnat s"
conT0: shows "continuous_on (T - {0::'a}) (\<lambda>y. y /\<^sub>R norm y)"
weak_morphism: shows "weak_ring_morphism (eval_pmod q) (PIdl\<^bsub>poly_ring L\<^esub> q) (poly_ring L)"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes "a1 \<longrightarrow>\<^sub>c a2" obtains M :: "trm" and a :: "coname" and N :: "trm" and x :: "name" where "a1 = Cut <aa__>.Ma__ xa__.N__" and "a2 = Ma__{aa__:=xa__.N__}" and "\<not> fic Ma__ aa__" and "aa__ \<sharp> N__" and "xa__ \<sharp> Ma__"   | N :: "trm" and x :: "name" and a :: "coname" and M :: "trm" where "a1 = Cut <a>.M x.N" and "a2 = N{x:=<a>.M}" and "\<not> fin N x" and "a \<sharp> N" and "x \<sharp> M"
gg: shows "g = (\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}"
integral_density: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and g :: "'d \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (g x))) f = LINT x|M. g x *\<^sub>R f x"
Suc_1: fixes fs :: "int vec list" assumes "LLL_invariant True i fs"   and "\<mu>_small_row i fs j"   and "basis_reduction_add_rows_loop i fs j = fs'"   and "i < m"   and "j \<le> i" shows "LLL_invariant False i fs' \<and> LLL_measure i fs' = LLL_measure i fs"
simps_3: fixes f :: "form"   and g :: "form" shows "fv (FConj f g) = fv f @ fv g"
simps_5: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi>\<^sub>1 :: "'b formula"   and \<phi>\<^sub>2 :: "'b formula" assumes "formula \<Sigma>" shows "formula.satisfies \<Sigma> (w, I) (FOr \<phi>\<^sub>1 \<phi>\<^sub>2) = (formula.satisfies \<Sigma> (w, I) \<phi>\<^sub>1 \<or> formula.satisfies \<Sigma> (w, I) \<phi>\<^sub>2)"
Cd_single: fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> X"   and "inj_on Xd X"   and "x \<in> Field (Pd' d)" shows "x \<in> PdXXX.Cd d X"
wbt_del_max: fixes t :: "('b \<times> nat) tree" assumes "t \<noteq> \<langle>\<rangle>"   and "wbt t" shows "wbt (snd (del_max t))"
dim_mult_mat_vec: fixes A :: "'a mat"   and v :: "'a vec" shows "dim_vec (A *\<^sub>v v) = dim_row A"
assms: shows "K \<le> norm (f *\<^sub>V x) / norm x"
fv_subset_subterms: fixes t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "fv t \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
rel_distinct_4: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and y3 :: "'g" shows "\<not> rel_sum_bot R (sum_bot.Right y3) Bottom"
edge_from_def: fixes c :: "'form"   and "is" :: "nat list" shows "edge_from c is = ((c, 0 # is), Reg (iOutPort (tree_at (it' c) is)))"
fls_shift_eq_iff: fixes m :: "int"   and f :: "'a fls"   and g :: "'a fls" shows "(fls_shift m f = fls_shift m g) = (f = g)"
A: shows "(t, Xa) \<in> F P \<and> (u, Ya) \<in> F Q \<and> sa setinterleaves ((t, u), insert tick (ev ` A))"
lens_equiv_via_bij: fixes Z :: "'d \<Longrightarrow> 'e"   and X :: "'d \<Longrightarrow> 'f"   and Y :: "'e \<Longrightarrow> 'f" assumes "bij_lens Z"   and "X = Z ;\<^sub>L Y" shows "X \<approx>\<^sub>L Y"
V_def: fixes b :: "bool" shows "V b = (if b then B else C)"
add_closed_inverse_left: fixes g :: "'g"   and x :: "'g" assumes "g \<notin> G"   and "x \<in> G" shows "g + x \<notin> G"
equal_val_Zp: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp"   and "val_Zp a = val_Zp b"   and "val_Zp b < val_Zp (c \<ominus> a)" shows "val_Zp c = val_Zp b"
q: fixes a :: "int" shows "rat_of_int (g a) = poly q (rat_of_int a)"
irreducible_m_def: fixes m :: "int"   and f :: "int poly" shows "poly_mod.irreducible_m m f = (poly_mod.Mp m f \<noteq> poly_mod.Mp m 0 \<and> \<not> poly_mod.dvdm m f 1 \<and> (\<forall>a b. poly_mod.Mp m f = poly_mod.Mp m (a * b) \<longrightarrow> poly_mod.dvdm m a 1 \<or> poly_mod.dvdm m b 1))"
deg_sect_insert: fixes x :: "'b"   and X :: "'b set"   and d :: "nat" shows "deg_sect (insert x X) d = (\<Union>d0\<le>d. (+) (monomial (d - d0) x) ` deg_sect X d0)"
simps_1: fixes uu :: "'a" shows "lderiv uu Zero = Zero"
hyperpow_less_le: fixes r :: "real star"   and n :: "nat star"   and N :: "nat star" assumes "0 \<le> r"   and "r \<le> 1"   and "n < N" shows "r pow N \<le> r pow n"
None: fixes ord :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d option" shows "ord_option ord None x"
phi_complemented: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c" assumes "triple phi" shows "complement (phi x) (phi (- x))"
bij_Rep_perm: fixes p :: "'b perm" shows "bij (Rep_perm p)"
neville_aitken_interpolation_poly_def: fixes x_fs :: "('b \<times> 'b) list" shows "neville_aitken_interpolation_poly x_fs = Let (map (\<lambda>(xi, fi). (xi, xi, [:fi:])) x_fs) neville_aitken_impl_main"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "group_add_hom hom"   and "zero_hom_0 hom" shows "inj_group_add_hom hom"
rreq_def: shows "rreq \<equiv> \<lambda>(hops, rreqid, dip, dsn, dsk, oip, osn, sip). Rreq hops rreqid dip dsn dsk oip osn sip"
hom: shows "x \<in> hom (G\<lparr>carrier := generate G {a}\<rparr>) (H\<lparr>carrier := generate H {b}\<rparr>)"
merkle_interface_aux: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" shows "merkle_interface h bo m = merge_on UNIV h bo m"
r_inv_ex: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<exists>y\<in>carrier F. x \<oplus>\<^bsub>F\<^esub> y = \<zero>\<^bsub>F\<^esub>"
filt_equiv_sgt: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 \<le> q"   and "q \<le> 1"   and "0 < p"   and "p < 1"   and "filt_equiv nat_filtration M N" shows "0 < q"
upd_inj: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "(upd i = upd j) = (i = j)"
map_of_map_to_set: fixes l :: "('e \<times> 'f) list"   and m :: "'e \<Rightarrow> 'f option" assumes "distinct (map fst l)" shows "(map_of l = m) = (set l = map_to_set m)"
mono_sup: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b"   and B :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "order.mono less_eq f" shows "f A \<squnion> f B \<le> f (sup A B)"
b': shows "b' = e_append b (Partial_Recursive.list_encode (replicate (y - e_length b) 0 @ [1 - the (\<phi> (the (s b)) y)]))"
parCompose: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and R :: "pi"   and Rel' :: "(pi \<times> pi) set"   and S :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto>[Rel] Q"   and "R \<leadsto>[Rel'] S"   and "(P, Q) \<in> Rel"   and "(R, S) \<in> Rel'"   and "\<And>P' Q' R' S'. \<lbrakk>(P', Q') \<in> Rel; (R', S') \<in> Rel'\<rbrakk> \<Longrightarrow> (P' \<parallel> R', Q' \<parallel> S') \<in> Rel''"   and "\<And>S T x. (S, T) \<in> Rel'' \<Longrightarrow> (<\<nu>x>S, <\<nu>x>T) \<in> Rel''" shows "P \<parallel> R \<leadsto>[Rel''] Q \<parallel> S"
ord_iso_alt: fixes f :: "'a \<Rightarrow> 'b" shows "ord_iso f = (ord_embed f \<and> top_pres ((`) f))"
index_of_const_at_suc: fixes u :: "nat" shows "index_of_const (Suc u) = c_pair 4 (c_pair 2 (index_of_const u))"
dbm_int_inv_abstr: fixes A :: "('g \<times> ('e, 'f) cconstraint \<times> 'h \<times> 'e list \<times> 'g) set \<times> ('g \<Rightarrow> ('e, 'f) cconstraint)"   and n :: "nat"   and l :: "'g"   and v :: "'e \<Rightarrow> nat" assumes "\<forall>(x, m)\<in>clkp_set A. m \<in> \<nat>" shows "dbm_int (abstr (inv_of A l) (\<lambda>i j. \<infinity>) v) n"
simps_1: shows "del_min \<langle>\<rangle> = \<langle>\<rangle>"
count_tr_def: shows "count_tr = foldl (foldl (\<lambda>y x. if x then y else Suc y)) 0"
iT_Mult_Div_self: fixes k :: "nat"   and I :: "nat set" assumes "0 < k" shows "I \<otimes> k \<oslash> k = I"
list_all2_reorder_left_invariance: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and xs' :: "'b list" assumes "list_all2 R xs ys"   and "mset xs' = mset xs" shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
eqvtI: fixes P :: "pi"   and Q :: "pi"   and p :: "(name \<times> name) list" assumes "P \<simeq>\<^sup>s Q" shows "p \<bullet> P \<simeq>\<^sup>s p \<bullet> Q"
assms_2: fixes n :: "nat" assumes "0 < n" shows "real n * a \<le> u n"
set_child_nodes_locs_impl: shows "set_child_nodes_locs = local.a_set_child_nodes_locs"
convolution_distrib_right: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "convolution (f + g) h = convolution f h + convolution g h"
mtf2_forward_effect1: fixes q :: "'b"   and xs :: "'b list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q < i \<and> i < length xs" shows "index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index xs q < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < length xs"
select_convs_10: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getWatchList \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getWatchList"
vimage_diff: fixes C :: "'a set" shows "carrier\<^bsub>S\<^esub> - f -` (carrier - C) = carrier\<^bsub>S\<^esub> \<inter> f -` C"
valid_decompD_finite: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "finite X"   and "valid_decomp X ps"   and "(h, U) \<in> set ps" shows "finite U"
dvdm_iff_pdivmod0: fixes F :: "'a mod_ring poly"   and f :: "int poly"   and G :: "'a mod_ring poly"   and g :: "int poly" assumes "F = of_int_poly f"   and "G = of_int_poly g" shows "(g dvdm f) = (snd (pdivmod F G) = 0)"
zero_mat_mult_vector: fixes x :: "'b vec"   and nc :: "nat"   and nr :: "nat" assumes "x \<in> carrier_vec nc" shows "0\<^sub>m nr nc *\<^sub>v x = 0\<^sub>v nr"
has_tabulation_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "'b arrow_of_spans_data" assumes "identity_arrow_in_span_bicategory C prj0 prj1 r" shows "\<exists>g. bicategory.adjoint_pair (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) \<lparr>Chn = partial_magma.dom C (span_in_category.apex C (Dom r)), Dom = \<lparr>Leg0 = span_in_category.apex C (Dom r), Leg1 = Leg1 (Dom r)\<rparr>, Cod = \<lparr>Leg0 = span_in_category.apex C (Dom r), Leg1 = Leg1 (Dom r)\<rparr>\<rparr> g"
assms_2: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<le> ennreal r"
poly_rel_karatsuba_main: fixes x1 :: "'i list"   and x2 :: "'a list"   and y1 :: "'i list"   and y2 :: "'a list"   and n :: "nat"   and m :: "nat" assumes "list_all2 R x1 x2"   and "list_all2 R y1 y2" shows "poly_rel (karatsuba_main_i ops x1 n y1 m) (karatsuba_main x2 n y2 m)"
valid_plain_def: fixes plain :: "bool list" shows "valid_plain plain = (length plain = len)"
QCongNull_def: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool" shows "QCongNull l \<equiv> QCong l \<and> (\<exists>A. l A A)"
elims_3: fixes x :: "((nat \<times> nat) \<times> transition) fset"   and xa :: "nat"   and xb :: "nat \<Rightarrow>f value option"   and xc :: "(String.literal \<times> value list) list" assumes "\<not> recognises_prim x xa xb xc" obtains e :: "((nat \<times> nat) \<times> transition) fset" and s :: "nat" and r :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and t :: "(String.literal \<times> value list) list" where "x = e" and "xa = s" and "xb = r" and "xc = (l, i) # t" and "\<not> (let poss_steps = possible_steps e s r l i in \<exists>(s', T)|\<in>|poss_steps. recognises_prim e s' (evaluate_updates T i r) t)"
transp_equivclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "transp (equivclp r)"
next_plane0_incr_faces: fixes g :: "graph"   and g' :: "graph"   and p :: "nat" assumes "minGraphProps g"   and "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)" shows "|finals g'| = |finals g| + 1 \<and> |nonFinals g| - 1 \<le> |nonFinals g'|"
Let_def: fixes s :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "Let s f \<equiv> f s"
g: shows "(\<Sum>i\<in>set_pmf (mix_pmf a p q). pmf (mix_pmf a p q) i * f i) = (\<Sum>i\<in>set_pmf (mix_pmf a p q). (a * pmf p i + (1 - a) * pmf q i) * f i)"
mat_times_vec_leq_def: fixes A :: "'a mat"   and x :: "'a vec"   and b :: "'a vec" shows "[A*\<^sub>vx]\<le>b = ((\<forall>i<dim_vec b. (A *\<^sub>v x) $ i \<le> b $ i) \<and> dim_row A = dim_vec b \<and> dim_col A = dim_vec x)"
sm: shows "Polynomial.smult lc (Polynomial.smult x r) - monom (coeff (Polynomial.smult x r) dr) n * d = Polynomial.smult x (Polynomial.smult lc r - monom (coeff r dr) n * d)"
count_le_length: fixes xs :: "'b list"   and x :: "'b" shows "count_list xs x \<le> length xs"
inv_mult_group: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier C"   and "y \<in> carrier C" shows "inv\<^bsub>C\<^esub> (x \<otimes>\<^bsub>C\<^esub> y) = inv\<^bsub>C\<^esub> y \<otimes>\<^bsub>C\<^esub> inv\<^bsub>C\<^esub> x"
digit_to_char_simps_9: shows "digit_to_char 8 = CHR ''8''"
axioms_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and \<mu> :: "'b" assumes "arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>" shows "tabulation_in_maps V H \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1"