hcju/isabelleps huggingface dataset:
lm102: fixes x :: "'a"   and X :: "'a set"   and Y :: "'a set" assumes "x \<in> X \<union> Y" shows "(X <| Y) x = chi X x"
finite_f_out: fixes A :: "'v set" shows "\<integral>\<^sup>+y\<in>A. f (x, y)\<partial>count_space UNIV \<noteq> \<top>"
power_up_fl: fixes x :: "float"   and n :: "nat"   and p :: "nat" assumes "0 \<le> x" shows "x ^ n \<le> power_up_fl p x n"
assms: shows "P \<longmapsto>\<alpha> \<prec> P'"
simps_1: fixes f :: "'c \<Rightarrow> 'd" shows "remdups' f [] = []"
simps_3: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm"   and ds :: "dir list" shows "position_of (App t s) (dir.Left # ds) = position_of t ds"
mod_2_eq_odd: fixes a :: "'a" shows "a mod (2::'a) = of_bool (odd a)"
minGraphProps4: fixes g :: "graph" assumes "minGraphProps g" shows "|faceListAt g| = countVertices g"
l8_14_2_1b: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and Y :: "'p" assumes "X PerpAt A B C D "   and "Col Y A B"   and "Col Y C D" shows "X = Y"
A: shows "finite A \<and> maximal A (\<lambda>S. S \<subseteq> carrier (V.vs kerT) \<and> ker.lin_indpt S)"
f: shows "monotone (ord_spmf (=)) (ord_spmf (=)) F"
wt_subst_trm': fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "typed_model arity public Ana \<Gamma>"   and "typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<sigma>"   and "\<Gamma> s = \<Gamma> t" shows "\<Gamma> (s \<cdot> \<sigma>) = \<Gamma> (t \<cdot> \<sigma>)"
resChainPerm: fixes perm :: "(name \<times> name) list"   and lst :: "name list"   and P :: "pi" shows "perm \<bullet> resChain lst P = resChain (perm \<bullet> lst) (perm \<bullet> P)"
coeffs_def: fixes p :: "'b Polynomial.poly" shows "Polynomial.coeffs p = (if p = 0 then [] else map (poly.coeff p) [0..<Suc (Polynomial.degree p)])"
invertibleD_2: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "invertible A" shows "A ** matrix_inv A = mat (1::'a)"
env_restr_empty: fixes m :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "edom m \<inter> S = {}" shows "m f|` S = \<bottom>"
hequiv_proj_opening: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and a :: "'b" assumes "Hintikka A H"   and "Nom a at i in' H"   and "a \<in> A"   and "i \<in> A" shows "proj (hequiv_rel A H) a = proj (hequiv_rel A H) i"
linord_of_list_def: fixes xs :: "'a list" shows "linord_of_list xs \<equiv> {(x, y). linord_of_listP x y xs}"
assms_10: shows "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"
power_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "z \<squnion> x \<cdot> y \<le> y" shows "x ^ n \<cdot> z \<le> y"
finfun_All_except_def: fixes A :: "'a list"   and P :: "'a \<Rightarrow>f bool" shows "finfun_All_except A P \<equiv> \<forall>a. a \<in> set A \<or> P $ a"
rwp_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "rwp_on A f = {(x, y). \<exists>B\<subseteq>A. x \<in> B \<and> y \<in> f B}"
wls_substEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and Z :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var"   and z :: "'var" assumes "wlsEnv rho"   and "wls (asSort ys) Y"   and "wls (asSort zs) Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "rho &[Y / y]_ys &[Z / z]_zs = rho &[Z / z]_zs &[Y #[Z / z]_zs / y]_ys"
trans_rtrancl: fixes r :: "('g \<times> 'g) set" shows "trans (r\<^sup>*)"
step: fixes x :: "name"   and xs :: "name list"   and P :: "('d, 'e, 'f) psi" shows "\<lparr>\<nu>*(x # xs)\<rparr>P = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xs\<rparr>P)"
unsat: shows "\<not> satisfiable (grounding_of_clss (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)))"
on_circline_y_axis: fixes z :: "complex_homo" shows "on_circline y_axis z = ((\<exists>c. is_imag c \<and> z = of_complex c) \<or> z = \<infinity>\<^sub>h)"
effect_updateI: fixes h' :: "heap"   and r :: "'a ref"   and v :: "'a"   and h :: "heap"   and x :: "unit" assumes "h' = Ref.set r v h" shows "effect (r := v) h h' x"
SPLIT_5: shows "({#s#}, w1, ce1') \<in> trcl (ntr fg)"
that: fixes m :: "nat" assumes "n < m"   and "c * g m < f m" shows "thesis"
f: shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S l"
lspasl_empl_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "sep_algebra.sep_empty zero h" shows "h = zero"
rbl_mult_take2: fixes bla :: "bool list"   and blb :: "bool list" assumes "length bla \<le> length blb" shows "rbl_mult bla (take (length bla) blb) = rbl_mult bla blb"
create_initial_object_def: fixes allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set" shows "heap_base.create_initial_object allocate = (\<lambda>(h, ads, b) C. if b then let HA = allocate h (Class_type C) in if HA = {} then (h, ads, False) else let (h', a'') = SOME ha. ha \<in> HA in (h', ads @ [a''], True) else (h, ads, False))"
expand_term_prop_help: fixes n' :: "('b, 'c) node_scheme"   and ns' :: "('b, 'd) node_scheme set"   and n :: "('b, 'c) node_scheme"   and ns :: "('b, 'd) node_scheme set"   and \<phi> :: "'b ltlr"   and f :: "('b, 'c) node_scheme \<times> ('b, 'd) node_scheme set \<Rightarrow> 'e nres"   and P :: "'e \<Rightarrow> bool" assumes "((n', ns'), n, ns) \<in> expand_ord \<phi> \<and> expand_inv \<phi> (n', ns')"   and "\<lbrakk>expand_inv \<phi> (n', ns'); ((n', ns'), n, ns) \<in> expand_ord \<phi>\<rbrakk> \<Longrightarrow> f (n', ns') \<le> SPEC P" shows "f (n', ns') \<le> SPEC P"
edges_it_correct: fixes g :: "'G" assumes "invar g" shows "set_iterator (edges_it g) (edges (\<alpha> g))"
subtermeq_subterm_neq: fixes t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term"   and v :: "('a, 'b) Term.term" assumes "t \<sqsubseteq> u"   and "u \<sqsubset> v" shows "t \<noteq> v"
rupture_surj_norm_is_hom: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (A [X])"   and "p \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)" shows "UP_A.rupture_surj K p \<circ> UP_A.poly_of_const \<in> ring_hom ((A [X])\<lparr>carrier := K\<rparr>) (Rupt\<^bsub>A [X]\<^esub> K p)"
rep_eq: fixes x :: "'b fds" shows "fds_nth x = case_nat (0::'b) (Rep_fds x)"
CL_dualCL: shows "dual cl \<in> CompleteLattice"
4: fixes x :: "typ"   and a :: "(String.literal \<times> String.literal) list"   and b :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "alist_conds cto"   and "x \<in> ran (map_of cto)"   and "sa = (a, b)" shows "typ_ok_sig (map_of cto, map_of tao, set a, map_of (map (apsnd map_of) b)) x"
wfG': shows "wf_graph \<lparr>nodes = V, edges = E - {f}\<rparr>"
result_eq: shows "(rel, aux') = update_since args rel1 rel2 (\<tau> \<sigma> ne) aux"
hatom_of_fatom_bij: shows "bij_betw hatom_of_fatom ground_fatoms UNIV"
Fresh_def: fixes h :: "'d \<Rightarrow> 'c" shows "Fresh h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
card_lift: fixes x :: "V"   and y :: "V" shows "vcard (lift x y) = vcard y"
conforms_Err: fixes e :: "error"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option" shows "(Some (Error e), s)\<Colon>\<preceq>(G, L) = Norm s\<Colon>\<preceq>(G, L)"
Pxy: shows "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
find_consistent_signs_at_roots_thm: fixes p :: "real poly"   and qs :: "real poly list" shows "find_consistent_signs_at_roots p qs = snd (snd (calculate_data p qs))"
widen_Array: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S.[]\<preceq>T" shows "T = Class Object \<or> (\<exists>T'. T = T'.[] \<and> G\<turnstile>S\<preceq>T')"
complex_coneE: fixes z :: "complex"   and a :: "real"   and b :: "real" assumes "z \<in> complex_cone a b" obtains r :: "real" and \<alpha> :: "real" where "0 \<le> r" and "\<alpha> \<in> {a..b}" and "z = rcis r \<alpha>"
sinh_ln_real: fixes x :: "real" assumes "0 < x" shows "sinh (ln x) = (x - inverse x) / 2"
set_integral_mult_right: fixes M :: "'b measure"   and A :: "'b set"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a" shows "\<integral>t\<in>A. (a * f t)\<partial>M = a * set_lebesgue_integral M A f"
subst_ex_same': fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'c \<Rightarrow> 'b"   and v :: "('a, 'c) uexpr"   and P :: "'b upred" assumes "mwb_lens x" shows "\<sigma>(x \<mapsto>\<^sub>s v) \<dagger> (\<exists> &x \<bullet> P) = \<sigma> \<dagger> (\<exists> &x \<bullet> P)"
pmdl_proj_orig_basis': fixes n :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pmdl (set (proj_orig_basis n bs)) = proj_poly_syz n ` pmdl (set bs)"
set_sorted_list_of_set: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
less_1: fixes y :: "nat" assumes "y < i'"   and "y cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<And>k' i'. \<lbrakk>k' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k; P k'; i' icd\<^bsup>\<pi>\<^esup>\<rightarrow> k'\<rbrakk> \<Longrightarrow> P i'" shows "P y"
sym: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "symmetric A equiv"   and "equiv x y"   and "x \<in> A"   and "y \<in> A" shows "equiv y x"
assms_1: shows "insert_ops (xs @ [(oid, Some ref)] @ ys)"
assms: shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
Id_onI: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "(a, a) \<in> s2r A"
N1_Delta1_E2_disjoint: shows "N\<^bsub>\<V>1\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {}"
bre'_algo_def: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set" shows "bre'_algo Qi \<delta> \<equiv> \<lparr>wa_cond = bre'_cond Qi, wa_step = br'_step \<delta>, wa_initial = br'_initial \<delta>, wa_invar = br'_invar \<delta>\<rparr>"
coboundedI2: fixes b :: "'b multiset"   and c :: "'b multiset"   and a :: "'b multiset" assumes "b \<subseteq># c" shows "a \<inter># b \<subseteq># c"
refine: fixes E :: "'a \<Rightarrow> 'a list"   and onstack :: "('a, unit) HashMap.hashmap"   and V :: "('a, unit) HashMap.hashmap"   and u :: "'a" shows "RETURN (code_red_dfs_hash E onstack V u) \<le> impl_red_dfs_hash E onstack V u"
gt1: shows "prob {x \<in> space M. a < X x} = exp (- a * l)"
closedin_subset: fixes U :: "'c topology"   and S :: "'c set" assumes "closedin U S" shows "S \<subseteq> topspace U"
converse_galois2: fixes f :: "('e \<times> 'd) set \<Rightarrow> 'f"   and g :: "('d \<times> 'e) set \<Rightarrow> 'f" shows "(f \<circ> \<smile> = g) = (g \<circ> \<smile> = f)"
set_insert: fixes x :: "'g"   and A :: "'g set" assumes "x \<in> A" obtains B :: "'g set" where "A = insert x B" and "x \<notin> B"
Eq_RESUMPTION_eq: fixes r :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and r' :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "Eq_RESUMPTION r r' = (r = r' \<and> r \<in> resumption)"
A_descriptions: fixes v :: "i"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>" shows "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
assms_2: fixes x :: "'a"   and i :: "nat" assumes "x \<in> set xs"   and "P x" shows "p ((f ^^ i) x)"
Sup_upper2: fixes u :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set"   and v :: "'b \<times> 'c" assumes "u \<in> A"   and "less_eq_prod v u" shows "less_eq_prod v (Sup_prod A)"
take_bit_signed_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "m \<le> Suc n" shows "take_bit m (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a) = take_bit m a"
coalesce_eq_OK2_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and zs :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [\<squnion>\<^bsub>f\<^esub>] ys) = OK zs" shows "ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
nn_integral_spmf_def: fixes p :: "'a spmf" shows "nn_integral_spmf p = integral\<^sup>N (measure_spmf p)"
3: fixes m :: "int" shows "poly_mod_prime_type TYPE('b) m = (m = int CARD('b))"
openin_subset_trans: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "openin (top_of_set U) S"   and "S \<subseteq> T"   and "T \<subseteq> U" shows "openin (top_of_set T) S"
fmfilter_upd: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and m :: "('c, 'd) fmap" shows "fmfilter P (m(x $$:= y)) = (if P x then fmfilter P m(x $$:= y) else fmfilter P m)"
product_overlap: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CM"   and "P z x"   and "O z y" shows "O z (x \<otimes> y)"
wlconf_gext: fixes G :: "prog"   and s :: "st"   and l :: "'a \<Rightarrow> val option"   and L :: "'a \<Rightarrow> ty option"   and s' :: "st" assumes "G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L"   and "s\<le>|s'" shows "G,s'\<turnstile>l[\<sim>\<Colon>\<preceq>]L"
subdg_is_arr_vsubset: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
elims_3: fixes x :: "'c tree" assumes "\<not> complete x" obtains l :: "'c tree" and xa :: "'c" and r :: "'c tree" where "x = \<langle>l, xa, r\<rangle>" and "\<not> (height l = height r \<and> complete l \<and> complete r)"
closed_injective_linear_image_eq: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set" assumes "linear f"   and "inj f" shows "closed (f ` s) = closed s"
compat2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "gt x y"   and "z \<le> y" shows "gt x z"
ptensor_mat_mult: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat"   and m4 :: "'b mat" assumes "m1 \<in> carrier_mat ps2_P.d1 ps2_P.d1"   and "m2 \<in> carrier_mat ps2_P.d1 ps2_P.d1"   and "m3 \<in> carrier_mat ps2_P.d2 ps2_P.d2"   and "m4 \<in> carrier_mat ps2_P.d2 ps2_P.d2" shows "tensor_P (m1 * m2) (m3 * m4) = tensor_P m1 m3 * tensor_P m2 m4"
DynCom: fixes P :: "'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 \<Rightarrow> ('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c s Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P DynCom c Q,A"
updS_Rmi_g: fixes n :: "nat"   and i :: "nat ifex"   and s :: "bdd"   and x :: "nat \<times> nat \<times> nat"   and r :: "nat" shows "Rmi_g n i (updS s x r) = Rmi_g n i s"
out_var_semi_uvar: fixes x :: "'a \<Longrightarrow> 'b" assumes "mwb_lens x" shows "mwb_lens ($x\<acute>)"
prv_cnj_assoc_imp2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (cnj (cnj \<phi> \<chi>) \<psi>) (cnj \<phi> (cnj \<chi> \<psi>)))"
split_sum_ex: fixes P :: "'a + 'b \<Rightarrow> bool" shows "(\<exists>x. P x) = ((\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x)))"
simps: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and a :: "'c stream" shows "alw \<phi> a = (\<exists>xs. a = xs \<and> \<phi> xs \<and> alw \<phi> (stl xs))"
"4_3": fixes uu :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uu: t \<doteq> t'\<rangle> = fv t \<union> fv t'"
ttl_def: fixes tllist :: "('a, 'b) tllist" shows "ttl tllist = (case tllist of TNil b \<Rightarrow> TNil b | TCons x21 x22 \<Rightarrow> x22)"
RPDi_Times: fixes i :: "nat"   and r :: "mregex"   and s :: "mregex" shows "RPDi i (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesL r (\<Union>j\<le>i. RPDi j s) \<union> (\<Union>j\<le>i. RPDi j r)"
Listn_sl_aux: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "Semilat A r f" shows "semilat (Listn.sl n (A, r, f))"
true_cls_add_mset: fixes I :: "'a set"   and C :: "'a literal"   and D :: "'a literal multiset" shows "(I \<Turnstile> add_mset C D) = (I \<Turnstile>l C \<or> I \<Turnstile> D)"
step: fixes bn :: "'a poly"   and cn :: "'a poly" assumes "bn \<noteq> 1"   and "P (bn div gcd bn (cn - pderiv bn)) ((cn - pderiv bn) div gcd bn (cn - pderiv bn))" shows "P bn cn"
compatible_iff_seq: fixes g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "(S.VV.domains g \<inter> S.VV.codomains f \<noteq> {}) = S.VV.seq g f"
ptensor_mat_dim_col: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "dim_col (ptensor_mat m1 m2) = d0"
mom_select_correct_aux: fixes k :: "nat"   and xs :: "'b list" assumes "mom_select_dom (k, xs)"   and "k < length xs" shows "mom_select k xs = select k xs"
list_conj: fixes vs :: "'b list"   and bs :: "'b list"   and ps :: "fm list" shows "Ifm vs bs (list_conj ps) = (\<forall>p\<in>set ps. Ifm vs bs p)"
isContD: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "isCont f x" shows "f \<midarrow>x\<rightarrow> f x"
or_nat_def: fixes m :: "nat"   and n :: "nat" shows "m OR n = nat (int m OR int n)"
in_set_simps_3: fixes x :: "'b" shows "(x \<in> set []) = False"
rep_eq: fixes x :: "'b st_vec" shows "Rep_measure (measure_of_st_vec x) = (UNIV, UNIV, measure_of_st_vec' x)"
se_star_mono_for_sub: fixes c1 :: "('c, 'd) conf"   and ls :: "('c, 'd) label list"   and c1' :: "('c, 'd) conf"   and c2 :: "('c, 'd) conf"   and c2' :: "('c, 'd) conf" assumes "se_star c1 ls c1'"   and "se_star c2 ls c2'"   and "c2 \<sqsubseteq> c1" shows "c2' \<sqsubseteq> c1'"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r"
not_le_imp_less: fixes y :: "'a"   and x :: "'a" assumes "\<not> y \<sqsubseteq> x" shows "x \<sqsubset> y"
head_\<omega>_0: shows "head_\<omega> 0 = 0"
pdg_digraph_dg_prod: shows "digraph \<alpha> (dg_prod I \<AA>)"
ldrop_LNil: fixes n :: "enat" shows "ldrop n LNil = LNil"
sgn_not_eq_imp: 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"   and b :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "sgn b \<noteq> sgn a"   and "sgn a \<noteq> zero"   and "sgn b \<noteq> zero" shows "sgn a = uminus (sgn b)"
consec_intermediate2: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "consec i j \<and> consec (i \<squnion> j) k \<longrightarrow> consec j k"
infsetsum_product: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "countable A"   and "countable B"   and "f abs_summable_on A"   and "g abs_summable_on B" shows "(\<Sum>\<^sub>a(x, y)\<in>A \<times> B. f x * g y) = infsetsum f A * infsetsum g B"
fds_totient_times_zeta: shows "fds (\<lambda>n. of_nat (totient n)) * fds_zeta = fds of_nat"
sup_of_two_least: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "is_lub L (\<Squnion>\<^bsub>L\<^esub>{x, y}) {x, y}"
iter_widen_termination: fixes P :: "'a acom \<Rightarrow> bool"   and f :: "'a acom \<Rightarrow> 'a acom"   and m :: "'a acom \<Rightarrow> nat"   and C :: "'a acom" assumes "\<And>C. P C \<Longrightarrow> P (f C)"   and "\<And>C1 C2. \<lbrakk>P C1; P C2\<rbrakk> \<Longrightarrow> P (C1 \<nabla> C2)"   and "\<And>C1 C2. \<lbrakk>P C1; P C2; \<not> C2 \<le> C1\<rbrakk> \<Longrightarrow> m (C1 \<nabla> C2) < m C1"   and "P C" shows "\<exists>C'. iter_widen f C = Some C'"
ad0: shows "adjoint (tensor_P (1\<^sub>m N) M0) = tensor_P (1\<^sub>m N) M0"
eFreshAbs_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var"   and eA :: "'d withERR" assumes "\<not> eFreshAbs MOD ys y eA" shows "\<exists>us s A. eA = OK A \<and> \<not> igFreshAbs MOD ys y A \<and> igWlsAbs MOD (us, s) A"
neg_less_iff_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus b) (uminus a) = less a b"
vimage_inter_cong: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and y :: "'d set" assumes "\<And>w. w \<in> S \<Longrightarrow> f w = g w" shows "f -` y \<inter> S = g -` y \<inter> S"
deterministic: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a \<noteq> targetnode a'"   and "intra_kind (kind a)"   and "intra_kind (kind a')" shows "\<exists>Q Q'. kind a = (Q)\<^sub>\<surd> \<and> kind a' = (Q')\<^sub>\<surd> \<and> (\<forall>s. (Q s \<longrightarrow> \<not> Q' s) \<and> (Q' s \<longrightarrow> \<not> Q s))"
borel_prod: shows "borel \<Otimes>\<^sub>M borel = borel"
node_ptr_casts_commute3: fixes ptr :: "(_) object_ptr" assumes "is_node_ptr_kind ptr" shows "cast (the (cast ptr)) = ptr"
Sup_inter_less_eq: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (A \<inter> B)) (inf (Sup A) (Sup B))"
\<Delta>'_flow_attainability: assumes "\<alpha> \<noteq> \<top>" shows "flow_attainability \<Delta>'"
simps_1: fixes l :: "'d tree"   and v :: "'d"   and r :: "'d tree" shows "right \<langle>l, v, r\<rangle> = r"