hcju/isabelleps huggingface dataset:
inf_seq_distrib: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "seq_distrib_right (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) ((c\<^sub>0::'b::refinement_lattice) \<sqinter> (c\<^sub>1::'b::refinement_lattice)) (d::'b::refinement_lattice) = seq c\<^sub>0 d \<sqinter> seq c\<^sub>1 d"
compress_normalize_primitve_preserves_normalized_n_primitive: fixes disc2 :: "'c \<Rightarrow> bool"   and sel2 :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool"   and m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'e"   and C :: "'e \<Rightarrow> 'c"   and f :: "'e negation_type list \<Rightarrow> ('e list \<times> 'e list) option"   and m' :: "'c match_expr" assumes "normalized_n_primitive (disc2, sel2) P m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc2 (C a)"   and "normalized_nnf_match m"   and "compress_normalize_primitive (disc, sel) C f m = Some m'" shows "normalized_nnf_match m' \<and> normalized_n_primitive (disc2, sel2) P m'"
le_numeral_extra_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
lt_max_keys: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "v \<preceq>\<^sub>t lt p"
memberid_Base_foo_simp: shows "memberid (mdecl Base_foo) = mid foo_sig"
inv: shows "q.M (ilc * lead_coeff f) = 1"
simps_1: fixes uu :: "'b"   and acc :: "'b list" shows "remove_cycles [] uu acc = rev acc"
poincare_map_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.poincare_map f X P x = auto_ll_on_open.flow0 f X x (auto_ll_on_open.return_time f X P x)"
is_lub_bin_chain: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "below x y" shows "po.is_lub below (range (\<lambda>i. if i = 0 then x else y)) y"
a_eq: shows "Sup (emeasure M ` {Q \<in> sets M. emeasure N Q \<noteq> \<infinity>}) = emeasure M (\<Union>i. \<Union> (Q' ` {..i}))"
generated_ring_disjoint_Un: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> generated_ring"   and "b \<in> generated_ring"   and "a \<inter> b = {}" shows "a \<union> b \<in> generated_ring"
a_Hcarr: fixes h :: "'a" assumes "h \<in> H" shows "h \<in> carrier G"
sprod_mHom_closed: fixes N :: "('g, 'b, 'h) Module_scheme"   and a :: "'b"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "a \<in> carrier R"   and "f \<in> mHom R M N" shows "sprod_mHom R M N a f \<in> mHom R M N"
inj_on: shows "inj_on (\<lambda>I i. if i < n then sorted_list_of_set I ! i else i) {I. I \<subseteq> {0..<m} \<and> card I = n}"
assms: shows "fmdom' (snd a) \<inter> s = {}"
when_distrib: fixes b :: "bool"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "\<not> b \<Longrightarrow> f (0::'b) = (0::'a)" shows "f (a when b) = (f a when b)"
Mp_Mp_pow_is_Mp: fixes n :: "nat"   and p :: "int"   and f :: "int poly" assumes "n \<noteq> 0"   and "1 < p" shows "poly_mod.Mp p (poly_mod.Mp (p ^ n) f) = poly_mod.Mp p f"
convergent_prod_offset: fixes f :: "nat \<Rightarrow> 'e"   and m :: "nat" assumes "convergent_prod (\<lambda>n. f (n + m))" shows "convergent_prod f"
leq_antisym: fixes x :: "'c llist"   and y :: "'c llist" assumes "lprefix x y"   and "lprefix y x" shows "x = y"
prems_2: shows "le\<cdot>(length\<cdot>(x : xs) + n)\<cdot>0 = TT"
type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap" assumes "type_wf h" shows "ShadowRootClass.type_wf h"
bit_imp_le_length: fixes w :: "'a word"   and n :: "nat" assumes "bit w n" shows "n < LENGTH('a)"
pivot_positions_4: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat nr nc"   and "pivot_fun A f nc" shows "length (pivot_positions A) = card {i. i < nr \<and> row A i \<noteq> 0\<^sub>v nc}"
one_le_card_finite: shows "Suc 0 \<le> CARD('a)"
psubset_sphere_Compl_connected: fixes S :: "'a set"   and a :: "'a"   and r :: "real" assumes "S \<subset> sphere a r"   and "0 < r"   and "2 \<le> DIM('a)" shows "connected (- S)"
assms_5: shows "\<forall>pair\<in>a. finite (snd pair)"
IH: fixes x :: "('sf, 'sz) trm" assumes "x \<in> range x2a"   and "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT x. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)" shows "sterm_sem (local.adjoint I \<sigma> \<nu>) x = sterm_sem (local.adjoint I \<sigma> \<omega>) x"
iIN_union: fixes n :: "nat"   and n' :: "nat"   and d :: "nat"   and d' :: "nat" assumes "n \<le> n'"   and "n' \<le> Suc (n + d)"   and "n + d \<le> n' + d'" shows "[n\<dots>,d] \<union> [n'\<dots>,d'] = [n\<dots>,n' - n + d']"
sums_scaleR_right: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a"   and r :: "real" assumes "X sums a" shows "(\<lambda>n. r *\<^sub>R X n) sums (r *\<^sub>R a)"
measurable_bind_prob_space: fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'d \<Rightarrow> 'e measure"   and R :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M prob_algebra N"   and "g \<in> N \<rightarrow>\<^sub>M prob_algebra R" shows "(\<lambda>x. f x \<bind> g) \<in> M \<rightarrow>\<^sub>M prob_algebra R"
inject_2: fixes x31 :: "'b"   and x32 :: "'b"   and x33 :: "'c"   and y31 :: "'b"   and y32 :: "'b"   and y33 :: "'c" shows "(AllowPortFromTo x31 x32 x33 = AllowPortFromTo y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
length_list_of_eucl: fixes x :: "'c" shows "length (list_of_eucl x) = DIM('c)"
Un_in_lattice: fixes x :: "'b set"   and L :: "'b set set"   and y :: "'b set" assumes "x \<in> L"   and "y \<in> L"   and "lattice L" shows "x \<union> y \<in> L"
Tangle_Diagram_Equivalence_def: fixes T1 :: "Tangle_Diagram"   and T2 :: "Tangle_Diagram" shows "T1 ~T T2 \<equiv> Rep_Tangle_Diagram T1 ~ Rep_Tangle_Diagram T2"
dis_rews_3: fixes x :: "['b]" shows "(null\<cdot>x = \<bottom>) = (x = \<bottom>)"
dom_release_shared_eq: fixes sb :: "'b memref list"   and \<S>' :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat set" shows "dom (release sb \<S>' \<R>) = dom (release sb \<S> \<R>)"
pemute_minus_self: fixes p :: "perm" shows "- p \<bullet> p = p"
sources_are_isomorphic: fixes a :: "'a"   and \<mu> :: "'a"   and a' :: "'a" assumes "a \<in> sources \<mu>"   and "a' \<in> sources \<mu>" shows "a \<cong> a'"
fpairs_vsubset_fdomain_frange: fixes r :: "V" shows "fpairs r \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<bullet> r \<times>\<^sub>\<bullet> \<R>\<^sub>\<bullet> r"
disjoint_eq_subset_Compl: fixes A :: "'b set"   and B :: "'b set" shows "(A \<inter> B = {}) = (A \<subseteq> - B)"
succsr_isNode: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> succsr\<^sup>*"   and "x \<in> reach (set \<Sigma>) \<delta> q\<^sub>0" shows "y \<in> reach (set \<Sigma>) \<delta> q\<^sub>0"
E19: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<box>\<diamond>\<box>F"
indep_in_subset_carrier: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep_system.indep_in indep \<E> X" shows "X \<subseteq> \<E>"
dropWhile_filter': fixes xs :: "(nat \<times> 'a) list"   and nt :: "nat"   and c :: "nat" assumes "sorted (map fst xs)"   and "\<forall>t\<in>fst ` set xs. t \<le> nt" shows "dropWhile (\<lambda>(t, X). c \<le> nt - t) xs = filter (\<lambda>(t, X). nt - t < c) xs"
sel_1: fixes x21 :: "('c, 'd) tree"   and x22 :: "'c"   and x23 :: "'d"   and x24 :: "('c, 'd) tree" shows "lsub (Node x21 x22 x23 x24) = x21"
assms_2: fixes n :: "'a" shows "continuous_on (cbox a b) (f n)"
spr_sim_rep_abs: fixes ec :: "'s odlist"   and a :: "'a"   and t :: "'s Trace" assumes "spr_simAbs ec = SPRsingle.sim_equiv_class a t" shows "toSet ec = spr_abs t"
point_indices_elem_in: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and ps :: "'c set"   and t :: "nat" assumes "incidence_system point_set block_collection"   and "ps \<subseteq> point_set"   and "card ps = t" shows "int (block_collection index ps) \<in> incidence_system.point_indices point_set block_collection (int t)"
density_RN_deriv_density: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure" assumes "f \<in> borel_measurable M" shows "density M (RN_deriv M (density M f)) = density M f"
assms: fixes a :: "'a set"   and b :: "'b set" assumes "a \<in> sets A"   and "b \<in> sets B" shows "a \<times> b \<in> sets N"
Texp_def: fixes e :: "exp" shows "Texp e = (\<Lambda> a. ccTTree (edom (\<A>\<^bsub>a\<^esub> e)) (\<G>\<^bsub>a\<^esub> e))"
le_add_same_cancel1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus a b) = less_eq zero b"
primepow_def: fixes n :: "'b" shows "primepow n = (\<exists>p k. prime p \<and> 0 < k \<and> n = p ^ k)"
subcls_notin_has_fields2: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and D :: "char list" assumes "P \<turnstile> C has_fields FDTs"   and "C \<noteq> Object"   and "P \<turnstile> C \<prec>\<^sup>1 D" shows "(D, C) \<notin> (subcls1 P)\<^sup>*"
divide_diff_eq_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "minus (divide x z) y = divide (minus x (times y z)) z"
elements_diag_block_mat: fixes As :: "'b mat list" shows "elements_mat (diag_block_mat As) \<subseteq> {0::'b} \<union> \<Union> (set (map elements_mat As))"
exhaust: fixes y :: "('b, 'c, 'd) sum3" obtains x1 :: "'b" where "y = In1 x1"   | x2 :: "'c" where "y = In2 x2"   | x3 :: "'d" where "y = In3 x3"
min_dist_minD: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "dist v d v'" shows "min_dist v v' \<le> d"
CConcEnd: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and y :: "unit decision"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "Cp a x = \<lfloor>y\<rfloor>" shows "Cp (list2FWpolicy (xs @ [a])) x = \<lfloor>y\<rfloor>"
assms_2: shows "M' = M @ [(uLiteral, False)]"
check_precond_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o = (s\<^sub>e < s\<^sub>o \<and> 0 \<le> v\<^sub>e \<and> 0 \<le> v\<^sub>o \<and> a\<^sub>e < 0 \<and> a\<^sub>o < 0)"
less_add_one: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g"   and a :: "'g" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less a (plus a one)"
infinite_Un: fixes S :: "'c set"   and T :: "'c set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
bind_def: fixes action\<^sub>1 :: "'d io"   and action\<^sub>2 :: "'d \<Rightarrow> 'c io" shows "action\<^sub>1 \<bind> action\<^sub>2 = Abs_io (\<lambda>world\<^sub>0. let (a, world\<^sub>1) = Rep_io action\<^sub>1 world\<^sub>0; (b, world\<^sub>2) = Rep_io (action\<^sub>2 a) world\<^sub>1 in (b, world\<^sub>2))"
msetext_dersh_def: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" shows "msetext_dersh gt ys xs = (let N = mset ys; M = mset xs in \<exists>Y X. Y \<noteq> {#} \<and> Y \<subseteq># N \<and> M = N - Y + X \<and> (\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> gt y x)))"
dom_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VVV.dom f = (if VVV.arr f then VxVxV.dom f else VxVxV.null)"
simps_4: fixes uu :: "'sets set" shows "public (Abs uu) = False"
frechet_derivative_transform_within_open_ext: fixes f :: "'a \<Rightarrow> 'b"   and xa :: "'a"   and X :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f differentiable at xa"   and "open X"   and "xa \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x" shows "frechet_derivative f (at xa) x = frechet_derivative g (at xa) x"
card_w_len_m_bound: shows "real (card (cw_len_concat -` {m})) \<le> b ^ m"
prod_rw: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> F" shows "(\<Prod>i = 0..<n. B $$ (f i, i)) = (\<Prod>i = 0..<n. B $$ (f (\<pi> i), \<pi> i))"
pfp_pfp: fixes f :: "'a \<Rightarrow> 'a"   and x0 :: "'a"   and x :: "'a" assumes "pfp f x0 = Some x" shows "f x \<le> x"
mid_preserves_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and M :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Col A B C"   and "M Midpoint A A'"   and "M Midpoint B B'"   and "M Midpoint C C'" shows "Col A' B' C'"
assms_3: shows "s' \<in> set_pmf \<mu>"
pred_cong: fixes x :: "'a option"   and ya :: "'a option"   and P :: "'a \<Rightarrow> bool"   and Pa :: "'a \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set_option ya \<Longrightarrow> P z = Pa z" shows "pred_option P x = pred_option Pa ya"
l_get_owner_document_axioms: shows "l_get_owner_document get_owner_document"
cod_simps_2: shows "\<nu>\<pi>.cod.leg0 = \<nu>\<pi>.leg0"
conc_trans_additional_5: fixes A :: "'b nres"   and B :: "'b nres"   and C :: "'b nres" assumes "A \<le> \<Down> Id B"   and "B \<le> C" shows "A \<le> C"
rep_eq: fixes x :: "'e \<Longrightarrow> 'f"   and xa :: "('g, 'f) uexpr" shows "(x \<natural> xa) = (\<forall>b b'. \<lbrakk>xa\<rbrakk>\<^sub>e (b' \<triangleleft>\<^bsub>x\<^esub> b) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
s_top_commute_sg: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "K \<diamondop> H = H \<diamondop> K" shows "G \<guillemotright> H \<diamondop> K "
gSubstPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstPresGWls MOD \<equiv> \<forall>ys Y' Y y s X' X. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> gWls MOD s (gSubst MOD ys Y' Y y X' X)"
distinct_36: fixes x31 :: "Syntax.var"   and x32 :: "Syntax.term"   and x2 :: "Syntax.var" shows "Lam x31 x32 \<noteq> Var x2"
simps_6: fixes uz :: "nat" shows "(extN.N uz \<le> extN.Bot) = False"
MM10: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ F = G" shows "\<turnstile> \<box>[F]_v = \<box>[G]_v"
Rep_List_cases: fixes y :: "('b, unit) Old_Datatype.node set" assumes "y \<in> List" obtains x :: "'b SList.list" where "y = Rep_List x"
INT_subset_iff: fixes B :: "'c set"   and A :: "'d \<Rightarrow> 'c set"   and I :: "'d set" shows "(B \<subseteq> \<Inter> (A ` I)) = (\<forall>i\<in>I. B \<subseteq> A i)"
elims_3: fixes x :: "'e list"   and xa :: "'e list set"   and xb :: "'e list" assumes "\<not> mcp x xa xb" obtains z :: "'e list" and W :: "'e list set" and p :: "'e list" where "x = z" and "xa = W" and "xb = p" and "\<not> (prefix p z \<and> p \<in> W \<and> (\<forall>p'. prefix p' z \<and> p' \<in> W \<longrightarrow> length p' \<le> length p))"
lens_indep_comm: fixes x :: "'e \<Longrightarrow> 'f"   and y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'g"   and u :: "'e" assumes "x \<bowtie> y" shows "put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"
fun_upds_append2_drop: fixes xs :: "'a list"   and ys :: "'b list"   and m :: "'a \<Rightarrow> 'b option"   and zs :: "'b list" assumes "length xs = length ys" shows "m(xs [\<mapsto>] ys @ zs) = m(xs [\<mapsto>] ys)"
assms_3: fixes i :: "'a" assumes "i \<in> B" shows "i \<bullet> x = i \<bullet> y"
l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes \<and> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"
conv_star_commute: fixes x :: "'a" shows "x\<^sup>\<star>\<^sup>T = x\<^sup>T\<^sup>\<star>"
iadd_SUP_le_iff: shows "(n + \<Squnion> (f ` A) \<le> y) = (if A = {} then n \<le> y else \<forall>x\<in>A. n + f x \<le> y)"
f: shows "finite {x. f x \<noteq> (0::'a)}"
eq_upto_BAD_mem: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "f \<in> eq_upto (BAD P) g i" shows "f j \<in> A"
supp_perm_singleton: fixes p :: "perm"   and b :: "atom" shows "(supp p \<subseteq> {b}) = (p = 0)"
star_if_lang: fixes w :: "'a list"   and A :: "'a list set" assumes "w \<in> A" shows "w \<in> star A"
invertible_inverse_closed: fixes u :: "'b" assumes "A.additive.invertible u"   and "u \<in> A" shows "A.additive.inverse u \<in> A"
frechet_vec_lambda: fixes x :: "real"   and T :: "real set"   and f :: "real \<Rightarrow> ('c, 'b) vec"   and f' :: "real \<Rightarrow> ('c, 'b) vec" assumes "\<forall>i. ((\<lambda>y. (f y $ i - f (netlimit (at x within T)) $ i - (y - netlimit (at x within T)) *\<^sub>R f' x $ i) /\<^sub>R (\<parallel>y - netlimit (at x within T)\<parallel>)) \<longlongrightarrow> (0::'c)) (at x within T)" shows "((\<lambda>y. (f y - f (netlimit (at x within T)) - (y - netlimit (at x within T)) *\<^sub>R f' x) /\<^sub>R (\<parallel>y - netlimit (at x within T)\<parallel>)) \<longlongrightarrow> 0) (at x within T)"
len_lexext_rev_trans_strong: fixes zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" 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 "len_lexext_rev gt zs ys"   and "len_lexext_rev gt ys xs" shows "len_lexext_rev gt zs xs"
t1_space_nhds: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.t1_space open"   and "x \<noteq> y" shows "\<forall>\<^sub>F x in topological_space.nhds open x. x \<noteq> y"
foundation18': fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<upsilon> X) = (X \<tau> \<noteq> \<bottom>)"
repr_independence2: fixes U :: "'a set"   and g :: "'a" assumes "Group.group G"   and "U \<in> rcosets H"   and "g \<in> U" shows "U = H #> g"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "??.Sublist.prefix b a"   and "??.Sublist.prefix c b" shows "??.Sublist.prefix c a"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c" shows "(case Point x21 x22 of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
equivp_inf: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "equivp R"   and "equivp S" shows "equivp (inf R S)"
empty_Collect_eq: fixes P :: "'h \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
case_nat_unfold: fixes x :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "case n of 0 \<Rightarrow> x | Suc x \<Rightarrow> f x \<equiv> if n = 0 then x else f (n - 1)"
typing_implies_valid: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : T" shows "valid \<Gamma>"
not_proj2_set_Col_iff_span: fixes S :: "proj2 set" assumes "card S = 3" shows "(\<not> proj2_set_Col S) = (span (proj2_rep ` S) = UNIV)"
basis_def: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme"   and A :: "'i set" assumes "vectorspace K V" shows "vectorspace.basis K V A = (\<not> LinearCombinations.module.lin_dep K V A \<and> LinearCombinations.module.span K V A = carrier V \<and> A \<subseteq> carrier V)"
foldl_conv_foldr: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b"   and xs :: "'c list" shows "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
diff_self_eq_0: fixes m :: "nat" shows "(m::nat) - m = 0"
weakBisimParPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<rhd> P \<parallel> R \<approx> Q \<parallel> R"
subfield_Units: fixes K :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subfield K R" shows "Units (R\<lparr>carrier := K\<rparr>) = K - {\<zero>\<^bsub>R\<^esub>}"
Nml_HcompD_5: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "\<not> is_Prim\<^sub>0 u"
Form_quot_fm: fixes e :: "(name, hf) finfun"   and A :: "fm" shows "Form \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"
TokensAt_items_le: fixes k :: "nat"   and I :: "item set" shows "TokensAt k (items_le k I) = TokensAt k I"
simplex_map_def: fixes p :: "nat"   and g :: "'c \<Rightarrow> 'b"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" shows "simplex_map p g c \<equiv> restrict (g \<circ> c) (standard_simplex p)"