hcju/isabelleps huggingface dataset:
infer_unitE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> Unit = Some \<tau>" shows "\<tau> = TUnit"
map_preserves_comp: fixes g :: "'c"   and f :: "'c" assumes "Cop.seq g f" shows "local.map (g \<cdot>\<^sup>o\<^sup>p f) = local.map g \<cdot>\<^sub>S local.map f"
poly_add: fixes p1 :: "'a list"   and p2 :: "'a list"   and x :: "'a" shows "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
rel_frontier_of_polyhedron: fixes S :: "'b set" assumes "polyhedron S" shows "rel_frontier S = \<Union> {F. F facet_of S}"
cut_cut_less: fixes i :: "'a set"   and a :: "'a"   and b :: "'a" shows "i \<down>< a \<down>< b = i \<down>< min a b"
35: shows "i < length y \<longrightarrow> vec_vec_Tensor (z # zs) y ! i = z * y ! i"
appendPath2_rev: 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 C :: "char list"   and Cs :: "char list list"   and Ds :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "Subobjs P (last Cs) (last Cs # rev Ds)"   and "wf_prog wf_md P" shows "Subobjs P C (Cs @ tl (last Cs # rev Ds))"
pmf_eq_iff: fixes M :: "'a pmf"   and N :: "'a pmf" shows "(M = N) = (\<forall>i. pmf M i = pmf N i)"
Int_UNIV_left: fixes B :: "'b set" shows "UNIV \<inter> B = B"
bounded_norm_comp: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
smult_dvd: fixes p :: "'b poly"   and q :: "'b poly"   and a :: "'b" assumes "p dvd q"   and "a \<noteq> (0::'b)" shows "smult a p dvd q"
hyps: fixes i :: "nat" shows "i < length xs \<longrightarrow> local.times a xs ! i = a * xs ! i"
FrameParPres: fixes \<Psi>' :: "'b"   and \<Psi>\<^sub>U :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi"   and A\<^sub>U :: "name list" assumes "(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel"   and "extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>"   and "A\<^sub>U \<sharp>* \<Psi>'"   and "A\<^sub>U \<sharp>* S"   and "A\<^sub>U \<sharp>* T" shows "(\<Psi>', U \<parallel> S, U \<parallel> T) \<in> Rel"
defined_args_valid: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and y :: "'a state \<times> 'a state \<Rightarrow> Set('b)" shows "(\<tau> \<Turnstile> \<delta> (x \<doteq> y)) = (\<tau> \<Turnstile> \<upsilon> x \<and> \<tau> \<Turnstile> \<upsilon> y)"
integral_unique: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and k :: "'a set" assumes "(f has_integral y) k" shows "integral k f = y"
elims: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "'c list"   and xb :: "'b list"   and y :: "'b list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'c \<Rightarrow> 'b" and bs :: "'b list" where "x = f__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'c \<Rightarrow> 'b" and a :: "'c" and as :: "'c list" and bs :: "'b list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
finite_dom_map_of: fixes l :: "('d \<times> 'e) list" shows "finite (dom (map_of l))"
ans3: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.antidomain_near_semiring antidomain_op plus times one zero less_eq less" shows "plus (antidomain_op (antidomain_op x)) (antidomain_op x) = one"
nodes_\<alpha>g_aux: fixes g :: "'m" assumes "invar g" shows "nodes (\<alpha>g g) = \<alpha>nodes_aux g"
poly_nth_root_def: fixes n :: "nat"   and p :: "'a poly" shows "poly_nth_root n p = p \<circ>\<^sub>p Polynomial.monom (1::'a) n"
N_2: shows "fds_nth h N \<noteq> (0::'a)"
act_rews_1: fixes v :: "state \<Rightarrow> 'c"   and s :: "state"   and t :: "state" shows "($v) (s, t) \<equiv> v s"
eNorm_def: fixes N :: "'b quasinorm"   and x :: "'b" shows "eNorm N x = snd (Rep_quasinorm N) x"
set_mult_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and K :: "'c set" shows "H <#>\<^bsub>G\<^esub> K = (\<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes>\<^bsub>G\<^esub> k})"
monomial_eq_itself: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" shows "monomial (punit.lc p) (punit.lt p) = p"
pref_prod_long: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v \<cdot> w"   and "\<^bold>|v\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "v \<le>p u"
bind_pmf_const: fixes M :: "'d pmf"   and c :: "'c pmf" shows "M \<bind> (\<lambda>x. c) = c"
cyclic_on_singleI: fixes s :: "'e"   and S :: "'e set"   and f :: "'e \<Rightarrow> 'e" assumes "s \<in> S"   and "S = orbit f s" shows "cyclic_on f S"
aux_2_2: fixes m :: "'node" assumes "m \<noteq> n" shows "Mapping.lookup u' m = Mapping.lookup (fst (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))) m"
insertion_num_params_cong: fixes p :: "'a poly"   and e :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "\<And>i. i < num_params p \<Longrightarrow> e i = f i" shows "insertion e p = insertion f p"
path_take_from: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and m :: "nat"   and l2' :: "'a" assumes "graph weights"   and "graph.path weights l1 l2 xs"   and "m < length xs"   and "(case xs ! m of (s, l, t) \<Rightarrow> s) = l2'" shows "graph.path weights l1 l2' (take m xs)"
resInputRight: fixes x :: "name"   and a :: "name"   and y :: "name"   and Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> y"   and "eqvt Rel"   and "Id \<subseteq> Rel" shows "a<y>.(<\<nu>x>P) \<leadsto>[Rel] <\<nu>x>a<y>.P"
ereal_of_enat_SUP: fixes A :: "'a set"   and f :: "'a \<Rightarrow> enat" assumes "A \<noteq> {}" shows "ereal_of_enat (Sup (f ` A)) = (SUP a\<in>A. ereal_of_enat (f a))"
inv_comp_right_1: fixes g :: "'d"   and f :: "'d" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso f" shows "D.inv (g \<cdot>\<^sub>D f) = D.inv f \<cdot>\<^sub>D D.inv g"
infer_appE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and A :: "'a trm"   and B :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> (App A B) = Some \<tau>" shows "\<exists>\<sigma>. infer \<Gamma> A = Some (TArr \<sigma> \<tau>) \<and> infer \<Gamma> B = Some \<sigma>"
r_field: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and Cf :: "varDef list"   and es :: "exp list"   and fi :: "nat"   and ei :: "exp" assumes "fields(CT,C) = Cf"   and "lookup2 Cf es (\<lambda>fd. vdName fd = fi) = Some ei" shows "CT \<turnstile> FieldProj (New C es) fi \<rightarrow> ei"
assms_3: shows "ind_in_set A a < ind_in_set A b"
fpxs_val_leI: fixes f :: "'b fpxs"   and r :: "rat" assumes "fpxs_nth f r \<noteq> (0::'b)" shows "fpxs_val f \<le> r"
assms_2: shows "B \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
subsums_refl: fixes c :: "('d, 'e) conf" shows "c \<sqsubseteq> c"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (cod \<nu> \<star> cod \<pi>)"
TS_inv'_det: fixes s :: "'a list \<times> 'a list"   and x :: "'a"   and i :: "'a list" shows "TS_inv' s x i = ((\<exists>hs. s = (if x = hd i then i else rev i, [x, x] @ hs)) \<or> s = (if x = hd i then i else rev i, []))"
E0'_def: shows "E0' = (\<Sum>\<^sub># (mset Cs0') + D0') \<cdot> \<tau>"
bit_word_scast_iff: fixes w :: "'c word"   and n :: "nat" shows "bit (scast w) n = (n < LENGTH('b) \<and> (bit w n \<or> LENGTH('c) \<le> n \<and> bit w (LENGTH('c) - Suc 0)))"
integrable_null_measure_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "integrable (null_measure M) f = (f \<in> borel_measurable M)"
mask_eqs_2: fixes a :: "'b word"   and b :: "'b word"   and n :: "nat" shows "a + (b AND mask n) AND mask n = a + b AND mask n"
hermitian_mat_lim_is_hermitian: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat" assumes "limit_mat X A m"   and "\<forall>n. hermitian (X n)" shows "hermitian A"
strategyproofness: shows "pmf (sds R12) c + pmf (sds R12) a \<le> pmf (sds R23) c + pmf (sds R23) a"
prefix_to_wordinterval_def2: fixes pfx :: "'b prefix_match" shows "prefix_to_wordinterval pfx = iprange_interval (pfxm_prefix pfx, pfxm_prefix pfx || pfxm_mask pfx)"
id: shows "{..<CARD('n)} = {0..<CARD('n)}"
op_fmap_is_empty_def: shows "op_fmap_is_empty \<equiv> (=) fmempty"
disjoint_Ax_mod2: shows "Ax \<inter> modRules2 = {}"
star_closed_unfold: fixes x :: "'a" assumes "x\<^sup>\<star> = x" shows "x = (1::'a) + x \<cdot> x"
path_continuous_image: fixes g :: "real \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "path g"   and "continuous_on (path_image g) f" shows "path (f \<circ> g)"
rep_eq: fixes x :: "nat"   and xa :: "float"   and xb :: "nat" shows "real_of_float (power_down_fl x xa xb) = power_down x (real_of_float xa) xb"
must_sync_def: fixes t :: "'t"   and x :: "'x"   and m :: "'m" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong> = (\<exists>taa x' m' s. t \<turnstile> \<langle>x, m\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle> \<and> shr s = m \<and> actions_ok s t taa)"
get_M_pure: fixes ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w"   and h :: "(_) heap" shows "pure (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h"
is_group: shows "Group.group (G\<lparr>carrier := conj.stabilizer P\<rparr>)"
assms_3: shows "snd (cs (S t i) cid) \<noteq> Done"
valid_refs_invD_4: fixes x :: "'e"   and m :: "'f"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and y :: "'e" assumes "x \<in> mut_m.tso_store_refs m s"   and "(x reaches y) s"   and "valid_refs_inv s" shows "\<exists>obj. sys_heap s y = Some obj"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y \<cdot> d z = x \<cdot> (y \<cdot> d z)"
rid: fixes x :: "'b \<times> real" assumes "x \<in> S'" shows "r x = x"
interp_ins_append_memb: fixes pre :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and suf :: "('a \<times> 'a option) list" assumes "insert_ops (pre @ [(oid, Some ref)] @ suf)"   and "ref \<in> set (interp_ins pre)" shows "oid \<in> set (interp_ins (pre @ [(oid, Some ref)] @ suf))"
cases: fixes a1 :: "nat"   and a2 :: "'a \<Rightarrow> 'a" assumes "swapidseq a1 a2" obtains "a1 = 0" and "a2 = id"   | n :: "nat" and p :: "'a \<Rightarrow> 'a" and a :: "'a" and b :: "'a" where "a1 = Suc n" and "a2 = a \<rightleftharpoons>\<^sub>F b \<circ> p" and "swapidseq n p" and "a \<noteq> b"
eq_shift: fixes w :: "'c stream"   and u :: "'c list"   and v :: "'c stream" shows "(w = u @- v) = (stake (length u) w = u \<and> sdrop (length u) w = v)"
esize_finite: fixes x :: "'a" assumes "esize x \<noteq> \<infinity>" shows "finite {y. y \<le> x}"
shconf_def: fixes P :: "'b prog"   and h :: "heap"   and sh :: "sheap" shows "P,h \<turnstile>\<^sub>s sh \<surd> \<equiv> \<forall>C sfs i. sh C = \<lfloor>(sfs, i)\<rfloor> \<longrightarrow> P,h,C \<turnstile>\<^sub>s sfs \<surd>"
cCons_def: fixes x :: "'b"   and xs :: "'b list" shows "cCons x xs = (if xs = [] \<and> x = (0::'b) then [] else x # xs)"
concatMapL_strict: fixes f :: "'c \<rightarrow> 'b LList" shows "concatMapL\<cdot>f\<cdot>\<bottom> = \<bottom>"
path_restrict_tl: fixes u :: "'b"   and R :: "'b set"   and E :: "('b \<times> 'b) set"   and p :: "'b list"   and v :: "'b" assumes "u \<notin> R"   and "path (E \<inter> UNIV \<times> - R) u p v" shows "path (rel_restrict E R) u p v"
less_float_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "(a < b) = flt a b"
fresh_PairD_1: fixes a :: "atom"   and x :: "'e"   and y :: "'f" assumes "a \<sharp> (x, y)" shows "a \<sharp> x"
rcosets_subset_PowG: fixes H :: "'a set" assumes "subgroup H H'" shows "rcosets\<^bsub>H'\<^esub> H \<subseteq> Pow (carrier H')"
elims: fixes x :: "term"   and y :: "name option" assumes "unconst x = y" obtains name :: "name" where "x = Const name" and "y = Some name"   | v :: "name" where "x = Free v__" and "y = None"   | v :: "term" where "x = \<Lambda> va__" and "y = None"   | v :: "nat" where "x = Bound vb__" and "y = None"   | v :: "term" and va :: "term" where "x = v $ va" and "y = None"
apath_Cons_iff: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and w :: "'a" shows "apath u (e # es) w = (e \<in> arcs G \<and> tail G e = u \<and> apath (head G e) es w \<and> tail G e \<notin> set (awalk_verts (head G e) es))"
3: shows "((F \<circ> real_of_ereal) \<longlongrightarrow> 0) (at_right (ereal 0))"
inner_Basis_sup_left: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (sup x y) i = sup (inner x i) (inner y i)"
IH: assumes "\<not> n < 10" shows "string_of_word True 10 0 (n div 10) = string_of_nat (unat (n div 10))"
qtables: shows "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length aux) \<phi>)) \<phi>s rels"
less_eq_prefix_match_def: fixes a :: "'a prefix_match"   and b :: "'a prefix_match" shows "(a \<le> b) = (if pfxm_length a = pfxm_length b then pfxm_prefix a \<le> pfxm_prefix b else pfxm_length b < pfxm_length a)"
well_order_on_Field: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "well_order_on A r" shows "A = Field r"
simps_2: fixes uv :: "'a infty" shows "(uv \<le> Infty) = True"
app_vimage_set_def: fixes r :: "V"   and A :: "V" shows "r `\<^sub>\<circ> A = set {b. \<exists>a\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
RestrictedP_LstSeqP: fixes s :: "tm"   and k :: "tm"   and s' :: "tm"   and y :: "tm" shows "{RestrictedP s (SUCC k) s', LstSeqP s k y} \<turnstile> LstSeqP s' k y"
split_sel_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "\<langle>'b\<rangle>\<^sub>\<bottom>" shows "P (case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (\<not> (option = \<bottom> \<and> \<not> P f1 \<or> option = \<lfloor>the option\<rfloor> \<and> \<not> P (f2 (the option))))"
div_diff1_eq_if: fixes b :: "nat"   and a :: "nat"   and m :: "nat" shows "(b - a) div m = b div m - a div m - (if a mod m \<le> b mod m then 0 else Suc 0)"
l_delete_shadow_root_get_tag_name_def: fixes get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_tag_name get_tag_name_locs \<equiv> \<forall>h shadow_root_ptr h' r ptr'. h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr' \<longrightarrow> r h h'"
\<pi>_pos_iff: fixes x :: "real" shows "(0 < \<pi> x) = (2 \<le> x)"
convex_halfspace_gt: fixes b :: "real"   and a :: "'b" shows "convex {x. b < a \<bullet> x}"
bv_not_Cons: fixes b :: "bit"   and bs :: "bit list" shows "bv_not (b # bs) = (\<not>\<^sub>b b) # bv_not bs"
sublattice_def: shows "sublattice \<equiv> SIGMA cl:CompleteLattice. {S. S \<subseteq> pset cl \<and> \<lparr>pset = S, order = induced S (order cl)\<rparr> \<in> CompleteLattice}"
sym_digraphI: fixes G :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "symmetric G" shows "sym_digraph G"
fSum_zero1: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "aGroup A"   and "n < m"   and "\<forall>j\<in>nset (Suc n) m. f j = \<zero>\<^bsub>A\<^esub>" shows "\<Sigma>\<^sub>f A f Suc n m = \<zero>\<^bsub>A\<^esub>"
conn2_4: shows "gs.lin_indpt (set (map of_int_hom.vec_hom fs'))"
simps_7: fixes v :: "'b poly" shows "isnpolyh (Neg v) = (\<lambda>k. False)"
fps_divide_shift_denom: fixes n :: "nat"   and g :: "'a fps"   and f :: "'a fps" assumes "n \<le> subdegree g"   and "subdegree g \<le> subdegree f" shows "f / fps_shift n g = Abs_fps (\<lambda>k. if k < n then 0::'a else fps_nth (f / g) (k - n))"
iMOD_cut_greater1: fixes t :: "nat"   and r :: "nat"   and m :: "nat" assumes "t \<in> [ r, mod m ]" shows "[ r, mod m ] \<down>> t = (if m = 0 then {} else [ t + m, mod m ])"
pochhammer_double: fixes z :: "'a"   and n :: "nat" shows "pochhammer ((2::'a) * z) (2 * n) = of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + (1::'a) / (2::'a)) n"
disc_1: fixes x1 :: "'h" shows "isl (Inl x1)"
join_rtrancl_join: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> A\<^sup>\<down>"   and "(c, b) \<in> A\<^sup>*" shows "(a, c) \<in> A\<^sup>\<down>"
llength_lmap: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "llength (lmap f xs) = llength xs"
higher_differentiable_on_imp_differentiable_on: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and k :: "nat" assumes "higher_differentiable_on S f k"   and "0 < k" shows "f differentiable_on S"
S_def: shows "S = \<Union> {set (a i) |i. True}"
pdg_digraphs: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "pdigraph_base \<alpha> I \<AA>"   and "i \<in>\<^sub>\<circ> I" shows "digraph \<alpha> (\<AA> i)"
sum_over_subset_pmf_support: fixes T :: "'b set"   and p :: "'b pmf"   and a :: "real"   and f :: "'b \<Rightarrow> real" assumes "finite T"   and "set_pmf p \<subseteq> T" shows "(\<Sum>i\<in>T. a * pmf p i * f i) = (\<Sum>i\<in>set_pmf p. a * pmf p i * f i)"
read_def: fixes c :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'a process" shows "read c A P \<equiv> Mprefix (c ` A) (P \<circ> inv c)"
BF: fixes \<alpha> :: "freeRes"   and Q' :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'"   and "x \<sharp> \<alpha>" shows "\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', <\<nu>x>Q') \<in> Rel"
closure_Int_ball_not_empty: fixes S :: "'a set"   and T :: "'a set"   and x :: "'a"   and r :: "real" assumes "S \<subseteq> closure T"   and "x \<in> S"   and "0 < r" shows "T \<inter> ball x r \<noteq> {}"
l4_5: 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 C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Cong A C A' C'" shows "\<exists>B'. Bet A' B' C' \<and> Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'"
finite_fmdom': fixes m :: "('a, 'b) fmap" shows "finite (fmdom' m)"
IFset_natural: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e IF1"   and y :: "'e IF2" shows "f ` IF1set x = IF1set (IF1map f x) \<and> f ` IF2set y = IF2set (IF2map f y)"
finite_atLeastLessThan_integer: fixes l :: "integer"   and u :: "integer" shows "finite {l..<u}"
comp_arr_dom: fixes f :: "'a1"   and a :: "'a1" assumes "C1.arr f"   and "C1.dom f = a" shows "f \<cdot>\<^sub>1 a = f"
digraphs_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "ZFC_in_HOL.set {\<CC>. digraph \<alpha> \<CC>} \<in>\<^sub>\<circ> Vset \<beta>"
dims_tensor: fixes v :: "'b list"   and d :: "nat list" assumes "length v = prod_list d" shows "dims (tensor_from_vec d v) = d"
pdom_pfun_graph_finite: fixes f :: "('a, 'b) pfun" assumes "finite (pdom f)" shows "finite (pfun_graph f)"
narrow_st_def: shows "(\<triangle>) \<equiv> map_fun rep_st (map_fun rep_st St) (map2_st_rep (\<triangle>))"
bheight_size_bound: fixes t :: "('b \<times> color) tree" assumes "invc t"   and "invh t" shows "2 ^ bheight t \<le> size1 t"
values_set: fixes q :: "('c, 'd) pq" shows "set |q| = fst ` set (alist_of q)"
trivial_derived_seq_imp_solvable: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set"   and n :: "nat" assumes "Group.group G"   and "subgroup H G"   and "(derived G ^^ n) H = {\<one>\<^bsub>G\<^esub>}" shows "solvable_seq G H"
units_of_pow: fixes x :: "nat \<times> nat"   and n :: "nat" assumes "x \<in> Units S" shows "x [^]\<^bsub>units_of S\<^esub> n = x [^]\<^bsub>S\<^esub> n"
axiom_addimp_def: fixes p :: "fol fm"   and q :: "fol fm" shows "axiom_addimp p q \<equiv> Thm (Imp p (Imp q p))"
indRelRTPO_impl_TRel_weakly_reflects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelRTPO TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs TRel TWB"
FAssNone: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
proj_atomize_poly: fixes k :: "nat"   and p :: "nat \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (atomize_poly p) = lookup p k"
remdups_adj_Cons: fixes x :: "'a"   and xs :: "'a list" shows "remdups_adj (x # xs) = (case remdups_adj xs of [] \<Rightarrow> [x] | y # xs \<Rightarrow> if x = y then y # xs else x # y # xs)"
assms: shows "subgroup U (G\<lparr>carrier := H\<rparr>)"
bind_right_unit: fixes m :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E m unit\<^sub>S\<^sub>E = m"
uses'_all_def: fixes g :: "'g" shows "uses'_all g \<equiv> fst (substAll g)"
Sum_any_right_distrib: fixes g :: "'c \<Rightarrow> 'd"   and r :: "'d" assumes "finite {a. g a \<noteq> (0::'d)}" shows "r * Sum_any g = (\<Sum>n. r * g n)"
A\<^sub>0B: shows "PAC_Format\<^sup>*\<^sup>* (\<V>\<^sub>0, A\<^sub>0) (\<V>, B)"
same_fw_behaviour_one_equi_1: fixes x :: "'a word"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list" shows "same_fw_behaviour_one x x c rs"
marked_not_white: fixes r :: "'d"   and s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'd) Model.local_state" assumes "white r s" shows "\<not> marked r s"
tendsto_minus_cancel_left: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and F :: "'c filter" shows "(f \<longlongrightarrow> - y) F = ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
edges_of_comp2: fixes ys :: "'a list"   and xs :: "'a list" shows "H.edges_of_walk ys \<subseteq> H.edges_of_walk (xs @ ys)"