hcju/isabelleps huggingface dataset:
Impl1: fixes A :: "triple set"   and P :: "'a \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Cm :: "cname \<times> mname"   and Q :: "'a \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Z :: "'a" assumes "\<forall>Z. A \<union> (\<Union>x. {(P x Cm, Impl Cm, Q x Cm)}) \<turnstile> {P Z Cm} body Cm {Q Z Cm}" shows "A \<turnstile> {P Z Cm} Impl Cm {Q Z Cm}"
RBTi1_axioms: shows "RBTi1 TYPE('b) (\<lambda>x xa. real xa \<le> log 2 (real (x + 1)) + 1)"
in_homE: fixes f :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" assumes "V.in_hom f a b" obtains "V.arr f" and "V.dom f = a" and "V.cod f = b"
Suc_double_half: fixes n :: "nat" shows "Suc (2 * n) div 2 = n"
TAtom_term_cases: fixes t :: "('fun, 'var) Term.term"   and \<tau> :: "'atom" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<Gamma> t = Var \<tau>" shows "(\<exists>v. t = Var v) \<or> (\<exists>f. t = Fun f [])"
path_entry_gt: fixes E :: "(nat \<times> nat) set"   and n :: "nat"   and d :: "nat"   and n' :: "nat" assumes "\<forall>p. path_entry E p n \<longrightarrow> d \<in> set p"   and "\<forall>p. path_entry E p n' \<longrightarrow> n \<in> set p" shows "\<forall>p. path_entry E p n' \<longrightarrow> d \<in> set p"
components_are_iso: fixes a :: "'a" assumes "ide a" shows "local.iso (R a)"
2: shows "(- (x ^ 3) + 12 * x\<^sup>2 - 96 * x + 384) ^ 4 / 21743271936 \<le> inverse (exp x)"
ev_order: shows "\<forall>x. cmod x = 1 \<longrightarrow> order x (map_poly complex_of_real (char_poly A)) \<le> d"
simps_5: fixes a :: "nat rexp"   and b :: "nat rexp"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "subst (Times a b) \<sigma> = Times (subst a \<sigma>) (subst b \<sigma>)"
inverse_unique: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "a +\<^sub>o\<^sub>w b = 0\<^sub>o\<^sub>w" shows "-\<^sub>o\<^sub>w a = b"
prems_4: shows "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> p \<and> occurs_on (t ! k) \<noteq> q"
hyps_2: shows "ik\<^sub>e\<^sub>s\<^sub>t A \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t' \<cdot> \<I>"
fmrel_drop: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'g \<rightharpoonup>\<^sub>f 'e"   and n :: "'g \<rightharpoonup>\<^sub>f 'f"   and a :: "'g" assumes "fmrel P m n" shows "fmrel P (fmdrop a m) (fmdrop a n)"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "S.arr f"   and "S.dom f = a" shows "f \<cdot>\<^sub>S a = f"
simple_match_valid_alt: shows "simple_match_valid = (\<lambda>m. (let c = \<lambda>(s, e). s \<noteq> 0 \<or> e \<noteq> - 1 in if c (sports m) \<or> c (dports m) then proto m = Proto TCP \<or> proto m = Proto UDP \<or> proto m = Proto L4_Protocol.SCTP else True) \<and> valid_prefix_fw (src m) \<and> valid_prefix_fw (dst m))"
nt_zero0: fixes a :: "'a" assumes "a \<in> carrier A" shows "0\<triangleright>a\<^bsub>A\<^esub> = \<zero>"
T1: fixes a :: "letter"   and ws :: "letter list list"   and zs :: "letter list list"   and w :: "letter list" assumes "T a ws zs" shows "T a (w # ws) ((a # w) # zs)"
Rep: fixes x :: "'a word" shows "unat x \<in> unats LENGTH('a)"
exhaust: fixes y :: "('c, 'd) list_op" obtains x11 :: "'c option" and x12 :: "'d" where "y = Insert x11 x12"   | x2 :: "'c" where "y = Delete x2"
test_inf_right_below: fixes x :: "'a"   and y :: "'a" assumes "test x" shows "y \<sqinter> x \<le> y"
f4: shows "rk {a, b, c, \<beta>, \<gamma>} = 3"
poly_root_factor_1: fixes r :: "'b"   and q :: "'b poly"   and k :: "'b" shows "(poly ([:r, 1::'b:] * q) k = (0::'b)) = (k = - r \<or> poly q k = (0::'b))"
extend_2: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "f \<in> sub.diff_fun_space"   and "x \<in> closure B" shows "f x = extend f x"
distinct_87: fixes x6 :: "nat"   and x101 :: "nat"   and x102 :: "msg" shows "Passwd x6 \<noteq> Crypt x101 x102"
lookup_eq: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list" shows "lookup env xs = (case xs of [] \<Rightarrow> Some env | x # xs \<Rightarrow> case env of Val x \<Rightarrow> Map.empty x | Env b es \<Rightarrow> case es x of None \<Rightarrow> None | Some e \<Rightarrow> lookup e xs)"
has_domain_iff_arr: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(domains f \<noteq> {}) = arr f"
fvi_Suc: fixes x :: "nat"   and b :: "nat"   and \<phi> :: "formula" shows "(x \<in> fvi (Suc b) \<phi>) = (Suc x \<in> fvi b \<phi>)"
redT_updWs_WokenUp_SuspendD: fixes t :: "'a"   and ws :: "'a \<Rightarrow> 'b wait_set_status option"   and was :: "('a, 'b) wait_set_action list"   and ws' :: "'a \<Rightarrow> 'b wait_set_status option"   and w :: "'b wait_set_status" assumes "redT_updWs t ws was ws'"   and "Notified \<in> set was \<or> WokenUp \<in> set was"   and "ws' t = \<lfloor>w\<rfloor>" shows "\<exists>w. Suspend w \<in> set was"
passt: shows "{j. Suc j \<in> S} \<subseteq> {..<length as}"
s: shows "Partial_Cost_Model.config'_rand BIT (type1 init x y) b = type0 init x y"
keys_single: fixes k :: "'c"   and v :: "'d" shows "Poly_Mapping.keys (Poly_Mapping.single k v) = (if v = (0::'d) then {} else {k})"
assms: shows "(u \<longlongrightarrow> 0) F"
e: shows "steps0 (Suc 0 + off, l, r) (A @ shift B off) (Suc n) = (0, l', r')"
mset_tree_nonempty: fixes t :: "'b tree" shows "mset_tree t \<noteq> {#}"
simps_6: fixes c :: "poly"   and s :: "tm" shows "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
distinct_11: fixes x3 :: "'b" shows "Rule x3 \<noteq> Helper"
card_singleton_union: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite S" shows "card (\<Union>x\<in>S. {f x}) \<le> card S"
elims_2: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b list" assumes "\<guillemotleft>x\<guillemotright> xa" obtains p :: "'b \<Rightarrow> bool" and a :: "'b" and S :: "'b list" where "x = p" and "xa = a # S" and "p a"
pp_dist_sup: fixes 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 uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.stone_algebra inf less_eq less sup bot top uminus" shows "uminus (uminus (sup x y)) = sup (uminus (uminus x)) (uminus (uminus y))"
ack_le_mono2: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j \<le> k" shows "ack i j \<le> ack i k"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.arr f"   and "C.dom f = a"   and "C.cod f = b" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>"
pick_alphaAbs_iff_equal: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and B :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodAbs A"   and "goodAbs B" shows "(pick A $= pick B) = (A = B)"
arg_min_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" shows "ord.arg_min less f P = (SOME x. ord.is_arg_min less f P x)"
G\<^sub>1_def: shows "G\<^sub>1 = (SOME G. \<exists>\<eta>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G F\<^sub>1 (fst \<eta>\<epsilon>) (snd \<eta>\<epsilon>))"
exhaust_sel: fixes option :: "'a option" obtains "option = None"   | "option = pure (the option)"
c: fixes i :: "nat" shows "emeasure M (B i) < \<infinity>"
f_le_p: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "f x y n \<le> p x y n"
AE_pmfI: fixes M :: "'b pmf"   and P :: "'b \<Rightarrow> bool" assumes "\<And>y. y \<in> set_pmf M \<Longrightarrow> P y" shows "almost_everywhere (measure_pmf M) P"
g3mod2_def: shows "g3mod2 \<equiv> {(xa, x). g3mod2p xa x}"
Dlt: shows "{x \<in> S. \<bar>det (matrix (g' x))\<bar> * f (g x) < 0} \<in> sets lebesgue"
per_union_impl: fixes i :: "'a"   and j :: "'a"   and R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "(i, j) \<in> R" shows "(i, j) \<in> per_union R a b"
witness_prop_def: fixes \<delta> :: "('d, 'e) ta_rule set"   and m :: "'d \<Rightarrow> 'e tree option" shows "witness_prop \<delta> m \<equiv> \<forall>q t. m q = Some t \<longrightarrow> accs \<delta> t q"
induced_automorphism_chamber_image_fg: fixes C :: "'a set" assumes "folding_g.chamber C" shows "\
 ` C = (if C \<in> f \<turnstile> folding_g.\<C> then g ` C else f ` C)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) \<eta>' \<epsilon>'"
ivl_disj_un_two_3: fixes l :: "'b"   and m :: "'b"   and u :: "'b" assumes "l \<le> m"   and "m \<le> u" shows "{l..<m} \<union> {m..<u} = {l..<u}"
Diag_Diagonalize_1: fixes t :: "'c local.term" assumes "Arr t" shows "Diag \<^bold>\<lfloor>t\<^bold>\<rfloor>"
ipurge_ref_aux_append: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and ys :: "'b list"   and X :: "'b set" shows "ipurge_ref_aux I D U (xs @ ys) X = ipurge_ref_aux I D (sinks_aux I D U xs) ys X"
HInv3_L_def: fixes s :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" shows "HInv3_L s p q d = (phase s p \<in> {1, 2} \<and> phase s q \<in> {1, 2} \<and> hasRead s p d q \<and> hasRead s q d p)"
flip_at_simps_1: fixes a :: "'b"   and b :: "'b" shows "(a \<leftrightarrow> b) \<bullet> a = b"
assms_6: shows "d \<le> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>"
OC_iff_indRelRTPO_is_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_corresponding (TRel\<^sup>*) \<and> weak_reduction_bisimulation (TRel\<^sup>+) Target) = weak_reduction_bisimulation (indRelRTPO TRel) (STCal Source Target)"
sat_push_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat" shows "sat_push_precond f l \<equiv> \<lambda>(u, v). (0::'capacity) < excess f u \<and> cf_of f (u, v) \<le> excess f u \<and> (u, v) \<in> cfE_of f \<and> l u = l v + 1"
simps_6: fixes p :: "char list"   and es :: "expr list"   and rets :: "char list list" shows "#:Call p es rets = 2"
Rf_mono: fixes N :: "'a literal multiset set"   and N' :: "'a literal multiset set" assumes "N \<subseteq> N'" shows "Rf N \<subseteq> Rf N'"
Reverse_nfa_def: fixes MS :: "'b dfa" shows "Reverse_nfa MS = \<lparr>nfa.states = dfa.states MS, init = dfa.final MS, final = {dfa.init MS}, nxt = \<lambda>q x. {q' \<in> dfa.states MS. q = dfa.nxt MS q' x}, eps = {}\<rparr>"
subgroup_inverse_equality: fixes u :: "'a \<Rightarrow> 'a" assumes "u \<in> Translations" shows "transformation.symmetric.inverse u = transformation.sub.inverse u"
welldefined_subsets: shows "all_list_constr_R subsets (length qs)"
ennreal_zero_divide: fixes x :: "ennreal" shows "0 / x = 0"
echelon_form_fold_Hermite_of_row_i: fixes A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat" assumes "echelon_form A"   and "ass_function ass"   and "res_function res" shows "echelon_form (foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<k]))"
d_approx_main: fixes fs :: "int vec list"   and ii :: "nat" assumes "LLL_invariant_weak fs"   and "g_bound fs"   and "ii \<le> m"   and "m \<noteq> 0" shows "rat_of_int (d fs ii) \<le> rat_of_nat (N ^ ii)"
rep_abs_mndet: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}" shows "Rep_process (Abs_process (\<Union>x\<in>A. F (x \<rightarrow> P x), \<Union>x\<in>A. D (x \<rightarrow> P x))) = (\<Union>x\<in>A. F (x \<rightarrow> P x), \<Union>x\<in>A. D (x \<rightarrow> P x))"
g': shows "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))"
simps_30: fixes vd :: "'a"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "(SKIP {vd} \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
reduction_deletes_bad_sign_conds_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "set (characterize_consistent_signs_at_roots p qs) = set (reduction_signs_R p qs signs subsets (M_mat_R signs subsets))"
cycle_def: fixes G :: "('c, 'd) pre_digraph"   and p :: "'d list" shows "pre_digraph.cycle G p \<equiv> \<exists>u. pre_digraph.awalk G u p u \<and> distinct (tl (pre_digraph.awalk_verts G u p)) \<and> p \<noteq> []"
set_Cons_def: fixes A :: "'g set"   and XS :: "'g list set" shows "set_Cons A XS = {x # xs |x xs. x \<in> A \<and> xs \<in> XS}"
to_bl_eq_rev: fixes w :: "'a word" shows "to_bl w = map (bit w) (rev [0..<LENGTH('a)])"
sorted2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and zs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # y # zs) = (less_eq x y \<and> linorder.sorted less_eq (y # zs))"
Dom_cod: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Dom (cod f) = Cod f"
update_by_fun_gr_def: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "'x" shows "update_by_fun_gr k f xs = oalist_of_list (update_by_fun_gr_raw k f (list_of_oalist xs))"
a2: fixes t :: "'a" assumes "t \<in> keys p" shows "lookup p t = lookup q t"
find_closest_bf_code_eq: fixes ps :: "(int \<times> int) list"   and c :: "int \<times> int"   and p :: "int \<times> int"   and \<delta>' :: "int"   and c' :: "int \<times> int" assumes "0 < length ps"   and "c = find_closest_bf p ps"   and "(\<delta>', c') = find_closest_bf_code p ps" shows "c = c'"
simps_5: fixes Vs :: "char list list"   and V :: "char list" shows "compE\<^sub>1 Vs (Var V) = Var (last_index Vs V)"
initial_error: shows "dist x0 x0' \<le> euler_C TYPE('a) / B' * (exp 1 - 1) * stepsize 0"
lt_\<omega>_imp_ex_of_nat: fixes M :: "hmultiset" assumes "M < \<omega>" shows "\<exists>n. M = of_nat n"
abs_eq: fixes x :: "int poly" shows "roots_of_real_alg x = map Real_Alg_Quotient (roots_of_3 x)"
word_ops_nth_2: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" shows "(x AND y) !! n = (x !! n \<and> y !! n)"
Hash_LtK: fixes X :: "msg"   and X' :: "ltkey" shows "Hash X \<noteq> LtK X'"
simps_1: fixes x :: "'a" shows "pp_first (Node x) = x"
rep_eq: fixes x :: "('b, 'c) mapping" shows "lookup x = rep x"
same_uptoD: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and i' :: "nat"   and j' :: "nat" assumes "same_upto n j A B"   and "i' < n"   and "j' < j" shows "A $$ (i', j') = B $$ (i', j')"
esssup_AE: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "AE x in M. f x \<le> esssup M f"
iso_\<beta>\<^sub>\<tau>: shows "local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])"
true_lit_simps_2: fixes I :: "'a set"   and A :: "'a" shows "(I \<Turnstile>l Neg A) = (A \<notin> I)"
mulex_on_multisetsD_2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset" assumes "mulex_on P F M N" shows "N \<in> multisets F"
tauChainInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "(P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"   and "F P"   and "\<And>P'' P'''. \<lbrakk>(P, P'') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*; P'' \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'''; F P''\<rbrakk> \<Longrightarrow> F P'''" shows "F P'"
WT_converter_bot1: fixes \<I> :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" shows "\<bottom>, \<I> \<turnstile>\<^sub>C conv \<surd>"
match_term_list_None: fixes P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option" assumes "match_term_list P \<sigma> = None" shows "matchers_map \<sigma> \<inter> matchers (set P) = {}"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_uminus" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_uminus" and "-\<^sub>\<int> a = b"
less_imp_inverse_less: fixes a :: "'a"   and b :: "'a" assumes "a < b"   and "(0::'a) < a" shows "inverse b < inverse a"
IH: assumes "boundary_chain (set l)"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "valid_chain_list l" shows "valid_path (rec_join l)"
mset_poly_rel_def: fixes \<phi> :: "char list \<Rightarrow> nat" assumes "poly_embed \<phi>" shows "poly_embed.mset_poly_rel \<phi> = {(a, b). b = poly_embed.polynomial_of_mset \<phi> a}"
prefixes_snoc: fixes xs :: "'c list"   and x :: "'c" shows "prefixes (xs @ [x]) = prefixes xs @ [xs @ [x]]"
all_derived_clauses_are_wellconstrained: fixes C :: "'a eclause"   and S :: "'a eclause set" assumes "derivable_ecl C S"   and "Ball S well_constrained" shows "well_constrained C"
d1_def: shows "ps_P.d1 = prod_list ps_P.dims1"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c list" shows "map g (map f v) = map (g \<circ> f) v"
in_idem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "min x (Min A) = Min A"
nsum_0: fixes R :: "('d, 'e) aGroup_scheme"   and f :: "nat \<Rightarrow> 'd" shows "\<Sigma>\<^sub>e R f 0 = f 0"
par: shows "S.par (\<Psi> (\<Phi> f)) f"
iFROM_Suc: fixes x :: "nat"   and n :: "nat" assumes "x \<in> [n\<dots>]" shows "Suc x \<in> [n\<dots>]"
inverse_eq_divide: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse a = divide one a"
x_in_dom': shows "x \<in> dom (to_partial mem |` Y)"
rel_spmf_try_spmf: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d spmf"   and p' :: "'e spmf"   and q :: "'d spmf"   and q' :: "'e spmf" assumes "rel_spmf R p p'"   and "\<not> lossless_spmf p' \<Longrightarrow> rel_spmf R q q'" shows "rel_spmf R (TRY p ELSE q) (TRY p' ELSE q')"
integral: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes>\<^bsub>K\<^esub> b = \<zero>\<^bsub>K\<^esub>"   and "a \<in> carrier K"   and "b \<in> carrier K" shows "a = \<zero>\<^bsub>K\<^esub> \<or> b = \<zero>\<^bsub>K\<^esub>"
bilateral_substitutes_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Bossy.bilateral_substitutes_on A f = (\<not> (\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A \<and> Xd a \<notin> Xd ` B \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<and> b \<in> f (B \<union> {a, b})))"
suffix_ConsD: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x # xs \<le>s ys" shows "xs \<le>s ys"
group_homomorphism_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" shows "group_homomorphism \<eta> G composition unit G' composition' unit' \<equiv> monoid_homomorphism \<eta> G composition unit G' composition' unit' \<and> Group_Theory.group G composition unit \<and> Group_Theory.group G' composition' unit'"
supto_split1: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "j \<le> k" shows "supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>k) = supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>(j - 1)) :@ supto\<cdot>(MkI\<cdot>j)\<cdot>(MkI\<cdot>k)"
insertI: fixes A :: "'e set"   and a :: "'e" assumes "finite A" shows "finite (Set.insert a A)"
subringI: fixes H :: "'a set" assumes "H \<subseteq> carrier R"   and "\<one> \<in> H"   and "\<And>h. h \<in> H \<Longrightarrow> \<ominus> h \<in> H"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H\<rbrakk> \<Longrightarrow> h1 \<otimes> h2 \<in> H"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H\<rbrakk> \<Longrightarrow> h1 \<oplus> h2 \<in> H" shows "subring H R"
cases: fixes x :: "('a, 'l, 'm) prog" obtains F :: "'a" and H :: "'l" and f :: "'m" where "x = Prog F H f"
abelian_groupE_2: fixes R :: "('e, 'f) ring_scheme" assumes "abelian_group R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
weakening: fixes \<Gamma>\<^sub>1 :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and \<Gamma>\<^sub>2 :: "(name \<times> ty) list" assumes "\<Gamma>\<^sub>1 \<turnstile> t : T"   and "valid \<Gamma>\<^sub>2"   and "\<Gamma>\<^sub>1 \<sqsubseteq> \<Gamma>\<^sub>2" shows "\<Gamma>\<^sub>2 \<turnstile> t : T"
rep_eq: fixes x :: "'a covector"   and xa :: "'a covector" shows "Rep_covector (x \<sqinter> xa) = Rep_covector x \<sqinter> Rep_covector xa"
prems: shows "l = map (\<lambda>x. (x, a)) (x # xs)"
subject_reduction': fixes t :: "dB"   and t' :: "dB"   and e :: "nat \<Rightarrow> LambdaType.type"   and T :: "LambdaType.type" assumes "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'"   and "e \<turnstile> t : T" shows "e \<turnstile> t' : T"
freshResidual: fixes P :: "pi"   and Rs :: "residual"   and x :: "name" assumes "P \<longmapsto> Rs"   and "x \<sharp> P" shows "x \<sharp> Rs"
welltyped_lc: fixes t :: "expr"   and T :: "Unary_PCF.type" assumes "t ::: T" shows "lc t"
PAC_checker_l_alt_def: fixes p :: "(char list list \<times> int) list"   and \<V>A :: "char list set \<times> (nat, (char list list \<times> int) list) fmap"   and status :: "char list code_status"   and steps :: "((char list list \<times> int) list, char list, nat) pac_step list" shows "PAC_checker_l p \<V>A status steps = (let (\<V>, A) = \<V>A in PAC_checker_l' p \<V> A status steps)"
1: shows "x = real (card G) - real (card (pseudoAllocation aa \<inter> pseudoAllocation a))"
refine: fixes A :: "('c \<times> 'd) set"   and Xi :: "'c list"   and X :: "'d set"   and P_impl :: "'c \<Rightarrow> bool nres"   and P :: "'d \<Rightarrow> bool nres"   and I :: "'d \<Rightarrow> bool" assumes "single_valued A"   and "(Xi, X) \<in> \<langle>A\<rangle>list_wset_rel"   and "(P_impl, P) \<in> A \<rightarrow> \<langle>bool_rel\<rangle>nres_rel" shows "(WEAK_ALL_list Xi P_impl, WEAK_ALL\<^bsup>I\<^esup> X P) \<in> \<langle>bool_rel\<rangle>nres_rel"
simps_2: fixes A :: "(('b, 'c) vec, 'd) vec"   and n :: "nat"   and i :: "'d"   and j :: "'c"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "bezout_iterate A (Suc n) i j bezout = (if Suc n \<le> mod_type_class.to_nat i then A else bezout_iterate (bezout_matrix A i (mod_type_class.from_nat (Suc n)) j bezout ** A) n i j bezout)"
partial_order_onD_2: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "partial_order_on A r" shows "trans r"
G_def: shows "G = {(i, j). A $h i $h j \<noteq> 0}"
fds_nth_one: shows "fds_nth 1 = (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
direct_decompI_alt: fixes ss :: "'a set list"   and A :: "'a set" assumes "\<And>qs. qs \<in> listset ss \<Longrightarrow> sum_list qs \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>!qs. qs \<in> listset ss \<and> a = sum_list qs" shows "direct_decomp A ss"
UP_cring: fixes R :: "('a, 'b) ring_scheme" assumes "UP_cring R" shows "cring (UP R)"
tendsto_rabs_zero_cancel: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"