hcju/isabelleps huggingface dataset:
xor_closed: fixes x :: "'e"   and y :: "'e" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus> y \<in> carrier L"
absi: fixes i :: "'N" assumes "i \<in> Basis" shows "\<exists>g. g absolutely_integrable_on cbox a b \<and> ((\<forall>x\<in>cbox a b. f x \<bullet> i \<le> g x) \<or> (\<forall>x\<in>cbox a b. g x \<le> f x \<bullet> i))"
lift_def: fixes return :: "'e \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'e"   and x :: "'d" shows "monad_base.lift return bind f x = bind x (\<lambda>x. return (f x))"
arg_min_natI: fixes P :: "'d \<Rightarrow> bool"   and k :: "'d"   and m :: "'d \<Rightarrow> nat" assumes "P k" shows "P (arg_min m P)"
dim_is_n: fixes n :: "nat" shows "vectorspace.dim class_ring (module_vec TYPE('b) n) = n"
bij_is_inj: fixes f :: "'h \<Rightarrow> 'i" assumes "bij f" shows "inj f"
CDMode_CTRL_Action2_def: shows "CDMode_CTRL_Action2 = ({}, CDMode_CTRL_Update2)"
nu_exchange_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq g"   and "ord.isotone less_eq h"   and "order.has_greatest_postfixpoint less_eq (u \<circ> h)"   and "order.has_greatest_postfixpoint less_eq (h \<circ> g)"   and "order.has_greatest_fixpoint less_eq (g \<circ> h)"   and "ord.lifted_less_eq less_eq (g \<circ> h \<circ> u) (u \<circ> h \<circ> g)" shows "less_eq (order.the_greatest_fixpoint less_eq (g \<circ> h)) (order.the_greatest_fixpoint less_eq (u \<circ> h))"
inf_sup_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> y = x"
simps_2: fixes e :: "nat exp"   and es :: "nat exp list" shows "compEs\<^sub>2 (e # es) = compE\<^sub>2 e @ compEs\<^sub>2 es"
Invariants_in_sets: fixes A :: "'a set" assumes "A \<in> sets Invariants" shows "A \<in> sets M"
meeting_paths_produce_cycle: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G (v # xs) \<and> distinct (v # xs)"   and "xs \<noteq> []"   and "Graph.walk G (v # ys) \<and> distinct (v # ys)"   and "ys \<noteq> []"   and "last xs = last ys"   and "hd xs \<noteq> hd ys" shows "\<exists>zs. Graph.cycle G zs \<and> hd zs = v"
wf_gglobal_def: fixes gc :: "(cmd, local_state, global_state) Gen_Scheduler.global_config" shows "li'.wf_gglobal gc \<equiv> \<forall>lc. lc \<in># global_config.processes gc \<longrightarrow> ty_local (command lc) (local_config.state lc) (global_config.state gc)"
h: shows "continuous_map (prod_topology (top_of_set {0..1}) X1) Y1 h"
cupcake_pmatch_preserve0: fixes pat :: "pat"   and v0 :: "v"   and env :: "(char list \<times> v) list"   and envC :: "(char list, char list, nat \<times> tid_or_exn) namespace" assumes "is_cupcake_pat pat"   and "is_cupcake_value v0"   and "list_all (is_cupcake_value \<circ> snd) env"   and "cupcake_c_ns envC" shows "if_match (list_all (is_cupcake_value \<circ> snd)) (cupcake_pmatch envC pat v0 env)"
lookupa_unicity: fixes x :: "name"   and a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and y :: "name"   and b :: "coname" assumes "lookupa x a \<theta>_c = Ax y b"   and "b \<sharp> \<theta>_c"   and "y \<sharp> \<theta>_c" shows "x = y \<and> a = b"
val_of_nonzero_1: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "f \<in> padic_set p"   and "f \<noteq> padic_zero p"   and "prime p" shows "f (nat (padic_val p f) + 1) \<noteq> \<zero>\<^bsub>residue_ring (p ^ (nat (padic_val p f) + 1))\<^esub>"
subsetI: fixes A :: "'e set"   and B :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
proj5_def: shows "\<pi>\<^sub>5 \<equiv> \<lambda>(uu_, uu_, uu_, hops, uu_, uu_). hops"
eventually_at_filter: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" shows "eventually P (at a within s) = (\<forall>\<^sub>F x in nhds a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
Ord_\<phi>: fixes \<gamma> :: "V"   and k :: "nat" assumes "\<gamma> \<in> elts (\<omega> \<up> \<alpha>s ! k)"   and "k < length \<alpha>s" shows "Ord (\<phi> (\<omega>_sum (take k \<alpha>s) + \<gamma>))"
assms_4: shows "Res_ivl \<subseteq> Csafe"
S1: shows "c_len u = Suc (c_len u - 1)"
degree_var_i: fixes x :: "nat"   and i :: "nat" shows "MPoly_Type.degree (Var x ^ i) x = i"
ang_vec_c_sym: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle>c z1 z2 = \<angle>c z2 z1"
lm112: fixes Z :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "Z \<subseteq> X \<union> Y"   and "finite Z" shows "sum (X <| Y) Z = card (Z \<inter> X)"
psubstAbs_compose: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodAbs A"   and "goodEnv rho"   and "goodEnv rho'" shows "A $[rho] $[rho'] = A $[rho &[rho']]"
assms: shows "\<forall>x\<in>set xs. \<forall>y\<in>set xs. (f1 x = f1 y) = (f2 x = f2 y)"
CONST_INTF_def: fixes c :: "'g"   and I :: "interface" shows "c ::\<^sub>i I \<equiv> True"
g_2: shows "permute_list g (map fst ys) = map fst xs"
real_eqI: fixes a :: "real"   and b :: "real" assumes "a - b = 0" shows "a = b"
prefix_elem_to_carriers: fixes history :: "nat \<Rightarrow> 'c list"   and xs :: "'c list"   and i :: "nat"   and x :: "'c" assumes "node_histories history"   and "node_histories.prefix_of_node_history history xs i"   and "x \<in> set xs" shows "x \<in> set (history i)"
vsubset_vsingleton_left: fixes a :: "V"   and A :: "V" shows "(ZFC_in_HOL.set {a} \<subseteq>\<^sub>\<circ> A) = (a \<in>\<^sub>\<circ> A)"
lless_double_def: fixes x :: "double"   and y :: "double" shows "lless_double x y = (lcompare_double x y < 0)"
5: shows "a - (X \<union> {i}) \<times> Range a = {} \<or> {(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})} = {} \<longrightarrow> a - (X \<union> {i}) \<times> Range a \<union> ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})}) \<in> allocationsUniverse \<and> \<Union> (Range (a - (X \<union> {i}) \<times> Range a \<union> ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})}))) = \<Union> (Range a)"
echelon_form_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "echelon_form A = echelon_form_upt_k A (ncols A)"
dim_mono: fixes V :: "'a set"   and W :: "'a set" assumes "V \<subseteq> span W" shows "dim V \<le> dim W"
Gamma_1: shows "Gamma (1::'a) = (1::'a)"
that_1: assumes "p = []"   and "t = s" shows "thesis"
implies_antitone_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<leadsto> z \<le> x \<leadsto> z"
simps_1: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b" shows "Partial_Cost_Model.T_on' A s [] = 0"
iT_Div_0_finite: fixes I :: "nat set" shows "finite (I \<oslash> 0)"
simps_2: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('a \<times> 'b) spmf"   and p :: "'b spmf"   and a :: "'c"   and b :: "'a"   and xs :: "('c \<times> 'a) list"   and x :: "'c" shows "trace_callee callee p ((a, b) # xs) x = trace_callee callee (cond_spmf_fst (p \<bind> (\<lambda>s. callee s a)) b) xs x"
conv_contrav: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<rhd> (1::'a) = (y \<rhd> (1::'a)) \<cdot> (x \<rhd> (1::'a))"
preserves_ide: fixes a :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.ide a" shows "B.ide (B.HoVH a)"
grid_def: shows "grid \<equiv> \<lambda>b ds. {x. gridp b (\<lambda>x. x \<in> ds) x}"
a: shows "{i. i < length qs \<and> qs ! i \<in> {y, x}} = {..<length qs} \<inter> {i. qs ! i \<in> {y, x}}"
one_dim_iso_of_zero: shows "one_dim_iso (0::'c) = (0::'b)"
c: shows "summable (\<lambda>x. (\<MM> (real x) - ln (real x) + c) / real x)"
poly_map_poly_eval_poly: fixes p :: "int poly" shows "poly (of_int_poly p) = eval_poly of_int p"
abs_eq: shows "circline_point_0 = abs_circline circline_point_0_clmat"
periodic_integer_multiple: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "(\<forall>x. f (x + a) = f x) = (\<forall>x. \<forall>n\<in>\<int>. f (x + n * a) = f x)"
CD_on_domain: fixes d :: "'b"   and ds :: "'b set"   and A :: "'a set" assumes "d \<in> ds" shows "Singleton_for_d.CD_on ds A \<inter> Field (Pd_singletons_for_ds X {Xd x} d) = Singleton_for_d.Cd d (A \<inter> Field (Pd_singletons_for_ds X {Xd x} d))"
R_implies_SR_for_modified_view: fixes \<V>' :: "'a V_rec" assumes "R \<V>' Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "SR \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
enc_length: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list" assumes "w \<noteq> []"   and "enc (w, I) = enc (w', I')" shows "length I = length I'"
cases: fixes a :: "'b llist"   and A :: "'b set" assumes "a \<in> A\<^sup>\<star>" obtains "a = LNil"   | l :: "'b llist" and aa :: "'b" where "a = aa ## l" and "l \<in> A\<^sup>\<star>" and "aa \<in> A"
prv_ldsj_mono_imp: fixes \<phi>1s :: "'fmla list"   and \<phi>2s :: "'fmla list" assumes "set \<phi>1s \<subseteq> fmla"   and "set \<phi>2s \<subseteq> fmla"   and "\<forall>\<phi>1\<in>set \<phi>1s. \<exists>\<phi>2\<in>set \<phi>2s. prv (imp \<phi>1 \<phi>2)" shows "prv (imp (ldsj \<phi>1s) (ldsj \<phi>2s))"
assms: fixes a :: "'b"   and c :: "'a" assumes "P a" shows "hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)"
ih: shows "iexp x = (\<Sum>k\<le>n. (\<i> * complex_of_real x) ^ k / fact k) + \<i> ^ Suc n / fact n * (CLBINT s=0..ereal x. f s n * iexp s)"
assigns_mk_I_FO: fixes m :: "nat"   and I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset"   and bs1 :: "WS1S_Nameful.fo list"   and bs2 :: "WS1S_Nameful.so list" shows "m\<^bsup>mk_I I1 I2 bs1 bs2\<^esup>FO = (if m < length bs1 then {|I1 (bs1 ! m)|} else {||})"
simps_2: fixes x :: "char list"   and e :: "aexp"   and P :: "'b" shows "strip (x ::= e {P}) = x ::= e"
mono_filter_plus': fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'a"   and b2 :: "'a"   and r :: "'a"   and r' :: "'a" assumes "a1 \<sqsubseteq> b1"   and "a2 \<sqsubseteq> b2"   and "r \<sqsubseteq> r'" shows "filter_plus' r a1 a2 \<sqsubseteq> filter_plus' r' b1 b2"
simps_2: fixes x :: "complex"   and i :: "nat"   and xs :: "(complex \<times> nat) list" shows "complex_roots_to_real_factorization ((x, i) # xs) = (if x \<in> \<real> then ([:- Re x, 1:], Suc i) # complex_roots_to_real_factorization xs else let xx = cnj x; ys = delete_cnj xx (Suc i) xs; p = map_poly Re ([:- x, 1:] * [:- xx, 1:]) in (p, Suc i) # complex_roots_to_real_factorization ys)"
distinct_3: fixes x1 :: "rat"   and x31 :: "int poly"   and x32 :: "nat" shows "Rat_Info x1 \<noteq> Real_Alg_Info x31 x32"
OrdDom_hrestrict: fixes s :: "hf"   and A :: "hf" assumes "OrdDom s" shows "OrdDom (hrestrict s A)"
h0a: shows "dim_row (take_rows_from_matrix (alt_matrix_A_R signs subsets) S) = length (take_indices subsets S)"
I: fixes c :: "sequent"   and F :: "char list"   and x :: "NominalSequents.var"   and A :: "form"   and ps :: "'a list" assumes "NominalSequents.mset c = \<LM> F \<nabla> [x].A  \<RM>"   and "ps \<noteq> []" shows "(ps, c) \<in> nprovRules"
insert_absorb: fixes a :: "'a"   and A :: "'a set" assumes "s2p A a" shows "insert a A = A"
\<delta>ssp_\<delta>ss_eq: shows "\<delta>ssp = (\<lambda>x. x \<in> \<delta>ss)"
is_natural_1: fixes f :: "'a" assumes "A.arr f" shows "A.map f \<cdot>\<^sub>A A.map (A.dom f) = A.map f"
B: fixes ys :: "'a list"   and zs :: "'a list" shows "xs \<simeq> {ys, zs, P} \<longrightarrow> map f xs \<simeq> {map f ys, map f zs, \<lambda>w ws. P (inv f w) (map (inv f) ws)}"
unifiers_insert: fixes p :: "('b, 'a) Term.term \<times> ('b, 'a) Term.term"   and t :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "unifiers (insert p t) = {\<sigma>. fst p \<cdot> \<sigma> = snd p \<cdot> \<sigma>} \<inter> unifiers t"
even_mask_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one) = (n = 0)"
Rep_G_cases: fixes y :: "'b" assumes "y \<in> G" obtains x :: "'b G" where "y = Rep_G x"
V3_def: shows "V3 = filter (\<lambda>v. vertextype g v = (5, 0, 1)) V2"
hom_mult_eq_zero: fixes x :: "int"   and y :: "int" assumes "x * y = 0" shows "of_int x * of_int y = (0::'a)"
iso_in_homI: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] b\<guillemotright>"   and "Cop_S.iso f" shows "Cop_S.iso_in_hom f a b"
neg_equal_iff_equal: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(uminus a = uminus b) = (a = b)"
induct: fixes P :: "('a list \<Rightarrow> bool) \<Rightarrow> 'a list \<times> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list \<Rightarrow> bool"   and a1 :: "'a list \<times> 'a list"   and a2 :: "'a list"   and a3 :: "'a list" assumes "\<And>Pa res ps. P Pa res ps []"   and "\<And>Pa res ps q qs. P Pa (if Pa ps then (ps, q # qs) else res) (ps @ [q]) qs \<Longrightarrow> P Pa res ps (q # qs)" shows "P a0 a1 a2 a3"
mk_safe_coll_def: fixes ode_ops :: "ode_ops"   and X :: "'c set" shows "mk_safe_coll ode_ops X = sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>x. mk_safe ode_ops x \<bind> (\<lambda>s. RETURN (mk_coll s))) (\<lambda>b c. RETURN (b \<union> c)))"