hcju/isabelleps huggingface dataset:
add_decreasing2: fixes c :: "'k multiset"   and a :: "'k multiset"   and b :: "'k multiset" assumes "c \<subseteq># {#}"   and "a \<subseteq># b" shows "a + c \<subseteq># b"
LcmI: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and b :: "'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times a b"   and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times a c) \<Longrightarrow> dvd.dvd times b c"   and "normalize b = b" shows "b = Lcm A"
intros_4: fixes \<Gamma> :: "binding list"   and t :: "trm"   and S :: "ty"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : S"   and "\<Gamma>\<turnstile>S<:T" shows "\<Gamma> \<turnstile> t : T"
Rep_complex_homo_coords: fixes x :: "complex_homo_coords" shows "Rep_complex_homo_coords x \<in> {v. v \<noteq> vec_zero}"
interval_inter_times_UNIVI: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and c :: "'c"   and d :: "'c"   and e :: "'c \<times> 'd"   and f :: "'c \<times> 'd" assumes "{fst a..fst b} \<inter> {c..d} = {fst e..fst f}"   and "{snd a..snd b} = {snd e..snd f}" shows "{a..b} \<inter> {c..d} \<times> UNIV = {e..f}"
invrules_1: shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
linearize_def: shows "linearize \<equiv> map_fun Rep_queue id (\<lambda>q. case q of (fs, ls) \<Rightarrow> fs @ rev ls)"
oalist_inv_raw_tl: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs" shows "oalist_inv_raw (tl xs)"
H2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "\<turnstile> F" shows "\<turnstile> \<box>[F]_v"
only_call_get_return_edges: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>Q r p fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
absorb2: fixes a :: "'e"   and b :: "'e" assumes "a \<le> b" shows "max a b = b"
unwrap_def: shows "unwrap \<equiv> \<Lambda> f r. f\<cdot>(a2c\<cdot>r)"
strict_mono_compose: fixes r :: "'b \<Rightarrow> 'c"   and s :: "'d \<Rightarrow> 'b" assumes "strict_mono r"   and "strict_mono s" shows "strict_mono (\<lambda>x. r (s x))"
simps_4: fixes \<Gamma> :: "char list set"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "wfe \<Gamma> (Add e\<^sub>1 e\<^sub>2) = (wfe \<Gamma> e\<^sub>1 \<and> wfe \<Gamma> e\<^sub>2)"
r_mul: fixes a :: "nat"   and b :: "nat" shows "eval r_mul [a, b] \<down>= a * b"
floor_minus: fixes x :: "'a" shows "\<lfloor>- x\<rfloor> = - \<lceil>x\<rceil>"
prime_sum_of_squares_unique: fixes p :: "nat" assumes "prime p"   and "p = 2 \<or> [p = 1] (mod 4)" shows "\<exists>!(x, y). x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>A B. A < B \<Longrightarrow> f A < f B" shows "OrdinalInduct.strict_mono f"
compact_segment: fixes a :: "'a"   and b :: "'a" shows "compact (closed_segment a b)"
neq_top: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<noteq> \<top>" shows "x \<noteq> \<top>"
eq_Unit_def: shows "eq = (\<Lambda> \<langle>\<rangle> \<langle>\<rangle>. TT)"
convex_on_def: fixes s :: "'b set"   and f :: "'b \<Rightarrow> ereal" shows "Lower_Semicontinuous.convex_on s f = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> ereal u * f x + ereal v * f y)"
SPEC_rule_leof_conjI: fixes m :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool"   and \<Psi> :: "'b \<Rightarrow> bool" assumes "m \<le>\<^sub>n SPEC \<Phi>"   and "m \<le>\<^sub>n SPEC \<Psi>" shows "m \<le>\<^sub>n SPEC (\<lambda>x. \<Phi> x \<and> \<Psi> x)"
finite_lin_indpt2: fixes A :: "'e set" assumes "finite A"   and "A \<subseteq> carrier W"   and "\<And>a. \<lbrakk>a \<in> A \<rightarrow> carrier K; W.module.lincomb a A = \<zero>\<^bsub>W\<^esub>\<rbrakk> \<Longrightarrow> \<forall>v\<in>A. a v = \<zero>\<^bsub>K\<^esub>" shows "W.module.lin_indpt A"
nth_mem_mset: fixes i :: "nat"   and ls :: "'k list" assumes "i < length ls" shows "ls ! i \<in># mset ls"
inducts: fixes P :: "'d option \<Rightarrow> bool"   and option :: "'d option" assumes "P \<bottom>"   and "\<And>x. P (\<lfloor>x\<rfloor>)" shows "P option"
before_in_mtf: fixes z :: "'b"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "z \<in> set xs" shows "x < y in mtf z xs = (y \<noteq> z \<and> (if x = z then y \<in> set xs else x < y in xs))"
conjugate_sprod_vec: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (v \<bullet> w) = conjugate v \<bullet>c w"
complex_norm_square: fixes z :: "complex" shows "complex_of_real ((cmod z)\<^sup>2) = z * cnj z"
tt_def: shows "tt \<equiv> if t0 \<le> t_ne then min (t_max + et) t_ne else max (t_max - et) t_ne"
homomorphism: shows "group_hom (G\<lparr>carrier := H\<rparr>) (G\<lparr>carrier := N <#> H\<rparr> Mod N) f"
ee_wfactors: fixes as :: "'a list"   and a :: "'a"   and bs :: "'a list"   and b :: "'a" assumes "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim> b = essentially_equal G as bs"
of_nat_1_eq_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(one = semiring_1.of_nat one plus zero n) = (n = 1)"
ide_self_inverse: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "category.inverse_arrows C a a"
not_knows_conc_not_knows_elem_tail: fixes P :: "specID"   and a :: "Expression"   and e :: "Expression list" assumes "\<not> knows P (a # e)" shows "\<not> knows P [a] \<or> \<not> knows P e"
coprime_n_idealsTr1: fixes C :: "'a set"   and n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "ideal R C" shows "(\<forall>k\<le>n. ideal R (J k)) \<and> (\<forall>i\<le>n. coprime_ideals R (J i) C) \<longrightarrow> coprime_ideals R (i\<Pi>\<^bsub>R,n\<^esub> J) C"
maximal_independent_subset: fixes V :: "'b set" assumes "V \<subseteq> U\<^sub>V\<^sub>S" obtains B :: "'b set" where "B \<subseteq> U\<^sub>V\<^sub>S" and "B \<subseteq> V" and "\<not> local.dependent B" and "V \<subseteq> local.span B"
get_offending_flows_un: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "\<Union> (get_offending_flows M G) = (\<Union>m\<in>set M. \<Union> (c_offending_flows m G))"
simps_2: fixes i :: "nat" shows "local.worst_chain (Suc i) = (SOME x. local.bad x \<and> p (local.worst_chain i) x \<and> (\<forall>y. local.bad y \<and> p (local.worst_chain i) y \<longrightarrow> \<not> gt x y))"
euler_witness_mod: fixes a :: "int"   and p :: "nat" shows "euler_witness (a mod int p) p = euler_witness a p"
Merge_upper: fixes A :: "transaction\<^sub>m set"   and x :: "transaction\<^sub>m"   and y :: "transaction\<^sub>m" assumes "transaction.Merge A = Some x"   and "y \<in> A" shows "blinding_of_transaction y x"
inducts: fixes p :: "'c \<times> 'd"   and x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "fstsp p x"   and "P (fst p)" shows "P x"
Abs_poly_inverse: fixes y :: "nat \<Rightarrow> 'a" assumes "y \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'a)}" shows "coeff (Abs_poly y) = y"
ttree_restr_noop: fixes t :: "'a ttree"   and S :: "'a set" assumes "carrier t \<subseteq> S" shows "ttree_restr S t = t"
rv: fixes i :: "'i" shows "random_variable (M' i) (X i)"
initial_time_in: shows "t_max \<in> cball t_max et"
assms: shows "x\<^sup>2 + y\<^sup>2 = n"
that: fixes A :: "int"   and B :: "int" assumes "0 < A"   and "real_of_int A * Re (zeta 3) + real_of_int B \<in> {0<..2 * Re (zeta 3) * real (Lcm {1..n} ^ 3) / 27 ^ n}" shows "thesis"
f_Suc_eq: fixes x :: "'s"   and n :: "nat" shows "f x x (Suc n) = 0"
Vfin_imp_Efin: assumes "finite cf.V" shows "finite cf.E"
cases: fixes x :: "'b tree" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>"
RID\<^sub>GI_1: fixes s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option"   and r :: "'e"   and v :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" assumes "s r = Some v" shows "r \<in> RID\<^sub>G s"
iteratei_rule: fixes m :: "'s" assumes "invar m" shows "map_iterator (iteratei m) (\<alpha> m)"
convex_imp_AR: fixes S :: "'b set" assumes "convex S"   and "S \<noteq> {}" shows "AR S"
u_simps_2: shows "hseq (tab\<^sub>0 s) p\<^sub>0"
simps: fixes a1 :: "'c set"   and a2 :: "'c set"   and F :: "'c program" shows "((a1, a2) \<in> leads F) = ((\<exists>A B. a1 = A \<and> a2 = B \<and> F \<in> A ensures B) \<or> (\<exists>A B C. a1 = A \<and> a2 = C \<and> (A, B) \<in> leads F \<and> (B, C) \<in> leads F) \<or> (\<exists>S B. a1 = \<Union> S \<and> a2 = B \<and> (\<forall>A\<in>S. (A, B) \<in> leads F)))"
NullPointerC_def: shows "NullPointerC \<equiv> (NullPointer, [], [], [])"
hyps_2: shows "(c, s\<^sub>1_) \<Rightarrow> s\<^sub>2_"
right_neutral: fixes a :: "'b upred" shows "(a \<and> true) = a"
borel_measurable_lebesgue_preimage_borel: fixes f :: "'a \<Rightarrow> 'b" shows "(f \<in> borel_measurable lebesgue) = (\<forall>T. T \<in> sets borel \<longrightarrow> {x. f x \<in> T} \<in> sets lebesgue)"
loop_super_orientable_def: fixes x :: "'a" shows "bounded_distrib_allegory.loop_super_orientable top (\<sqinter>) (\<le>) (\<squnion>) id x \<equiv> \<exists>y. x \<le> y \<squnion> id y \<and> y \<sqinter> id y \<le> top"
cod_pr1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod \<pp>\<^sub>1[a, b] = a"
tabulate2_def: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c" shows "tabulate2 m n f = IArray.of_fun (\<lambda>i. IArray.of_fun (f i) n) m"
edom_AnalBinds: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "EdomArityAnalysis Aexp" shows "edom (ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>ae) \<subseteq> fv \<Gamma>"
nonconstant: fixes p :: "int poly" assumes "p \<in> P" shows "0 < Polynomial.degree p"
hyps: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i < length xs" shows "xs ! i \<in> set (drop j xs)"
predicate_Zorn: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "partial_order_on A (relation_of P A)"   and "\<And>C. C \<in> Chains (relation_of P A) \<Longrightarrow> \<exists>u\<in>A. \<forall>a\<in>C. P a u" shows "\<exists>m\<in>A. \<forall>a\<in>A. P m a \<longrightarrow> a = m"
eq_fold: fixes g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "sum g A = Finite_Set.fold ((+) \<circ> g) (0::'b) A"
lc_def: fixes cmp_term :: "('b, 'c) pp nat_term_order"   and p :: "('b, 'c) pp \<Rightarrow>\<^sub>0 'a" shows "lc_punit cmp_term p = lookup p (lt_punit cmp_term p)"
simps_3: fixes lformula0 :: "'a \<Rightarrow> bool"   and \<phi> :: "('a, 'b) aformula" shows "Formula_Operations.lformula lformula0 (FNot \<phi>) = Formula_Operations.lformula lformula0 \<phi>"
c_top_welldefTr1: fixes N :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "x \<in> N \<bullet> a"   and "y \<in> N \<bullet> b" shows "x \<cdot> y \<in> N \<bullet> (a \<cdot> b)"
Conseq: fixes A :: "triple set"   and P' :: "'a \<Rightarrow> assn"   and c :: "stmt"   and Q' :: "'a \<Rightarrow> assn"   and P :: "assn"   and Q :: "assn" assumes "\<forall>Z. A \<turnstile> {P' Z} c {Q' Z}"   and "\<forall>s t. (\<forall>Z. P' Z s \<longrightarrow> Q' Z t) \<longrightarrow> P s \<longrightarrow> Q t" shows "A \<turnstile> {P} c {Q}"
congruent_triangleI_sas: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "congruent_triangle a1 b1 c1 a2 b2 c2"
subset_imageE_inj: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "B \<subseteq> f ` A" obtains C :: "'c set" where "C \<subseteq> A" and "B = f ` C" and "inj_on f C"
Lim_within_le: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and a :: "'b"   and S :: "'b set" shows "(f \<longlongrightarrow> l) (at a within S) = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
eq3: shows "exH_n * (2 \<cdot>\<^sub>m pre) * exH_n = 2 \<cdot>\<^sub>m (exH_n * pre * exH_n)"
inorder_baldR: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.inorder (baldR l a r) = Tree2.inorder l @ a # Tree2.inorder r"
subspace_kernel: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec.subspace {x. A *v x = 0}"
preferred_alts_subset_alts: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and x :: "'b" assumes "pref_profile_wf agents alts R" shows "preferred_alts (R i) x \<subseteq> alts"
NotI: fixes v :: "i"   and \<phi> :: "\<o>" assumes "\<And>\<psi>. [\<phi> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
cod_src_eq_dom_src: shows "\<mu>.cod.src = \<mu>.dsrc"
asymp_equiv_imp_bigo: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "f \<in> O[F](g)"
Dlist_def: fixes xs :: "'a list" shows "Dlist xs = Abs_dlist (remdups xs)"
complex_quadratic_equation_monic_distinct_roots: fixes b :: "complex"   and c :: "complex" assumes "b\<^sup>2 - 4 * c \<noteq> 0" shows "\<exists>k\<^sub>1 k\<^sub>2. k\<^sub>1 \<noteq> k\<^sub>2 \<and> k\<^sub>1\<^sup>2 + b * k\<^sub>1 + c = 0 \<and> k\<^sub>2\<^sup>2 + b * k\<^sub>2 + c = 0"
scaleR_add_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and x :: "'a" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR (a + b) x = plus (scaleR a x) (scaleR b x)"
rep_eq: fixes x :: "\<kappa>" shows "rep x = the (eval\<kappa> x)"
directional_derivative_inner: fixes k :: "enat"   and a :: "'c"   and x :: "'c"   and j :: "'c" shows "directional_derivative k a x (\<lambda>x. x \<bullet> j) = x \<bullet> j"
simps: fixes p :: "('a, 'b) form" shows "dest_Forall (Forall p) = p"
l1_l2_def_1: shows "l1 = L ! 0"
Abs_ExpList_eq: fixes y :: "freeExp list"   and z :: "freeExp list" assumes "(y, z) \<in> listrel exprel" shows "Abs_ExpList y = Abs_ExpList z"
inj_on_image_Fpow: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "inj_on ((`) f) (Fpow A)"
iteration_idep: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y = (0::'a)" shows "x \<cdot> y\<^sup>\<infinity> = x"
ccBinds_reorder1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and v :: "Vars.var"   and e :: "exp" assumes "map_of \<Gamma> v = Some e" shows "ccBinds \<Gamma> = ccBind v e \<squnion> ccBinds (delete v \<Gamma>)"
nonempty_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "Basis \<noteq> {}"
dual_assertion_prop: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "sup (times (dual x) bot) one = dual x"
fls_finite_nonzero_neg_nth: fixes x :: "'a fls" shows "finite {xa. fls_nth x (- int xa) \<noteq> (0::'a)}"
number_partition_def: fixes n :: "nat"   and N :: "nat multiset" shows "number_partition n N = (\<Sum>\<^sub># N = n \<and> 0 \<notin># N)"
bij_exists_def: fixes P :: "'a set set"   and V :: "'a set" shows "bij_exists P V = (\<exists>f. bij_betw f P V \<and> (\<forall>B\<in>P. real c < W B + w (f B)))"
Mp_ident_iff': fixes f :: "int poly" shows "(Mp f = f) = (set (coeffs f) \<subseteq> {0..<p})"
Fimpl_rule: fixes s :: "bdd" assumes "bdd_sane s" shows "ospec (fmi' s) (\<lambda>(ni, s'). (ni, Falseif) \<in> Rmi s' \<and> bdd_sane s' \<and> mi.les s s')"
size_mult_mono: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "euclidean_size a \<le> euclidean_size (times a b)"
right_minus: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus a) = zero"
comp_ide_self: fixes a :: "'d" assumes "D.ide a" shows "a \<cdot>\<^sub>D a = a"
right_unit: fixes a :: "'a" assumes "a \<in> G" shows "a \<cdot> \<one> = a"
poly_mapping_eq_iff: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and b :: "'b \<Rightarrow>\<^sub>0 'c" shows "(a = b) = (lookup a = lookup b)"
lincomb_append_left: fixes rs :: "'r list"   and ss :: "'r list"   and ms :: "'m list" shows "(rs @ ss) \<bullet>\<cdot> ms = rs \<bullet>\<cdot> ms + ss \<bullet>\<cdot> drop (length rs) ms"
simps_2: fixes x21 :: "'c"   and x22 :: "'c Tree"   and x23 :: "'c Tree" shows "E \<noteq> T x21 x22 x23"
redT_updW_None_implies_None: fixes t' :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and t :: "'b" assumes "redT_updW t' ws wa ws'"   and "ws t = None"   and "t \<noteq> t'" shows "ws' t = None"
bin_nth_numeral_unfold_1: fixes x :: "num"   and n :: "nat" shows "bit (numeral (num.Bit0 x)) n = (0 < n \<and> bit (numeral x) (n - 1))"
ivl_disj_int_one_4: fixes l :: "'a"   and u :: "'a" shows "{..<l} \<inter> {l..u} = {}"
x1: shows "x1 \<in> poincare_circle (of_complex z1) r1 \<inter> poincare_circle (of_complex z2) r2"
bisimResNil: fixes \<Psi> :: "'b"   and x :: "name" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>\<zero> \<sim> \<zero>"
R: fixes s1 :: "'b"   and s2 :: "'a"   and a :: "'e"   and b :: "'e" assumes "bad1 s1 = bad2 s2"   and "\<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2"   and "bad2 s2 \<Longrightarrow> X_bad s1 s2" shows "R (a, s1) (b, s2)"
InitNoneReds: fixes sh :: "sheap"   and C :: "char list"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "sh C = None"   and "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh(C \<mapsto> (sblank P C, Prepared))),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
psubst_id: fixes e :: "exp"   and \<rho> :: "(nat \<times> exp) list" assumes "FV e \<inter> sdom \<rho> = {}" shows "psubst \<rho> e = e"
one_assn_rule: fixes h :: "pheap" shows "(h \<Turnstile> emp) = (addrOf h = {})"
sturm_seq_sturm: fixes p :: "real poly" assumes "rsquarefree p" shows "sturm_seq (sturm p) p"
valid_to_sound: fixes A :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and B :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list" assumes "seq_valid A" shows "sound (B, A)"
ord_mul_divides: fixes x :: "'a"   and y :: "'a" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "ord (x \<otimes> y) dvd ord x * ord y"
a_remove_child_locs_def: fixes ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" shows "a_remove_child_locs ptr owner_document = set_child_nodes_locs ptr \<union> set_disconnected_nodes_locs owner_document"
SComplex_SReal_hcomplex_of_hypreal: fixes x :: "real star" assumes "x \<in> \<real>" shows "hcomplex_of_hypreal x \<in> SComplex"
x_def: shows "x = x1 @\<^sub>v x2"
as_bs_2: shows "u = as \<bullet>\<cdot> us"
pow_ord_eq_1: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> group.ord G x = \<one>\<^bsub>G\<^esub>"
dom_def: fixes m :: "'d \<Rightarrow> 'e option" shows "dom m = {a. m a \<noteq> \<bottom>}"
grid_level: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set" assumes "p \<in> grid b ds" shows "level b \<le> level p"
abs_eq: fixes xa :: "'b"   and x :: "'b poly" shows "smult xa x = Abs_poly (\<lambda>n. xa * coeff x n)"
winding_number_cindex_pathE: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "finite_ReZ_segments g z"   and "valid_path g"   and "z \<notin> path_image g"   and "pathfinish g = pathstart g" shows "winding_number g z = complex_of_real (- cindex_pathE g z / 2)"
nonempty_hd_tinres: fixes r :: "cell list"   and r' :: "cell list" assumes "tinres r r'"   and "r \<noteq> []"   and "r' \<noteq> []" shows "hd r = hd r'"
lin_data_prev': shows "linearize data_prev' = filter (\<lambda>(t, X). enat (nt - t) \<le> right I) (linearize data_prev)"
is_join_and_compatible: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.cpo below"   and "below x z"   and "below y z"   and "\<And>a. \<lbrakk>below x a; below y a\<rbrakk> \<Longrightarrow> below z a" shows "cpo.compatible below x y \<and> cpo.join below x y = z"
ba_iso_inv_regular: fixes xf :: "('c, 'd) lifted_boolean_algebra" shows "regular (Abs_lifted_pair (\<lambda>f. (Rep_lifted_boolean_algebra xf f, Rep_phi f (- Rep_lifted_boolean_algebra xf f))))"