hcju/isabelleps huggingface dataset:
coeff_powsum_mpoly_singleton: fixes X :: "nat set"   and k :: "nat"   and x :: "nat" assumes "finite X"   and "0 < k" shows "MPoly_Type.coeff (powsum_mpoly X k) (Poly_Mapping.single x k) = (if x \<in> X then 1::'b else (0::'b))"
hom_add: fixes x :: "'a"   and y :: "'a" shows "f (x + y) = f x + f y"
rtrancl_rel_chain: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" shows "((x, y) \<in> R\<^sup>*) = (\<exists>l. l \<noteq> [] \<and> hd l = x \<and> last l = y \<and> rel_chain l R)"
orderI: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "semilattice_order f less_eq less"   and "a = f a b" shows "less_eq a b"
nth_prime_partition'': fixes x :: "real" assumes "2 \<le> x" shows "x \<in> {real (nth_prime (nat \<lfloor>\<pi> x\<rfloor> - 1))..<real (nth_prime (nat \<lfloor>\<pi> x\<rfloor>))}"
G_transpose: fixes A :: "(('d, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "fixed_mat.G (Finite_Cartesian_Product.transpose A) = (fixed_mat.G A)\<inverse>"
monad_optionT: fixes return :: "'b option \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b option \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad return bind" shows "monad (return_option return) (bind_option return bind)"
simps_1: fixes uu :: "'b word" shows "routing_table_semantics [] uu = routing_action undefined"
coreE: fixes x :: "'b"   and s :: "'b set" assumes "x \<in> core\<^bsub>T\<^esub> s" obtains m :: "'b set" where "m open\<^bsub>T\<^esub>" and "m \<subseteq> s" and "x \<in> m"
fmdrop_empty: fixes a :: "'c" shows "fmdrop a fmempty = fmempty"
fps_XD'_prod: fixes a :: "'b fps"   and b :: "'b fps"   and c :: "'b fps" shows "fps_XD' a (b * c) = fps_XD' a b * c + b * fps_XD' a c"
ln_fact_diff_bounds: fixes n :: "nat" shows "\<bar>ln (fact n) - 2 * ln (fact (n div 2)) - real n * ln 2\<bar> \<le> 4 * ln (real (if n = 0 then 1 else n)) + 3"
hom_lead_coeff: fixes p :: "int poly" shows "lead_coeff (of_int_poly p) = of_int (lead_coeff p)"
UN_ball_bex_simps_1: fixes A :: "'d set set"   and P :: "'d \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> A. P x) = (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
is_bal_r_bal: fixes l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0"   and n :: "'a" assumes "is_bal l"   and "is_bal r"   and "height r = height l + 2" shows "is_bal (r_bal\<^sub>0 n l r)"
simps_2: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and env :: "(name, term) fmap" shows "subst (t\<^sub>1 $ t\<^sub>2) env = subst t\<^sub>1 env $ subst t\<^sub>2 env"
countable_web_web_of_network: assumes "\<And>x. \<not> edge \<Delta> x (source \<Delta>)"   and "\<And>y. \<not> edge \<Delta> (sink \<Delta>) y"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> (\<exists>z. edge \<Delta> y z) \<or> (\<exists>z. edge \<Delta> z x)"   and "\<not> edge \<Delta> (source \<Delta>) (sink \<Delta>)"   and "\<And>x. \<not> edge \<Delta> x x" shows "countable_web (web_of_network \<Delta>)"
simps_1: fixes xs :: "('c, 'd) bintree option list" shows "add None xs = xs"
lunit_char_2: fixes f :: "'a" assumes "B.ide f" shows "B.L (B.lunit f) = (\<i>\<^sub>B[trg\<^sub>B f] \<star>\<^sub>B f) \<cdot>\<^sub>B B.\<a>' (trg\<^sub>B f) (trg\<^sub>B f) f"
awalk_verts_reachable_from: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "awalk u p v"   and "w \<in> set (awalk_verts u p)" shows "u \<rightarrow>\<^sup>* w"
prod_encode_prod_decode_aux: fixes k :: "nat"   and m :: "nat" shows "prod_encode (prod_decode_aux k m) = triangle k + m"
simps_47: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Await x71 x72"
toplevel_summands_REV: fixes r :: "'d rexp" shows "toplevel_summands (REV r) = REV ` toplevel_summands r"
induct: fixes x :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "hb_consistent x"   and "P []"   and "\<And>xs y. \<lbrakk>hb_consistent xs; P xs; \<forall>x\<in>set xs. \<not> y \<prec> x\<rbrakk> \<Longrightarrow> P (xs @ [y])" shows "P x"
intros_6: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and C :: "'a"   and e :: "real" assumes "ms_closure P xs f (b # basis)"   and "is_expansion C basis" shows "ms_closure P (scale_shift_ms_aux (C, e) xs) (\<lambda>x. eval C x * b x powr e * f x) (b # basis)"
fourier_primitive_1: fixes m :: "nat" shows "\<chi> m = gauss_sum 1 / complex_of_real (sqrt (real n)) / complex_of_real (sqrt (real n)) * (\<Sum>k = 1..n. cnj (\<chi> k) * unity_root n (- int m * int k))"
borel_measurable_euclidean_space: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
bij_to_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" shows "bij_to f A B = (surj_to f A B \<and> inj_on f A)"
measurable_completion2: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "complete_measure M"   and "f \<in> M \<rightarrow>\<^sub>M N"   and "null_sets N \<subseteq> null_sets (distr M N f)" shows "f \<in> M \<rightarrow>\<^sub>M completion N"
p2r_simps_2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<lceil>P\<rceil> = \<lceil>Q\<rceil>) = (\<forall>s. P s = Q s)"
listsum_def: fixes ds :: "nat list"   and As :: "'b tensor list" shows "listsum ds As = foldr (+) As (tensor0 ds)"
change_on_1_init_tau: fixes i :: "nat"   and t :: "nat" assumes "change_on_1 (block i t) (gap i t)" shows "\<tau> i \<triangleright> (t + 1) = e_update (block i t) (gap i t) 1"
perm_compose: fixes pi2 :: "(name \<times> name) list"   and pi1 :: "(name \<times> name) list"   and x :: "'d" shows "pi2 \<bullet> pi1 \<bullet> x = (pi2 \<bullet> pi1) \<bullet> pi2 \<bullet> x"
tl_prefixes_idx: fixes i :: "nat"   and p :: "'c list" assumes "i < length p" shows "tl (prefixes p) ! i = take (Suc i) p"
Interruptible_Separation_Kernel_axioms: shows "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting"
assms: shows "a \<notin>\<^sub>\<circ> A"
sort_leq_ex: fixes s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "sort_leq cs s1 s2" shows "sort_ex cs s2"
mult_sv_ex_id1: fixes v :: "complex \<times> complex" shows "\<exists>k. k \<noteq> 0 \<and> k *\<^sub>s\<^sub>v v = v"
invertible_def: fixes r :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set" shows "invertible r R \<equiv> \<forall>n S. r \<in> R \<and> (snd (extendRule S r), n) \<in> derivable R* \<longrightarrow> (\<forall>p\<in>set (fst (extendRule S r)). \<exists>m\<le>n. (p, m) \<in> derivable R*)"
trancl_converse: fixes r :: "('a \<times> 'a) set" shows "(r\<inverse>)\<^sup>+ = (r\<^sup>+)\<inverse>"
roots_of_complex_poly_def: fixes p :: "complex poly" shows "roots_of_complex_poly p \<equiv> let (c, pis) = yun_factorization gcd p in if c \<noteq> 0 \<and> (\<forall>(p, i)\<in>set pis. degree p \<le> 2 \<or> (\<forall>x\<in>set (coeffs p). x \<in> \<rat>)) then Some (concat (map (roots_of_complex_main \<circ> fst) pis)) else None"
pickE_def: fixes rho :: "'f \<Rightarrow> 'g \<Rightarrow> 'h set option" shows "pickE rho \<equiv> \<lambda>xs. lift pick (rho xs)"
divisors_pos_fun_def: fixes df :: "'b \<Rightarrow> 'b list" shows "divisors_pos_fun df \<equiv> (\<forall>x. x \<noteq> (0::'b) \<longrightarrow> set (df x) = {d. d dvd x \<and> (0::'b) < d}) \<and> (\<forall>x. distinct (df x))"
lnth_iterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" shows "iterates f x ?! n = (f ^^ n) x"
euler_mascheroni_convergent: shows "convergent (\<lambda>n. harm n - ln (real n))"
prefix_conc_length: fixes w :: "'a list"   and w' :: "nat \<Rightarrow> 'a" shows "prefix (length w) (w \<frown> w') = w"
Q_init_Q_2: shows "Q_lookup Q_initQ = initQ"
pdivmod_then_div_mod_int_poly: fixes q :: "int poly"   and p :: "int poly"   and r :: "int poly"   and m :: "int poly" assumes "q \<noteq> 0"   and "(of_int_poly p div of_int_poly q, of_int_poly p mod of_int_poly q) = (of_int_poly r, of_int_poly m)" shows "div_mod_int_poly p q = Some (r, m)"
RSubmoduleI: fixes N :: "'m set" assumes "VecGroup.Subgroup N"   and "\<And>r n. \<lbrakk>r \<in> R; n \<in> N\<rbrakk> \<Longrightarrow> r \<cdot> n \<in> N" shows "RSubmodule N"
elims_2: fixes x :: "'v"   and xa :: "'v list" assumes "adj_path x xa" obtains v :: "'v" where "x = vb__" and "xa = []" and "vb__ \<in> V"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "adjacent v u \<and> adj_path u us"
less_multiset\<^sub>H\<^sub>O: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M < N) = (M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x)))"
bv_not_def: fixes w :: "bit list" shows "bv_not w = map bitnot w"
card_insert_disjoint: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
append_eq_Cons: fixes x :: "'b"   and xs :: "'b list" shows "[x] @ xs = x # xs"
hs_def: shows "hs = (case e_hs of (x, hs) \<Rightarrow> hs)"
assms_4: shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
E31: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> \<box>F) \<and> \<diamond>F \<longrightarrow> \<diamond>\<box>F"
pseudo_simplex_like_has_bottom: fixes P :: "'a set"   and f :: "'a \<Rightarrow> 'c set"   and A :: "'c set" assumes "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f"   and "f ` P = Pow A" shows "has_bottom P"
inv_equality: fixes y :: "'a"   and x :: "'a" assumes "y \<oplus> x = \<zero>"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x = y"
kad_def: fixes f :: "'a \<Rightarrow> 'b set" shows "kad f = (\<lambda>x. if f x = {} then \<eta> x else {})"
infinite: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "infinite A" shows "sum g A = (0::'e)"
disc_eq_case_1: fixes option :: "'f option" shows "(option = None) = (case option of None \<Rightarrow> True | Some uu_ \<Rightarrow> False)"
cover_of_subset_def: fixes X :: "'c set"   and U :: "'c set"   and index :: "real set"   and cover :: "real \<Rightarrow> 'c set" shows "cover_of_subset X U index cover \<equiv> U \<subseteq> X \<and> (\<forall>i. i \<in> index \<longrightarrow> cover i \<subseteq> X) \<and> U \<subseteq> \<Union> (cover ` index)"
abs_eq: fixes x :: "'a \<Rightarrow> 'b option" shows "pdom (Abs_pfun x) = dom x"
lb_exp_pos: fixes x :: "float"   and prec :: "nat" assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
simps_8: fixes n :: "nat"   and p :: "fm"   and q :: "fm" shows "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
ex_filter_syzygy_basis_adds_lt: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_pot_ord"   and "distinct bs"   and "is_Groebner_basis (set gs)"   and "pmdl (set gs) = pmdl (set (init_syzygy_list bs))"   and "f \<in> pmdl (syzygy_module_list bs)"   and "f \<noteq> 0" shows "\<exists>g\<in>set (filter_syzygy_basis (length bs) gs). g \<noteq> 0 \<and> lt g adds\<^sub>t lt f"
F1set3_JF2set_JF1set: fixes a' :: "'d JF2"   and a :: "'d JF1"   and c :: "'d" assumes "a' \<in> F1set3 (dtor1 a)"   and "c \<in> JF2set a'" shows "c \<in> JF1set a"
pCons_eq_iff: fixes a :: "'a"   and p :: "'a poly"   and b :: "'a"   and q :: "'a poly" shows "(pCons a p = pCons b q) = (a = b \<and> p = q)"
fully_abstract_impl_symcl_source_target_relation_is_preorder: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> (symcl (SRel\<^sup>=))\<^sup>+) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> (symcl (TRel\<^sup>=))\<^sup>+)" shows "\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (symcl (SRel\<^sup>=))\<^sup>+ = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> symcl Rel) \<and> (symcl (TRel\<^sup>=))\<^sup>+ = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> symcl Rel) \<and> preorder (symcl Rel)"
maximal_def: fixes S :: "'a set"   and C :: "'a set set" shows "maximal S C \<equiv> \<forall>S'\<in>C. S \<subseteq> S' \<longrightarrow> S = S'"
unify_Some_UNIF: fixes E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and bs :: "('c \<times> ('d, 'c) Term.term) list"   and cs :: "('c \<times> ('d, 'c) Term.term) list" assumes "unify E bs = Some cs" shows "\<exists>ds ss. cs = ds @ bs \<and> subst_of ds = compose ss \<and> UNIF ss (mset E) {#}"
map_nth: fixes xs :: "'e list" shows "map ((!) xs) [0..<length xs] = xs"
epi_def: fixes f :: "'a" shows "epi f = (arr f \<and> inj_on (\<lambda>g. g \<cdot> f) {g. seq g f})"
continuous_on_domain: fixes c :: "('a, 'b) chart" shows "continuous_on (domain c) (apply_chart c)"
extreme_point_of_stillconvex: fixes S :: "'b set"   and x :: "'b" assumes "convex S" shows "(x extreme_point_of S) = (x \<in> S \<and> convex (S - {x}))"
\<alpha>_def: fixes \<mu>\<nu>\<tau> :: "'b \<times> 'b \<times> 'b" shows "\<alpha> \<mu>\<nu>\<tau> \<equiv> \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
subsumption_preserves_redundancy: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set"   and S :: "'b Literal set set"   and D :: "'b Literal set" assumes "propositional_atoms atom_ordering"   and "redundant C S"   and "subsumes C D" shows "redundant D S"
simps_1: fixes var :: "nat"   and p :: "real mpoly" shows "freeIn var (fm.Atom (Eq p)) = (var \<notin> vars p)"
monom_of_atom_coeff: fixes ns :: "QDelta ns_constraint"   and a :: "QDelta atom"   and v :: "nat" assumes "is_monom (poly ns)"   and "a = qdelta_constraint_to_atom ns v" shows "monom_coeff (poly ns) *R nsc_of_atom a = ns"
fmdom_empty_fmempty_iff: fixes m :: "('a, 'b) fmap" shows "(fmdom m = {||}) = (m = fmempty)"
ass_function_def: fixes f :: "'b \<Rightarrow> 'b" shows "ass_function f = ((\<forall>a. normalize a = normalize (f a)) \<and> pairwise (\<lambda>a b. normalize a \<noteq> normalize b) (range f))"
sine_law_triangle'': fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<noteq> c" shows "sin (angle c b a) = sin (angle b a c) * dist a c / dist b c"
space_subgroup_props_5: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set"   and k :: "'a"   and v :: "'a" assumes "subfield K R"   and "dimension n K E"   and "k \<in> K"   and "v \<in> E" shows "k \<otimes> v \<in> E"
vdisjnt_vsubset_left: fixes X :: "V"   and Y :: "V"   and Z :: "V" assumes "vdisjnt X Y"   and "Z \<subseteq>\<^sub>\<circ> X" shows "vdisjnt Z Y"
inv_comp_left_2: fixes g :: "'a"   and f :: "'a" assumes "local.iso (g \<cdot> f)"   and "local.iso g" shows "local.iso f"
RTS_safe_axioms: shows "RTS_safe out equiv_out deselect progs tests"
timpl_closure_Fun_inv'_3: fixes g :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and f :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set" assumes "Fun g S \<in> timpl_closure (Fun f T) TI"   and "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> TI\<^sup>+"
dirichlet_inverse_0: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b" shows "dirichlet_inverse f i 0 = (0::'b)"
const_def: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and c :: "nat" assumes "bob_fun f n" shows "bob_fun.const f n c = (\<forall>x\<in>{i. i < 2 ^ n}. f x = c)"
q': shows "q' = (tObsC_ii_abs p, vq)"
double_complement: fixes A :: "'a set" shows "- (- A) = A"
mergeable_envs_set_Cons: fixes n :: "nat"   and S :: "'c list set" assumes "mergeable_envs n S" shows "mergeable_envs (Suc n) (set_Cons UNIV S)"
sclose_FV: fixes n :: "nat"   and s :: "char list"   and p :: "char list"   and t :: "sterm" shows "FV ({n \<leftarrow> [s,p]} t) = FV t - {s} - {p}"
B_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and QA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and QB :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Simulation_Invariants A B sim PA QA PB QB"   and "PB a"   and "B a b" shows "QB b"
less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubset> z" shows "x \<sqsubset> z"
hyps_1: shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"
permutation_of_cycle: fixes cs :: "'b list" shows "permutation (cycle_of_list cs)"
4: shows "\<forall>i. F (v (Y i)) \<subseteq> F (u (Y i))"
neq: fixes x :: "complex" assumes "0 < Im x" shows "\<i> + x \<noteq> 0"
simps_4: fixes xpos' :: "nat \<Rightarrow> nat"   and xpos :: "nat"   and ypos :: "nat"   and more :: "'a" shows "xpos_update xpos' \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = \<lparr>xpos = xpos' xpos, ypos = ypos, \<dots> = more\<rparr>"
holo1: shows "(\<lambda>w. f (u * w)) holomorphic_on S"
almost_full_on_UNIV: shows "almost_full_on (\<lambda>_ _. True) UNIV"
apx_mono: fixes Z' :: "('c \<Rightarrow> real) set"   and Z :: "('c \<Rightarrow> real) set" assumes "Z' \<subseteq> V"   and "Z \<subseteq> Z'" shows "Approx\<^sub>\<beta> Z \<subseteq> Approx\<^sub>\<beta> Z'"
filterlim_tendsto_neg_mult_at_top_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c < 0" shows "(LIM x F. f x * g x :> at_top) = filterlim g at_bot F"
span_closed: fixes U :: "'c set"   and v :: "'c" assumes "U \<subseteq> carrier V"   and "v \<in> span U" shows "v \<in> carrier V"
subset_Above_Under: fixes B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "B \<subseteq> Field r" shows "B \<subseteq> Above r (Under r B)"
D2_ALL: fixes P :: "D2 \<Rightarrow> bool" shows "(\<forall>d. P d) = (\<forall>d\<in>{D1, D2}. P d)"
xor_self: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl x x = zero"
invert_side_of_triangle_1: fixes h :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<phi>.B.arr h"   and "f \<cdot>\<^sub>B g = h"   and "\<phi>.B.iso f" shows "\<phi>.B.seq (\<phi>.B.inv f) h \<and> g = \<phi>.B.inv f \<cdot>\<^sub>B h"
CollectionPartsNilTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and x :: "'a collection_literal_part_expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>C [x] : \<tau>" obtains "\<Gamma> \<turnstile>\<^sub>P x : \<tau>"
orbit_stabiliser: fixes x :: "Vertex" assumes "finite (carrier tetrahedral_group)" shows "order tetrahedral_group = card (tetrahedral.orbit x) * card (tetrahedral.stabiliser x)"
endpow_list_apply_closed: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and v :: "'b"   and k :: "nat" assumes "VectorSpaceEnd smult V T"   and "v \<in> V" shows "set (map (\<lambda>S. S v) (map (VectorSpaceEnd.endpow V T) [0..<k])) \<subseteq> V"
pure_app_eq: fixes R :: "('a \<times> 'b) set"   and a :: "'b"   and c :: "'a" shows "pure R a c = \<up> ((c, a) \<in> R)"
INFM_nat_le: fixes P :: "nat \<Rightarrow> bool" shows "(\<exists>\<^sub>\<infinity>n. P n) = (\<forall>m. \<exists>n\<ge>m. P n)"
unlex_lexlist: fixes xs :: "'b list" shows "unlex (lexlist xs) = xs"
mdl_is_ag: fixes H :: "'a set" assumes "submodule R M H" shows "aGroup (mdl M H)"
bit_Suc: fixes a :: "'a"   and n :: "nat" shows "bit a (Suc n) = bit (a div (2::'a)) n"
outside_frontier_eq_complement_closure: fixes S :: "'a set" assumes "bounded S"   and "convex S" shows "outside (frontier S) = - closure S"
nn_integral_map_pmf: fixes g :: "'d \<Rightarrow> 'c"   and M :: "'d pmf"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf (map_pmf g M)) f = \<integral>\<^sup>+ x. f (g x) \<partial>measure_pmf M"
length_take: fixes n :: "nat"   and xs :: "'d list" shows "length (xs \<down> n) = min (length xs) n"
simps_3: fixes x2 :: "'b" shows "Msg x2 \<noteq> NoMsg"
lhd'_simps_2: fixes x :: "'b"   and xs :: "'b llist" shows "lhd' (x ## xs) = Some x"
simps_1: fixes f :: "nat list \<Rightarrow> nat"   and g :: "nat list \<Rightarrow> nat" shows "PREC f g [] = 0"
ge_gt_tpoly_trans: fixes r :: "('v pvar, hmultiset) tpoly"   and q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" assumes "r \<ge>\<^sub>p q"   and "q >\<^sub>p p" shows "r >\<^sub>p p"
weak_axioms: shows "weak substTerm substAssert substCond (\<otimes>) SImp' (\<one>) SChanEq'"
wf_Act: fixes x :: "('idx, 'pred, 'act) formula"   and \<alpha> :: "'act" assumes "weak_formula x" shows "weak_formula \<langle>\<langle>\<alpha>\<rangle>\<rangle>x"
closedin_Euclidean_space_iff: fixes m :: "nat"   and S :: "(nat \<Rightarrow> real) set" shows "closedin (Euclidean_space m) S = (closed S \<and> S \<subseteq> topspace (Euclidean_space m))"
update_append_some: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and e :: "('a, 'b, 'c) env"   and y :: "'c"   and ys :: "'c list"   and opt :: "('a, 'b, 'c) env option" assumes "lookup env xs = Some e" shows "lookup (update (xs @ y # ys) opt env) xs = Some (update (y # ys) opt e)"
nformsTerminal: fixes f :: "'a \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and n :: "'a" assumes "nforms (f n) = []" shows "terminal subs (f n)"
eqvt_uncurry: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "eqvt f" shows "eqvt (\<lambda>(x, y). f x y)"
int: fixes b :: "real"   and z :: "complex"   and e :: "real"   and c :: "real" assumes "real_of_int a \<le> b"   and "z \<in> U" shows "(\<lambda>t. complex_of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}"
HT'_partial_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT'_partial \<pi> P c Q \<equiv> \<forall>s\<^sub>0. P s\<^sub>0 \<longrightarrow> wlp \<pi> (c s\<^sub>0) (Q s\<^sub>0) s\<^sub>0"
closed_transdb_inv: fixes e :: "db"   and \<Gamma> :: "nat \<Rightarrow> nat" assumes "closed e" shows "free (transdb_inv e \<Gamma> 0 0) = []"
Step1_4_3a: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCFwd crCh rmCh rst p \<and> $S3 rmhist p \<and> S4 rmhist p$ \<and> unchanged (e p, c p, m p, rmhist!p) \<longrightarrow> unchanged (rtrner memCh!p, resbar rmhist!p)"
assms_1: shows "\<forall>p\<in>path_image \<gamma>. Im z \<le> Im p"
u_simps_2: shows "arr t\<^sub>0u\<^sub>1.p\<^sub>0"
skip_empty_aux: fixes n :: "nat"   and l :: "'a list list"   and x :: "nat" assumes "concat (take (Suc n) l) = concat (take (Suc x) l)"   and "Suc n \<le> length l"   and "x \<le> n" shows "\<forall>i. x < i \<and> i \<le> n \<longrightarrow> l ! i = []"
map_eq_append_conv: fixes f :: "'b \<Rightarrow> 'a"   and ls :: "'b list"   and fl :: "'a list"   and fl' :: "'a list" shows "(map f ls = fl @ fl') = (\<exists>l l'. ls = l @ l' \<and> map f l = fl \<and> map f l' = fl')"