hcju/isabelleps huggingface dataset:
lfilter_ltake: fixes xs :: "'a llist"   and i :: "nat"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>n. enat n \<le> llength xs \<longrightarrow> i \<le> n \<longrightarrow> \<not> P (lnth xs n)" shows "lfilter P xs = lfilter P (ltake (enat i) xs)"
sets_Collect_restrict_space_iff: fixes S :: "'c set"   and M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "S \<in> sets M" shows "({x \<in> space (restrict_space M S). P x} \<in> sets (restrict_space M S)) = ({x \<in> space M. x \<in> S \<and> P x} \<in> sets M)"
\<psi>'\<psi>: shows "\<forall>x\<in>V. \<psi>' (\<psi> x) = x"
sub: fixes es :: "nat set set"   and n :: "nat" assumes "es \<in> space (MGn p n)"   and "P n es" shows "Q n es"
errMOD_igFreshIGVarSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igVarIPresIGWls MOD"   and "igFreshIGVar MOD" shows "igFreshIGVar (errMOD MOD)"
reflp_tllist: shows "reflp (\<lambda>(xs, a) (ys, b). xs = ys \<and> (lfinite ys \<longrightarrow> a = b))"
mult_commute_imp_mult_inverse_commute: 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 y :: "'b"   and x :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "times y x = times x y" shows "times (inverse y) x = times x (inverse y)"
cobounded1: fixes a :: "'e"   and b :: "'e" shows "inf a b \<le> a"
hyps: fixes bs1 :: "WS1S_Nameful.fo list"   and bs2 :: "WS1S_Nameful.so list"   and \<AA> :: "interp" assumes "set (fos \<phi>) \<subseteq> set bs1"   and "set (sos \<phi>) \<subseteq> set bs2"   and "ok_I \<AA> bs1" shows "WS1S.sat \<AA> (\<epsilon> bs1 bs2 \<phi>) = satisfies (dec_I1 \<AA> bs1) (dec_I2 \<AA> bs2) \<phi>"
length_filter_conv_card: fixes p :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "length (filter p xs) = card {i. i < length xs \<and> p (xs ! i)}"
cod_cod: fixes f :: "('o, 'a) arr" assumes "arr f" shows "cod (cod f) = cod f"
elims_3: fixes x :: "'a tree" assumes "\<not> is_empty x" obtains t :: "'a tree" where "x = t" and "t \<noteq> \<langle>\<rangle>"
Un_UNIV_left: fixes B :: "'e set" shows "UNIV \<union> B = UNIV"
antisym_restrictPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "antisymp r" shows "antisymp (r |` A)"
sheaf_spec_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.sheaf_spec R addition multiplication zero unit U \<equiv> {s \<in> \<Pi>\<^sub>E \<pp>\<in>U. R \<^bsub>\<pp> addition multiplication zero\<^esub>. comm_ring.is_regular R addition multiplication zero unit s U}"
T8_1: fixes w :: "i"   and \<psi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>" shows "[\<^bold>\<forall>\<^sub>1x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
that: fixes b :: "V" assumes "\<R>\<^sub>\<circ> r = set {b}" shows "thesis"
insert_code_1: fixes x :: "'a"   and xs :: "'a list" shows "insert x (set xs) = set (List.insert x xs)"
in_dgrad_p_set_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "(p \<in> punit.dgrad_p_set d m) = (\<forall>v\<in>keys p. d (local.punit.pp_of_term v) \<le> m)"
cpi: shows "coeff p i = ff (coeff q' i) / ff x'"
le_add_same_cancel1: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus a b) = less_eq zero b"
simps_3: fixes x1 :: "'a"   and x21 :: "'a"   and x22 :: "nat" shows "PWt x1 \<noteq> PCoef x21 x22"
rreq_rrep_nsqn_fresh_any_step_invariant: fixes i :: "nat"   and A :: "state \<Rightarrow> state \<Rightarrow> bool" shows "opaodv i \<Turnstile>\<^sub>A (act (recvmsg rreq_rrep_sn), other A {i} \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<sigma>, uu_), a, uu_). anycast (msg_fresh \<sigma>) a)"
that: fixes sdf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set" assumes "weak_pareto sdf A Is universal_domain"   and "iia sdf A Is"   and "\<nexists>j. dictator sdf A Is j"   and "SDF sdf A Is universal_domain" shows "thesis"
degree_monic_char_poly: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "degree (char_poly A) = n \<and> coeff (char_poly A) n = (1::'b)"
dBC_1: shows "dim_row (FB B C) = dim_row B + dim_row C"
beta_one_star: fixes t1 :: "lam"   and t2 :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta> t2" shows " t1 \<longrightarrow>\<^sub>1\<^sup>* t2"
direct_decomp_map_Int_zero: fixes A :: "'b set"   and f :: "'c \<Rightarrow> 'b set"   and ss :: "'c list"   and s1 :: "'c"   and s2 :: "'c" assumes "direct_decomp A (map f ss)"   and "s1 \<in> set ss"   and "s2 \<in> set ss"   and "s1 \<noteq> s2"   and "\<And>s. s \<in> set ss \<Longrightarrow> (0::'b) \<in> f s" shows "f s1 \<inter> f s2 = {0::'b}"
semilattice_axioms: shows "semilattice (\<^bold>*)"
bot_eq_sup_iff: fixes x :: "'a multiset"   and y :: "'a multiset" shows "({#} = x \<union># y) = (x = {#} \<and> y = {#})"
nn_integral_has_integral: fixes f :: "'c \<Rightarrow> real"   and r :: "real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal r"   and "0 \<le> r" shows "(f has_integral r) UNIV"
fds_converges_truncate: fixes n :: "nat"   and f :: "'a fds"   and s :: "'a" shows "fds_converges (fds_truncate n f) s"
inner_scaleR_right: 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 x :: "'b"   and r :: "real"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (scaleR r y) = r * inner x y"
simps_36: fixes x61 :: "Unary_PCF.type"   and x62 :: "expr"   and x51 :: "expr"   and x52 :: "expr" shows "\<Lambda>\<langle>x61\<rangle> x62 \<noteq> x51 \<cdot> x52"
eq_fold: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "sum g A = Finite_Set.fold ((+) \<circ> g) (0::'a) A"
semiring_norm_75: fixes m :: "num" shows "(m < num.One) = False"
less_eq_basic_type_code: shows "(\<le>) = (\<lambda>x y. basic_subtype_fun x y \<or> x = y)"
lower_rank_of_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>' \<subseteq> \<E>" shows "indep_system.lower_rank_of (\<E>.indep_in \<E>) \<E>' = \<E>.rank_of \<E>'"
mod2_eq_if: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = (if dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a then zero else one)"
below_monic_indexed: fixes f :: "'a \<rightarrow> 'b" assumes "below_monic_cfun f" shows "below_monic (\<lambda>y i. f\<cdot>(y i))"
Units_l_inv: fixes x :: "'a" assumes "x \<in> Units (add_monoid R)" shows "inv\<^bsub>add_monoid R\<^esub> x \<oplus> x = \<zero>"
merge_coeffs0_is_normalize_poly_p: fixes xs :: "(char list list \<times> int) list"   and ys :: "(char list multiset \<times> int) multiset" assumes "(xs, ys) \<in> sorted_repeat_poly_list_rel_with0 (rel2p (Id \<union> term_order_rel))" shows "\<exists>r. (merge_coeffs0 xs, r) \<in> sorted_poly_rel \<and> normalize_poly_p\<^sup>*\<^sup>* ys r"
interchange: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and f :: "'b"   and k :: "'b"   and g :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C (C h f)"   and "partial_magma.arr C (C k g)" shows "C (elementary_category_with_binary_products.prod C pr0 pr1 h k) (elementary_category_with_binary_products.prod C pr0 pr1 f g) = elementary_category_with_binary_products.prod C pr0 pr1 (C h f) (C k g)"
GSubspace_is_FinGroupRep: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "FinGroupRepresentation G (\<cdot>) U"
\<theta>_fv: fixes v :: "'a" shows "fv (\<theta> v) = {}"
pbernpoly_over_power_tendsto_0: fixes s :: "complex"   and n :: "nat" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "0 < n" shows "(\<lambda>x. complex_of_real (pbernpoly (Suc n) (real x)) / (of_nat (Suc n) * (s + of_nat x) ^ n)) \<longlonglongrightarrow> 0"
de_morgan3: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "!(!p + !q) = p \<cdot> q"
respects_hcomp: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B g = trg\<^sub>B f" shows "(\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>C \<a>\<^sub>C[\<tau>\<^sub>0 (trg\<^sub>B g), F g, F f] \<cdot>\<^sub>C (\<tau>\<^sub>1 g \<star>\<^sub>C F f) \<cdot>\<^sub>C C.inv \<a>\<^sub>C[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f] \<cdot>\<^sub>C (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>C \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)] = \<tau>\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>C (\<Phi>\<^sub>G (g, f) \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))"
cases: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "semiconnex A less"   and "x \<in> A"   and "y \<in> A" obtains "less x y"   | "x = y"   | "less y x"
L5_Controller_Zero: fixes y :: "nat \<Rightarrow> nat list"   and lIn :: "nat \<Rightarrow> bit"   and lOut :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y lIn lOut z"   and "lOut t = Zero"   and "z t = []" shows "lIn t = Zero"
power_inductl: fixes z :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and n :: "nat" assumes "z \<union> x O y \<subseteq> y" shows "rel_dioid.power x n O z \<subseteq> y"
sorted_append: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (xs @ ys) = (linorder.sorted less_eq xs \<and> linorder.sorted less_eq ys \<and> (\<forall>x\<in>list.set xs. \<forall>y\<in>list.set ys. less_eq x y))"
distinct_19: fixes x41 :: "'a"   and x42 :: "('a, 'b) com"   and x51 :: "('a, 'b) com"   and x52 :: "('a, 'b) com" shows "while x41 do x42 \<noteq> x51 | x52"
minus_order: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'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 a :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq (uminus a) (uminus b)" shows "less_eq b a"
ordered_fmap_set_eq: fixes S :: "('c \<times> 'd) fset" assumes "is_fmap S" shows "fset_of_list (ordered_fmap S) = S"
conn: fixes U :: "'a set" assumes "U \<subset> T" shows "connected_component (- U) a b"
simps_5: fixes f1 :: "'f"   and f2 :: "nat \<Rightarrow> 'f"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
sup_distrib_inf_le: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> (x \<squnion> z) \<le> x \<squnion> y \<sqinter> z"
assms: shows "\<forall>j\<le>D + z. str (t + j) = x"
inject_5: fixes x51 :: "CTypeId"   and x52 :: "nat"   and y51 :: "CTypeId"   and y52 :: "nat" shows "(objV x51 x52 = objV y51 y52) = (x51 = y51 \<and> x52 = y52)"
NotL: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F, \<Delta>" shows "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta>"
strict_prefix_simps_2: fixes x :: "'c"   and xs :: "'c list" shows "strict_prefix [] (x # xs) = True"
prefixes_are_paths': fixes p :: "(symbol \<times> character list) list"   and q :: "(symbol \<times> character list) list" assumes "p \<in> \<PP>"   and "is_prefix q p" shows "q \<in> \<PP>"
not_eventually_impI: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "eventually P F"   and "\<not> eventually Q F" shows "\<not> (\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x)"
game_sem_loop_unwind_reduce: fixes I :: "interp"   and \<alpha> :: "game" shows "(\<lambda>X. X \<union> game_sem I (\<alpha> ;; \<alpha>**) X) = game_sem I \<alpha>**"
epm_6: fixes e :: "'a"   and w :: "'a"   and v :: "'a" assumes "e \<le> w"   and "injective w"   and "w * v \<le> v"   and "e \<le> v * - v\<^sup>T"   and "vector v" shows "prim_E w v e = e"
con_comp_ipurge_ref_filter: fixes I :: "('f \<times> 'f) set"   and D :: "'g \<Rightarrow> 'f"   and E :: "'h \<Rightarrow> 'f"   and p :: "'g \<Rightarrow> 'e"   and q :: "'h \<Rightarrow> 'e"   and u :: "'f option"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref (con_comp_pol I) (con_comp_map D E p q) u (filter (\<lambda>x. x \<in> range p \<union> range q) xs) X = ipurge_ref (con_comp_pol I) (con_comp_map D E p q) u xs X"
join_le: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and z :: "'c"   and y :: "'c" assumes "weak_upper_semilattice L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> z"   and "y \<sqsubseteq>\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> y \<sqsubseteq>\<^bsub>L\<^esub> z"
v11_injective: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "v11 r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = r\<lparr>b\<rparr>" shows "a = b"
adjoint_ode_vars: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and I :: "('b, 'c, 'a) interp"   and \<sigma> :: "('b, 'c, 'a) subst"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and ODE :: "('b, 'a) ODE"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ODE_vars (ids.adjoint I \<sigma> \<nu>) ODE = ODE_vars (ids.adjoint I \<sigma> \<omega>) ODE"
sccs_verts_subsets: fixes S :: "'a set" assumes "S \<in> G.sccs_verts" shows "S \<subseteq> verts G"
rev_replicate: fixes n :: "nat"   and x :: "'c" shows "rev (replicate n x) = replicate n x"
gR_rel_def: fixes x :: "'a"   and y :: "'a" shows "(x \<preceq>\<^sub>R y) = (\<exists>z. D x z \<and> x \<oplus> z = y)"
assms: shows "pair_wf_digraph (list_digraph G_list)"
mem_permute_iff: fixes p :: "perm"   and x :: "'a"   and X :: "'a set" shows "(p \<bullet> x \<in> p \<bullet> X) = (x \<in> X)"
hcomp_arr_obj_1: fixes \<mu> :: "'b"   and a :: "'b" assumes "D.arr \<mu>"   and "D.obj a"   and "src\<^sub>D \<mu> = a" shows "\<mu> \<star>\<^sub>D a = \<r>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>] \<cdot>\<^sub>D \<mu> \<cdot>\<^sub>D \<r>\<^sub>D[D.dom \<mu>]"
linear: fixes ko :: "'b key_order"   and x :: "'b"   and y :: "'b" shows "ko.le ko x y \<or> ko.le ko y x"
foundation3: fixes \<tau> :: "'b state \<times> 'b state" shows "\<tau> |\<noteq> invalid"
intpow_zero: fixes x :: "'b" shows "x ^\<^sub>Z 0 = (1::'b)"
cr_envT_stateT_def: fixes s :: "'d"   and M :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m1 :: "('d, 'e) envT"   and m2 :: "('d, 'f) stateT" shows "cr_envT_stateT s M m1 m2 = (eq_onp ((=) s) ===> M) (run_env m1) (run_state m2)"
fix_eqI: fixes F :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "F\<cdot>x = x"   and "\<And>z. F\<cdot>z = z \<Longrightarrow> x \<sqsubseteq> z" shows "fix\<cdot>F = x"
IHP_1: fixes ll :: "('b, 'c) label list"   and u :: "'a"   and c :: "'a list multiset" assumes "length ll \<le> length w"   and "isolated_ret fg p"   and "proc_of fg u = p"   and "u \<noteq> return fg p"   and "(([u], c), ll, [return fg p'], c') \<in> trcl (trss fg)" shows "False"
set_val_get_disconnected_nodes: fixes ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
atLeast_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a" shows "ord.atLeast less_eq l \<equiv> {x. less_eq l x}"
simps_10: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_node R SOURCE SOURCE"
inf_sup_ord_2: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) y"
X_poly_def: fixes R :: "('c, 'd) ring_scheme" shows "X_poly R = monom (UP R) \<one>\<^bsub>R\<^esub> 1"
mult_unit_dvd_iff': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "(a * b dvd c) = (b dvd c)"
subst_vsubst_compose2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Z :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and y1 :: "'var" assumes "good X"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "X #[y1 // y]_ys #[Z / z]_zs = X #[Z / z]_zs #[Var ys y1 #[Z / z]_zs / y]_ys"
assms: shows "\<forall>(i, p)\<in>insert x (set xs). \<forall>(i', p')\<in>insert x (set xs). p = p' \<longrightarrow> i = i'"
GroupHom: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom R smult M smult' T" shows "GroupHom M T"
map_add_dom_app_simps_2: fixes m :: "'b"   and l1 :: "'b \<Rightarrow> 'c option"   and l2 :: "'b \<Rightarrow> 'c option" assumes "m \<notin> dom l1" shows "(l1 ++ l2) m = l2 m"
word_div_less: fixes m :: "'b word"   and n :: "'b word" assumes "m < n" shows "m div n = 0"
multiplicity_eq_imp_eq: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero"   and "\<And>p. normalization_semidom.prime zero times one normalize p \<Longrightarrow> factorial_semiring.multiplicity times one p x = factorial_semiring.multiplicity times one p y" shows "normalize x = normalize y"
topspace_discrete_topology: fixes U :: "'b set" shows "topspace (discrete_topology U) = U"
freshBinp_def: fixes xs :: "'b"   and x :: "'c"   and binp :: "'d \<Rightarrow> ('e, 'f, 'b, 'c, 'g) qAbs set option" shows "freshBinp xs x binp \<equiv> liftAll (freshAbs xs x) binp"
ih1: fixes t :: "'a trm" shows "vars_of (t \<lhd> \<theta>1) \<subseteq> vars_of M \<union> vars_of M' \<union> vars_of t"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit (mask m) n = ((2::'c) ^ n \<noteq> (0::'c) \<and> n < m)"
with_proj_simps_2: fixes G :: "'c pair_pre_digraph" shows "arcs (with_proj G) = parcs G"
positive_def: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "positive M = (Measure.measure M {} = 0 \<and> (\<forall>A. A \<in> measurable_sets M \<longrightarrow> 0 \<le> Measure.measure M A))"
simps_2: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" shows "successively P [x] = True"
sel_2: fixes x1 :: "('f \<times> 'e \<times> 'e) set"   and x2 :: "'e set" shows "vertices (LG x1 x2) = x2"
sg_mult_closedr: fixes H :: "'a set"   and x :: "'a"   and h :: "'a" assumes "G \<guillemotright> H "   and "x \<in> carrier G"   and "h \<in> H" shows "x \<cdot> h \<in> carrier G"
abs_induct: fixes P :: "real \<Rightarrow> bool"   and x :: "real" assumes "\<And>y. realrel y y \<Longrightarrow> P (Real.Real y)" shows "P x"
view_determination: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and u :: "'b"   and \<rho> :: "'c"   and v :: "'b" assumes "weak_lens x"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "u = v"
beta_reduction_left_fstE: fixes P :: "'a trm"   and M' :: "'a trm" assumes "Fst P \<rightarrow>\<beta> M'" shows "(\<exists>P'. P \<rightarrow>\<beta> P' \<and> M' = Fst P') \<or> (\<exists>A B. P = SimplyTyped.Pair A B \<and> M' = A)"
fin_long_chain_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and Q :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.fin_long_chain betw f x y z Q \<equiv> x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z \<and> finite Q \<and> MinkowskiBetweenness.long_ch_by_ord betw f Q \<and> f 0 = x \<and> y \<in> Q \<and> f (card Q - 1) = z"
next_thread_eq_None_iff: fixes active :: "'s_t"   and seed :: "natural \<times> natural" assumes "thr'_invar active"   and "True" shows "(next_thread seed active = None) = (thr'_\<alpha> active = {})"
assms_3: shows "orient_lit_inst L t1 s polarity \<sigma>'"
elims: fixes x :: "nat"   and y :: "nat \<Rightarrow> int" assumes "ns x = y" obtains "x = 0" and "y = a"   | n :: "nat" where "x = Suc n" and "y = newton_step (ns n)"
freshOutputTransition: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
a_get_dom_component_def: fixes ptr :: "(_) object_ptr" shows "local.a_get_dom_component ptr = Heap_Error_Monad.bind (get_root_node ptr) to_tree_order"
inv_fg_def: fixes l :: "(bool \<times> 'd) list" shows "inv_fg l = rev (map inv1 l)"
healthy_wp_Bind: fixes p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. healthy (wp (p (f s)))" shows "healthy (wp (Bind f p))"
intros: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "simple_function M f"   and "emeasure M {y \<in> space M. f y \<noteq> (0::'c)} \<noteq> \<infinity>" shows "Bochner_Integration.simple_bochner_integrable M f"
list_all_mp: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and Q :: "'a \<Rightarrow> bool" assumes "list_all P xs"   and "\<And>x. \<lbrakk>x \<in> set xs; P x\<rbrakk> \<Longrightarrow> Q x" shows "list_all Q xs"
diff_union_swap: fixes a :: "'a"   and b :: "'a"   and M :: "'a multiset" assumes "a \<noteq> b" shows "add_mset b (remove1_mset a M) = remove1_mset a (add_mset b M)"