hcju/isabelleps huggingface dataset:
BINDINGc_decreasing: fixes X :: "ntrm set"   and Y :: "ntrm set"   and B :: "ty" assumes "X \<subseteq> Y" shows "BINDINGc B Y \<subseteq> BINDINGc B X"
finite_basis_exists: assumes "V.fin_dim" shows "\<exists>\<beta>. finite \<beta> \<and> V.basis \<beta>"
simps_5: fixes b :: "bexp"   and c :: "com"   and t :: "char list \<Rightarrow> int option" shows "Fold.fold (WHILE b DO c) t = WHILE b DO Fold.fold c (t |` (- lvars c))"
InitStatuses_def: shows "InitStatuses \<equiv> fst \<circ> snd \<circ> Rep_kripke"
cast\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes shadow_root :: "('m, 'n, 'o, 'u option) RShadowRoot_scheme" shows "cast shadow_root = RObject.extend (RObject.truncate shadow_root) (Inr (Inr (Inl (RObject.more shadow_root))))"
zoom_def: fixes l :: "'e \<Longrightarrow> 'c"   and m :: "('e, 'd) state" shows "l \<rhd> m = State (\<lambda>b. case run_state m (get\<^bsub>l\<^esub> b) of (c, a) \<Rightarrow> (c, put\<^bsub>l\<^esub> b a))"
A'_def2: shows "A' = Hermite_of_upt_row_i A k ass res"
distinct_19: fixes x31 :: "nat"   and x32 :: "nat"   and x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list" shows "recf.Id x31 x32 \<noteq> Cn x41 x42 x43"
A: fixes V :: "'a set" shows "(\<exists>u\<in>U. \<exists>v\<in>sources_aux I D V xs. (u, v) \<in> I) \<longrightarrow> (\<exists>u\<in>sinks_aux I D U xs. \<exists>v\<in>V. (u, v) \<in> I)"
class_type_of'_widenD: fixes T :: "ty"   and C :: "String.literal"   and P :: "'a prog" assumes "class_type_of' T = \<lfloor>C\<rfloor>" shows "P \<turnstile> T \<le> Class C"
simps_1: shows "New_max.merge_option (None, None) = None"
dioph_ineq_dfs_mono: fixes ks :: "int list"   and l :: "int"   and z :: "nat option list \<times> int list"   and xs :: "int list"   and i :: "nat"   and k :: "nat"   and n :: "nat" assumes "dioph_invariant ks l z"   and "list_all (dioph_is_node ks l) xs"   and "fst z ! i = Some k" shows "fst (dioph_ineq_dfs.dfs n ks z xs) ! i = Some k"
connected_verts: assumes "Digraph_Component.connected G"   and "arcs G \<noteq> {}" shows "verts G = tail G ` arcs G \<union> head G ` arcs G"
FG_map: fixes r :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "m \<in> V" shows "T (r \<cdot> m) = r \<star> T m"
subst_trans: fixes s1 :: "('b \<times> 'b trm) list"   and s2 :: "('b \<times> 'b trm) list"   and s3 :: "('b \<times> 'b trm) list" assumes "s1 \<doteq> s2"   and "s2 \<doteq> s3" shows "s1 \<doteq> s3"
assms_3: shows "\<forall>n w. qty n w = qty (Suc n) w"
DAxioms_def: shows "DAxioms = {z. \<exists>p vs. z = ([FAtom Neg p vs, FAtom Pos p vs], {})}"
simps_1: fixes s :: "nat \<Rightarrow> 'a list"   and x :: "nat" shows "join_ti s x 0 = s x"
simple_path2: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>m" and "distinct ns'" and "set ns' \<subseteq> set ns" and "length ns' \<le> length ns" and "n \<notin> set (tl ns')" and "m \<notin> set (butlast ns')"
l_set_disconnected_nodes_get_disconnected_nodes_axioms: shows "l_set_disconnected_nodes_get_disconnected_nodes ShadowRootClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
domintros_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and z :: "nat" shows "rb_aux_dom ((bs, ss, []), z)"
res: shows "(if degree f = 0 then (lead_coeff f, []) else (sgn (lead_coeff f) * content f, square_free_factorization_int_main (sdiv_poly f (sgn (lead_coeff f) * content f)))) = (d, fs)"
l_unit: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<one> \<cdot> a = a"
prod_coprime_right: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and A :: "'d set"   and a :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>i. i \<in> A \<Longrightarrow> algebraic_semidom.coprime times one a (f i)" shows "algebraic_semidom.coprime times one a (comm_monoid_mult.prod times one f A)"
Collect_mono: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
cycle_free_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "cycle_free m n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (\<forall>j\<le>n. len m i j (rem_cycles i j xs) \<le> len m i j xs) \<and> (0::'a) \<le> len m i i xs"
moebius_pt_rotation_inf_iff: fixes v :: "real"   and x :: "complex_homo" shows "(moebius_pt (moebius_rotation v) x = \<infinity>\<^sub>h) = (x = \<infinity>\<^sub>h)"
prod_fun_def: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and k :: "'c" shows "prod_fun f1 f2 k = (\<Sum>l. f1 l * (\<Sum>q. f2 q when k = l + q))"
simps_3: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg" shows "bicategorical_language.Cod V src trg (t \<^bold>\<star> u) = bicategorical_language.Cod V src trg t \<^bold>\<star> bicategorical_language.Cod V src trg u"
rot_circle_left_edge_def: shows "rot_circle_left_edge = (- 1, \<lambda>t. (- d * circle_y (x_coord t), x_coord t * d))"
\<phi>: shows "\<guillemotleft>\<phi> : src f \<Rightarrow> g \<star> f\<guillemotright> \<and> local.iso \<phi>"
im_CP_V: fixes v :: "'v" assumes "v \<in> V" shows "(\<lambda>g. CP g v) ` G \<subseteq> V"
assms_1: shows "integrable M (\<lambda>x. (X x)\<^sup>2)"
3: assumes "mat nr (length (w # N)) (w # N)" shows "nr = length w"
eq_classes_def: fixes S :: "('d, 'e) eq_object_scheme" shows "classes\<^bsub>S\<^esub> = {class_of\<^bsub>S\<^esub> x |x. x \<in> carrier S}"
equivpartpCI: fixes x :: "'b"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x \<noteq> y \<Longrightarrow> sympartp r x y" shows "equivpartp r x y"
assms: shows "cone (h1, U1) \<subseteq> cone (h2, U2)"
order_iff_strict: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term" shows "b \<unrhd> a = (b \<rhd> a \<or> a = b)"
assms_1: fixes x :: "nat \<Rightarrow> 'a" shows "A x \<in> carrier_mat m n"
ex_minimum: shows "\<forall>X. X \<subseteq> carrier D \<and> X \<noteq> {} \<longrightarrow> (\<exists>x. minimum_elem D X x)"
flt_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "flt a b = (fcompare a b = ccode.Lt)"
arrow_from_functor_axioms: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a d x"
bij_betw_sequence_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A" shows "bij_betw (sequence_of A enum) (A \<rightarrow>\<^sub>E B) {xs. set xs \<subseteq> B \<and> length xs = card A}"
simple_match_port_code: fixes s :: "16 word"   and e :: "16 word"   and p_p :: "16 word" shows "simple_match_port (s, e) p_p = (s \<le> p_p \<and> p_p \<le> e)"
s\<^sub>0: shows "s\<^sub>0 \<in> A"
lspasl_magicr_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h2 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "(\<not> sep_algebra.sep_impl plus sep_disj A B h2) = (\<exists>h1 h0. heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> A h1 \<and> \<not> B h0)"
orthogonal_leave_RF: fixes x :: "'v"   and y :: "'v" assumes "edge \<Delta> x y"   and "x \<in> local.cut_of_sep S"   and "y \<notin> local.cut_of_sep S" shows "(x, y) \<in> S"
respectfulness_10: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "Unf\<^sub>G \<phi> \<equiv>\<^sub>P Unf\<^sub>G \<psi>"
inf_def: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and cfg :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" shows "\<Gamma>\<turnstile> cfg \<rightarrow> \<dots>(\<infinity>) \<equiv> \<exists>f. f 0 = cfg \<and> (\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (i + 1))"
hyps_1: fixes xs :: "bool list" assumes "n = harry_meas xs" shows "True \<notin> set ((harry_step ^^ harry_meas xs) xs)"
smult_assoc1: fixes x :: "'a vec"   and a :: "'a"   and b :: "'a" assumes "True"   and "True"   and "x \<in> carrier_vec n" shows "a * b \<cdot>\<^sub>v x = a \<cdot>\<^sub>v (b \<cdot>\<^sub>v x)"
hom_eq: fixes f :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and f' :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c" assumes "f \<in> hom (free_Abelian_group T) H"   and "\<And>x. x \<in> carrier (free_Abelian_group T) \<Longrightarrow> f' x = f x" shows "f' \<in> hom (free_Abelian_group T) H"
sees_method_decl_above: fixes P :: "'b prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b option"   and D :: "String.literal" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
top_add: fixes x :: "'k" assumes "(0::'k) \<le> x" shows "\<top> + x = \<top>"
subFDual: fixes theta :: "vbl \<Rightarrow> vbl"   and p :: "signs \<Rightarrow> signs"   and q :: "signs \<Rightarrow> signs"   and r :: "signs \<Rightarrow> signs"   and A :: "formula" shows "subF theta (dual p q r A) = dual p q r (subF theta A)"
finite_incidence_system_axioms: shows "finite_incidence_system (\<V> \<union> bl) (add_block bl)"
eq_\<I>_converter_reflI: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv"
weak_lower_semilattice_axioms: shows "weak_lower_semilattice (division_rel G)"
rel_witness_spmf_def: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_witness_spmf A = map_pmf rel_witness_option \<circ> rel_witness_pmf (rel_option A)"
G1: shows "hs.\<alpha> (fst loop) = b_accessible (ls.\<alpha> \<delta>)"
protocol_def: fixes M :: "'a \<times> 'a"   and v :: "bool" shows "protocol M v = (let (m0, m1) = M in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>v = a * b mod order \<G> in sample_uniform (order \<G>) \<bind> (\<lambda>c\<^sub>v'. sample_uniform_units (order \<G>) \<bind> (\<lambda>r0. sample_uniform_units (order \<G>) \<bind> (\<lambda>s0. let w0 = (\<^bold>g [^] a) [^] s0 \<otimes> \<^bold>g [^] r0; z0' = (\<^bold>g [^] (if v then c\<^sub>v' else c\<^sub>v)) [^] s0 \<otimes> (\<^bold>g [^] b) [^] r0 in sample_uniform_units (order \<G>) \<bind> (\<lambda>r1. sample_uniform_units (order \<G>) \<bind> (\<lambda>s1. let w1 = (\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1; z1' = (\<^bold>g [^] (if v then c\<^sub>v else c\<^sub>v')) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1; enc_m0 = z0' \<otimes> m0; enc_m1 = z1' \<otimes> m1; out_2 = if v then enc_m1 \<otimes> inv (w1 [^] b) else enc_m0 \<otimes> inv (w0 [^] b) in return_spmf ((), out_2)))))))))"
bprop: shows "\<forall>i\<in>A. [b i = 1] (mod m i) \<and> [b i = 0] (mod prod m (A - {i}))"
finsert_ident: fixes x :: "'b"   and A :: "'b fset"   and B :: "'b fset" assumes "x |\<notin>| A"   and "x |\<notin>| B" shows "(finsert x A = finsert x B) = (A = B)"
lift_le_self: fixes x :: "V"   and y :: "V" shows "(lift x y \<le> x) = (y = 0)"
Map_in_Hom: fixes f :: "('a, 'a) CC.arr" assumes "CC.arr f" shows "CC.Map f \<in> hom (CC.Dom f) (CC.Cod f)"
coboundedI1: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c \<le> a" shows "c \<le> sup a b"
v_nS: fixes cfg :: "'s cfg" assumes "cfg \<notin> valid_cfg" shows "v cfg = undefined"
complex_i_not_zero: shows "\<i> \<noteq> 0"
simps_8: fixes f :: "'d \<Rightarrow> 'c" shows "map f [] = []"
antisym: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> r"   and "(y, x) \<in> r" shows "x = y"
trace_induct_final: fixes x1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and x2 :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and final :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and P :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> ((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool" assumes "x1 \<Rightarrow>\<^sup>*\<^bsub>x2\<^esub> final"   and "P final [] final"   and "\<And>conf' T conf. \<lbrakk>conf' \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> final; P conf' T final; conf \<Rightarrow> conf'\<rbrakk> \<Longrightarrow> P conf (conf' # T) final" shows "P x1 x2 final"
ereal_sgn_abs: fixes l :: "ereal" shows "sgn l * l = \<bar>l\<bar>"
gcd_proj2_if_dvd: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "gcd a b = normalize b"
distributed_AE2: fixes N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal"   and P :: "'b \<Rightarrow> bool" assumes "distributed M N X f"   and "Measurable.pred N P" shows "(AE x in M. P (X x)) = (AE x in N. 0 < f x \<longrightarrow> P x)"
mult_rel_well_defined: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b"   and z :: "'a set"   and z' :: "'a set" assumes "x \<in> Sigma I \<FF>1"   and "y \<in> Sigma I \<FF>1"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y"   and "z' \<in> I"   and "z' \<subseteq> fst x"   and "z' \<subseteq> fst y" shows "stk1.class_of z (mult_str1 z (\<rho>1 (fst x) z (snd x)) (\<rho>1 (fst y) z (snd y))) = stk1.class_of z' (mult_str1 z' (\<rho>1 (fst x) z' (snd x)) (\<rho>1 (fst y) z' (snd y)))"
dfs_def: shows "local.dfs \<equiv> REC\<^sub>T (\<lambda>dfs (V, v). if v = tgt then RETURN (V, True) else let V = insert v V in FOREACH\<^sub>C (E `` {v}) (\<lambda>(V, brk). \<not> brk) (\<lambda>v' (V, brk). if v' \<notin> V then dfs (V, v') else RETURN (V, False)) (V, False)) ({}, src) \<bind> (\<lambda>(uu_, r). RETURN r)"
7: shows "(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F"
star_inductl: fixes z :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" assumes "z \<union> x O y \<subseteq> y" shows "x\<^sup>* O z \<subseteq> y"
open_surjective_linear_image: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "open A"   and "linear f"   and "surj f" shows "open (f ` A)"
is_extensional: fixes f :: "'t" assumes "\<not> S'.arr f" shows "\<Psi>\<Phi>.map f = S'.null"
dbl_def: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus x = plus x x"
term_ok'_decr: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and i :: "nat" assumes "term_ok' \<Sigma> t" shows "term_ok' \<Sigma> (decr i t)"
one_ge_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "SN_Orders.class.ordered_semiring_1 one times plus zero less_eq" shows "less_eq zero one"
simps_1: fixes "is" :: "nat list"   and f :: "'var \<Rightarrow> 'var"   and n :: "('form, 'rule) graph_node"   and p :: "'form"   and i :: "nat"   and s :: "'subst" shows "globalize_node is f (I n p i s) = I n p (isidx is) (subst_renameLCs f s)"
map_tailrec_def: fixes f :: "'c \<Rightarrow> 'b"   and as :: "'c list" shows "map_tailrec f as = rev (map_tailrec_rev f as \<epsilon>)"
usubst_upd_comm: fixes x :: "'c \<Longrightarrow> 'd"   and y :: "'e \<Longrightarrow> 'd"   and \<sigma> :: "'f \<Rightarrow> 'd"   and u :: "('c, 'f) uexpr"   and v :: "('e, 'f) uexpr" assumes "x \<bowtie> y" shows "\<sigma>(x \<mapsto>\<^sub>s u, y \<mapsto>\<^sub>s v) = \<sigma>(y \<mapsto>\<^sub>s v, x \<mapsto>\<^sub>s u)"
s_id_om: shows "1\<^sub>\<sigma>\<^sup>\<omega> = U"
Inf_superset_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'a set"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "B \<subseteq> A" shows "less_eq (Inf A) (Inf B)"
accepts_def: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and w :: "'d list" shows "NA.accepts A w = (\<exists>q\<in>NA.delta A w (start A). fin A q)"
dickson_grading_varnum: fixes X :: "'a set" assumes "finite X" shows "dickson_grading (varnum X)"
ev_alw_impl: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "ev \<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs) xs" shows "ev \<psi> xs"
invariant_after_set_error_code: shows "\<forall>s a. invariant s \<longrightarrow> invariant (set_error_code s a)"
vec0_def: fixes n :: "nat" shows "vec0 n = replicate n (0::'c)"
has_vector_derivative_complex_iff: fixes f :: "real \<Rightarrow> complex"   and x :: "complex"   and F :: "real filter" shows "(f has_vector_derivative x) F = (((\<lambda>x. Re (f x)) has_real_derivative Re x) F \<and> ((\<lambda>x. Im (f x)) has_real_derivative Im x) F)"
bst_insert: fixes t :: "('a \<times> 'b) tree"   and x :: "'a" assumes "Tree2.bst t" shows "Tree2.bst (local.insert x t)"
elims_3: fixes x :: "'v"   and xa :: "('v \<times> 'w \<times> 'v) list"   and xb :: "'v" assumes "\<not> is_path x xa xb" obtains v :: "'v" and v' :: "'v" where "x = vb__" and "xa = []" and "xb = v'a__" and "\<not> (vb__ = v'a__ \<and> v'a__ \<in> V)"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and p :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, w, v2) # p" and "xb = v'" and "\<not> (v = v1 \<and> (v1, w, v2) \<in> E \<and> is_path v2 p v')"
multiplicity_distinct_prime_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "factorial_semiring.multiplicity times one p (power.power one times q n) = 0"
\<omega>_def: shows "\<omega> = L (\<lambda>n. (S ^^ n) Z)"
smult_smult_assoc: fixes a :: "'c"   and b :: "'c"   and v :: "'c vec" shows "a \<cdot>\<^sub>v (b \<cdot>\<^sub>v v) = a * b \<cdot>\<^sub>v v"
at_left_eq: fixes y :: "'a"   and x :: "'a" assumes "y < x" shows "at_left x = (INF a\<in>{..<x}. principal {a<..<x})"
finite_Image: fixes R :: "('b \<times> 'c) set"   and A :: "'b set" assumes "finite R" shows "finite (R `` A)"
zero_closed: shows "0\<^sub>v dA \<in> carrier_vec dA"
order_less_not_sym: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "\<not> y < x"
ImpR: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and G :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "F, \<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta>"
closedin_diff: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "closedin U S"   and "openin U T" shows "closedin U (S - T)"
58: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> (y \<sqinter> z) \<squnion> (x \<sqinter> (y \<sqinter> u) \<squnion> x \<sqinter> (y \<sqinter> (z \<squnion> u))) = x \<sqinter> (y \<sqinter> (z \<squnion> u))"
u: shows "u \<equiv> getFr [x, y, z] [] [\<phi>]"
prems: shows "fst ` set (a # xs) \<subseteq> fst ` upds"
hyps_2: fixes k :: "nat" assumes "k \<le> length xs"   and "\<And>i. i < length xs \<longrightarrow> k \<le> i \<longrightarrow> xs ! i = ys ! i" shows "drop k xs = drop k ys"
pmf_prob_code_unfold: fixes p :: "'a pmf" shows "pmf_prob p = measure_pmf.prob p"
assms_1: shows "is_heap_except_up (\<le>) heap l r x"
wpo_s: fixes i :: "nat" assumes "i \<in> set (\<sigma> (f, length ss))" shows "Fun f ss \<succ> ss ! i"
comm_append_are_replicate: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<cdot> ys = ys \<cdot> xs" shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
limsup_finite_then_bounded: fixes u :: "nat \<Rightarrow> real" assumes "limsup (\<lambda>x. ereal (u x)) < \<infinity>" shows "\<exists>C. \<forall>n. u n \<le> C"
yPx: assumes "y \<^bsub>r\<^esub>\<prec> x" shows "thesis"
teq: shows "T = S \<inter> {x. a \<bullet> x = b}"
strict_implies_not_eq: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subset># b" shows "a \<noteq> b"
strategyproofness_1: shows "pmf (sds R28) b + pmf (sds R28) a \<le> pmf (sds R2) c + pmf (sds R2) a"
dims0_def: shows "a.dims0 = nths dims a.vars0"
assms: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
fset_exhaust: fixes S :: "'b fset" obtains "S = {||}"   | x :: "'b" and S' :: "'b fset" where "S = insert_fset x S'"
norm_signed_length: fixes w :: "bit list" shows "length (norm_signed w) \<le> length w"
prv_all_imp_gen: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and x :: "'a"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "x \<notin> Fvars \<phi>"   and "prv (imp \<phi> \<chi>)" shows "prv (imp \<phi> (all x \<chi>))"
ds_trans_contrapos: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "trans r"   and "x \<notin> r \<down>s S"   and "(x, y) \<in> r" shows "y \<notin> r \<down>s S"
reduce_system_matrix_subsets_helper_aux: fixes signs :: "rat list list"   and subsets :: "nat list list"   and S :: "nat list"   and p :: "real poly" assumes "length signs \<le> length subsets"   and "\<forall>x. List.member S x \<longrightarrow> x < length subsets"   and "p \<noteq> 0" shows "alt_matrix_A signs (take_indices subsets S) = take_rows_from_matrix (alt_matrix_A signs subsets) S"
seq_comp_list_ref_union_closed: fixes P :: "'c option process"   and PS :: "'c option process list" assumes "\<And>x. x \<in> set (butlast (P # PS)) \<Longrightarrow> weakly_sequential x"   and "\<And>x. x \<in> set (P # PS) \<Longrightarrow> ref_union_closed x" shows "ref_union_closed (foldl (;) P PS)"