hcju/isabelleps huggingface dataset:
ordIso_ordLeq_trans: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and r'' :: "('f \<times> 'f) set" assumes "r =o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
ab: shows "vinsert a b = set {a} \<union>\<^sub>\<circ> b"
silent_move_return: fixes f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ms :: "'node list"   and S :: "'node SDG_node set"   and ms' :: "'node list" assumes "pred (f a) s"   and "transfer (f a) s = s'"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "Ball (set (tl ms)) return_node"   and "length ms = length s"   and "length s = Suc (length s')"   and "s' \<noteq> []"   and "hd ms = sourcenode a"   and "hd (tl ms) = targetnode a"   and "ms' = tl ms" shows "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"
prod_zero_iff': fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "(prod f A = (0::'c)) = (\<exists>x\<in>A. f x = (0::'c))"
ab_diff_conv_add_uminus': fixes a :: "'a"   and b :: "'a" shows "a - b = - b + a"
index_mult_mat_3: fixes A :: "'c mat"   and B :: "'c mat" shows "dim_col (A * B) = dim_col B"
Allowed_extend: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program" assumes "Extend h" shows "Allowed (extend h F) = project h UNIV -` Allowed F"
n: fixes m :: "nat" assumes "n \<le> m" shows "path_strategies P $ n = path_strategies P $ m"
invertible_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "invertible A = (\<exists>A'. A ** A' = mat (1::'c) \<and> A' ** A = mat (1::'c))"
simps_1: fixes uu :: "iface \<Rightarrow> ('a word \<times> nat) list option" shows "iiface_rewrite uu MatchAny = MatchAny"
lossless_spmf_of_set: fixes A :: "'a set" shows "lossless_spmf (spmf_of_set A) = finite' A"
MonotModel_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "MonotModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP \<equiv> MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi> \<and> CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
anonymous_profile_def: shows "anonymous_profile R = image_mset (weak_ranking \<circ> R) (mset_set agents)"
butterfly_scaleC_left: fixes c :: "complex"   and \<psi> :: "'b"   and \<phi> :: "'a" shows "butterfly (c *\<^sub>C \<psi>) \<phi> = c *\<^sub>C butterfly \<psi> \<phi>"
AssertAllState''_sat_atoms_equiv_bounds: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "\<not> \<U> (assert_bound_loop ats (init t)) \<longrightarrow> (snd ` set ats) \<doteq> \<B> (assert_bound_loop ats (init t))"
simps: fixes f :: "'e \<Rightarrow> 'd option"   and g :: "'e \<Rightarrow> 'd option"   and x :: "'e" shows "(f;;g) x = (if g x = None then f x else g x)"
card0: shows "card B = dim {x. a \<bullet> x = 0}"
change_respecting_subset': fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set"   and \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "(cms, mem) \<leadsto> (cms', mem')"   and "\<forall>\<sigma>. dom \<sigma> = X \<longrightarrow> (cms, subst \<sigma> mem) \<leadsto> (cms', subst \<sigma> mem')"   and "dom \<sigma> \<subseteq> X" shows "(cms, subst \<sigma> mem) \<leadsto> (cms', subst \<sigma> mem')"
pfun_plus_minus: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "f \<subseteq>\<^sub>p g" shows "g - f + f = g"
rtrancl_induct3: fixes ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and r :: "(('a \<times> 'b \<times> 'c) \<times> 'a \<times> 'b \<times> 'c) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "((ax, ay, az), bx, by, bz) \<in> r\<^sup>*"   and "P ax ay az"   and "\<And>a aa b ab ac ba. \<lbrakk>((ax, ay, az), a, aa, b) \<in> r\<^sup>*; ((a, aa, b), ab, ac, ba) \<in> r; P a aa b\<rbrakk> \<Longrightarrow> P ab ac ba" shows "P bx by bz"
simps_3: fixes p :: "real mpoly" shows "aNeg (Leq p) = Less (- p)"
empty_rels: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "\<R> = Map.empty"
inv_is_inverse: fixes f :: "'a" assumes "C.iso f" shows "C.inverse_arrows f (C.inv f)"
b_e_check_weaken_top: fixes \<C> :: "t_context"   and es :: "b_e list"   and tn :: "t list"   and tm :: "ct list"   and ts :: "t list" assumes "check \<C> es (Type tn) = TopType tm" shows "check \<C> es (Type (ts @ tn)) = TopType tm"
GroupByPresentationInducedFun_S_R_signed_lconjaction: shows "GroupByPresentationInducedFun S (map (charpair S) ` R) signed_lconjpermutation"
hom_one: fixes x :: "'c"   and y :: "'c" shows "poly2 1 x y = (1::'c)"
do_app_no_runtime_error: fixes s :: "'b state"   and op0 :: "op0"   and vs :: "v list"   and refs' :: "v store_v list"   and ffi' :: "'b ffi_state"   and res :: "(v, v) result" assumes "do_app (refs s, ffi s) op0 (rev vs) = Some ((refs', ffi'), res)" shows "res \<noteq> Rerr (Rabort Rtimeout_error)"
take_nths_drop: fixes i :: "nat"   and k :: "nat"   and j :: "nat"   and xs :: "'b list" shows "take_nths i k (drop j xs) = take_nths (i + j) k xs"
filtermap_sequentually_ne_bot: fixes f :: "nat \<Rightarrow> 'b" shows "filtermap f sequentially \<noteq> bot"
call_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (call_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
simps_2: fixes n :: "int \<times> int" shows "headconst (C n) = n"
Units_one_closed: shows "\<one>\<^bsub>S\<^esub> \<in> Units S"
assms: shows "\<forall>y\<in>set ys. x \<le> y"
assms_2: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q I a b" shows "Q I b a"
sel_3: fixes x21 :: "'g"   and x22 :: "('g, 'h) tllist" shows "thd (TCons x21 x22) = x21"
integral_by_parts_in_bounded_set: fixes f :: "'b \<Rightarrow> real"   and I :: "real"   and a :: "'b"   and b :: "'b"   and g :: "'b \<Rightarrow> 'c"   and P :: "'c" assumes "(f has_integral I) (cbox a b)"   and "((\<lambda>x. f x *\<^sub>R g x) has_integral P) (cbox a b)"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> 0 \<le> f x"   and "bounded (g ` cbox a b)"   and "cbox a b \<noteq> {}" shows "P \<in> (*\<^sub>R) I ` closure (convex hull g ` cbox a b)"
insertion_fun_single: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and v :: "nat"   and n :: "nat" shows "insertion_fun f (\<lambda>m. a when Poly_Mapping.single v n = m) = a * f v ^ n"
lookup_plus_fun: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (s + t) = lookup s + lookup t"
arc_subpath_eq: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "arc (subpath u v g) = (path (subpath u v g) \<and> u \<noteq> v \<and> inj_on g (closed_segment u v))"
card_B: shows "|characteristic_weak_formula ` S| <o natLeq +c |UNIV|"
get_owner_document_def: shows "Shadow_DOM.get_owner_document \<equiv> l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document Shadow_DOM.get_root_node get_disconnected_nodes"
dualization_unl_2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'b" shows "(\<not> (\<diamond>F) w) = (\<box>\<not> F) w"
abs_eq: fixes x :: "complex_homo_coords" shows "conjugate (abs_complex_homo x) = abs_complex_homo (conjugate_hcoords x)"
foldli_foldli_prod_conv: fixes l2 :: "'e list"   and ctd :: "'d \<Rightarrow> bool"   and l1 :: "'f list"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d" shows "foldli l2 ctd (\<lambda>i. foldli l1 ctd (f i)) s = foldli (List.product l2 l1) ctd (\<lambda>(i, j). f i j) s"
tseytin_assmt_has: fixes G :: "'a formula"   and F :: "'a formula" assumes "G \<in> set (subformulae F)" shows "\<exists>n. (n, G) \<in> set (tseytin_assmt F)"
strengthening: fixes \<Gamma> :: "binding list"   and x :: "vrs"   and Q :: "ty"   and \<Delta> :: "binding list"   and S :: "ty"   and T :: "ty" assumes "(\<Gamma> @ VarB x Q # \<Delta>)\<turnstile>S<:T" shows "(\<Gamma> @ \<Delta>)\<turnstile>S<:T"
setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p_member_iff: fixes t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term"   and x :: "('c, 'd) stateful_strand_step" shows "((t, s) \<in> setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p x) = (x = insert\<langle>t,s\<rangle> \<or> x = delete\<langle>t,s\<rangle> \<or> (\<exists>ac. x = \<langle>ac: t \<in> s\<rangle>) \<or> (\<exists>X F F'. x = \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> \<and> (t, s) \<in> set F'))"
scale_zero_right: fixes a :: "complex" shows "a *\<^sub>C (0::'d) = (0::'d)"
bv_leqs_trans: fixes xs :: "bool list"   and ys :: "bool list"   and zs :: "bool list" assumes "xs \<preceq>\<^sub>b ys"   and "ys \<preceq>\<^sub>b zs" shows "xs \<preceq>\<^sub>b zs"
S17: shows "g_step (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair n x) = g_pair (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair n x)"
stream_rel_sconst: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" shows "stream_all2 P (sconst x) (sconst y) = P x y"
continuous_at_Inf_mono: fixes f :: "'b \<Rightarrow> 'd"   and S :: "'b set" assumes "mono f"   and "continuous (at_right (Inf S)) f"   and "S \<noteq> {}"   and "bdd_below S" shows "f (Inf S) = Inf (f ` S)"
assms_2: shows "A = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
Px_nn: fixes x :: "'b" assumes "x \<in> space S" shows "0 \<le> Px x"
S\<^sub>e_E1: fixes s :: "'s"   and t :: "'s" assumes "s \<in> S\<^sub>e \<inter> S1"   and "(s, t) \<in> E" shows "t \<in> S\<^sub>e"
dimI: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K R"   and "dimension n K E" shows "(local.dim over K) E = n"
termination_variant_invar: fixes guard :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> nat"   and bound :: "nat"   and p :: "real"   and body :: "'a \<Rightarrow> 'a spmf"   and s :: "'a" assumes "\<And>s. \<lbrakk>guard s; I s\<rbrakk> \<Longrightarrow> f s \<le> bound"   and "\<And>s. \<lbrakk>guard s; I s\<rbrakk> \<Longrightarrow> p \<le> spmf (map_spmf (\<lambda>s'. f s' < f s) (body s)) True"   and "0 < p"   and "\<And>s. \<lbrakk>guard s; I s\<rbrakk> \<Longrightarrow> lossless_spmf (body s)"   and "\<And>s s'. \<lbrakk>s' \<in> set_spmf (body s); I s; guard s\<rbrakk> \<Longrightarrow> I s'"   and "I s" shows "lossless_spmf (loop_spmf.while guard body s)"
sinks_aux_elem: fixes u :: "'e"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and U :: "'e set"   and xs :: "'f list" assumes "u \<in> sinks_aux I D U xs" shows "u \<in> U \<or> (\<exists>x\<in>set xs. u = D x)"
verticesFrom_splitAt_v_snd: fixes f :: "face"   and v :: "nat" assumes "distinct (verticesFrom f v)" shows "snd (splitAt v (verticesFrom f v)) = tl (verticesFrom f v)"
len_greater_imp_nonempty: fixes x :: "nat"   and l :: "'c list" assumes "x < length l" shows "l \<noteq> []"
list_of_dlist_eq_Nil_iff: fixes dxs :: "'c set_dlist" shows "(list_of_dlist dxs = []) = (dxs = DList_Set.empty)"
item_origin_finished_item: fixes \<alpha> :: "symbol list" shows "item_origin (finished_item \<alpha>) = 0"
constraints_of_mat_vec_solution: fixes A :: "rat mat"   and nr :: "nat"   and nc :: "nat"   and b :: "rat vec" assumes "A \<in> carrier_mat nr nc"   and "b \<in> carrier_vec nr" shows "(\<exists>x\<in>carrier_vec nc. A *\<^sub>v x \<le> b) = (\<exists>v. \<forall>c\<in>constraints_of_mat_vec A b. v \<Turnstile>\<^sub>l\<^sub>e c)"
trss_drop_context: fixes s :: "'f list"   and c :: "'f list multiset"   and ce :: "'f list multiset"   and w :: "('g, 'h) label list"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c + ce), w, s', c' + ce) \<in> trcl (trss fg)" shows "((s, c), w, s', c') \<in> trcl (trss fg) \<and> mon_w fg w \<inter> mon_c fg ce = {}"
simps_19: fixes uy :: "'a"   and v :: "'a rexp" shows "local.le_rexp (Atom uy) (Star v) = True"
index_of_bound: fixes y :: "'b list"   and x :: "'b" assumes "y \<noteq> []"   and "x \<in> set y" shows "index_of x y < length y"
opp_chambers_subset_opp_half_apartment: shows "order.greater_eq folding_f.Y (folding_g.\<C> - f \<turnstile> folding_g.\<C>)"
trans_Restr: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "trans r" shows "trans (Restr r A)"
assms_5: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path \<gamma>"   and "path_image \<gamma> \<subseteq> S"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "z \<notin> S" shows "winding_number \<gamma> z = 0"
SCallRedsNone: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,False\<rangle>"   and "\<nexists>b Ts T m D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow>* \<langle>THROW NoSuchMethodError,s\<^sub>2,False\<rangle>"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" shows "((a1, a2) \<in> encoding.indRelLPO Enc) = ((\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S) \<or> (\<exists>S. a1 = SourceTerm S \<and> a2 = SourceTerm S) \<or> (\<exists>T. a1 = TargetTerm T \<and> a2 = TargetTerm T) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> (P, Q) \<in> encoding.indRelLPO Enc \<and> (Q, R) \<in> encoding.indRelLPO Enc))"
vid_on_eq_atI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "a = b"   and "a \<in>\<^sub>\<circ> A" shows "vid_on A\<lparr>a\<rparr> = b"
mask_Suc_rec: fixes n :: "nat" shows "mask (Suc n) = 2 * mask n + 1"
pair_commute_spmf: fixes p :: "'a spmf"   and q :: "'b spmf" shows "pair_spmf p q = map_spmf (\<lambda>(y, x). (x, y)) (pair_spmf q p)"
pgallery_map: fixes X :: "'c set set"   and Y :: "'d set set"   and f :: "'c \<Rightarrow> 'd"   and Cs :: "'c set list" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.pmaxsimpchain X Cs" shows "SimplicialComplex.pmaxsimpchain Y (f \<Turnstile> Cs)"
pdivides_imp_roots_incl: fixes p :: "'a list list"   and q :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "q \<in> carrier (poly_ring (poly_ring R))"   and "q \<noteq> []"   and "p pdivides\<^bsub>poly_ring R\<^esub> q" shows "UP.roots p \<subseteq># UP.roots q"
real_of_def: shows "real_of \<equiv> (rep_real_alg ---> id) real_of_3"
kernel_def: fixes nr :: "nat"   and nc :: "nat"   and A :: "'b mat" shows "Matrix_Kernel.kernel nr nc A \<equiv> A \<in> carrier_mat nr nc"
finite_UNIV_num0_def: shows "finite_UNIV = Phantom(num0) False"
gtr_preserve: fixes c :: "'f multiset"   and w :: "'g list"   and c' :: "'f multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set"   and P :: "'f multiset \<Rightarrow> bool" assumes "(c, w, c') \<in> trcl (gtr T)"   and "P c"   and "\<And>s c s' c' e. \<lbrakk>P (add_mset s c); ((s, c), e, s', c') \<in> T\<rbrakk> \<Longrightarrow> P (add_mset s' c')" shows "P c'"
NegPos_map_append: fixes C :: "'d \<Rightarrow> 'c"   and as :: "'d negation_type list"   and bs :: "'d negation_type list" shows "NegPos_map C (as @ bs) = NegPos_map C as @ NegPos_map C bs"
simps_2: fixes k :: "'b" shows "form_of_lit (k\<inverse>) = \<^bold>\<not> (Atom k)"
assms_2: shows "range p \<subseteq> {0..<q}"
WHILE_le_WHILEI: fixes b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c"   and I :: "'c \<Rightarrow> bool" shows "WHILE b f s \<le> WHILE\<^bsup>I\<^esup> b f s"
id_cblinfun_apply: fixes \<psi> :: "'c" shows "id_cblinfun *\<^sub>V \<psi> = \<psi>"
inf_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
retraction_if_Img_eq_Cod_2: fixes g :: "'s" assumes "arr g"   and "Img g = Cod g" shows "ide (g \<cdot> mkArr (Cod g) (Dom g) (inv_into (Dom g) (Fun g)))"
assms_2: fixes exp :: "exp"   and c :: "'a"   and var :: "Vars.var" assumes "\<And>c. P1 c exp" shows "P1 c (App exp var)"
expectation_gpv_cong_fail: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and fail :: "ennreal"   and f :: "'j \<Rightarrow> ennreal"   and fail' :: "ennreal" assumes "colossless_gpv \<I> gpv" shows "expectation_gpv fail \<I> f gpv = expectation_gpv fail' \<I> f gpv"
in_set_tlD: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
consistentD: fixes x :: "real \<Rightarrow> 'b"   and t :: "real"   and T :: "real"   and B :: "real"   and p :: "nat"   and incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "real" assumes "consistent x t T B p incr"   and "0 \<le> h"   and "t + h \<le> T" shows "dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \<le> B * h ^ (p + 1)"
program_equality_iff: fixes F :: "'c program"   and G :: "'c program" shows "(F = G) = (Init F = Init G \<and> Acts F = Acts G \<and> AllowedActs F = AllowedActs G)"
sigma_sets_mono: fixes A :: "'c set set"   and X :: "'c set"   and B :: "'c set set" assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
stl_def: fixes stream :: "'c stream" shows "stl stream = (case stream of x1 ## x2 \<Rightarrow> x2)"
slength_0_conv_4: fixes xs :: "[:'a:]" shows "eq\<cdot>(slength\<cdot>xs)\<cdot>0 = snull\<cdot>xs"
isometry_cblinfun_image_inf_distrib': fixes U :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b ccsubspace"   and C :: "'b ccsubspace" shows "U *\<^sub>S B \<sqinter> C \<le> U *\<^sub>S B \<sqinter> (U *\<^sub>S C)"
IH: assumes "vs @ xs \<in> L M1 - L M2"   and "vs \<in> L M2 \<inter> L M1" shows "\<exists>xs'. prefix xs' xs \<and> sequence_to_failure M1 M2 (vs @ xs')"
induct: fixes x1 :: "'a"   and x2 :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "hb x1 x2"   and "\<And>m1 i m2. Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2 \<Longrightarrow> P m1 m2"   and "\<And>m1 i m2. Deliver m1 \<sqsubset>\<^sup>i Broadcast m2 \<Longrightarrow> P m1 m2"   and "\<And>m1 m2 m3. \<lbrakk>hb m1 m2; P m1 m2; hb m2 m3; P m2 m3\<rbrakk> \<Longrightarrow> P m1 m3" shows "P x1 x2"
consequent_in_maximal: fixes C :: "'b fm set set"   and V :: "'b fm set"   and p :: "'b fm"   and q :: "'b fm" assumes "consistency C"   and "V \<in> C"   and "maximal V C"   and "p \<in> V"   and "(p \<^bold>\<longrightarrow> q) \<in> V" shows "q \<in> V"
ecut_apply: fixes y :: "hf"   and x :: "hf"   and f :: "hf \<Rightarrow> 'a" assumes "y \<^bold>\<in> eclose x" shows "ecut f x y = f y"
length_lang_pow_lb: fixes X :: "'b list set"   and k :: "nat"   and x :: "'b list"   and n :: "nat" assumes "\<forall>x\<in>X. k \<le> |x|"   and "x \<in> X\<^bsup>n\<^esup>" shows "k \<cdot> n \<le> |x|"
N2_leadsto_a: shows "\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2) \<and> SF(N2)_(x, y, sem, pc1, pc2) \<and> \<box>#True \<longrightarrow> (pc2 = #a \<or> pc2 = #b \<or> pc2 = #g \<leadsto> pc2 = #a)"
step_z_V': fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "\<forall>c\<in>clk_set A. v c \<le> n"   and "Z \<in> V'" shows "Z' \<in> V'"
a_get_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr" shows "CD.a_get_disconnected_nodes document_ptr = get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes"
Ramsey_def: fixes \<F> :: "nat set set"   and r :: "nat" shows "Ramsey \<F> r \<equiv> \<forall>f\<in>\<F> \<rightarrow> {..<r}. \<forall>M. infinite M \<longrightarrow> (\<exists>N i. N \<subseteq> M \<and> infinite N \<and> i < r \<and> (\<forall>j<r. j \<noteq> i \<longrightarrow> f -` {j} \<inter> \<F> \<inter> Pow N = {}))"
h_def: shows "h = Abs_fps (\<lambda>n. norm (fps_nth f n))"
UNpart1_UNpart01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n" shows "UNpart1 cl dl P = UNpart01 cl dl P - {BrnFT cl dl}"
skel_def: fixes X :: "('d, 'e, 'f, 'g, 'h) qTerm set" shows "skel X \<equiv> qSkel (pick X)"
QTrans: shows "Q \<Longrightarrow>\<^sub>l \<alpha> \<prec> Q'"
get_integrable_path: fixes s :: "complex set"   and pts :: "complex set"   and f :: "complex \<Rightarrow> complex"   and a :: "complex"   and b :: "complex" assumes "open s"   and "connected (s - pts)"   and "finite pts"   and "f holomorphic_on s - pts"   and "a \<in> s - pts"   and "b \<in> s - pts" obtains g :: "real \<Rightarrow> complex" where "valid_path g" and "pathstart g = a" and "pathfinish g = b" and "path_image g \<subseteq> s - pts" and "f contour_integrable_on g"
p: shows "p = (\<lambda>n. 16 ^ n * fact n ^ 4 / (fact (2 * n))\<^sup>2 / (2 * real n + 1))"
lnths_lappend_lfinite: fixes xs :: "'c llist"   and k :: "nat"   and ys :: "'c llist"   and A :: "nat set" assumes "llength xs = enat k" shows "lnths (lappend xs ys) A = lappend (lnths xs A) (lnths ys {n. n + k \<in> A})"
cond_or_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool"   and S :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<or> Q \<triangleleft> b \<triangleright>  R \<or> S) = ((P \<triangleleft> b \<triangleright>  R) \<or> (Q \<triangleleft> b \<triangleright>  S))"
qml_3: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<diamond>\<phi>]]"