hcju/isabelleps huggingface dataset:
bisim1Sync4: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and V :: "nat"   and e1 :: "'addr expr1"   and a :: "'addr" assumes "P,e2,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (insync\<^bsub>V\<^esub> (a) e', xs) \<leftrightarrow> (stk, loc, Suc (Suc (Suc (length (compE2 e1) + pc))), xcp)"
beta_l_Suc_l_derive: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "2 * (grover_state.alpha_l n f l * complex_of_real (cos (grover_state.\<theta> n f / 2)) - grover_state.beta_l n f l * complex_of_real (sin (grover_state.\<theta> n f / 2))) * complex_of_real (sin (grover_state.\<theta> n f / 2)) + grover_state.beta_l n f l = grover_state.beta_l n f (l + 1)"
step: fixes t :: "('c, 'd) tm"   and u :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"   and "u \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
subfrmlsn_sf_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "sf \<psi> \<subseteq> sf \<phi>"
similar_matD: fixes A :: "'b mat"   and B :: "'b mat" assumes "similar_mat A B" shows "\<exists>n P Q. {A, B, P, Q} \<subseteq> carrier_mat n n \<and> P * Q = 1\<^sub>m n \<and> Q * P = 1\<^sub>m n \<and> A = P * B * Q"
states_Accessible_dfa: shows "states Accessible.Accessible_dfa = Accessible.accessible"
adm_not_mem: fixes f :: "'d \<Rightarrow> 'e set"   and y :: "'e" assumes "cont f" shows "adm (\<lambda>x. y \<notin> f x)"
deg_sectD_1: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and d :: "nat" assumes "t \<in> deg_sect X d" shows "t \<in> .[X]"
"1_1": fixes x :: "'a option list set list"   and xa :: "'a option list set list"   and x2 :: "((nat set list \<times> nat set \<times> nat set list) \<times> nat set list \<times> nat set \<times> nat set list) \<times> ('a option list set list \<times> 'a option list set \<times> 'a option list set list) \<times> 'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xb :: "(nat set list \<times> nat set \<times> nat set list) \<times> nat set list \<times> nat set \<times> nat set list"   and y :: "('a option list set list \<times> 'a option list set \<times> 'a option list set list) \<times> 'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xc :: "nat set list \<times> nat set \<times> nat set list"   and ya :: "nat set list \<times> nat set \<times> nat set list"   and xd :: "nat set list"   and yb :: "nat set \<times> nat set list"   and xe :: "nat set"   and yc :: "nat set list"   and xf :: "nat set list"   and yba :: "nat set \<times> nat set list"   and xg :: "nat set"   and yaa :: "nat set list"   and xh :: "'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and yab :: "'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xi :: "'a option list set list"   and yd :: "'a option list set \<times> 'a option list set list"   and xj :: "'a option list set"   and ye :: "'a option list set list"   and xk :: "'a option list set list"   and yf :: "'a option list set \<times> 'a option list set list"   and xl :: "'a option list set"   and yg :: "'a option list set list" assumes "\<not> length A_pos + length A_neg \<noteq> length L"   and "x = take (length A_pos) L"   and "xa = drop (length A_pos) L"   and "dominate_True A_pos x = None"   and "dominate_False A_pos x A_neg xa = Some x2"   and "(xb, y) = x2"   and "(xc, ya) = xb"   and "(xd, yb) = xc"   and "(xe, yc) = yb"   and "(xf, yba) = ya"   and "(xg, yaa) = yba"   and "(xh, yab) = y"   and "(xi, yd) = xh"   and "(xj, ye) = yd"   and "(xk, yf) = yab"   and "(xl, yg) = yf"   and "xd @ xe # yc \<noteq> []"   and "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((xd @ xe # yc) @ xf @ yaa) ((xi @ bin_join n xe xj False xg xl # ye) @ xk @ yg)" shows "mmulti_join n (xd @ xe # yc) (xf @ yaa) ((xi @ bin_join n xe xj False xg xl # ye) @ xk @ yg) = mmulti_join' (xd @ xe # yc) (xf @ yaa) ((xi @ bin_join n xe xj False xg xl # ye) @ xk @ yg)"
extremum_uniqueI: fixes a :: "'c" assumes "a \<le> bot" shows "a = bot"
IntI: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
star_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x ; y \<le> y" shows "x\<^sup>\<star> ; z \<le> y"
par_beta_cases_2: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and t :: "sterm" assumes "Obj f T => t" obtains f' :: "Label \<Rightarrow> sterm option" and L :: "char list set" where "t = Obj f' T" and "dom f' = dom f" and "finite L" and "\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (f' l) = \<sigma>[s,p] t)" and "\<forall>l\<in>dom f. body (the (f l))"
cases: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" obtains e :: "'a" and a :: "'b" and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "x = BsSkewBinomialHeapStruc.Element e a q"
part: shows "List.partition (\<lambda>gi. p.dvdm gi h) gs = (gs1, gs2)"
PDG_BS_s_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and S :: "'g set" assumes "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "StandardControlDependencePDG.PDG_BS_s sourcenode targetnode valid_edge Def Use Exit S \<equiv> PDG.PDG_BS sourcenode targetnode valid_edge Def Use (Postdomination.standard_control_dependence sourcenode targetnode valid_edge Exit) S"
rep_eq: fixes x :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fset (fmdom x) = dom (fmlookup x)"
VN_disjoint_def: fixes v :: "'a V_rec" shows "VN_disjoint v \<equiv> V\<^bsub>v\<^esub> \<inter> N\<^bsub>v\<^esub> = {}"
preprocess_unsat_indices: fixes cs :: "('i \<times> 'a ns_constraint) list"   and t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and trans_v :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and U :: "'i list"   and i :: "'i" assumes "preprocess cs = (t, as, trans_v, U)"   and "i \<in> set U" shows "\<nexists>v. ({i}, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set cs"
right_minus_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b" shows "c \<ominus> a = c \<ominus> b"
simps_2: fixes l1 :: "'a tree"   and x :: "'a"   and r1 :: "'a tree"   and t2 :: "'a tree" shows "mrbst_inter \<langle>l1, x, r1\<rangle> t2 = (case split_bst' x t2 of (sep, l2, r2) \<Rightarrow> mrbst_inter l1 l2 \<bind> (\<lambda>l. mrbst_inter r1 r2 \<bind> (\<lambda>r. if sep then return_pmf \<langle>l, x, r\<rangle> else mrbst_join l r)))"
HFail_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and b :: "nat"   and v :: "InputsOrNi" assumes "HFail s s' q"   and "b \<in> \<Union> (range Ballot)"   and "maxBalInp s b v" shows "maxBalInp s' b v"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> certified_mail"   and "KAB \<noteq> priEK TTP" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
short_cut'': shows "(\<two> \<doteq> \<one>) = false"
are_all_operators_applicable_def: fixes s :: "'c \<Rightarrow> bool option"   and ops :: "'c strips_operator list" shows "are_all_operators_applicable s ops \<equiv> list_all (STRIPS_Representation.is_operator_applicable_in s) ops"
rep_eq: fixes x :: "nat"   and xa :: "uint8" shows "Rep_uint8 (push_bit x xa) = push_bit x (Rep_uint8 xa)"
smult_0_right: fixes p :: "'b" shows "MPoly_Type.smult p 0 = 0"
DBM_reset'_complete: fixes v :: "'e \<Rightarrow> nat"   and u :: "'e \<Rightarrow> 'f"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and n :: "nat"   and cs :: "'e list"   and d :: "'f" assumes "u \<turnstile>\<^bsub>v,n\<^esub> M"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n" shows "[cs\<rightarrow>d]u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"
map_spmf_conv_bind_spmf: fixes f :: "'q \<Rightarrow> 'p"   and p :: "'q spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
Restrp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and a :: "'c"   and b :: "'c" shows "(r \<restriction> A) a b \<equiv> a \<in> A \<and> b \<in> A \<and> r a b"
index_add_vec_2: fixes v\<^sub>1 :: "'a Matrix.vec"   and v\<^sub>2 :: "'a Matrix.vec" shows "dim_vec (v\<^sub>1 + v\<^sub>2) = dim_vec v\<^sub>2"
refl: fixes a :: "'b" shows "a \<sqsubseteq> a"
span_closed: fixes U :: "complex vec set"   and v :: "complex vec" assumes "U \<subseteq> carrier_vec d"   and "v \<in> local.span U" shows "v \<in> carrier_vec d"
rmlab_same_label: fixes l :: "'b"   and A :: "'b \<Rightarrow> 'a set" shows "rmlab l A l = UNIV"
intro: fixes X :: "'b set set" assumes "\<And>C z. \<lbrakk>SimplicialComplex.maxsimp X C; z \<lhd> C; \<exists>D\<in>X - {C}. z \<lhd> D\<rbrakk> \<Longrightarrow> \<exists>!D. D \<in> X - {C} \<and> z \<lhd> D" shows "ThinishChamberComplex_axioms X"
g_def: shows "g = (\<lambda>x. ennreal c - f x)"
length_Basis_list: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and Basis :: "'d set"   and abs :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Basis_list :: "'d list"   and eucl_down :: "int \<Rightarrow> 'd \<Rightarrow> 'd"   and eucl_truncate_down :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and eucl_truncate_up :: "nat \<Rightarrow> 'd \<Rightarrow> 'd" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "length Basis_list = card Basis"
diagonal_step_PQ: fixes P :: "(('a, 'b) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and p :: "'a"   and q :: "'a"   and u :: "'a"   and v :: "'a"   and d :: "'a" assumes "(P, Q) = diagonal_step_PQ A i k bezout"   and "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)"   and "i < min (nrows A) (ncols A)"   and "k < min (nrows A) (ncols A)"   and "i \<noteq> k"   and "is_bezout_ext bezout"   and "isDiagonal A" shows "diagonal_step A i k d v = P ** A ** Q"
MakeCatObj: fixes C :: "('g, 'h, 'i) Category_scheme" shows "obj\<^bsub>MakeCat C\<^esub> = obj\<^bsub>C\<^esub>"
id: shows "L\<lbrakk>Vx : A \<turnstile> d : B\<rbrakk>\<^bsub>CI T\<^esub> \<rightarrow> IMor [A,d,B]\<^bsub>T\<^esub>"
map_default_insert: fixes b :: "'b"   and f :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and b' :: "'b" shows "map_default b (f(a \<mapsto> b')) = (map_default b f)(a := b')"
assms_3: shows "g \<turnstile> Entry g-ns\<rightarrow>m"
f_def: shows "f = (\<lambda>A. SOME x. card x = s \<and> x \<subseteq> A)"
IH_2: fixes t :: "real" assumes "t \<le> 0" shows "nth_derivative k f t 1 = 0"
simps_1: fixes x :: "'a"   and y :: "'a" shows "destNet (DenyAllFromTo x y) = y"
\<phi>_in_hom: fixes y :: "'b"   and f :: "'a"   and x :: "'a" assumes "B.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>A x\<guillemotright>" shows "\<guillemotleft>\<phi> y f : y \<rightarrow>\<^sub>B G x\<guillemotright>"
assms_1: shows "(v, v') \<in> \<langle>Ra\<rangle>option_rel"
fin_cut_same_replicate: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "fin_cut_same x (xs @ replicate n x) = fin_cut_same x xs"
map_add_assoc: fixes m1 :: "'d \<Rightarrow> 'e option"   and m2 :: "'d \<Rightarrow> 'e option"   and m3 :: "'d \<Rightarrow> 'e option" shows "m1 ++ (m2 ++ m3) = m1 ++ m2 ++ m3"
mult_zero_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.mult_zero times zero" shows "times zero a = zero"
simps_4: fixes x :: "quat"   and y :: "quat" shows "Im3 (x + y) = Im3 x + Im3 y"
B_trusts_YM4: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and NB :: "nat"   and evs :: "event list"   and NA :: "nat" assumes "Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, Crypt K (Nonce NB)\<rbrace> \<in> set evs"   and "Says B Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>na nb. Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"
axis_index_axis: fixes u :: "'a" shows "axis_index (axis u 1) = u"
le_ceiling_iff: fixes z :: "int"   and x :: "'b" shows "(z \<le> \<lceil>x\<rceil>) = (of_int z - (1::'b) < x)"
f_inv_into_f: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
set_plus_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "order.greater_eq D C" shows "order.greater_eq (a +o D) (a +o C)"
Suc: shows "\<exists>k. inf_concat_simple f k = (i, 0)"
meet_idem: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> x = x"
intro: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" assumes "Prim2.prim_invar2_ctd w g r Q \<pi>" shows "Prim_Invar2_ctd_Presentation_Loc w g r Q \<pi>"
sublist_append_front: fixes l1 :: "'a list"   and l2 :: "'a list" shows "subseq l1 (l1 @ l2)"
one_fun_def: shows "1 = (\<lambda>x. 1::'d)"
llength_ldrop: fixes n :: "enat"   and xs :: "'b llist" shows "llength (ldrop n xs) = (if n = \<infinity> then 0 else llength xs - n)"
standard_lifting_axioms: shows "standard_lifting Inf_FL Bot_G (Inf_G_q q) (entails_q q) (Red_I_q q) (Red_F_q q) Bot_FL (\<G>_F_L_q q) (\<G>_I_L_q q)"
flag_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, kno, flg, hops, sip, {})" shows "the (flag (update rt dip (dsn, kno, flg, hops, sip, {})) dip) = flg"
Semi: fixes P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and d :: "com"   and R :: "state \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {P} c {Q}"   and "\<turnstile>\<^sub>t {Q} d {R}" shows "\<turnstile>\<^sub>t {P} c; d {R}"
elims_3: fixes x :: "'a extB"   and xa :: "'a extB" assumes "\<not> x < xa" obtains x :: "'a extB" and y :: "'a extB" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
disj: shows "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
ge0: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
mod_eqE: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a mod c = b mod c" obtains d :: "'a" where "b = a + c * d"
transfer_unstar: fixes p :: "bool star"   and P :: "nat \<Rightarrow> bool" assumes "p \<equiv> star_n P" shows "unstar p \<equiv> eventually P \<U>"
hyps: shows "\<Psi> \<rhd> \<tau>.Q \<longmapsto> \<tau> \<prec> Q'"
map_val_conv: fixes xs :: "char list exp list"   and ys :: "val list" shows "(xs = map Val ys) = map_val xs ys"
odd_even_add: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b)"
nonzeros_fun_upd: fixes f :: "'b \<Rightarrow> 'c"   and n :: "'b"   and a :: "'c" shows "{i. (f(n := a)) i \<noteq> (0::'c)} \<subseteq> {i. f i \<noteq> (0::'c)} \<union> {n}"
ltl_ldropn: fixes n :: "nat"   and xs :: "'b llist" shows "ltl (ldropn n xs) = ldropn n (ltl xs)"
codomain_unique: fixes b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b' :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "b \<in> Cop_S.codomains f"   and "b' \<in> Cop_S.codomains f" shows "b = b'"
cos_limit_1: fixes \<theta> :: "nat \<Rightarrow> real" assumes "(\<lambda>j. cos (\<theta> j)) \<longlonglongrightarrow> 1" shows "\<exists>k. (\<lambda>j. \<theta> j - real_of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0"
triangle_in_hom_2: shows "\<guillemotleft>(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) : trg g \<star> g \<Rightarrow> g \<star> src g\<guillemotright>"
path_length: fixes xs :: "'a list" assumes "path xs" shows "length xs \<le> card V"
preserves_cod: fixes f :: "'a" assumes "C.arr f" shows "D.cod (F f) = F (C.cod f)"
sset_defined: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<in> sset xs" shows "x \<noteq> \<bottom>"
orderE: fixes a :: "'b upred"   and b :: "'b upred" assumes "b \<sqsubseteq> a" obtains "a = (a \<and> b)"
assms_1: shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"
subgroup_self: shows "subgroup (carrier (G\<lparr>carrier := H\<rparr>)) (G\<lparr>carrier := H\<rparr>)"
set_child_nodes_get_child_nodes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr"   and "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr \<rightarrow>\<^sub>r children"
remove_shadow_root_def: shows "remove_shadow_root \<equiv> l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root Shadow_DOM.get_child_nodes get_shadow_root set_shadow_root"
block_complement_subset_points: fixes ps :: "'a set"   and bl :: "'a set" assumes "ps \<subseteq> bl\<^sup>c" shows "ps \<subseteq> \<V>"
square_eq_iff: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.idom times one plus zero minus uminus" shows "(times a a = times b b) = (a = b \<or> a = uminus b)"
id_FSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'a, 'b, 'd) model_scheme" shows "FSbImorph id id MOD MOD"
simps_2: fixes c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and rc :: "color"   and rlc :: "('b, 'c) RBT_Impl.rbt"   and rk :: "'b"   and rv :: "'c"   and rrt :: "('b, 'c) RBT_Impl.rbt" shows "rbt_max_opt (Branch c lt k v (Branch rc rlc rk rv rrt)) = rbt_max_opt (Branch rc rlc rk rv rrt)"
R: shows "(\<lambda>(X, Y). X \<times> Y) \<in> carrier (G Mod H) \<times> carrier (K Mod N) \<rightarrow> carrier (G \<times>\<times> K Mod H \<times> N)"
update_convs_3: fixes sys_reg' :: "(sys_reg \<Rightarrow> word32) \<Rightarrow> sys_reg \<Rightarrow> word32"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "sys_reg_update sys_reg' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg' sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
EM_remainder_converges_of_real_iff: fixes a :: "int"   and f :: "real \<Rightarrow> real"   and n :: "nat" assumes "continuous_on {real_of_int a..} f" shows "EM_remainder_converges n (\<lambda>x. of_real (f x)) a = EM_remainder_converges n f a"
assms: shows "\<forall>v\<in>vars_clause S. \<sigma> v = \<tau> v"
continuous: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_antilinear f"   and "continuous F g" shows "continuous F (\<lambda>x. f (g x))"
bisim1_callD: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and M' :: "String.literal"   and n0 :: "nat" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)"   and "call1 e' = \<lfloor>(a, M, vs)\<rfloor>"   and "compE2 e ! pc = Invoke M' n0" shows "M = M'"
mult_poly_0_right: fixes p :: "'a poly" shows "p * 0 = 0"
ordLess_imp_ordLeq: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r <o r'" shows "r \<le>o r'"
SeqRed: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e';; e\<^sub>2,s'\<rangle>"
simps_1: fixes "is" :: "'b set" shows "interrupt_actions_ok' is [] = True"