hcju/isabelleps huggingface dataset:
dest_in_sdnets: fixes x :: "((int \<times> int) set set, int) Combinators"   and p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "\<not> member DenyAll x"   and "p \<in> dom (C x)" shows "subnetsOfAdr (dest p) \<inter> snd_set (sdnets x) \<noteq> {}"
top_elim: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and strong_iteration :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.dra plus times one zero less_eq less star strong_iteration" shows "less_eq (times x y) (times x (strong_iteration one))"
simps: fixes x :: "'a"   and xs :: "'a list"   and i :: "int" shows "(x # xs) !! i = (if i = 0 then x else xs !! (i - 1))"
hyps_3: assumes "F \<noteq> {}" shows "IArray.length (sum f F) = Max {IArray.length (f x) |x. x \<in> F}"
cxt_inv_skip: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and E :: "('a, 'c, 'd) Stmt list"   and c :: "('a, 'c, 'd) Stmt" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "sifum_lang.cxt_to_stmt E c = Skip" shows "c = Skip \<and> E = []"
vpeq_ns_t: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and execs :: "'d \<Rightarrow> 'b list list"   and u :: "'d"   and t :: "'a" assumes "Separation_Kernel kstep output_f current cswitch kprecondition control kinvolved ifp vpeq"   and "Kernel.precondition kprecondition (Kernel.next_state current control s execs) (Kernel.next_action current control s execs)"   and "\<not> ifp\<^sup>*\<^sup>* (current s) u"   and "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s t" shows "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v (Kernel.step kstep (Kernel.next_state current control s execs) (Kernel.next_action current control s execs)) t"
content_cbox_if: fixes a :: "'a"   and b :: "'a" shows "content (cbox a b) = (if cbox a b = {} then 0 else \<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i)"
hyps_2: fixes e'' :: "'a expr" assumes "is_lub,P,E \<turnstile> e1 \<leadsto> e''" shows "e1' = e''"
isnormNum_unique: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "isnormNum x"   and "isnormNum y" shows "(INum x = INum y) = (x = y)"
vars_def: fixes g :: "'g" shows "vars g \<equiv> fold (\<union>) (map (uses g) (\<alpha>n g)) {}"
set2_FGf_def: fixes x :: "('b, 'a, 'e, 'f, 'g, 'h, 'i, 'j, 'k, ('l, 'e, 'm, 'h, 'n, 'k, 'o) G) F" shows "set2_FGf x = set2_F x"
arr_Rel_comp_Rel_id_Rel_right: fixes \<alpha> :: "V"   and F :: "V"   and A :: "V" assumes "arr_Rel \<alpha> F"   and "F\<lparr>ArrDom\<rparr> = A" shows "F \<circ>\<^sub>R\<^sub>e\<^sub>l id_Rel A = F"
dgrad_max_set_closed_sig_red_single: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_red :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "dickson_grading d"   and "p \<in> dgrad_max_set d"   and "f \<in> dgrad_max_set d"   and "sig_red_single sing_red top_tail p q f t" shows "q \<in> dgrad_max_set d"
p_def: shows "p = Poly (map (\<lambda>k. (\<Sum>i\<le>min N k. cs ! (N - i) * f (k - i)) - g k) [0..<N + m])"
hyps_2: fixes xs :: "'a rexp list" shows "(Inters s = Some xs) = (Zero \<notin> set xs \<and> xs = sorted_list_of_set (toplevel_inters s - {Full}))"
piecewise_C1_imp_differentiable: fixes f :: "real \<Rightarrow> 'b"   and i :: "real set" assumes "f piecewise_C1_differentiable_on i" shows "f piecewise_differentiable_on i"
exists_trace_for_any_i: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and i :: "nat" assumes "\<exists>c'. trace c t c'" shows "trace c (take i t) (s c t i)"
ik: shows "A $h k $h j \<noteq> (0::'a)"
coprime_minus_right_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "algebraic_semidom.coprime times one a (uminus b) = algebraic_semidom.coprime times one a b"
in_span: fixes S :: "'c set"   and V :: "('c, 'g) partial_object_scheme"   and v :: "'c" assumes "S \<subseteq> carrier V"   and "v \<in> V.module.span S" shows "\<exists>a A. A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier K \<and> V.module.lincomb a A = v"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> certified_mail" shows "K \<notin> keysFor (parts (knows Spy evs))"
proj_g: fixes i :: "'i" assumes "i \<in> domain g" shows "(\<lambda>n. (f n)\<^sub>F i) \<longlonglongrightarrow> (g)\<^sub>F i"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "a1 >\<^sub>t a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tb__" and "a2 = sb__" and "is_App tb__" and "fun tb__ \<ge>\<^sub>t sb__ \<or> arg tb__ \<ge>\<^sub>t sb__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tc__" and "a2 = sc__" and "head tc__ >\<^sub>h\<^sub>d head sc__" and "chkvar tc__ sc__" and "chksubs (>\<^sub>t) tc__ sc__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "head t = head s" and "chksubs (>\<^sub>t) t s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
ltl_to_rabin_correct_exposed': shows "\<PP>\<^sub>\<infinity> \<phi> \<G> w = accept\<^sub>R'"
zero_order_1: fixes x :: "'d multiset" shows "{#} \<subseteq># x"
comp_red_basis_def: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "comp_red_basis xs = comp_red_basis_aux (comp_min_basis xs) []"
im_of_max_lideal_is_max: fixes I :: "'c set"   and A :: "'c set"   and addA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroA :: "'c"   and oneA :: "'c"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and addB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroB :: "'d"   and oneB :: "'d" assumes "max_lideal I A addA multA zeroA oneA"   and "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "max_lideal (f ` I) B addB multB zeroB oneB"
eq2: shows "\<Union> \<D> = \<Union> (G ` \<D>)"
dual\<^sub>s\<^sub>t_append: fixes A :: "('a, 'b) strand_step list"   and B :: "('a, 'b) strand_step list" shows "dual\<^sub>s\<^sub>t (A @ B) = dual\<^sub>s\<^sub>t A @ dual\<^sub>s\<^sub>t B"
star_nd_fun_def: fixes f :: "'a nd_fun" shows "f\<^sup>\<star> = qstar f"
lift_CFG: 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" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "CFG src trg (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry"
ta_seq_consist_most_recent_write_for: fixes P :: "'a prog"   and E :: "('c \<times> ('b, 'c) obs_event action) llist"   and r :: "nat" assumes "ta_seq_consist P Map.empty (lmap snd E)"   and "r \<in> read_actions E"   and "\<And>adal a a'. \<lbrakk>a \<in> new_actions_for P E adal; a' \<in> new_actions_for P E adal\<rbrakk> \<Longrightarrow> a = a'" shows "\<exists>i. P,E \<turnstile> r \<leadsto>mrw i \<and> i < r"
filterlim_base: fixes J :: "'a set"   and i :: "'a \<Rightarrow> 'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd set"   and f :: "'d \<Rightarrow> 'e"   and G :: "'a \<Rightarrow> 'e set" assumes "\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I"   and "\<And>m x. \<lbrakk>m \<in> J; x \<in> F (i m)\<rbrakk> \<Longrightarrow> f x \<in> G m" shows "filterlim f (INF j\<in>J. principal (G j)) (INF i\<in>I. principal (F i))"
PO_a0i_agreement_trans: shows "{a0i_agreement} TS.trans a0i {> a0i_agreement}"
upds_append: fixes S :: "stack_elem list"   and S' :: "stack_elem list" shows "upds (S @ S') = upds S \<union> upds S'"
Rf: shows "f \<in> {..n + m} \<rightarrow> carrier R"
orbits_as_classes: shows "classes\<^bsub>\<lparr>carrier = E, eq = \<lambda>x y. y \<in> orbit G \<phi> x\<rparr>\<^esub> = orbits G E \<phi>"
sums_if: fixes g :: "nat \<Rightarrow> real"   and x :: "real"   and f :: "nat \<Rightarrow> real"   and y :: "real" assumes "g sums x"   and "f sums y" shows "(\<lambda>n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
InvariantWatchListsContainOnlyClausesFromFNotifyWatchesLoop: fixes state :: "State"   and Wl :: "nat list"   and newWl :: "nat list"   and literal :: "Literal" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<or> c \<in> set newWl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl newWl state in InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
bounded_hashcode_def: fixes n :: "uint32"   and x :: "'a" shows "bounded_hashcode n x = hashcode x mod n"
disjoint_symmetry: fixes x :: "i"   and y :: "i" assumes "PM"   and "\<not> O x y" shows "\<not> O y x"
prime_power_product_in_Pi: fixes n :: "nat" shows "(\<lambda>g. \<Prod>p\<in>{p. p \<le> n \<and> prime p}. p ^ g p) \<in> ({p. p \<le> n \<and> prime p} \<rightarrow>\<^sub>E UNIV) \<rightarrow> {m. 0 < m \<and> prime_factors m \<subseteq> {..n}}"
opeTV: shows "openin (top_of_set ({0..1} \<times> T)) V"
NSLIMSEQ_diff: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a"   and Y :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a"   and "Y \<longlonglongrightarrow>\<^sub>N\<^sub>S b" shows "(\<lambda>n. X n - Y n) \<longlonglongrightarrow>\<^sub>N\<^sub>S a - b"
bi_unique_def: fixes R :: "'j \<Rightarrow> 'k \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
inv_comm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<oplus> y = \<zero>"   and "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "y \<oplus> x = \<zero>"
empty_eff_not_in_as_proj: fixes vs :: "'d set"   and a :: "('d, 'f) fmap \<times> ('d, 'e) fmap"   and as :: "(('d, 'f) fmap \<times> ('d, 'e) fmap) list" assumes "fmdom' (fmrestrict_set vs (snd a)) = {}" shows "as_proj (a # as) vs = as_proj as vs"
map_le_implies_dom_le: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m g" shows "dom f \<subseteq> dom g"
carrier_polynomial_shell: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "p \<in> carrier (poly_ring R)"
main: shows "quadratic_hensel_main C p S T D H n = (D1, H1)"
mkPoint_in_hom: fixes a :: "'s"   and t :: "'s" assumes "S.ide a"   and "t \<in> S.set a" shows "\<guillemotleft>S.mkPoint a t : S.unity \<rightarrow>\<^sub>S a\<guillemotright>"
hom: shows "homotopic_with (\<lambda>k. k ` S \<subseteq> S) X X f id"
bij_lens_pbij: fixes x :: "'d \<Longrightarrow> 'e" assumes "bij_lens x" shows "pbij_lens x"
Pinp_def: shows "Pinp \<equiv> subst P (Var inp) xx"
demod: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "(d p \<le> wp x q) = (d p \<cdot> x \<le> x \<cdot> d q)"
4: fixes i :: "nat"   and j :: "nat" assumes "i < length (seq' j)" shows "seq ` {0..<i} = set (take i (rev (seq' j)))"
stat_is_stat_red: shows "statically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
assms_2: shows "\<forall>x\<in>S. \<exists>y. Q x y"
vlrestriction_vdomain: shows "xs \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> xs = xs"
t_lim: shows "Liminf_llist (lmap fst TNs) = {}"
finite_imp_foldSetD: fixes A :: "'b set"   and e :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "finite A"   and "e \<in> D"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D" shows "\<exists>x. (A, x) \<in> foldSetD D f e"
not_2: shows "(y < 0 \<or> n < 0) = False"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "alternate f g 0 = id"
Invariants_func_is_invariant_n: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and n :: "nat" assumes "f \<in> borel_measurable Invariants"   and "x \<in> space M" shows "f ((T ^^ n) x) = f x"
scaleR_fds_def: fixes c :: "real"   and f :: "'b fds" shows "c *\<^sub>R f = fds (\<lambda>n. c *\<^sub>R fds_nth f n)"
normalize_length_le: fixes p :: "'a list" shows "length (local.normalize p) \<le> length p"
sin_cos_eq: fixes x :: "'b" shows "sin x = cos (of_real pi / (2::'b) - x)"
T_get_min_rest_bound: fixes ts :: "'b tree list" assumes "invar ts"   and "ts \<noteq> []" shows "real (T_get_min_rest ts) \<le> log 2 (real (size (mset_heap ts) + 1))"
meval_sched_one: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')" shows "(cms, mem) \<rightarrow>\<^bsub>[k]\<^esub> (cms', mem')"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "hom'.in_hom f a b" shows "hom'.in_hom (\<psi>.map f) (FG.map a) (hom'.map b)"
ws_def: fixes i :: "nat" shows "ws i = (THE w. P,E \<turnstile> i \<leadsto>mrw w)"
AI_wn_def: shows "AI_ivl' = pfp_wn (step_ivl \<top>)"
nsets_2_eq: fixes A :: "'a set" shows "[A]\<^bsup>2\<^esup> = (\<Union>x\<in>A. \<Union>y\<in>A - {x}. {{x, y}})"
underS_I: fixes i :: "'d"   and j :: "'d"   and R :: "('d \<times> 'd) set" assumes "i \<noteq> j"   and "(i, j) \<in> R" shows "i \<in> underS R j"
natLeq_on_wo_rel: fixes n :: "nat" shows "wo_rel (natLeq_on n)"
dagger_annil: fixes x :: "'a" shows "(1::'a) + x \<cdot> (0::'a) = (x \<cdot> (0::'a))\<^sup>\<dagger>"
diff_less_0_iff_less: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) zero = less a b"
minus_eq: fixes x :: "'a"   and y :: "'a" shows "x - y = x + - y"
iso_class_eqI: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.isomorphic f g" shows "Maps.iso_class f = Maps.iso_class g"
zero_eq_add_iff_both_eq_0: fixes x :: "'a"   and y :: "'a" shows "((0::'a) = x + y) = (x = (0::'a) \<and> y = (0::'a))"
that: fixes k :: "'a" assumes "k \<in> fst ` set xs \<union> fst ` set ys"   and "aux = f k (lookup_pair xs k) (lookup_pair ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set xs \<union> fst ` set ys; lt k' k\<rbrakk> \<Longrightarrow> f k' (lookup_pair xs k') (lookup_pair ys k') = Some Eq" shows "thesis"
uniqOneElementCharacterization: fixes l :: "'a list"   and a :: "'a" assumes "uniq l" shows "(l = [a]) = (set l = {a})"
case_2: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and x2 :: "'d" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
permutation_inverse: fixes x :: "'a permutation" shows "Abs_permutation ((\<rightarrow>) x) = x"
in_smallo_zero_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f \<in> o[F](\<lambda>_. 0::'b)) = (\<forall>\<^sub>F x in F. f x = (0::'b))"
assms_3: shows "components (- h \<sqinter> g) \<le> F"
3: shows "h \<cdot> g \<cdot> f = mkarr (rep h \<^bold>\<cdot> rep (mkarr (rep g \<^bold>\<cdot> rep f)))"
fun_type_id_eq: fixes f :: "'fun"   and X :: "('fun, 'var) Term.term list"   and g :: "'fun"   and Y :: "('fun, 'atom) Term.term list" assumes "\<Gamma> (Fun f X) = Fun g Y" shows "f = g"
simps_3: fixes va :: "'c \<times> nat"   and vb :: "('c \<times> nat) tree" shows "skew \<langle>\<langle>\<rangle>, va, vb\<rangle> = \<langle>\<langle>\<rangle>, va, vb\<rangle>"
foldD_insert: fixes B :: "'b set"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and e :: "'c" assumes "LCD B D f"   and "finite A"   and "x \<notin> A"   and "x \<in> B"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D f e (insert x A) = f x (foldD D f e A)"
lookup_update: fixes k :: "'c"   and v :: "'b"   and m :: "('c, 'b) mapping" shows "Mapping.lookup (Mapping.update k v m) k = Some v"
fst_quot_of_fract_eq_0_iff: fixes x :: "'b fract" shows "(numerator x = (0::'b)) = (x = 0)"
hyps_2: assumes "\<forall>a\<in>ds'. length b \<le> a" shows "p' \<in> grid b ds"
homeomorphic_convex_compact_sets: fixes S :: "'a set"   and T :: "'b set" assumes "convex S"   and "compact S"   and "convex T"   and "compact T"   and "aff_dim S = aff_dim T" shows "S homeomorphic T"
cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a set['c]"   and a2 :: "'b set['c]" assumes "rel_bset R a1 a2" obtains x :: "('a \<times> 'b) set['c]" where "a1 = map_bset fst x" and "a2 = map_bset snd x" and "set_bset x \<subseteq> {(x, y). R x y}"
arrow_in_bicategory_of_spans_axioms: shows "arrow_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg r r r"
assms_3: shows "\<forall>x. eval r_phi [i, x] = eval r_phi [j, x]"
x: shows "x \<in> elements_mat A1 \<union> elements_mat A2 \<union> elements_mat A3 \<union> elements_mat A4"
b: shows "{i. i = b \<and> i < dim_col A} = {b}"
FAccRedsThrow: 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 a :: "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 F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>"
subst_floatarith_Var_DERIV_floatarith: fixes n :: "nat"   and s :: "nat \<Rightarrow> nat"   and fa :: "floatarith" assumes "\<And>x. (x = n) = (s x = n)" shows "subst_floatarith (\<lambda>x. Var (s x)) (DERIV_floatarith n fa) = DERIV_floatarith n (subst_floatarith (\<lambda>x. Var (s x)) fa)"
aff_dim_cball: fixes e :: "real"   and a :: "'a" assumes "0 < e" shows "aff_dim (cball a e) = int DIM('a)"
extensional_funcset_extend_domain_inj_on_eq: fixes x :: "'c"   and S :: "'c set"   and T :: "'d set" assumes "x \<notin> S" shows "{f \<in> insert x S \<rightarrow>\<^sub>E T. inj_on f (insert x S)} = (\<lambda>(y, g). g(x := y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E T - {y} \<and> inj_on g S}"
reduced_row_echelon_form_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "reduced_row_echelon_form_upt_k A k = ((\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)) \<and> (\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (1::'a)) \<and> (\<forall>i. i < i + (1::'c) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'c)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ (i + (1::'c)) $ n \<noteq> (0::'a))) \<and> (\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) = (0::'a))))"
ideal_implies_subset: 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 A :: "'c set"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "comm_ring R addition multiplication zero unit"   and "ideal A R add mult zero unit" shows "A \<subseteq> R"
LIMSEQ_SUP: fixes X :: "nat \<Rightarrow> 'b" assumes "incseq X" shows "X \<longlonglongrightarrow> Sup (range X)"
MakeCatCod: fixes f :: "'d"   and C :: "('e, 'd, 'f) Category_scheme" assumes "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> f = cod\<^bsub>MakeCat C\<^esub> f"
right_residual: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x * a \<le> b) = (x \<le> a l\<rightarrow> b)"
divide_mult_eq: fixes a :: "ennreal"   and x :: "ennreal"   and b :: "ennreal" assumes "a \<noteq> 0"   and "a \<noteq> \<infinity>" shows "x * a / (b * a) = x / b"
rep_var_def: shows "rep_var \<equiv> quot_type.rep Rep_var"
map_comp_map: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "map f \<circ> map g = map (f \<circ> g)"
elims_2: fixes x :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and xa :: "'varSort \<times> 'sort"   and xb :: "('varSort, 'f) sAbs" assumes "sWlsAbs x xa xb" obtains SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme" and xs :: "'varSort" and s :: "'sort" and xs' :: "'varSort" and sF :: "'f \<Rightarrow> 'f" where "x = SEM" and "xa = (xs, s)" and "xb = sAbs xs' sF" and "isInBar (xs, s) \<and> xs = xs' \<and> (\<forall>sX. if sWls SEM (asSort xs) sX then sWls SEM s (sF sX) else sF sX = sDummy SEM s)"
qfN_cube_prime: fixes p :: "int"   and N :: "int"   and q :: "int"   and a :: "int"   and b :: "int" assumes "prime (p\<^sup>2 + N * q\<^sup>2) \<and> 1 < N \<and> (p\<^sup>2 + N * q\<^sup>2) ^ 3 = a\<^sup>2 + N * b\<^sup>2 \<and> coprime a b" shows "\<bar>a\<bar> = \<bar>p ^ 3 - 3 * N * p * q\<^sup>2\<bar> \<and> \<bar>b\<bar> = \<bar>3 * p\<^sup>2 * q - N * q ^ 3\<bar>"
simps_4: fixes A :: "dbfm" shows "quot_dbfm (DBNeg A) = Q_Neg (quot_dbfm A)"
C_merge_ub: fixes xs :: "'b list"   and ys :: "'b list" shows "C_merge xs ys \<le> length xs + length ys"
Inr_iff: fixes a :: "hf"   and b :: "hf" shows "(HF.Inr a = HF.Inr b) = (a = b)"
psubsetI: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "A \<noteq> B" shows "A \<subset> B"
path_component_of_sym: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" assumes "path_component_of X x y" shows "path_component_of X y x"
assms_1: fixes x :: "'a"   and y :: "'b"   and z :: "'c" assumes "phi x y z" shows "chi x y z"