hcju/isabelleps huggingface dataset:
continuous_map_of_snd: fixes X :: "'b topology"   and Y :: "'c topology"   and Z :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map (prod_topology X Y) Z (f \<circ> snd) = (topspace X = {} \<or> continuous_map Y Z f)"
cs_select_is_cs_1: fixes \<pi> :: "nat \<Rightarrow> 'n"   and xs :: "'n list"   and k :: "nat" assumes "is_path \<pi>"   and "xs \<noteq> []"   and "xs \<prec> cs\<^bsup>\<pi>\<^esup> k" shows "cs\<^bsup>\<pi>\<^esup> (\<pi> \<exclamdown> xs) = xs"
c: fixes \<Gamma> :: "char list \<Rightarrow> 'b rule list option"   and \<gamma> :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c"   and rs :: "'b rule list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
monom_mult_zero_left: fixes t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult (0::'c) t p = 0"
prj_in_hom_6: shows "\<guillemotleft>\<mu>\<nu>\<pi>.Prj\<^sub>0\<^sub>0 : H\<mu>H\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<pi>.apex\<guillemotright>"
AE_integrable_fst: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f x y)" shows "AE x in lborel. integrable lborel (f x)"
DERIV_powr: fixes g :: "real \<Rightarrow> real"   and m :: "real"   and x :: "real"   and f :: "real \<Rightarrow> real"   and r :: "real" assumes "(g has_real_derivative m) (at x)"   and "0 < g x"   and "(f has_real_derivative r) (at x)" shows "((\<lambda>x. g x powr f x) has_real_derivative g x powr f x * (r * ln (g x) + m * f x / g x)) (at x)"
subseteq_guards_refl: fixes c :: "('d, 'e, 'f) com" shows "c \<subseteq>\<^sub>g c"
Mprefix_refines_Mndet: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "mndet A P \<le> Mprefix A P"
height_size1_exp: fixes t :: "'b tree" assumes "wbt t"   and "t \<noteq> \<langle>\<rangle>" shows "2 \<le> (1 - \<alpha>) ^ (height t - 1) * real (size1 t)"
pred_rel: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b uprod" shows "pred_uprod P x = rel_uprod (eq_onp P) x x"
minf_2: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and Q' :: "'a \<Rightarrow> bool" assumes "\<exists>z. \<forall>x<z. P x = P' x"   and "\<exists>z. \<forall>x<z. Q x = Q' x" shows "\<exists>z. \<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
unl_lift: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'e \<Rightarrow> 'd"   and w :: "'e" shows "(f<x>) w \<equiv> f (x w)"
limsup_shift: fixes u :: "nat \<Rightarrow> 'a" shows "limsup (\<lambda>n. u (n + 1)) = limsup u"
suffix_conc_fst: fixes n :: "nat"   and u :: "'b list"   and v :: "nat \<Rightarrow> 'b" assumes "n \<le> length u" shows "suffix n (u \<frown> v) = drop n u \<frown> v"
poly_of_pm_times: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_of_pm x (p * q) = poly_of_pm x p * poly_of_pm x q"
perm_count: fixes A :: "'b list"   and B :: "'b list" assumes "A <~~> B" shows "\<forall>x. PermutationLemmas.count x A = PermutationLemmas.count x B"
ring_record_simps_12: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
A_def: shows "A = (\<Sum>n. norm (fds_nth h (n + Suc N)) / nat_power (n + Suc N) c) * nat_power (Suc N) c"
continuous_imp_closed_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X Y f"   and "compact_space X"   and "Hausdorff_space Y" shows "closed_map X Y f"
asymD: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "asym R"   and "(x, y) \<in> R" shows "(y, x) \<notin> R"
subst_dismantle: fixes x :: "name"   and v :: "trm"   and t :: "trm"   and k :: "stack" shows "(t \<star> k)[x::=v] = t[x::=v] \<star> ssubst x v k"
assms_3: shows "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs' uses' phis' var'"
inj_Some: fixes A :: "'a set" shows "inj_on Some A"
BotE: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> F"
that: fixes U :: "nat \<Rightarrow> 'a set" assumes "\<Union> (range U) = carrier"   and "\<And>i. open (U i)"   and "\<And>i. compact (closure (U i))"   and "\<And>i. closure (U i) \<subseteq> carrier" shows "thesis"
RN_2: fixes \<psi> :: "\<o>"   and \<phi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<psi> in v] \<Longrightarrow> [\<phi> in v]"   and "[\<^bold>\<box>\<psi> in v]" shows "[\<^bold>\<box>\<phi> in v]"
eq_iff: fixes a :: "'b list"   and b :: "'b list" shows "(a = b) = (prefix b a \<and> prefix a b)"
frechet_derivative_inverse: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f differentiable at x"   and "f x \<noteq> (0::'c)" shows "frechet_derivative (\<lambda>x. inverse (f x)) (at x) = (\<lambda>h. - (1::'c) / (f x)\<^sup>2 * frechet_derivative f (at x) h)"
minus_assn_def: fixes a :: "assn"   and b :: "assn" shows "a - b \<equiv> inf a (- b)"
typ_of_eta_norm: fixes t :: "Core.term"   and ty :: "typ" assumes "typ_of t = Some ty" shows "typ_of (eta_norm t) = Some ty"
cod_simps_3: shows "\<mu>.cod.leg1 = \<mu>.leg1"
A: shows "((s, c), e, s', c') \<in> trss fg"
word_sless_def: shows "(<s) \<equiv> map_fun Word.rep (map_fun Word.rep id) (\<lambda>k l. signed_take_bit (LENGTH('a) - Suc 0) k < signed_take_bit (LENGTH('a) - Suc 0) l)"
effective_scalar_product_append: fixes zs :: "'a list"   and ws :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "length zs = length ws"   and "length xs = length ys" shows "scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws"
assms: shows "exec_step_input P C M pc ics = StepC C' Cs"
termi_impl_no_inf_chain: fixes c :: "com"   and s :: "state" assumes "c \<down> s" shows "\<nexists>f. f 0 = ([c], s) \<and> (\<forall>i. (f i, f (i + 1)) \<in> exec1\<^sup>+)"
k_def: shows "k = Min {k. k \<le> n \<and> \<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len m i i xs)}"
gen_llength_cons_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and n :: "enat"   and s :: "'c" shows "gen_llength_cons g n s = n + llength (lunstream g s)"
multiplicity_eq_zero_iff: shows "(multiplicity p x = 0) = (\<not> p dvd x)"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>P. eventually P F \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> dist x y < e)"
rel_spmf_reflI: fixes p :: "'c spmf"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> P x x" shows "rel_spmf P p p"
simps_4: fixes v :: "('fun, 'var) Term.term" shows "tfr\<^sub>s\<^sub>t\<^sub>p (receive\<langle>v\<rangle>\<^sub>s\<^sub>t) = True"
assms_1: shows "Frame' (P \<and>* P' \<and>* $ (n - m)) \<box> Q F"
replete_set_category_axioms: shows "replete_set_category (\<cdot>\<^sub>S)"
euclidean_size_int_def: shows "euclidean_size = nat \<circ> abs"
line_integral_on_path: shows "one_chain_line_integral F {j} \<gamma> = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
insert_ops_rem_prefix: fixes pre :: "('b \<times> 'b option) list"   and suf :: "('b \<times> 'b option) list" assumes "insert_ops (pre @ suf)" shows "insert_ops suf"
\<phi>'_nonneg: shows "0 \<le> \<phi>'"
singleton_space_subset_in_sets: fixes J :: "'d set"   and I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<in> I"   and "finite J" shows "space (Pi\<^sub>F {J} M) \<in> sets (Pi\<^sub>F I M)"
lunit_hcomp_2: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<a>\<^sup>-\<^sup>1[trg f, f, g] \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> g] = \<l>\<^sup>-\<^sup>1[f] \<star> g"
rev_morph_def: fixes f :: "'b list \<Rightarrow> 'c list" shows "rev_morph f = rev \<circ> f \<circ> rev"
hyps: assumes "markedElements M' \<noteq> []" shows "prefixBeforeLastMarked M' = prefixToLevel (currentLevel M' - 1) M'"
add_set_mono: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "A \<subseteq> B" shows "add_set f A n \<subseteq> add_set f B n"
host_apply_preserve_store: fixes t1s :: "t list"   and vs :: "v list"   and s :: "s"   and t2s :: "t list"   and f :: "host"   and hs :: "host_state"   and s' :: "s"   and vs' :: "v list" assumes "list_all2 types_agree t1s vs"   and "host_apply s (t1s _> t2s) f vs hs = Some (s', vs')" shows "store_extension s s'"
Transset_sup: fixes x :: "V"   and y :: "V" assumes "Transset x"   and "Transset y" shows "Transset (x \<squnion> y)"
symmetric: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<cong> Q" shows "Q \<cong> P"
distinct_1: fixes x1 :: "nat"   and x2 :: "nat" shows "EVar x1 \<noteq> ENat x2"
iTILL_inter': fixes n :: "nat"   and n' :: "nat" shows "[\<dots>n] \<inter> [\<dots>n'] = [\<dots>min n n']"
InvariantClCharacterization_def: fixes Cl :: "Literal"   and C :: "Literal list"   and M :: "(Literal \<times> bool) list" shows "InvariantClCharacterization Cl C M \<equiv> isLastAssertedLiteral Cl (oppositeLiteralList C) (elements M)"
subspace_isomorphism: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'd) vec set" assumes "vec.subspace S"   and "vec.subspace T"   and "vec.dim S = vec.dim T" shows "\<exists>f. Vector_Spaces.linear (*s) (*s) f \<and> f ` S = T \<and> inj_on f S"
Gamma_gbinomial: fixes z :: "'a" shows "(\<lambda>n. (z + of_nat n gchoose n) * exp (- z * of_real (ln (real n)))) \<longlonglongrightarrow> rGamma (z + (1::'a))"
B: fixes x :: "complex" assumes "x \<in> path_image (part_circlepath z r s t) - k" shows "cmod (f x) \<le> B"
map_lens_def: fixes x :: "'b" shows "map_lens x = \<lparr>lens_get = \<lambda>f. the (f x), lens_put = \<lambda>f u. f(x \<mapsto> u)\<rparr>"
coeff_pderiv: fixes p :: "'b poly"   and n :: "nat" shows "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
test_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "test (x \<squnion> y)"
dense_list_of_pdevs_def: fixes x :: "'b pdevs" shows "dense_list_of_pdevs x = map (pdevs_apply x) [0..<degree x]"
interleaves_equal_snd: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and zs' :: "'b list" assumes "xs \<simeq> {ys, zs, P}"   and "xs \<simeq> {ys, zs', P}" shows "zs = zs'"
rel_set_UNION: fixes Q :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'f set"   and B :: "'g set"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'h set"   and g :: "'g \<Rightarrow> 'i set" assumes "rel_set Q A B"   and "rel_fun Q (rel_set R) f g" shows "rel_set R (\<Union> (f ` A)) (\<Union> (g ` B))"
mersenne_mod2_def: fixes k :: "int"   and n :: "nat" shows "mersenne_mod2 k n = mersenne_mod (mersenne_mod k n) n"
foundation10': fixes \<tau> :: "'d state \<times> 'd state"   and A :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> A and B) = (\<tau> \<Turnstile> A \<and> \<tau> \<Turnstile> B)"
suminf_zero_on: fixes f :: "nat \<Rightarrow> 'c"   and X :: "nat set" assumes "summable f"   and "finite X" shows "suminf (zero_on f X) = suminf f - sum f X"
emeasure_std_simplex: shows "emeasure lborel (convex hull insert (0::'a) Basis) = ennreal (1 / fact DIM('a))"
Act_def: shows "Formula.Act \<equiv> (id ---> Formula.formula.Rep_formula ---> Formula.formula.Abs_formula) Formula.Act\<^sub>\<alpha>"
assms_2: shows "bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0"
in_rel_Collect_case_prod_eq: fixes X :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "in_rel {(x, y). X x y} = X"
int_pow_inv: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier G" shows "inv x [^] i = inv (x [^] i)"
IH_1: fixes w :: "nat \<Rightarrow> real" shows "remove_weights (insert_weights s m1 w) = m1"
Rel_match_Rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "Rel_match R x y" shows "Transfer.Rel R x y"
subspace_eq_real: fixes S :: "'c set" assumes "real_vector_space_on S" shows "module_on.subspace (*\<^sub>R) = subspace"
negativity_constraint: fixes \<phi> :: "i \<Rightarrow> bool" assumes "atomic \<phi>" shows "\<not> \<phi> n"
ii: fixes s :: "(char list \<Rightarrow> int option) \<times> nat" shows "(case s of (s, n) \<Rightarrow> I (s, n) \<and> lmaps_to_axpr b False s) \<longrightarrow> Q s"
list_ordered_it_correct: fixes x :: "'s" assumes "invar x" shows "set_iterator_linord (list_ordered_it x) (\<alpha> x)"
assms_1: shows "C \<in> f \<turnstile> folding_g.\<C>"
Fun_def: fixes A :: "ZF"   and B :: "ZF" shows "Fun A B \<equiv> Sep (PFun A B) (\<lambda>f. HOLZF.Domain f = A)"
z_prop: shows "- r < z \<and> a * z\<^sup>2 + b * z + c < 0"
lnull_ldrop: fixes n :: "enat"   and xs :: "'a llist" shows "lnull (ldrop n xs) = (llength xs \<le> n)"
dgrad_set_closed_plus: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and m :: "nat"   and t :: "'a" assumes "dickson_grading d"   and "s \<in> dgrad_set d m"   and "t \<in> dgrad_set d m" shows "s + t \<in> dgrad_set d m"
data_refinement_choice2: fixes D :: "'a \<Rightarrow> 'b"   and S1 :: "'c \<Rightarrow> 'a"   and D' :: "'c \<Rightarrow> 'd"   and S2 :: "'d \<Rightarrow> 'b"   and S1' :: "'c \<Rightarrow> 'a"   and S2' :: "'d \<Rightarrow> 'b" assumes "mono D"   and "DataRefinement S1 D D' S2"   and "DataRefinement S1' D D' S2'" shows "DataRefinement (S1 \<sqinter> S1') D D' (S2 \<sqinter> S2')"
range_of_not_empty: fixes \<Psi> :: "('b, 'a) sas_plus_problem"   and v :: "'b" shows "(range_of \<Psi> v \<noteq> None \<and> range_of \<Psi> v \<noteq> Some []) = (\<R>\<^sub>+ \<Psi> v \<noteq> {})"
pseudominimal_inI: fixes x :: "'b"   and P :: "'b set" assumes "x \<in> P"   and "x \<noteq> codomain.bottom P"   and "\<And>z. \<lbrakk>z \<in> P; codomain.greater x z\<rbrakk> \<Longrightarrow> z = codomain.bottom P" shows "codomain.pseudominimal_in P x"
lookup_override_on_eq: fixes m1 :: "'d \<Rightarrow> 'c"   and m2 :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and x :: "'d" shows "(m1 ++\<^bsub>S\<^esub> m2) x = (if x \<in> S then m2 x else m1 x)"
if_SE_split_asm: fixes \<sigma> :: "'d"   and P :: "'d \<Rightarrow> bool"   and B\<^sub>1 :: "'d \<Rightarrow> (bool \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> (bool \<times> 'd) option" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2) = (P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>1) \<or> \<not> P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>2))"
L_def: shows "L = (\<lambda>x. {y \<in> A. y < x})"
at_calc_3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "at TYPE('a)" shows "swap (a, b) c = (if a = c then b else if b = c then a else c)"
is_lfair_def: fixes ls :: "'b \<Rightarrow> 'l"   and ss :: "'b \<Rightarrow> 'c" shows "C.is_lfair ls ss \<equiv> \<forall>l i. \<exists>j\<ge>i. \<not> C.can_step l (ss j) \<or> ls j = l"
Max_lt_imp_lt_mset: fixes N :: "'a multiset"   and M :: "'a multiset" assumes "N \<noteq> {#}"   and "Max_mset M < Max_mset N" shows "M < N"
fundfoldpairs_def: shows "fundfoldpairs \<equiv> {(f, g). OpposedThinChamberComplexFoldings X f g C0}"
multi_union_self_other_eq: fixes A :: "'k multiset"   and X :: "'k multiset"   and Y :: "'k multiset" assumes "A + X = A + Y" shows "X = Y"
echelon_form_of_upt_k_def: fixes A :: "(('e, 'f) vec, 'g) vec"   and k :: "nat"   and bezout :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<times> 'e \<times> 'e \<times> 'e \<times> 'e" shows "echelon_form_of_upt_k A k bezout = fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
bind_mono: fixes B :: "('c \<Rightarrow> 'd option) \<Rightarrow> 'e option"   and C :: "'e \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'f option" assumes "monotone (flatf_ord None) option_ord B"   and "\<And>y. monotone (flatf_ord None) option_ord (C y)" shows "monotone (flatf_ord None) option_ord (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
Sup_set_def: fixes A :: "'a set set" shows "\<Union> A = {x. Sup ((\<in>) x ` A)}"
plus_adds_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a + b adds c" shows "b adds c"
cast_DEFL: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and liftemb :: "'e\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'e\<^sub>\<bottom>"   and liftdefl :: "'e itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'e \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'e"   and defl :: "'e itself \<Rightarrow> udom defl" assumes "class.domain below liftemb liftprj liftdefl emb prj defl" shows "cast\<cdot>(defl TYPE('e)) = emb oo prj"
assms_2: shows "\<pp>\<^sub>0[a, b] \<cdot> h = \<pp>\<^sub>0[a, b] \<cdot> h'"
subgame_maximal_path: fixes V' :: "'a set"   and P :: "'a llist" assumes "V' \<subseteq> V\<^bsub>subgame V'\<^esub>"   and "G'.maximal_path P"   and "lset P \<subseteq> V'" shows "Digraph.maximal_path (G'.subgame V') P"
parTrace_snoc: fixes cfl :: "(('test, 'atom) com \<times> 'state) list"   and cf :: "('test, 'atom) com \<times> 'state" shows "parTrace (cfl @ [cf]) = (cfl \<noteq> [] \<longrightarrow> parTrace cfl \<and> last cfl \<rightarrow>c cf)"
construct_lhs_vector_cleaner_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" assumes "p \<noteq> 0" shows "w_vec_R p qs signs = vec_of_list (map (\<lambda>s. rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) signs)"
reachable_step: fixes s :: "'e"   and A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and a :: "'f"   and s' :: "'e" assumes "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "s' \<in> reachable A I"
get_child_nodes_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_child_nodes ptr) h"
width: fixes n :: "nat" shows "B n - A n = (b - a) / 2 ^ n"
comp_fun_commute_Pow_fold: shows "comp_fun_commute (\<lambda>x A. A \<union> insert x ` A)"
dfree_Plus_simps: fixes a :: "('a, 'b) trm"   and b :: "('a, 'b) trm" shows "dfree (Plus a b) = (dfree a \<and> dfree b)"
cA0: shows "charpoly (HMA_Connect.map_matrix complex_of_real A) \<noteq> 0"
preserves_inv: fixes f :: "'a" assumes "local.iso f" shows "L (local.inv f) = local.inv (L f)"
semiring_norm_10: fixes m :: "num"   and n :: "num" shows "num.Bit1 m + num.Bit1 n = num.Bit0 (m + n + num.One)"
cong_dvd_modulus: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and m :: "'a"   and n :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment"   and "unique_euclidean_semiring.cong modulo x y m"   and "dvd.dvd times n m" shows "unique_euclidean_semiring.cong modulo x y n"
evalid_def2: fixes P :: "assn"   and e :: "expr"   and Q :: "vassn" shows "\<Turnstile>\<^sub>e {P} e {Q} = (\<forall>n. \<Turnstile>n:\<^sub>e (P, e, Q))"
star_left_induct_mult_equal: fixes x :: "'a"   and y :: "'a" assumes "x * y = y" shows "x\<^sup>\<star> * y \<le> y"
vwalk_join_non_Nil: fixes p :: "'a list"   and q :: "'a list" assumes "p \<noteq> []" shows "p \<oplus> q \<noteq> []"
monom_mult_dist_right: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult c t (p + q) = monom_mult c t p + monom_mult c t q"
mult_BIT_simps_2: fixes x :: "int"   and y :: "int" shows "x * y BIT False = (x * y) BIT False"
eq2: shows "y2 \<bullet> y1 = x \<bullet> y2"
det_fold_sorted_set: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c' :: "'g \<Rightarrow> bool"   and f' :: "'f \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma>' :: "'g"   and result :: "'f set \<Rightarrow> 'g"   and Rk :: "('h \<times> 'f) set"   and Rs :: "('h \<times> 'f) set \<Rightarrow> ('i \<times> 'f set) set"   and tsl :: "'i \<Rightarrow> 'h list"   and s :: "'i"   and s' :: "'f set"   and c :: "'j \<Rightarrow> bool"   and R\<sigma> :: "('j \<times> 'g) set"   and f :: "'h \<Rightarrow> 'j \<Rightarrow> 'j"   and \<sigma> :: "'j" assumes "det_fold_set ordR c' f' \<sigma>' result"   and "is_set_to_sorted_list ordR Rk Rs tsl"   and "(s, s') \<in> \<langle>Rk\<rangle>Rs"   and "(c, c') \<in> R\<sigma> \<rightarrow> bool_rel"   and "(f, f') \<in> Rk \<rightarrow> R\<sigma> \<rightarrow> R\<sigma>"   and "(\<sigma>, \<sigma>') \<in> R\<sigma>" shows "(foldli (tsl s) c f \<sigma>, result s') \<in> R\<sigma>"
recursive_compile_to_tm_correct2: fixes recf :: "recf"   and args :: "nat list" assumes "terminate recf args" shows "\<exists>stp m l. steps0 (Suc 0, [Bk, Bk], <args>) (tm_of_rec recf) stp = (0, Bk \<up> Suc (Suc m), Oc \<up> Suc (rec_exec recf args) @ Bk \<up> l)"
that: assumes "\<sigma>' = Some (snd (Array.alloc (replicate n x) (the_state \<sigma>)))"   and "r = fst (Array.alloc (replicate n x) (the_state \<sigma>))"   and "Array.get (the_state \<sigma>') r = replicate n x" shows "thesis"
id3_2: shows "(\<lambda>x y z. y) \<in> PrimRec3"
equalOn_vblcaseI: fixes A :: "vbl set"   and x :: "'a"   and y :: "'a"   and f :: "vbl \<Rightarrow> 'a"   and g :: "vbl \<Rightarrow> 'a" assumes "zeroX \<in> A \<longrightarrow> x = y"   and "equalOn (preImage nextX A) f g" shows "equalOn A (vblcase x f) (vblcase y g)"
connected_clopen: fixes S :: "'c set" shows "connected S = (\<forall>T. openin (top_of_set S) T \<and> closedin (top_of_set S) T \<longrightarrow> T = {} \<or> T = S)"
retraction_maps_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "retraction_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace Y. f (g x) = x)"
distinct_remove1_removeAll: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs" shows "remove1 x xs = removeAll x xs"
iso_reduced_homology_group_lower_hemisphere: fixes k :: "nat"   and n :: "nat"   and p :: "int" assumes "k \<le> n" shows "hom_induced p (nsphere n) {} (nsphere n) {x. x k \<le> 0} id \<in> Group.iso (reduced_homology_group p (nsphere n)) (relative_homology_group p (nsphere n) {x. x k \<le> 0})"
subgroup_self: shows "subgroup (carrier (G Mod N)) (G Mod N)"
v\<^sub>0_1: shows "v = substNext g v\<^sub>0"
eigenvalue_root_char_poly: fixes A :: "'b mat"   and n :: "nat"   and k :: "'b" assumes "A \<in> carrier_mat n n" shows "eigenvalue A k = (poly (char_poly A) k = (0::'b))"
shiftr_div_2n_w: fixes n :: "nat"   and w :: "'a word" assumes "n < size w" shows "w >> n = w div 2 ^ n"
supp_aezfun_times: fixes x :: "('b, 'a) aezfun"   and y :: "('b, 'a) aezfun" shows "supp (aezfun (x * y)) \<subseteq> supp (aezfun x) + supp (aezfun y)"
simps_22: fixes x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly"   and x1 :: "int \<times> int" shows "CN x81 x82 x83 \<noteq> C x1"
primerec_rec_minus_2: shows "primerec rec_minus (Suc (Suc 0))"
aug_ipp_closedTr: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i" shows "\<forall>x. x \<in> add_set f (aug_pm_set z i A) n \<longrightarrow> \<^sub>i- x \<in> add_set f (aug_pm_set z i A) n"