hcju/isabelleps huggingface dataset:
INV_from_refinement: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme"   and I :: "'f set" assumes "PO_refines R Ta Tc"   and "Range R \<subseteq> I" shows "reach Tc \<subseteq> I"
e_def: shows "e = (c - b\<^sup>2 / ((3::'a) * a)) / a"
valid_shift_mmuaux': fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and nt :: "nat" assumes "valid_mmuaux' args cur cur aux auxlist"   and "cur \<le> nt" shows "valid_mmuaux' args cur nt (shift_mmuaux args nt aux) auxlist \<and> (\<forall>ts\<in>set (lin_ts_mmuaux (shift_mmuaux args nt aux)). \<not> enat ts + right (args_ivl args) < enat nt)"
d_def: fixes x :: "'a" shows "d x = (1::'a) \<sqinter> x \<cdot> \<top>"
infinite_cartesian_product: fixes A :: "'b set"   and B :: "'c set" assumes "infinite A"   and "infinite B" shows "infinite (A \<times> B)"
inv_comp_left_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "category.iso C (C g f)"   and "category.iso C g" shows "category.iso C f"
preserves_cod: fixes f :: "'j" assumes "J.arr f" shows "C.cod (A'.F.map f) = A'.F.map (J.cod f)"
rows_def: fixes A :: "'b mat" shows "rows A = map (row A) [0..<dim_row A]"
map_fun_apply: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'g"   and h :: "'i \<Rightarrow> 'j"   and x :: "'h" shows "(f ---> g) h x = g (h (f x))"
Ord_sup: fixes x :: "V"   and y :: "V" assumes "Ord x"   and "Ord y" shows "Ord (x \<squnion> y)"
facesAt_distinctI: fixes g :: "graph" assumes "\<And>v. v \<in> \<V> g \<Longrightarrow> distinct (normFaces (facesAt g v))" shows "facesAt_distinct g"
reduction_inv_4: fixes u :: "db"   and t' :: "db"   and P :: "bool" assumes "DBAbsV u \<rightarrow>\<^sub>v t'" shows "P"
Abrupt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e" shows "\<Gamma>\<turnstile>c \<down> Abrupt s"
B': shows "B = (B - N) \<inter># M + B \<inter># N"
distinct_85: fixes x31 :: "Syntax.var"   and x32 :: "Syntax.term"   and x13 :: "Syntax.term" shows "Lam x31 x32 \<noteq> Roll x13"
simps_2: fixes v :: "int \<times> int" shows "Reflected_Multivariate_Polynomial.degree (C v) = 0"
neighbourhood_base_at_mono: fixes x :: "'c"   and P :: "'c set \<Rightarrow> bool"   and X :: "'c topology"   and Q :: "'c set \<Rightarrow> bool" assumes "neighbourhood_base_at x P X"   and "\<And>S. \<lbrakk>P S; x \<in> S\<rbrakk> \<Longrightarrow> Q S" shows "neighbourhood_base_at x Q X"
has_bottom_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ordering less_eq less" shows "ordering.has_bottom less_eq P \<equiv> \<exists>z\<in>P. \<forall>x\<in>P. less_eq z x"
fin_make_untimed_append_empty: fixes z :: "'a list list" shows "fin_make_untimed (z @ [[]]) = fin_make_untimed z"
Ideal1_def: fixes m1 :: "nat"   and m2 :: "nat"   and out1 :: "nat" shows "local.sim_non_det_def.Ideal1 m1 m2 out1 = S1 m1 out1 \<bind> (\<lambda>view1. Out1 m1 m2 out1 \<bind> (\<lambda>out1. return_spmf (view1, out1)))"
some_in_eq: fixes A :: "'d set" shows "((SOME x. x \<in> A) \<in> A) = (A \<noteq> {})"
eval_Mn': fixes n :: "nat"   and f :: "recf"   and xs :: "nat list" assumes "recfn (length xs) (Mn n f)" shows "eval (Mn n f) xs = (if \<exists>z. eval f (z # xs) \<down>= 0 \<and> (\<forall>y<z. eval f (y # xs) \<down>) then Some (LEAST z. eval f (z # xs) \<down>= 0) else None)"
commutative_squares_5: shows "C.commutative_square assoc\<mu>\<nu>\<pi>.cod.leg0 \<rho>.cod.leg1 (assoc\<mu>\<nu>\<pi>.chine \<cdot> \<p>\<^sub>1[assoc\<mu>\<nu>\<pi>.dom.leg0, \<rho>.leg1]) (\<rho>.chine \<cdot> \<p>\<^sub>0[assoc\<mu>\<nu>\<pi>.dom.leg0, \<rho>.leg1])"
arg_min_least: fixes S :: "'c set"   and y :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite S"   and "S \<noteq> {}"   and "y \<in> S" shows "f (arg_min_on f S) \<le> f y"
I: shows "C + 2 * E \<le> 187 * lambda\<^sup>2 * (C + deltaG TYPE('a) + deltaG TYPE('b))"
to_list: shows "SIDE_GEN_ALGO (is_set_to_list A Rs tol)"
ba_Suc: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j \<le> k" shows "b k (j, i) \<lless> a (Suc k)"
Inf_sat_def: shows "Inf = semilattice_neutr_set.F min top"
c\<^sub>0_def: shows "(c\<^sub>0, c\<^sub>1) = closest_pair_bf (p\<^sub>2 # p\<^sub>3 # ps)"
absorb_iff2: fixes a :: "'e"   and b :: "'e" shows "(a \<le> b) = (sup a b = b)"
joinable_simp: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'c) set" shows "((x, y) \<in> joinable R) = (\<exists>z. (x, z) \<in> R \<and> (y, z) \<in> R)"
integrable_mult_left_iff: fixes M :: "'b measure"   and c :: "real"   and f :: "'b \<Rightarrow> real" shows "integrable M (\<lambda>x. c * f x) = (c = 0 \<or> integrable M f)"
fun_when: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and P :: "bool" assumes "f (0::'b) = (0::'a)" shows "f (a when P) = (f a when P)"
Setcompr_eq_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "{f x |x. s2p A x} = \<P> f A"
T8: fixes w :: "i"   and \<psi> :: "'b \<Rightarrow> \<o>" shows "[\<^bold>\<forall>x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
iso_runit: fixes f :: "'a" assumes "Right.ide f" shows "Right.iso (Right.runit f)"
continuous_on_components: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "locally connected S"   and "\<And>C. C \<in> components S \<Longrightarrow> continuous_on C f" shows "continuous_on S f"
IH: fixes V :: "'a set" assumes "wf\<^sub>s\<^sub>t V (S @ \<langle>ac: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"   and "mgu t t' = Some \<delta>"   and "bvars\<^sub>s\<^sub>t (S @ \<langle>ac: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S') \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "wf\<^sub>s\<^sub>t (fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)) (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>)"
TensorDiag_preserves_Diag_2: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "r\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
U: fixes s :: "'s"   and f :: "'o" assumes "inv s" shows "T f s + \<Phi> (nxt f s) - \<Phi> s \<le> U f s"
hyps_2: shows "n_nodes (reflect t2) = n_nodes t2"
Un_Image: fixes R :: "('b \<times> 'a) set"   and S :: "('b \<times> 'a) set"   and A :: "'b set" shows "(R \<union> S) `` A = R `` A \<union> S `` A"
sup_loc_trans: fixes P :: "'b prog"   and a :: "ty err list"   and b :: "ty err list"   and c :: "ty err list" assumes "P \<turnstile> a [\<le>\<^sub>\<top>] b"   and "P \<turnstile> b [\<le>\<^sub>\<top>] c" shows "P \<turnstile> a [\<le>\<^sub>\<top>] c"
simps_2: fixes G :: "('e, 'd) labeled_graph"   and a :: "'e allegorical_term"   and b :: "'e allegorical_term" shows ":G:\<lbrakk>A_Cmp a b\<rbrakk> = (:G:\<lbrakk>a\<rbrakk>) O (:G:\<lbrakk>b\<rbrakk>)"
Mp_lift_modulus: fixes m :: "int"   and f :: "int poly"   and g :: "int poly"   and k :: "int" assumes "poly_mod.Mp m f = poly_mod.Mp m g" shows "poly_mod.Mp (m * k) (Polynomial.smult k f) = poly_mod.Mp (m * k) (Polynomial.smult k g)"
rel_cong: fixes x :: "'b list"   and ya :: "'b list"   and y :: "'c list"   and xa :: "'c list"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set ya; yb \<in> set xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "list_all2 R x y = list_all2 Ra ya xa"
br_table: fixes \<C> :: "t_context"   and ts :: "t list"   and "is" :: "nat list"   and i :: "nat"   and t1s :: "t list"   and t2s :: "t list" assumes "list_all (\<lambda>i. i < length (label \<C>) \<and> label \<C> ! i = ts) (is @ [i])" shows "\<C> \<turnstile> [Br_table is i] : t1s @ ts @ [T_i32] _> t2s"
assms_1: shows "\<Z>_sgrp_basis \<alpha> \<SS>"
is_ground_lit_def: fixes L :: "'a literal" shows "is_ground_lit L = is_ground_atm (atm_of L)"
c''s'': shows "(c'', s'') \<rightarrow>t s'"
map: fixes f :: "'e \<Rightarrow> 'd"   and x1 :: "'e"   and x2 :: "'e stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
target: shows "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"
absolutely_continuous_AE: fixes M' :: "'b measure"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "sets M' = sets M"   and "absolutely_continuous M M'"   and "almost_everywhere M P" shows "almost_everywhere M' P"
function_ring_car_memI: fixes f :: "'a \<Rightarrow> 'a" assumes "\<And>a. a \<in> carrier R \<Longrightarrow> f a \<in> carrier R"   and "\<And>a. a \<notin> carrier R \<Longrightarrow> f a = undefined" shows "f \<in> carrier (function_ring (carrier R) R)"
augment_alt: fixes p :: "(nat \<times> nat) list" assumes "f.isAugmentingPath p" shows "cf_of (f.augment (f.augmentingFlow p)) = f.cf.augment_cf (set p) (f.resCap p)"
elims: fixes x :: "nat set"   and xa :: "nat set \<times> 'a option list set"   and y :: "nat set \<times> 'a option list set" assumes "projectTable x xa = y" obtains V :: "nat set" and s :: "nat set" and t :: "'a option list set" where "x = V" and "xa = (s, t)" and "y = (s \<inter> V, restrict V ` t)"
Abs_eq_iff_1: fixes bs :: "atom set"   and x :: "'a"   and bs' :: "atom set"   and y :: "'a" shows "([bs]set. x = [bs']set. y) = (\<exists>p. (bs, x) \<approx>set (=) supp p (bs', y))"
OAlist_tc_lookup_eq_valueI: fixes k :: "'a"   and v :: "'b"   and xs :: "('a, 'b) oalist_tc" assumes "(k, v) \<in> set (list_of_oalist_tc xs)" shows "OAlist_tc_lookup xs k = v"
test_1_def: fixes x :: "'d"   and y :: "'c" shows "test_1 x (y =: z__) = y + z__"
Set_filter_cong: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> P x = Q x"   and "A = B" shows "Set.filter P A = Set.filter Q B"
square_free_factorization_def: fixes p :: "'b poly"   and cbs :: "'b \<times> ('b poly \<times> nat) list" shows "square_free_factorization p cbs \<equiv> case cbs of (c, bs) \<Rightarrow> p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i) \<and> (p = 0 \<longrightarrow> c = (0::'b) \<and> bs = []) \<and> (\<forall>a i. (a, i) \<in> set bs \<longrightarrow> square_free a \<and> 0 < degree a) \<and> (\<forall>a i b j. (a, i) \<in> set bs \<longrightarrow> (b, j) \<in> set bs \<longrightarrow> (a, i) \<noteq> (b, j) \<longrightarrow> coprime a b) \<and> distinct bs"
CConcStartaux: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and aa :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "Cp a x = \<bottom>" shows "(Cp a \<Oplus> Cp aa) x = Cp aa x"
distinct_15: fixes x91 :: "'b ltln"   and x92 :: "'b ltln" shows "true\<^sub>n \<noteq> x91 R\<^sub>n x92"
ffold_finsert: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'a"   and A :: "'a fset"   and z :: "'b" assumes "comp_fun_commute f"   and "x |\<notin>| A" shows "ffold f z (finsert x A) = f x (ffold f z A)"
backflows_finite: fixes E :: "('a \<times> 'a) set" assumes "finite E" shows "finite (backflows E)"
d: shows "\<forall>j\<ge>d. pdevs_apply x j = (0::'a)"
f_def: fixes k :: "nat" shows "f k = cdf (\<mu> (s k))"
matching_def: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set" shows "matching V E M = (M \<subseteq> E \<and> finite_graph V E \<and> (\<forall>e1\<in>M. \<forall>e2\<in>M. e1 \<noteq> e2 \<longrightarrow> disjoint_edges e1 e2))"
square_test_code: fixes n :: "nat" shows "square_test n = (q64_array !! (n mod 64) \<and> (let r = n mod 45045 in q63_array !! (r mod 63) \<and> q65_array !! (r mod 65) \<and> q11_array !! (r mod 11) \<and> n = (Discrete.sqrt n)\<^sup>2))"
uminus_in_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "((\<lambda>x. - f x) \<in> O[F](g)) = (f \<in> O[F](g))"
hyps_4: shows "(hs, data') = add_indices aux (snd data)"
take_Cons': fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
arctan_0_1_bounds_le: fixes x :: "real"   and xl :: "float"   and xu :: "float"   and p1 :: "nat"   and n :: "nat"   and p2 :: "nat" assumes "0 \<le> x"   and "x \<le> 1"   and "0 < real_of_float xl"   and "real_of_float xl \<le> x * x"   and "x * x \<le> real_of_float xu"   and "real_of_float xu \<le> 1" shows "arctan x \<in> {x * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu)..x * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)}"
empty: shows "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>s\<^sub>b \<longrightarrow> ts\<^sub>s\<^sub>b ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []"
WV: shows "\<forall>i. i \<in> I \<longrightarrow> W i \<subseteq> V \<and> s i \<in> local.ind_sheaf (W i)"
Act_eq_iff_perm: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c, 'd) formula"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c, 'd) formula" shows "(Act f1 \<alpha>1 x1 = Act f2 \<alpha>2 x2) = (f1 = f2 \<and> (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2))"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> nat \<Rightarrow> int \<times> int \<times> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int"   and a2 :: "nat"   and a3 :: "int \<times> int \<times> int" assumes "\<And>m a n p. \<lbrakk>\<lbrakk>n \<noteq> 0; n mod 2 = 0\<rbrakk> \<Longrightarrow> P m a (n div 2) p; \<lbrakk>n \<noteq> 0; n mod 2 \<noteq> 0\<rbrakk> \<Longrightarrow> P m a (n div 2) p\<rbrakk> \<Longrightarrow> P m a n p" shows "P a0 a1 a2 a3"
acc_def: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "acc A r = wf {(y, x). x \<in> A \<and> y \<in> A \<and> x \<sqsubset>\<^bsub>r\<^esub> y}"
trss_c_no_mon_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> trss fg" shows "mon_c fg c' = mon_c fg c"
monom_eq_0: fixes n :: "nat" shows "monom (0::'a) n = 0"
image_uminus_atLeastAtMost: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.atLeastAtMost less_eq x y = ord.atLeastAtMost less_eq (uminus y) (uminus x)"
token_run_merge: fixes x :: "nat"   and n :: "nat"   and y :: "nat"   and m :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "token_run x n = token_run y n" shows "token_run x (n + m) = token_run y (n + m)"
compute_butlast_\<ff>s_def: fixes s :: "'b list" shows "compute_butlast_\<ff>s s = (let \<ff>s = replicate (length s) 0; i = 0; j = 1 in WHILE\<^sub>T\<^bsup>I_out_cb (butlast s)\<^esup> (\<lambda>(b, i, j). j < length b) (\<lambda>(\<ff>s, i, j). ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. WHILE\<^sub>T\<^bsup>I_in_cb (butlast s) j\<^esup> (\<lambda>i. 0 < i \<and> s ! (i - 1) \<noteq> s ! (j - 1)) (\<lambda>i. ASSERT (i - 1 < length \<ff>s) \<bind> (\<lambda>_. Let (\<ff>s ! (i - 1)) RETURN)) i \<bind> (\<lambda>i. let i = i + 1 in ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let \<ff>s = \<ff>s[j := i]; j = j + 1 in RETURN (\<ff>s, i, j))))) (\<ff>s, i, j) \<bind> (\<lambda>(\<ff>s, uu_, uu_). RETURN \<ff>s))"
maximal_independent_subset_extend: fixes S :: "'b set"   and V :: "'b set" assumes "S \<subseteq> V"   and "local.independent S" obtains B :: "'b set" where "S \<subseteq> B" and "B \<subseteq> V" and "local.independent B" and "V \<subseteq> local.span B"
Setcompr_eq_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "{f x |x. x \<in> A} = f ` A"
add_less_same_cancel2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus a b) b = less a zero"
R2: fixes x :: "'b"   and y :: "'b"   and yl :: "'b bintree"   and yr :: "'b bintree"   and ts :: "'b bintree list"   and X :: "'b t_type" assumes "\<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr> \<in> t_ins_set X"   and "\<not> x \<le> y" shows "\<lparr>t_type.folding = False, item = x, subtrees = yr # Branch y yl yr # ts\<rparr> \<in> t_ins_set X"
mem_carrier: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "submonoid H G"   and "x \<in> H" shows "x \<in> carrier G"
hta_\<delta>_states_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_\<delta>_states H \<equiv> llh_set_xyy.g_Union_image id (ll_set_xy.g_image_filter (\<lambda>r. Some (rule_states_l r)) (hta_\<delta> H))"
mult_rng_of_frac_fundamental_lemma: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (r \<otimes> r' |\<^bsub>rel\<^esub> s \<otimes> s')"
image_restrict_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "restrict f A ` A = f ` A"
sorted_cons': fixes x :: "'b"   and xs :: "'b list" assumes "ASSUMPTION (sorted (x # xs))" shows "sorted xs"
equality: fixes r :: "('b, 'c) list_graph_scheme"   and r' :: "('b, 'c) list_graph_scheme" assumes "nodesL r = nodesL r'"   and "edgesL r = edgesL r'"   and "list_graph.more r = list_graph.more r'" shows "r = r'"
list_generator_def: shows "list_generator \<equiv> Generator (list_has_next, list_next)"
assms_2: shows "\<not> (\<forall>m. \<exists>n>m. f n = int (Suc k))"
h1: shows "List.member l1 m \<longrightarrow> (\<exists>n<length l1. l1 ! n = m)"
not_outside_2: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<succeq> q" shows "q \<in> \<P> outcomes"
raw_induct: fixes P :: "'s \<Rightarrow> nat \<Rightarrow> bool"   and s :: "'s"   and y :: "nat" assumes "\<And>terminates_within s sa. \<lbrakk>\<And>x y. terminates_within x = Some y \<Longrightarrow> P x y; (case g s of Done \<Rightarrow> Some 0 | Skip s' \<Rightarrow> map_option (\<lambda>n. n + 1) (terminates_within s') | Yield a s' \<Rightarrow> map_option (\<lambda>n. n + 1) (terminates_within s')) = Some sa\<rbrakk> \<Longrightarrow> P s sa"   and "local.terminates_within s = Some y" shows "P s y"
SUP_mono': fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` A))"
solvable_imp_subgroup: fixes H :: "'a set" assumes "solvable_seq G H" shows "subgroup H G"
run_one_step_break_imp_lfilled_prop_def: fixes s' :: "s"   and vs' :: "v list"   and n :: "nat"   and res :: "v list" shows "run_one_step_break_imp_lfilled_prop s' vs' n res = (\<lambda>d i (s, vs, ves, e). run_one_step d i (s, vs, ves, e) = (s', vs', RSBreak n res) \<longrightarrow> s = s' \<and> vs = vs' \<and> (res = ves \<and> e = $Br n \<or> (\<exists>n' lfilled es_c es les' ln. n < n' \<and> Lfilled_exact (n' - (n + 1)) lfilled (vs_to_es res @ [$Br n'] @ es_c) es \<and> e = Label ln les' es)))"
derived_set_incl: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H G" shows "derived_set G K \<subseteq> H"
closure_imp_adh: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier"   and "x \<in> closure A" shows "x adh A"
lower_3_def: fixes x :: "real_alg_3"   and i :: "nat" shows "lower_3 x i \<equiv> interval.lower (get_itvl_3 ((tighten_bounds_3 ^^ i) x))"
list_of_pdevs_def: fixes x :: "'b pdevs" shows "list_of_pdevs x = map (\<lambda>i. (i, pdevs_apply x i)) (rev (sorted_list_of_set (pdevs_domain x)))"
r2f_rtrancl_hom: fixes R :: "('a \<times> 'a) set" shows "\<F> (R\<^sup>*) = kstar (\<F> R)"
normalize_top: shows "normalize \<^bold>\<top> = \<^bold>\<top>"
assms: shows "\<exists>\<sigma>. fst3 e |\<union>| thd3 e |\<inter>| fmdom \<sigma> = {||} \<and> \<sigma>1 = [ fst3 e |=> Bot ] ++\<^sub>f \<sigma> \<and> \<sigma>2 = [ thd3 e |=> Top ] ++\<^sub>f \<sigma>"
d_def: shows "d__ = Polynomial.smult (unit_factor a__) c__"
absorb2: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "max a b = b"
closed_others_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "gen_refine.closed_others \<R> \<equiv> \<forall>c\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C mem\<^sub>C'. (\<langle>c\<^sub>A, gen_refine.mds\<^sub>A_of mds\<^sub>C, gen_refine.mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>x. mem\<^sub>C x \<noteq> mem\<^sub>C' x \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x) \<longrightarrow> (\<forall>x. dma mem\<^sub>C x \<noteq> dma mem\<^sub>C' x \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x) \<longrightarrow> (\<langle>c\<^sub>A, gen_refine.mds\<^sub>A_of mds\<^sub>C, gen_refine.mem\<^sub>A_of mem\<^sub>C'\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>"
list_all2_map2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and as :: "'a list"   and f :: "'c \<Rightarrow> 'b"   and bs :: "'c list" shows "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
\<Phi>_nneg: fixes t :: "'b tree" shows "0 \<le> \<Phi> t"
one_divides: fixes a :: "'a" assumes "a \<in> carrier R" shows "\<one> divides a"
fps_lr_inverse_eq_0_iff_2: fixes g :: "'a fps"   and y :: "'a" shows "(fps_right_inverse g y = 0) = (y = (0::'a))"
stut_imp: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F"   and "STUTINV G" shows "STUTINV (F \<longrightarrow> G)"
nextsuffix_def: fixes s :: "nat \<Rightarrow> 'b" shows "nextsuffix s \<equiv> s |\<^sub>s nextnat s"
apply_cm_commute: fixes c :: "('c, 't, 'd) Propagate.configuration_scheme"   and loc :: "'c"   and t :: "'t"   and n :: "int"   and loc' :: "'c"   and t' :: "'t"   and n' :: "int" shows "apply_cm (apply_cm c loc t n) loc' t' n' = apply_cm (apply_cm c loc' t' n') loc t n"
size_10: fixes x51 :: "nat"   and x52 :: "nat"   and x53 :: "v list"   and x54 :: "e list" shows "size (Local x51 x52 x53 x54) = size_list size x54 + Suc 0"
poly_eval_monomial: fixes a :: "'d \<Rightarrow> 'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "poly_eval a (monomial c t) = c * (\<Prod>x\<in>keys t. a x ^ lookup t x)"
member_in_class_relation: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_in C" shows "G\<turnstile>C\<preceq>\<^sub>C declclass m"