hcju/isabelleps huggingface dataset:
abs_summable_Im: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set" assumes "f abs_summable_on A" shows "(\<lambda>x. Im (f x)) abs_summable_on A"
mark_guards_simps_11: fixes f :: "'f"   and c\<^sub>1 :: "('d, 'e, 'g) com"   and c\<^sub>2 :: "('d, 'e, 'g) com" shows "mark_guards f (Catch c\<^sub>1 c\<^sub>2) = Catch (mark_guards f c\<^sub>1) (mark_guards f c\<^sub>2)"
sum: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and S :: "'e set" assumes "module_hom s1 s2 f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
coprime_heuristic_def: fixes f :: "int poly"   and g :: "int poly" shows "coprime_heuristic f g = (let lcf = lead_coeff f; lcg = lead_coeff g in find (\<lambda>p. (comm_monoid_mult_class.coprime lcf p \<or> comm_monoid_mult_class.coprime lcg p) \<and> coprime_approx_main p (finite_field_ops64 (uint64_of_int p)) f g) gcd_primes64 \<noteq> None)"
InterpTyE: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and i :: "('c, 'd) IType" assumes "L\<lbrakk>\<turnstile> A Type\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" obtains "i = IObj Ty\<lbrakk>A\<rbrakk>\<^bsub>I\<^esub>" and "Sig iS\<^bsub>I\<^esub> \<triangleright> \<turnstile> A Type"
isomorphic_relative_homology_groups_relboundary_contractible: fixes X :: "'a topology"   and T :: "'a set"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "contractible_space (subtopology X T)"   and "T \<subseteq> S"   and "topspace X \<inter> T \<noteq> {}" shows "relative_homology_group p X S \<cong> relative_homology_group (p - 1) (subtopology X S) T"
two_periodsN: fixes w :: "'b list"   and p :: "nat"   and q :: "nat" assumes "periodN w p"   and "periodN w q"   and "p + q \<le> \<^bold>|w\<^bold>|" shows "periodN w (gcd p q)"
fresh_PairD_2: fixes a :: "atom"   and x :: "'e"   and y :: "'f" assumes "a \<sharp> (x, y)" shows "a \<sharp> y"
enabled_disj1: fixes F :: "state \<times> state \<Rightarrow> bool"   and G :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled F \<longrightarrow> Enabled (F \<or> G)"
"2_6": fixes m :: "C" assumes "m \<in> hbase b"   and "evalC b (C ns) = evalC b m" shows "C ns = m"
finite_product_sigma_finite_axioms_def: fixes I :: "'d set" shows "finite_product_sigma_finite_axioms I \<equiv> finite I"
RPDs_alt: fixes r :: "mregex" shows "RPDs r = lfp (\<lambda>S. insert r S \<union> (insert r S \<bind> RPD))"
lunit_coherence2: fixes V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi> :: "'d \<times> 'd \<Rightarrow> 'e"   and f :: "'d" assumes "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"   and "partial_magma.ide V\<^sub>C f" shows "pseudofunctor.lF \<i>\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> f = V\<^sub>D (F (bicategory.lunit V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f)) (\<Phi> (trg\<^sub>C f, f))"
h: fixes p :: "real mpoly"   and xa :: "real" assumes "check_nonzero_const p" shows "insertion (nth_default 0 (xs @ xa # \<Gamma>)) p \<noteq> 0"
sglob_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "sglob s i j = s.globs s ! sglob_ind s i j"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list" shows "xs \<down> m \<down> n = xs \<down> min n m"
foldr_conv_foldl: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list"   and a :: "'a" shows "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
id4: shows "\<gamma> i = (- 1) ^ (\<delta> (i - 2) + 1 + (\<Sum>l\<leftarrow>[3..<i]. (n (l - 2) - (n (i - 1) - 1)) * (n (l - 1) - (n (i - 1) - 1)))) * (inverse (\<alpha> i) * (f (i - 1) ^ (\<delta> (i - 2) + 1) * \<beta> i) * (\<Prod>l\<leftarrow>[3..<i]. (\<beta> l / \<alpha> l) ^ (n (l - 1) - (n (i - 1) - 1))) * (\<Prod>l\<leftarrow>[3..<i]. f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1))))"
numeral_le_floor: fixes v :: "num"   and x :: "'b" shows "(numeral v \<le> \<lfloor>x\<rfloor>) = (numeral v \<le> x)"
f_sums_F: fixes x :: "'s"   and y :: "'s" shows "f x y sums F x y"
cnj_quad_form: fixes z :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "cnj (quad_form z H) = quad_form z (mat_adj H)"
added_ids_Deliver_Create_same_collapse: fixes i :: "'c"   and j :: "'c"   and e :: "'b" shows "added_ids [Deliver (i, Create j e)] e = [j]"
rel_funE: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and x :: "'e"   and y :: "'f" assumes "rel_fun A B f g"   and "A x y" obtains "B (f x) (g y)"
assms_5: shows "v \<sqinter> y\<^sup>\<star> = bot"
berlekamp_basis_i_def: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and u :: "'a list" shows "berlekamp_basis_i p ff_ops u = map (poly_of_list_i ff_ops \<circ> list_of_vec) (find_base_vectors_i ff_ops (berlekamp_resulting_mat_i p ff_ops u))"
inessential_eq_continuous_logarithm: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex" shows "(\<exists>a. homotopic_with_canon (\<lambda>h. True) S (- {0}) f (\<lambda>t. a)) = (\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (g x)))"
safe_max_empty_dest: fixes X :: "'a set" assumes "safe_max X = None" shows "X = {}"
wellFormedSubstEqvt: fixes p :: "(name \<times> name) list"   and \<sigma> :: "('c list \<times> 'f list) list" shows "p \<bullet> wellFormedSubst \<sigma> = wellFormedSubst (p \<bullet> \<sigma>)"
diag_0_eq: fixes x :: "'b" shows "(diag x = 0) = (x = (0::'b))"
map_simp: fixes f :: "(bool, unit) J.arr" assumes "J.arr f" shows "E.map f = C.dom p0"
cut_less_Diff: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A - B) \<down>< t = A \<down>< t - B \<down>< t"
hom_component_of_homogeneous: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "homogeneous p" shows "hom_component p n = (p when n = poly_deg p)"
unit_factor_simps_2: fixes n :: "nat" shows "unit_factor (Suc n) = 1"
closed_compactin: fixes X :: "'a topology"   and K :: "'a set"   and C :: "'a set" assumes "compactin X K"   and "C \<subseteq> K"   and "closedin X C" shows "compactin X C"
plusinf_uset: fixes p :: "fm"   and vs :: "'b list"   and a :: "'b"   and bs :: "'b list"   and x :: "'b" assumes "islin p"   and "\<not> Ifm vs (a # bs) (plusinf p)"   and "Ifm vs (x # bs) p" shows "\<exists>(c, s)\<in>set (uset p). x \<le> - Itm vs (a # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
LIMSEQ_power_zero: fixes x :: "'a" assumes "norm x < 1" shows "(^) x \<longlonglongrightarrow> (0::'a)"
project_guarantees: fixes F :: "'a program"   and X :: "'a program set"   and Y :: "'a program set"   and C :: "'c program \<Rightarrow> 'c set"   and X' :: "'c program set"   and Y' :: "'c program set" assumes "F \<in> X guarantees Y"   and "subset_closed (AllowedActs F)"   and "projecting C h F X' X"   and "extending C h F Y' Y" shows "extend h F \<in> X' guarantees Y'"
F_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "f \<in> Field leq"   and "x \<in> A \<Gamma>" shows "d_IN f x = 0"
simps_4: fixes Label :: "String.literal"   and Arity :: "nat"   and Guards :: "vname gexp list"   and Outputs :: "vname aexp list"   and Updates :: "(nat \<times> vname aexp) list"   and more :: "'a" shows "Outputs \<lparr>Label = Label, Arity = Arity, Guards = Guards, Outputs = Outputs, Updates = Updates, \<dots> = more\<rparr> = Outputs"
mult_mv_vv: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex" shows "A *\<^sub>m\<^sub>v v1 *\<^sub>v\<^sub>v v2 = v1 *\<^sub>v\<^sub>v (mat_transpose A *\<^sub>m\<^sub>v v2)"
circline_diagonalize: fixes H :: "circline" shows "\<exists>M H'. moebius_circline M H = H' \<and> circline_diag H'"
implies_not_e1: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>"
u_other_strict_ivt: fixes t :: "real" assumes "other.s t < u t" shows "collision_react {0..<t}"
SourcesS7_L2: shows "Sources level2 sS7 = {}"
map_raw_id: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "map_raw id xs = xs"
set_category_given_img_axioms: shows "set_category_given_img (\<cdot>) some_img \<AA>"
IH: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "t \<in> set T"   and "s \<in> timpl_closure t TI" shows "intruder_synth (timpl_closure_set M TI) s"
nn: shows "delta_bound d = (\<lambda>x. nat \<lceil>x * of_int N\<rceil>)"
at_top_within_def: fixes s :: "'b set" shows "at_top_within s = (INF k\<in>s. principal ({k..} \<inter> s))"
walk_edges_decomp: fixes xs :: "'a list"   and x :: "'a"   and xs' :: "'a list" shows "edges_of_walk (xs @ x # xs') = edges_of_walk (xs @ [x]) \<union> edges_of_walk (x # xs')"
gen_subset_gen2: fixes I :: "'a set"   and J :: "'a set"   and P :: "'a set set" assumes "{I, J} \<subseteq> P"   and "gen P I \<inter> gen P J \<noteq> {}" shows "gen P I \<subseteq> gen P J"
neg_1: shows "negligible {x \<in> S - T. f x \<noteq> (0::'a)}"
SDG_axioms: shows "SDG sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main (_Exit_) Def Use ParamDefs ParamUses"
Ands: fixes pos :: "formula list"   and neg :: "formula list"   and l :: "formula list" assumes "(pos, neg) = partition safe_formula l"   and "pos \<noteq> []"   and "list_all safe_formula pos"   and "list_all safe_formula (map remove_neg neg)"   and "\<Union> (fv ` set neg) \<subseteq> \<Union> (fv ` set pos)"   and "list_all P pos"   and "list_all P (map remove_neg neg)" shows "P (Ands l)"
a_in: shows "to_nat a \<in> {i..<min (nrows A) (ncols A)}"
fimageE: fixes v :: "'b set"   and F :: "'a set set" assumes "v \<in> fimage F" obtains u :: "'a set" where "v \<subseteq> carrier\<^bsub>T\<^esub>" and "u \<in> F" and "f ` u \<subseteq> v"
assms: shows "\<forall>v. freedb e v \<and> k \<le> v \<longrightarrow> closed (\<Gamma> (v - k))"
simps: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P \<turnstile> a1 \<prec>\<^sub>R a2 = (\<exists>C Bs rest D. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(Bs, rest)\<rfloor> \<and> Repeats D \<in> set Bs)"
elims_2: fixes x :: "16 word \<times> 16 word"   and xa :: "16 word" assumes "simple_match_port x xa" obtains s :: "16 word" and e :: "16 word" and p_p :: "16 word" where "x = (s, e)" and "xa = p_p" and "p_p \<in> {s..e}"
power_minus_odd: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.ring_1 one times plus zero minus uminus"   and "odd n" shows "power.power one times (uminus a) n = uminus (power.power one times a n)"
dom_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VVV.dom f = (if VVV.arr f then VxVxV.dom f else VxVxV.null)"
always_hyp_change_no_lim: fixes z :: "nat"   and i :: "nat"   and j :: "nat" assumes "\<forall>x>0. s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)" shows "\<not> learn_lim \<phi> {local.adverse z i j} s"
w1MinusBound: shows "- (2 ^ (size (SCAST(32 \<rightarrow> 64) w1) - 1)) \<le> sint (SCAST(32 \<rightarrow> 64) w1) + sint (- 0x7FFFFFFF) \<and> sint (SCAST(32 \<rightarrow> 64) w1) + sint (- 0x7FFFFFFF) \<le> 2 ^ (size (SCAST(32 \<rightarrow> 64) w1) - 1) - 1"
g_s\<^sub>0_def: shows "g_s\<^sub>0 \<equiv> Even"
alpha_Tree_supp_rel: fixes t1 :: "('a, 'b, 'c) Tree"   and t2 :: "('a, 'b, 'c) Tree" assumes "t1 =\<^sub>\<alpha> t2" shows "supp_rel (=\<^sub>\<alpha>) t1 = supp_rel (=\<^sub>\<alpha>) t2"
ucontk: shows "uniformly_continuous_on ({0..1} \<times> {0..1}) k"
PDG: shows "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit (Postdomination.standard_control_dependence sourcenode targetnode valid_edge Exit)"
rref_condition4: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) = (0::'a))"
minus_mod_eq_mult_div: fixes a :: "'a"   and b :: "'a" shows "a - a mod b = b * (a div b)"
im_jointfun: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and A :: "'c set"   and g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and B :: "'c set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> B" shows "jointfun n f m g ` {j. j \<le> Suc (n + m)} = f ` {j. j \<le> n} \<union> g ` {j. j \<le> m}"
all_in_i_def: fixes x :: "'a list"   and I :: "'a interval list" shows "(x all_in\<^sub>i I) = (length x = length I \<and> (\<forall>i<length I. x ! i \<in>\<^sub>i I ! i))"
chamber_distance_le: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "gallery (C # Cs @ [D])" shows "order.greater_eq (Suc (length Cs)) (chamber_distance C D)"
poles: shows "\<forall>p\<in>poles. is_pole f p"
cmp_inj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "inj_on f A"   and "inj_on g B" shows "inj_on (cmp g f) A"
map_concat: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list list" shows "map f (concat xs) = concat (map (map f) xs)"
assms_2: shows "z \<notin> {0, \<i>, - \<i>}"
comm_prim: fixes r :: "'b list"   and s :: "'b list" assumes "primitive r"   and "primitive s"   and "r \<cdot> s = s \<cdot> r" shows "r = s"
real_transfer_operator_def: fixes f :: "'a \<Rightarrow> real" shows "real_transfer_operator f = (\<lambda>x. enn2real (nn_transfer_operator (\<lambda>x. ennreal (f x)) x) - enn2real (nn_transfer_operator (\<lambda>x. ennreal (- f x)) x))"
in_idem: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<in>| A" shows "x \<^bold>* F A = F A"
assms_1: shows "\<forall>x y. P x y \<longrightarrow> R' x y \<longrightarrow> R x y"
has_codomain_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" shows "(A_BxA.codomains f \<noteq> {}) = (A_B.arr (fst f) \<and> A.arr (snd f))"
snapshot_stable: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "trace c t c'"   and "ps c p \<noteq> None" shows "ps c' p \<noteq> None"
prodag_pOp: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "(\<plusminus>\<^bsub>a\<Pi>\<^bsub>I\<^esub> A\<^esub>) = prod_pOp I A"
dg_eqI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "\<AA>\<lparr>Obj\<rparr> = \<BB>\<lparr>Obj\<rparr>"   and "\<AA>\<lparr>Arr\<rparr> = \<BB>\<lparr>Arr\<rparr>"   and "\<AA>\<lparr>Dom\<rparr> = \<BB>\<lparr>Dom\<rparr>"   and "\<AA>\<lparr>Cod\<rparr> = \<BB>\<lparr>Cod\<rparr>" shows "\<AA> = \<BB>"
two_comparisons_into_case_order_12: fixes x :: "'a"   and y :: "'a"   and P :: "'c"   and Q :: "'c"   and R :: "'c" shows "(if x = y then P else if le x y then Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
roots_of_coprime_r_capture_sgas_without_zeros: fixes qs :: "rat poly list"   and sga :: "rat list"   and x1 :: "real" assumes "pairwise_coprime_list qs"   and "\<And>q. q \<in> set qs \<Longrightarrow> rsquarefree q"   and "sga = sign_vec qs x1"   and "has_no_zeros sga" shows "\<exists>w\<in>roots_of_coprime_r (cast_rat_list qs). sga = sign_vec qs w"
sqrtu_4_eq: fixes x :: "real" assumes "0 \<le> x" shows "sqrtu x 4 = 1 / 32 * (225792840 * x ^ 6 + 64512240 * x ^ 5 + 601080390 * x ^ 8 + 471435600 * x ^ 7 + 496 * x + 1 + 35960 * x\<^sup>2 + 906192 * x ^ 3 + 10518300 * x ^ 4 + x ^ 16 + 906192 * x ^ 13 + 10518300 * x ^ 12 + 496 * x ^ 15 + 35960 * x ^ 14 + 471435600 * x ^ 9 + 225792840 * x ^ 10 + 64512240 * x ^ 11) / ((x + 1) * (x\<^sup>2 + 6 * x + 1) * (x ^ 4 + 28 * x ^ 3 + 70 * x\<^sup>2 + 28 * x + 1) * (1820 * x ^ 6 + 8008 * x ^ 5 + x ^ 8 + 120 * x ^ 7 + 120 * x + 1 + 1820 * x\<^sup>2 + 8008 * x ^ 3 + 12870 * x ^ 4))"
field_Units: shows "Units (residue_ring (int p)) = carrier (residue_ring (int p)) - {\<zero>\<^bsub>residue_ring (int p)\<^esub>}"
iso_Euclidean_complements_lemma1: fixes m :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and n :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "closedin (Euclidean_space m) S"   and "continuous_map (subtopology (Euclidean_space m) S) (Euclidean_space n) f" obtains g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" where "continuous_map (Euclidean_space m) (Euclidean_space n) g" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
valid_eqvt: fixes P :: "('state, 'effect) L_state"   and x :: "('a, 'pred, ('act, 'effect) L_action) Formula.formula"   and p :: "perm" assumes "L_transform.valid P x" shows "L_transform.valid (p \<bullet> P) (p \<bullet> x)"
ABinds_restr_subst: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var set"   and x :: "Vars.var"   and y :: "Vars.var"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "\<And>x' e a. (x', e) \<in> set \<Gamma> \<Longrightarrow> \<A>\<^bsub>a\<^esub> e[x::=y] f|` S = \<A>\<^bsub>a\<^esub> e f|` S"   and "x \<notin> S"   and "y \<notin> S"   and "domA \<Gamma> \<subseteq> S" shows "ABinds \<Gamma>[x::h=y]\<cdot>ae f|` S = ABinds \<Gamma>\<cdot>(ae f|` S) f|` S"
SC2: fixes p :: "('b, 'c) Combinators list" assumes "singleCombinators p" shows "singleCombinators (removeShadowRules2 p)"
map_poly_zero: fixes f :: "'c \<Rightarrow> 'd"   and p :: "'c poly" assumes "\<forall>c. f c = (0::'d) \<longrightarrow> c = (0::'c)" shows "(map_poly f p = 0) = (p = 0)"
h1: shows "\<forall>elem. List.member (subsets_smash_R (length qs1) subsets1 subsets2) elem \<longrightarrow> (\<exists>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<and> elem = (fst elem1 @ map ((+) (length qs1)) (fst elem2), snd elem1 @ map ((+) (length qs1)) (snd elem2)))"
Rep_inverse: fixes x :: "'a stream" shows "stream_of_llist (llist_of_stream x) = x"
ipresIGWlsAbs_def: fixes hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'ae, 'w, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ab, 'ai, 'aj, 'ak, 'x, 'al) model_scheme" shows "ipresIGWlsAbs hA MOD MOD' \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD' (us, s) (hA A)"
assms_2: shows "edges G1 \<union> edges G2 = edges G"
cont': fixes A :: "real set" assumes "A \<subseteq> {real_of_int a..real_of_int b}" shows "continuous_on A f"
det_identical_rows: fixes A :: "'a mat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "i \<noteq> j"   and "i < n"   and "j < n"   and "Matrix.row A i = Matrix.row A j" shows "Determinant.det A = (0::'a)"
lset_ltakeWhile_subset: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lset (ltakeWhile P xs) \<subseteq> lset xs \<inter> {x. P x}"
cltn2_inverse_def: fixes A :: "cltn2" shows "cltn2_inverse A \<equiv> cltn2_abs (matrix_inv (cltn2_rep A))"
irfb_def: shows "fb\<^sup>-\<^sub>\<R> = \<R> \<circ> fb\<^sup>-\<^sub>\<F>"
isometryD_4: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "1-lipschitz_on UNIV f"
SPEC_RES_ASSN_def: fixes uu :: "'h"   and uv :: "'i" shows "SPEC_RES_ASSN uu uv \<equiv> True"
assms_2: shows "\<D>\<^sub>\<circ> f = I"
adds_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" shows "plus.adds plus b a = (\<exists>k. a = plus b k)"
induct: fixes P :: "'a list list \<Rightarrow> 'a list list \<Rightarrow> bool"   and a0 :: "'a list list"   and a1 :: "'a list list" assumes "P [] []"   and "\<And>l. P [] [l]"   and "\<And>la acc2. P [] (la # acc2) \<Longrightarrow> P (la # acc2) []"   and "\<And>la acc2 l. P [] (l # la # acc2) \<Longrightarrow> P (la # acc2) [l]"   and "\<And>acc2 l1 l2 ls. P (merge l1 l2 # acc2) ls \<Longrightarrow> P acc2 (l1 # l2 # ls)" shows "P a0 a1"
step: shows "(pc', s') \<in> set (step pc (\<tau>s ! pc))"
prems_1: shows "are_all_operators_applicable s (op # op' # \<pi>)"
refine_basic: fixes Tc :: "('e, 'f) TS_scheme"   and R :: "('g \<times> 'e) set"   and Ta :: "('g, 'h) TS_scheme" assumes "init Tc \<subseteq> R `` init Ta"   and "{R} TS.trans Ta, TS.trans Tc {> R}" shows "PO_refines R Ta Tc"
closed_sum_zero_right: fixes A :: "'d set" shows "A +\<^sub>M {0::'d} = closure A"
size_insert: fixes x :: "'b"   and t :: "('b \<times> nat) tree" shows "size (Weight_Balanced_Trees.insert x t) = (if isin t x then size t else Suc (size t))"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "List_Supplement.embed (x # xs) = [x] # List_Supplement.embed xs"
upper_triangular_mat_delete_make_mat: fixes n :: "nat"   and k :: "nat"   and B :: "'a mat" shows "Matrix.upper_triangular (mat_delete (make_mat n k B) 0 0)"
L2_set_cong: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x" shows "L2_set f A = L2_set g B"
simps_1: fixes x1 :: "'g"   and x2 :: "'h"   and y1 :: "'g"   and y2 :: "'h" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
sup_of_singletonI: fixes x :: "'a" assumes "x \<in> carrier L" shows "is_lub L x {x}"
monotone_eadd2: fixes x :: "enat" shows "monotone (\<le>) (\<le>) ((+) x)"
antisym_conv: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
ord_pos: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "0 \<le> ord_Zp x"
qGoodEnv_iff_goodEnv_asEnv: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" shows "goodEnv (asEnv qrho) = qGoodEnv qrho"
g_def: shows "g \<equiv> circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
semiring_simprules_7: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
intros_6: fixes x :: "real" assumes "x \<in> radical_sqrt"   and "0 \<le> x" shows "sqrt x \<in> radical_sqrt"