hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'c"   and xa :: "('a, 'b, 'c) formula" shows "Rep_formula (Act x xa) = Act\<^sub>\<alpha> x (Rep_formula xa)"
vrestr_difference: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" shows "f --` (A - B) = f --` A - f --` B"
least_def: fixes P :: "'b \<Rightarrow> bool"   and po :: "'b potype" shows "least P po \<equiv> SOME x. x \<in> pset po \<and> P x \<and> (\<forall>y\<in>pset po. P y \<longrightarrow> (x, y) \<in> order po)"
assms_2: shows "run (v\<^sub>1 @- [a] @- v\<^sub>2) q"
steps_length: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and s :: "'c list" assumes "length qs = length as" shows "length (Partial_Cost_Model.steps s qs as) = length s"
elims_2: fixes x :: "'d ifex"   and xa :: "'d set" assumes "reduced x xa" obtains xb :: "'d" and t1 :: "'d ifex" and t2 :: "'d ifex" and X :: "'d set" where "x = IF xb t1 t2" and "xa = X" and "xb \<notin> X \<and> t1 \<noteq> t2 \<and> reduced t1 (insert xb X) \<and> reduced t2 (insert xb X)"   | uv_ :: "'d set" where "x = Trueif" and "xa = uv___"   | uv_ :: "'d set" where "x = Falseif" and "xa = uv_"
card_of_Sigma_mono2: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and J :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "inj_on f I"   and "f ` I \<subseteq> J" shows "|SIGMA i:I. A (f i)| \<le>o |Sigma J A|"
aperiodic_def: fixes C :: "'s option set" shows "K2.aperiodic C = (C \<in> UNIV // K2.communicating \<and> K2.period C = 1)"
SAStates_ChiPlus_ChiRel_trans: fixes S :: "'a"   and U :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and T :: "'a"   and SA :: "('a, 'b, 'c) seqauto" assumes "(S, U) \<in> ChiPlus A"   and "(U, T) \<in> ChiRel A"   and "S \<in> States SA"   and "SA \<in> SAs A" shows "T \<notin> States SA"
triangle_in_hom_3: shows "\<guillemotleft>\<l>[f] \<bullet> (\<epsilon> \<star> f) \<bullet> \<a>\<^sup>-\<^sup>1[f, g, f] \<bullet> (f \<star> \<eta>) \<bullet> \<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> f\<guillemotright>"
compl_mono: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>- y \<sqsubseteq> \<^bold>- x"
edge_preserving_on_graphI: fixes X :: "('a, 'b) labeled_graph"   and f :: "'b \<Rightarrow> 'c"   and Y :: "('a \<times> 'c \<times> 'c) set" assumes "\<And>l x y. \<lbrakk>(l, x, y) \<in> edges X; x \<in> vertices X; y \<in> vertices X\<rbrakk> \<Longrightarrow> (l, f x, f y) \<in> Y" shows "edge_preserving (on_graph X f) (edges X) Y"
const: fixes c :: "'a"   and F :: "'b filter" assumes "c \<noteq> (0::'a)" shows "\<omega>[F](\<lambda>_. c) = \<omega>[F](\<lambda>_. 1::'a)"
semiring_norm_64: fixes a :: "'a" shows "a * (0::'a) = (0::'a)"
ldrop_ldrop: fixes n :: "enat"   and m :: "enat"   and xs :: "'b llist" shows "ldrop n (ldrop m xs) = ldrop (n + m) xs"
fresh_3: fixes a :: "name"   and x1 :: "name"   and x2 :: "lam" shows "a \<sharp> Lam [x1].x2 = a \<sharp> [x1].x2"
Suc_1: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and fs :: "int vec list" assumes "LLL_impl_inv state i fs"   and "LLL_invariant True i fs"   and "\<mu>_small_row i fs j"   and "LLL_Impl.basis_reduction_add_rows_loop n state i j (map ((!) fs) (rev [0..<j])) = state'"   and "j \<le> i"   and "i < m" shows "LLL_impl_inv state' i fs' \<and> local.basis_reduction_add_rows_loop i fs j = fs'"
fun_eq_zeroI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> supp_fun f \<Longrightarrow> f x = (0::'b)" shows "f = 0"
accepts_or: fixes L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "accepts (or L R) w = (accepts L w \<or> accepts R w)"
norm_nth_le: fixes x :: "('b, 'c) vec"   and i :: "'c" shows "\<parallel>x $ i\<parallel> \<le> \<parallel>x\<parallel>"
prefix_trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs \<le> ys"   and "ys \<le> zs" shows "xs \<le> zs"
run_return_env: fixes return :: "'c \<Rightarrow> 'a"   and x :: "'c"   and r :: "'b" shows "run_env (return_env return x) r = return x"
pcr_nonzero_def: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_nonzero T \<equiv> T OO cr_nonzero"
linorder_rank_union: fixes A :: "'b set"   and B :: "'b set"   and R :: "('b \<times> 'b) set"   and x :: "'b" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "linorder_rank R (A \<union> B) x = linorder_rank R A x + linorder_rank R B x"
zcf_degree_zero: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "deg Zp f = 0" shows "f (deg Zp f) = zcf f"
finite_in_arcs: fixes v :: "'a" shows "finite (in_arcs (with_proj G) v)"
D_base_not_empty: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "{len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs} \<noteq> {}"
proj_preserves: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "P x" shows "(P /// \<phi>) (proj \<phi> x)"
scaleR_le_cancel_left_pos: fixes c :: "real"   and a :: "'b"   and b :: "'b" assumes "0 < c" shows "(c *\<^sub>R a \<le> c *\<^sub>R b) = (a \<le> b)"
tranclp_rtranclp_absorb: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "R\<^sup>+\<^sup>+\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
sorted_remdups: fixes xs :: "'d Word.word list" assumes "signed.sorted xs" shows "signed.sorted (remdups xs)"
naturality: fixes f :: "'a" assumes "A.arr f" shows "\<tau> (A.cod f) \<cdot>\<^sub>B G f = H f \<cdot>\<^sub>B \<tau> (A.dom f)"
length_output_witness: fixes rs :: "nat list" shows "dim_vec (tensors_from_net (witness_l rs)) = rs ! 0"
f2r_kstar_hom: fixes f :: "'a \<Rightarrow> 'a set" shows "\<R> (kstar f) = (\<R> f)\<^sup>*"
par_by_merge_choice_left: fixes P :: "('d, 'f) urel"   and Q :: "('d, 'f) urel"   and M :: "(('d, 'f, 'g) mrg, 'e) urel"   and R :: "('d, 'g) urel" shows "(P \<sqinter> Q) \<parallel>\<^bsub>M\<^esub> R = P \<parallel>\<^bsub>M\<^esub> R \<sqinter> Q \<parallel>\<^bsub>M\<^esub> R"
strand_step_id_subst: fixes x :: "('c, 'd) strand_step" shows "x \<cdot>\<^sub>s\<^sub>t\<^sub>p Var = x"
stationary_distribution_iterate: fixes N :: "'s pmf"   and y :: "'s"   and n :: "nat" assumes "stationary_distribution N" shows "ennreal (pmf N y) = \<integral>\<^sup>+ x. ennreal (p x y n) \<partial>measure_pmf N"
sibling_sibling_id: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t" shows "sibling t (sibling t a) = a"
assms_1: shows "\<forall>p\<in>path_image \<gamma>. Re z \<le> Re p"
poly_mult: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" shows "poly (p * q) x = poly p x * poly q x"
ode_e_def: fixes ode_ops :: "ode_ops" shows "ode_e ode_ops = ode_expression ode_ops"
length_coeffs_degree: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "length (coeffs p) = Suc (degree p)"
minus_mod_cancel: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "0 < a - b"   and "0 < a - b mod q" shows "[a - b + c = a - b mod q + c] (mod q)"
exec_normalize_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
right_gpv_Inr_transfer: fixes gpv :: "('h, 'j, 'l) gpv" shows "rel_gpv'' (=) (\<lambda>l r. l = Inr r) (\<lambda>l r. l = Inr r) (right_gpv gpv) gpv"
iso_comp: fixes less_eq'' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less'' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) less_eq'' less'' Q g"   and "order.greater_eq Q (f ` P)" shows "OrderingSetIso (\<^bold>\<le>) (\<^bold><) less_eq'' less'' P (g \<circ> f)"
mask_natural_def: shows "mask \<equiv> map_fun id natural_of_nat mask"
assms_2: shows "u \<in> subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<delta>) - range Var"
norm_normalize: fixes p :: "(real, 2) vec" shows "norm (normalize p) = 1"
subset_Compl_singleton: fixes A :: "'a set"   and b :: "'a" shows "(A \<subseteq> - {b}) = (b \<notin> A)"
simps_1: fixes u :: "nat" shows "pathVertices u [] = [u]"
undirected: fixes v :: "'a"   and w :: "'a" shows "v \<rightarrow> w = w \<rightarrow> v"
hseqE: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "trg \<nu> = src \<mu>"
raw_term_cases: fixes y :: "('a, 'b) trm" assumes "y \<in> {\<theta>. dfree \<theta>}" obtains x :: "('a, 'b) strm" where "y = raw_term x"
neq_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "(x \<noteq> y) = (less x y \<or> less y x)"
hyps_2: shows "(u, s) \<in> timpl_closure'_step c"
e1_def: shows "e1 = lcm d (g x) div d"
filter_upt: fixes l :: "nat"   and m :: "nat"   and n :: "nat" assumes "l \<le> m"   and "m < n" shows "filter ((\<le>) m) [l..<n] = [m..<n]"
div_less: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "a < b" shows "a div b = (0::'b)"
restrict_map_disj_right: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 \<bottom> h\<^sub>1 |` S"
wbt_balanceL: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree"   and l' :: "('b \<times> nat) tree"   and a' :: "'b" assumes "wbt \<langle>l, (a, n), r\<rangle>"   and "wbt l'"   and "size l = size l' + 1" shows "wbt (balanceL l' a' r)"
elims: fixes x :: "'b tree"   and y :: "real" assumes "\<Phi> x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>" and "y = log 2 (real (size \<langle>l, xa, r\<rangle>)) + \<Phi> l + \<Phi> r"
iprev_cut_greater_conv: fixes t :: "nat"   and n :: "nat"   and I :: "nat set" assumes "t < iprev n I" shows "iprev n (I \<down>> t) = iprev n I"
undual_inverse: fixes x :: "'a dual" shows "dual (undual x) = x"
timpls_transformable_to'_inv'_4: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to' TI (Fun f T) t"   and "f \<noteq> the_Fun t" shows "is_Abs f \<and> is_Abs (the_Fun t) \<and> in_trancl TI (the_Abs f) (the_Abs (the_Fun t))"
prfx: fixes p' :: "'a list" assumes "prefix (p @ [m]) p'" shows "augment_list (joint_strategy \<sigma> \<sigma>\<^sub>o) p' = augment_list (joint_strategy \<sigma> (\<sigma>\<^sub>o(p := m))) p'"
hd: shows "root \<midarrow>x\<rightarrow> root'"
SumComm: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<equiv>\<^sub>s Q \<oplus> P"
poly_add_normalize_2: fixes p1 :: "'a list list"   and p2 :: "'a list list" assumes "set p1 \<subseteq> carrier (K [X])"   and "set p2 \<subseteq> carrier (K [X])" shows "UP.poly_add p1 p2 = UP.poly_add p1 (UP.normalize p2)"
has_integral_improperE: fixes f :: "real \<Rightarrow> 'a"   and I :: "'a"   and a :: "real" assumes "(f has_integral I) {a..}"   and "f absolutely_integrable_on {a..}" obtains J :: "real \<Rightarrow> 'a" where "\<And>k. (f has_integral J k) {a..k}" and "(J \<longlongrightarrow> I) at_top"
is_span: shows "Maps.span SPN_fgh.\<nu>.leg0 SPN_fgh.\<nu>.leg1"
sum_character: shows "sum (dc2c \<chi>) (carrier G) = (if dc2c \<chi> = principal_char G then of_nat (order G) else 0)"
scalar_prod_is_Matrix_scalar_prod: fixes u :: "complex list"   and v :: "complex list" assumes "length u = length v" shows "plus_mult.scalar_product (*) 0 (+) u v = vec_of_list u \<bullet> vec_of_list v"
yoneda_lemma_axioms: shows "yoneda_lemma (\<cdot>) (\<cdot>\<^sub>S) \<phi> F a"
assms_1: shows "((\<lambda>x. f x / g x) \<longlongrightarrow> inverse c) F"
complete_less_not: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "complete A r"   and "hasw [x, y] A"   and "\<not> x \<^bsub>r\<^esub>\<prec> y" shows "y \<^bsub>r\<^esub>\<preceq> x"
uv_6: shows "\<not> (u1, u2) \<approx>\<^sub>v conjugate_cvec (reciprocal_cvec (v1, v2))"
weakBisimTransitive: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, Q, R) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(\<Psi>, P, R) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
validE: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn"   and Y :: "vals"   and s0 :: "state"   and Z :: "'b"   and n :: "nat"   and L :: "lenv"   and v :: "vals"   and s1 :: "state"   and accC :: "qtname"   and T :: "tys"   and C :: "assigned" assumes "G,A|\<Turnstile>\<Colon>{{P} t\<succ> {Q}}"   and "P Y s0 Z"   and "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"   and "s0\<Colon>\<preceq>(G, L)"   and "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"   and "normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> C" obtains "Q v s1 Z" and "s1\<Colon>\<preceq>(G, L)"
assert_def: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "assert P x = (if P x then return x else raise STR ''assert'')"
distinct_upt: fixes i :: "nat"   and j :: "nat" shows "cycle [i..<j]"
Mem_cong1: fixes t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" shows "{} \<turnstile> t EQ t' AND u EQ u' IMP t IN u IMP t' IN u'"
adds_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a adds b"   and "b adds c" shows "a adds c"
inv: fixes G :: "'b set"   and smult :: "('c, 'b) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and V :: "'d set"   and smult' :: "('c, 'b) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and T :: "'d \<Rightarrow> 'e"   and W :: "'e set" assumes "FGModuleIso G smult V smult' T W" shows "FGModuleIso G smult' W smult (the_inv_into V T \<down> W) V"
closureI: fixes T :: "'b set set"   and a :: "'b set"   and x :: "'b" assumes "\<And>c. \<lbrakk>c closed\<^bsub>T\<^esub>; a \<subseteq> c\<rbrakk> \<Longrightarrow> x \<in> c" shows "x \<in> closure\<^bsub>T\<^esub> a"
affected_fixate_no_binary_circle: fixes f :: "'b perm"   and a :: "'b" assumes "2 < order f a" shows "affected (fixate a f) = affected f - {a}"
isin_set: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> T h"   and "sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
last_message_one: fixes m :: "'b message_af" shows "last_message [m] = m"
h2a: shows "n mod 2 = 0 \<or> n mod 2 = 1"
is_distinguishing_formula_def: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state" shows "x distinguishes P from Q \<equiv> P \<Turnstile> x \<and> \<not> Q \<Turnstile> x"
x_1: shows "u *\<^sub>R x = cx *\<^sub>R xx"
ideD_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.arr C a"
nth_zip: fixes i :: "nat"   and xs :: "'c list"   and ys :: "'d list" assumes "i < length xs"   and "i < length ys" shows "zip xs ys ! i = (xs ! i, ys ! i)"
strict_strict_from_inj: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "inj_on f {0..<n}" shows "strict_mono_on (strict_from_inj n f) {0..<n}"
d_IN_current_of_bipartite: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal"   and x :: "'v" assumes "current (bipartite_web_of \<Gamma>) f" shows "d_IN (current_of_bipartite f) x = d_IN f (Inr x) - f (Inl x, Inr x)"
le_inf_reflp: fixes w :: "'action stream" shows "w \<preceq>\<^sub>I w"
mojmir_accept_token_set_def2: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "finite (mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F)"   and "finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i)"   and "infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i)" shows "mojmir_def.accept \<delta> q\<^sub>0 w F"
periodic_fun_simple_def: fixes f :: "'b \<Rightarrow> 'c"   and period :: "'b" shows "periodic_fun_simple f period \<equiv> \<forall>x. f (x + period) = f x"
inv_is_iD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = Aodv_Basic.inv" shows "ip \<in> iD rt"
hom_closed: fixes x :: "'b \<Rightarrow>\<^sub>0 int" assumes "x \<in> carrier (free_Abelian_group T)" shows "k x \<in> carrier (free_Abelian_group S)"
card_adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H'\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := H'\<rparr>)"   and "a \<notin> H" shows "card (adjoin (G\<lparr>carrier := H'\<rparr>) H a) = card H * subgroup_indicator (G\<lparr>carrier := H'\<rparr>) H a"
fres_def: fixes z :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "prod.fres z y = Sup_prod {x. less_eq_prod (times_prod' x y) z}"
minus_a: shows "- (real n * ln 2) \<le> 4 * ln (real n) + 3 - 2 * (1 + ln (real (n div 2))) - (1 + ln (real n)) - (real n * ln (real n) - real n - 2 * (real (n div 2) * ln (real (n div 2)) - real (n div 2)))"
simul_Un: fixes Retr :: "'b set \<Rightarrow> 'b set"   and theta1 :: "'b set"   and theta2 :: "'b set" assumes "Bisim.mono Retr"   and "simul Retr theta1"   and "simul Retr theta2" shows "simul Retr (theta1 \<union> theta2)"
countably_additive_eq: fixes \<mu>' :: "'a set \<Rightarrow> ennreal"   and \<mu> :: "'a set \<Rightarrow> ennreal" assumes "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu>' a = \<mu> a" shows "countably_additive (sigma_sets \<Omega> A) \<mu>' = countably_additive (sigma_sets \<Omega> A) \<mu>"
assms_1: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "P xs ys" shows "P (x # xs) (y # ys)"
y3: shows "labels (snd t', ts) = labels_conv (snd t', map (Pair True) ts)"
agunit_gone: fixes A :: "('e, 'f) aGroup_scheme" assumes "aGroup A" shows "\<one>\<^bsub>b_ag A\<^esub> = \<zero>\<^bsub>A\<^esub>"
fmrel_acyclic': fixes ym :: "'f \<rightharpoonup>\<^sub>f 'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "acyclicP_on (fmran' ym) R"   and "fmrel R\<^sup>+\<^sup>+ xm ym"   and "fmrel R ym xm" shows "xm = ym"
tt_min_keys: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "v \<in> keys p" shows "punit.tt p \<preceq> v"
mon_ww_empty: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme" shows "mon_ww fg [] = {}"
star_subsum: fixes x :: "'a" shows "x\<^sup>\<star> + x\<^sup>\<star> \<cdot> x = x\<^sup>\<star>"
nth_stirling_row: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "stirling_row n ! k = stirling n k"
STAR_sin_pi_divide_HNatInfinite_approx_pi: fixes n :: "nat star" assumes "n \<in> HNatInfinite" shows "(*f* sin) (hypreal_of_real pi / hypreal_of_hypnat n) * hypreal_of_hypnat n \<approx> hypreal_of_real pi"
std_dual_linear: shows "linear_on S (dual_space S) (*\<^sub>R) (*\<^sub>R) local.std_dual"
verit_comp_simplify_2: fixes a :: "'d" shows "a \<le> a"
joinable_refl: fixes R :: "('b \<times> 'b) set"   and x :: "'b" assumes "refl R" shows "(x, x) \<in> joinable R"
scast_ucast_id: fixes x :: "'b word" shows "scast (ucast x) = x"
sum_coeffs_smult_cmod: fixes v :: "complex"   and p :: "complex poly" shows "norm2 (Polynomial.smult v p) = (cmod v)\<^sup>2 * norm2 p"
d1: shows "column i (fst (QR_decomposition A)) = (1 / norm (\<chi>ia. Gram_Schmidt_matrix A $ ia $ i)) *\<^sub>R column i (Gram_Schmidt_matrix A)"
order_trans_rules_28: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a = b"   and "b < c" shows "a < c"
is_on_stack_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list"   and v :: "'a" assumes "GS_invar s" shows "is_on_stack_impl v s = (v \<in> \<Union> (set (GS.p_\<alpha> s)))"
j_le_\<ff>_le: fixes j :: "nat"   and s :: "'a list" assumes "j \<le> length s" shows "\<ff> s j \<le> j"
id_to_mor_locally_ringed_spaces: shows "morphism_locally_ringed_spaces S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S is_open \<FF> \<rho> b add_str mult_str zero_str one_str (identity S) (\<lambda>U. identity (\<FF> U))"
div_mult_self2: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and euclidean_size :: "'m \<Rightarrow> nat"   and b :: "'m"   and a :: "'m"   and c :: "'m" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times b c)) b = plus c (divide a b)"
tendsto_rabs: fixes f :: "'a \<Rightarrow> real"   and l :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
pos_lc_dom_quad: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "0 < a" shows "\<exists>w. \<forall>y>w. 0 < a * y\<^sup>2 + b * y + c"
merge_def: fixes I :: "'c set"   and J :: "'c set" shows "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
C0_2: shows "C0 \<inter> K \<noteq> {}"
Nil: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "listrelp r [] []"
one_unique: fixes u :: "nat \<Rightarrow> 'a" assumes "u \<in> carrier P"   and "\<And>x. x \<in> carrier P \<Longrightarrow> u \<oplus>\<^bsub>P\<^esub> x = x" shows "u = \<zero>\<^bsub>P\<^esub>"
U: shows "k < length (map the (fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma))"
llast_lmap: fixes xs :: "'b llist"   and f :: "'b \<Rightarrow> 'c" assumes "lfinite xs"   and "\<not> lnull xs" shows "llast (lmap f xs) = f (llast xs)"
add_0_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.monoid_add plus zero" shows "plus zero a = a"
SATAxiom_def: fixes fs :: "formula list" shows "SATAxiom fs = (\<exists>n vs. FAtom Pos n vs \<in> set fs \<and> FAtom Neg n vs \<in> set fs)"
coboundedI: fixes A :: "'t set"   and a :: "'t" assumes "finite A"   and "a \<in> A" shows "a \<preceq>\<^sub>t ord_term_lin.Max A"
root_cond_def: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" shows "root_cond plr x = (case plr of (p, l, r) \<Rightarrow> real_of_rat l \<le> x \<and> x \<le> real_of_rat r \<and> poly (real_of_int_poly p) x = 0)"
abs_lesseq_square: fixes x :: "real"   and y :: "real" shows "(\<bar>x\<bar> \<le> \<bar>y\<bar>) = (x * x \<le> y * y)"
restrict_nodes_def: fixes g :: "'b ugraph"   and V :: "'b set" shows "restrict_nodes g V \<equiv> graph (nodes g \<inter> V) (Restr (edges g) V)"