hcju/isabelleps huggingface dataset:
wf_trm_subst_rangeI: fixes \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term" assumes "\<And>x. wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
cop_F_def: fixes ds :: "'d set"   and XH :: "'x set" shows "cop_F ds XH = XH \<union> CD_on ds (- Rf CH XH)"
LIMSEQ_realpow_zero: fixes x :: "real" assumes "0 \<le> x"   and "x < 1" shows "(^) x \<longlonglongrightarrow> 0"
preprocess'_sat: fixes v :: "nat \<Rightarrow> QDelta"   and s :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" assumes "v \<Turnstile>\<^sub>a\<^sub>s (snd ` set (Atoms (preprocess' s start)))"   and "v \<Turnstile>\<^sub>t Tableau (preprocess' s start)"   and "set (UnsatIndices (preprocess' s start)) = {}" shows "v \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  (snd ` set s)"
summable': fixes A :: "nat set" shows "f abs_summable_on A"
reduced_def: fixes n :: "nat"   and fs :: "'a vec list"   and \<alpha> :: "'a"   and k :: "nat" shows "gram_schmidt_fs.reduced n fs \<alpha> k = (gram_schmidt_fs.weakly_reduced n fs \<alpha> k \<and> (\<forall>i j. i < k \<longrightarrow> j < i \<longrightarrow> \<bar>gram_schmidt_fs.\<mu> n fs i j\<bar> \<le> (1::'a) / (2::'a)))"
artanh_def: fixes x :: "'a" shows "artanh x = ln (((1::'a) + x) / ((1::'a) - x)) / (2::'a)"
subdg_refl: fixes \<alpha> :: "V"   and \<AA> :: "V" assumes "digraph \<alpha> \<AA>" shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"
min_union_def: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<union>\<^sub>m B = min_set (A \<union> B)"
zero_less_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (numeral.numeral one plus n)"
nodes_sim: fixes a :: "'a"   and b :: "'b" assumes "a \<in> a.V"   and "(a, b) \<in> R" shows "b \<in> b.V"
simps_2: fixes f :: "'a \<Rightarrow> 'b"   and ys :: "'a list" shows "Common.merge f [] ys = ys"
bA: shows "b \<in> local.underS (local.supr A)"
get_root_node_pure: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs" shows "pure (get_root_node ptr) h"
reflexive: fixes P :: "pi" shows "P \<simeq>\<^sup>s P"
simps: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b stream" assumes "class.order less_eq less" shows "order.sdescending less_eq a = (\<exists>aa b w. a = aa ## b ## w \<and> less_eq b aa \<and> order.sdescending less_eq (b ## w))"
comp2_conv: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'd" shows "f1 \<circ> f2 = (\<lambda>x. f1 (f2 x))"
typeableFreevars: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e :: "expr"   and T :: "secLevel" assumes "\<Gamma> \<turnstile> e : T" shows "fv e \<subseteq> dom \<Gamma>"
simps_2: fixes c :: "nat \<Rightarrow> ('a, 'b, 'c) com"   and p :: "'b"   and i :: "nat" shows "seq c p (Suc i) = subst_redex (seq c p i) (c i)"
pap: shows "path_image p \<inter> path_image (linepath (a - complex_of_real e) (a + complex_of_real e)) \<subseteq> {pathstart p, a - complex_of_real e}"
dotprod_is_0: fixes x :: "nat list"   and y :: "nat list" assumes "length x = length y" shows "(x \<bullet> y = 0) = (\<forall>i<length y. x ! i = 0 \<or> y ! i = 0)"
ub_ln_1: fixes y :: "float"   and prec :: "nat"   and x :: "float" assumes "Some y = ub_ln prec x" shows "ln (real_of_float x) \<le> real_of_float y"
midpoint_eq_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(midpoint a b = c) = (a + b = c + c)"
List_app: fixes h :: "'c \<Rightarrow> 'c"   and x :: "'c"   and as :: "'c list"   and bs :: "'c list" shows "List h x (as @ bs) = (\<exists>y. Path h x as y \<and> List h y bs)"
fspec_invar1: fixes aa :: "'b"   and b :: "'a set"   and a :: "'a"   and z :: "'b nres" assumes "I aa b"   and "a \<in> b"   and "SPEC (\<lambda>xb. c aa xb \<le> SPEC (\<lambda>r. I r (b - {a}))) \<le> z" shows "f a \<le> z"
tkd: fixes xs :: "int list" shows "decr (take_decr xs)"
simps_2: fixes x :: "'b set" shows "codomain.min_gallery [x] = codomain.chamber x"
attractor_def: fixes p :: "Player"   and W :: "'a set" shows "G'.attractor p W = lfp (\<lambda>S. W \<union> S \<union> G'.directly_attracted p S)"
case_two_imp_not_learn_lim_V: fixes i :: "nat" assumes "case_two i" shows "\<not> learn_lim \<phi> (V_bclim i) (\<sigma> i)"
inf_compl_bot_right: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (y \<sqinter> - x) = \<bottom>"
fls_lr_inverse_eq0_imp_starting0_2: fixes f :: "'a fls"   and x :: "'a" assumes "fls_right_inverse f x = 0" shows "x = (0::'a)"
pure_vec_def: fixes x :: "'c" shows "pure_vec x = (\<chi>_. x)"
map_bind_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'f spmf"   and g :: "'f \<Rightarrow> 'e spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
divide_le_eq_1_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less_eq (divide b a) one = less_eq a b"
factorize_polys_square_free_prod_list: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list" assumes "factorize_polys ps = (fs, data)" shows "square_free (prod_list fs)"
right_lang_def: fixes q :: "hf" shows "Accessible.right_lang q \<equiv> {u. Accessible.nextl q u \<in> final Accessible_dfa}"
simps_2: fixes cmp :: "'b comparator"   and y :: "'b"   and x :: "'b"   and xs :: "'b list" shows "Sorting_Algorithms.insort cmp y (x # xs) = (if compare cmp y x \<noteq> Greater then y # x # xs else x # Sorting_Algorithms.insort cmp y xs)"
HBL1_iff_prv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (PP \<langle>\<phi>\<rangle>) = prv \<phi>"
preserves_adjoint_pair: fixes f :: "'a"   and g :: "'a" assumes "adjoint_pair f g" shows "S.adjoint_pair (S.UP f) (S.UP g)"
wset_final_okD: fixes final :: "'d \<Rightarrow> bool"   and ws :: "'e \<Rightarrow> 'f wait_set_status option"   and ts :: "'e \<Rightarrow> ('d \<times> 'g \<Rightarrow>f nat) option"   and t :: "'e"   and w :: "'f wait_set_status" assumes "final_thread.wset_final_ok final ws ts"   and "ws t = \<lfloor>w\<rfloor>" shows "\<exists>x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x"
SA_refl_TVar: fixes \<Gamma> :: "binding list"   and X :: "tyvrs" assumes "\<turnstile> \<Gamma> ok"   and "X \<in> ty_dom \<Gamma>" shows "\<Gamma>\<turnstile>Tvar X<:Tvar X"
inf_make_untimed_def: fixes s :: "nat \<Rightarrow> 'd list" shows "inf_make_untimed s \<equiv> inf_make_untimed1 s"
trivial_factor_iso: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "the_elem \<in> Group.iso (G Mod {\<one>\<^bsub>G\<^esub>}) G"
trie_all_eq_set_of_trie: fixes t :: "('a, 'b list) trie"   and P :: "'b \<Rightarrow> bool" assumes "invar_trie t" shows "all_trie (list_all P) t = (\<forall>v\<in>set_tries t. P v)"
intros_4: fixes \<tau> :: "'a OCL_Types.type" shows "element_type (Bag \<tau>) \<tau>"
coneToFun_funToCone: fixes F :: "'j \<Rightarrow> 's" assumes "F \<in> Pi I (S.set \<circ> D) \<inter> S.extensional' I" shows "coneToFun (funToCone F) = F"
pdevs_val_pdevs_domain: fixes e :: "nat \<Rightarrow> real"   and X :: "'a pdevs" shows "pdevs_val e X = (\<Sum>i\<in>pdevs_domain X. e i *\<^sub>R pdevs_apply X i)"
order_consistent_sym: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "order_consistent r s" shows "order_consistent s r"
lstrict_prefix_def: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lstrict_prefix xs ys \<equiv> lprefix xs ys \<and> xs \<noteq> ys"
simps: fixes a1 :: "fmla set"   and a2 :: "fmla" shows "a1 \<turnstile> a2 = ((\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> H) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> Q_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> boolean_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> equality_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> special_axioms) \<or> (\<exists>H A B H'. a1 = H \<union> H' \<and> a2 = B \<and> H \<turnstile> A IMP B \<and> H' \<turnstile> A) \<or> (\<exists>H A B i. a1 = H \<and> a2 = exi i A IMP B \<and> H \<turnstile> A IMP B \<and> atom i \<sharp> B \<and> (\<forall>C\<in>H. atom i \<sharp> C)))"
nonbossy_Pd: fixes ds :: "'b set"   and \<phi> :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('c \<Rightarrow> 'a set \<Rightarrow> 'a set) \<Rightarrow> 'b set \<Rightarrow> 'a set"   and Pd' :: "'b \<Rightarrow> ('a \<times> 'a) set"   and Ch' :: "'c \<Rightarrow> 'a set \<Rightarrow> 'a set"   and d :: "'b"   and Pd'' :: "('a \<times> 'a) set" assumes "Bossy''.nonbossy ds \<phi>"   and "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd' Ch'"   and "ContractsWithBilateralSubstitutesAndIRC Xd Xh (Pd'(d := Pd'')) Ch'"   and "Bossy''.dX (\<phi> Pd' Ch' ds) d = Bossy''.dX (\<phi> (Pd'(d := Pd'')) Ch' ds) d"   and "d \<in> ds" shows "\<phi> Pd' Ch' ds = \<phi> (Pd'(d := Pd'')) Ch' ds"
field_poly_prod_mset_prime_factorization: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "\<Prod>\<^sub># (field_poly.prime_factorization p) = smult (inverse (lead_coeff p)) p"
sel_2: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "value \<langle>x21, x22, x23\<rangle> = x22"
d: shows "rational_proper_factor p = (if degree p \<le> 1 then None else if degree p = 2 then case rat_roots2 p of [] \<Rightarrow> None | x # xs \<Rightarrow> Some [:- x, 1:] else if degree p = 3 then case rational_root_test p of None \<Rightarrow> None | Some x \<Rightarrow> Some [:- x, 1:] else kronecker_factorization_rat p)"
Suc_1: fixes i :: "nat" assumes "k = j - i"   and "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<forall>x. x \<notin> tvars (\<T> (l ! i)) \<longrightarrow> Mapping.lookup (\<V> (l ! i)) x = Mapping.lookup (\<V> (l ! j)) x"
dord_pm_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less"   and "\<lbrakk>ord s t; ord t u\<rbrakk> \<Longrightarrow> ord s u"   and "linorder.dord_pm ord s t"   and "linorder.dord_pm ord t u" shows "linorder.dord_pm ord s u"
construct_witness_def: fixes Q :: "'e \<Rightarrow> 'd tree option"   and r :: "('e, 'd) ta_rule" shows "construct_witness Q r \<equiv> NODE (rhsl r) (map (\<lambda>q. the (Q q)) (rhsq r))"
poly_x_mult_y_def: fixes p :: "'a poly" shows "poly_x_mult_y p \<equiv> \<Sum>i\<le>degree p. monom (monom (coeff p i) i) i"
sample_uniform_one_time_pad: fixes c :: "'a" assumes "c \<in> carrier \<G>" shows "map_spmf (\<lambda>x. \<^bold>g [^] x \<otimes> c) (sample_uniform (order \<G>)) = map_spmf (([^]) \<^bold>g) (sample_uniform (order \<G>))"
rtranclp_conversep: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*\<inverse>\<inverse>"
trans: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and \<sigma> :: "nat \<Rightarrow> 'e"   and i :: "nat"   and s :: "'c"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and s' :: "'c"   and a :: "'d seq_action" assumes "openproc np onp sr"   and "\<sigma> i = fst (sr s)"   and "\<sigma>' i = fst (sr s')"   and "(s, a, s') \<in> automaton.trans (np i)" shows "((\<sigma>, snd (sr s)), a, \<sigma>', snd (sr s')) \<in> automaton.trans (onp i)"
List_app: fixes h :: "'c \<Rightarrow> 'c ref"   and x :: "'c ref"   and as :: "'c list"   and bs :: "'c list" shows "List h x (as @ bs) = (\<exists>y. Path h x as y \<and> List h y bs)"
sigma_algebra_imp_Dynkin_system: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "sigma_algebra \<Omega> M" shows "Dynkin_system \<Omega> M"
vec_extensionality: fixes v :: "('b, 'a) vec"   and w :: "('b, 'a) vec" assumes "\<And>i. v $ i = w $ i" shows "v = w"
dsn: shows "matrix_sum d f n \<in> carrier_mat d d"
is_sublist_nth1: fixes x :: "'a"   and y :: "'a"   and ls :: "'a list" assumes "is_sublist [x, y] ls" shows "\<exists>i j. i < |ls| \<and> j < |ls| \<and> ls ! i = x \<and> ls ! j = y \<and> Suc i = j"
not_iUntil: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<not> (P t1. t1 \<U> t2 I. Q t2)) = (\<box> t I. Q t \<longrightarrow> (\<diamond> t' I \<down>< t. \<not> P t'))"
s_def: fixes t :: "real" shows "ego2.s t = (if t \<le> 0 then ego.q \<delta> else if t \<le> ego2.t_stop then ego2.p t else ego2.p_max)"
CD_on_Chernoff: fixes ds :: "'b set" shows "Chernoff (CD_on ds)"
indis_trans: fixes s :: "'state"   and s' :: "'state"   and s'' :: "'state" assumes "s \<approx> s'"   and "s' \<approx> s''" shows "s \<approx> s''"
collapse: fixes prefix_match :: "'b prefix_match" shows "PrefixMatch (pfxm_prefix prefix_match) (pfxm_length prefix_match) = prefix_match"
cycleI: fixes xs :: "'a list" assumes "path xs"   and "2 < length xs"   and "last xs \<rightarrow> hd xs" shows "cycle xs"
image_constant: fixes x :: "'b"   and A :: "'b set"   and c :: "'c" assumes "x \<in> A" shows "(\<lambda>x. c) ` A = {c}"
cCons_not_0_eq: fixes x :: "'b"   and xs :: "'b list" assumes "x \<noteq> (0::'b)" shows "cCons x xs = x # xs"
s0_def: shows "s0 \<equiv> \<lparr>init = s0_init, trans = s0_trans, obs = id\<rparr>"
fps_conv_radius_cos: fixes c :: "'a" shows "fps_conv_radius (fps_cos c) = \<infinity>"
continuous_floor: fixes x :: "real" assumes "x \<notin> \<int>" shows "isCont (real_of_int \<circ> floor) x"
CollectD: fixes a :: "'a::type"   and P :: "'a::type \<Rightarrow> bool" assumes "(a::'a::type) \<in> {x::'a::type. (P::'a::type \<Rightarrow> bool) x}" shows "(P::'a::type \<Rightarrow> bool) (a::'a::type)"
add_eq_conv_ex: fixes a :: "'d"   and M :: "'d multiset"   and b :: "'d"   and N :: "'d multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
S2RPCUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RPCNext crCh rmCh rst p]_(r p) \<and> $S2 rmhist p \<longrightarrow> unchanged r p"
summands_equal: fixes a :: "rat vec"   and b :: "rat vec"   and c :: "rat vec"   and d :: "rat vec" assumes "a = b"   and "c = d" shows "a + c = b + d"
Exit_Def_empty: fixes V :: "'var" assumes "V \<in> Def (_Exit_)" shows "False"
assms_2: shows "\<forall>x. P x \<longrightarrow> (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (s n x) (l x) < e)"
factors_mult: fixes fa :: "'a list"   and a :: "'a"   and fb :: "'a list"   and b :: "'a" assumes "factors G fa a"   and "factors G fb b"   and "set fa \<subseteq> carrier G"   and "set fb \<subseteq> carrier G" shows "factors G (fa @ fb) (a \<otimes> b)"
leq_antisym: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and y :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord x y"   and "resumption_ord y x" shows "x = y"
nat_power_add: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.nat_power one times plus zero nat_power"   and "0 < n" shows "nat_power n (plus a b) = times (nat_power n a) (nat_power n b)"
tendsto_within_subset: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d filter"   and x :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "filterlim f l (at x within S)"   and "T \<subseteq> S" shows "filterlim f l (at x within T)"
insert_if: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "finite A" shows "sum g (insert x A) = (if x \<in> A then sum g A else g x + sum g A)"
list_emb_appendD: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "list_emb P (xs @ ys) zs" shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs"
subst_atm_mset_id_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and AA :: "'b multiset" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_atm_mset subst_atm AA id_subst = AA"
sum_phi'_factors: fixes n :: "nat" assumes "0 < n" shows "(\<Sum>d | d dvd n. phi' d) = n"
headn_0: fixes p :: "poly"   and n :: "nat"   and m :: "nat" assumes "isnpolyh p n"   and "m < n" shows "headn p m = p"
\<psi>_def: fixes x :: "'b"   and h :: "'c" shows "F'G'.\<eta>\<epsilon>.\<psi> x h = F'G'.\<epsilon> x \<cdot>\<^sub>B F' h"
Im_stereographic: fixes z :: "complex" shows "Im (2 * z / (1 + z * cnj z)) = 2 * Im z / (1 + (cmod z)\<^sup>2)"
update_im1_def: fixes ba :: "'b list \<times> 'b list"   and x :: "'b" shows "update_im1 ba x = (x # tl (fst ba), snd ba)"
funpow_dist1_least: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "0 < n"   and "n < funpow_dist1 f x y" shows "(f ^^ n) x \<noteq> y"
extended_predist_def: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_predist x y = real_of_ereal (min (esqrt (extended_Gromov_distance x y)) (eexp (ereal (- epsilonG TYPE('a)) * extended_Gromov_product_at basepoint x y)))"
simps_2: fixes N :: "nat"   and fas :: "floatarith list"   and xs :: "nat list"   and ds :: "nat list"   and vs :: "real list"   and n :: "nat" shows "isnFDERIV N fas xs ds vs (Suc n) = (isFDERIV N xs (FDERIV_n_floatariths fas xs (map Var ds) n) vs \<and> isnFDERIV N fas xs ds vs n)"
index_mat_addrow_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_addrow_gen ad mul a k l A $$ (i, j) = (if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
continuous_on_open_Union: fixes S :: "'b set set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>s. s \<in> S \<Longrightarrow> open s"   and "\<And>s. s \<in> S \<Longrightarrow> continuous_on s f" shows "continuous_on (\<Union> S) f"
distinct_append_butlast2: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list" assumes "distinct (map fst xs @ map fst ys)" shows "distinct (map fst xs @ map fst (butlast ys))"
lcond_split: fixes Q :: "'g set \<Rightarrow> bool"   and lp :: "'g set"   and lp' :: "'g set"   and b :: "'h \<Rightarrow> bool"   and s :: "'h" shows "Q (lcond lp lp' b s) = ((b s \<longrightarrow> Q lp) \<and> (\<not> b s \<longrightarrow> Q lp'))"
dis_log_alt_def: fixes \<G> :: "'a cyclic_group" shows "dis_log_alt \<G> \<equiv> 0 < order \<G>"
dghm_ArrMap_app_in_HomCod_Arr: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
timpl_apply_term_def: fixes a :: "'j set"   and b :: "'j set"   and t :: "(('h, 'i, 'j) prot_fun, 'k) Term.term" shows "\<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle> \<equiv> term_variants ((\<lambda>_. [])(Abs a := [Abs b])) t"
f6: shows "B * (1 / det B \<cdot>\<^sub>m adj_mat B) = 1\<^sub>m n"
lex_pm_alt: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "lex_pm s t = (s = t \<or> (\<exists>x. lookup s x < lookup t x \<and> (\<forall>y<x. lookup s y = lookup t y)))"
U_open: fixes p :: "'a" assumes "p \<in> carrier" shows "open (U p)"
pp_surjective: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.surjective (\<ominus> \<ominus> x) = matrix_p_algebra.dense (mtop \<odot> x)"
rewriteR_comp_comp2: fixes g :: "'c \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'c"   and r1 :: "'d \<Rightarrow> 'b"   and r2 :: "'a \<Rightarrow> 'd"   and f :: "'b \<Rightarrow> 'e"   and l :: "'d \<Rightarrow> 'e" assumes "g \<circ> h = r1 \<circ> r2"   and "f \<circ> r1 = l" shows "f \<circ> g \<circ> h = l \<circ> r2"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list"   and a2 :: "nat" assumes "\<And>x xs y ys. P xs ys 0 \<Longrightarrow> P (x # xs) (y # ys) 0"   and "\<And>xs y ys i. P xs ys i \<Longrightarrow> P xs (y # ys) (Suc i)"   and "\<And>ys. P [] ys 0"   and "\<And>v va uv_. P (v # va) [] uv_"   and "\<And>uu_ v. P uu_ [] (Suc v)" shows "P a0 a1 a2"
mat_ext_vars2: fixes A :: "'a mat" assumes "A \<in> carrier_mat K K" shows "mat_extension dims vars2 A = tensor_P (1\<^sub>m N) A"
disj: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm"   and q :: "fm" shows "Ifm vs bs (Parametric_Ferrante_Rackoff.disj p q) = Ifm vs bs (Or p q)"
replicate_eq_replicate: fixes m :: "nat"   and x :: "'d"   and n :: "nat"   and y :: "'d" shows "(x\<^bsup>m\<^esup> = y\<^bsup>n\<^esup>) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
infa: shows "\<forall>m. \<exists>n\<ge>m. x n = a"
jordan_nf_iff_linear_factorization: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(\<exists>n_as. jordan_nf A n_as) = (\<exists>as. char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'c:]))"
nLt_le_conv: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "(key_compare (rep_key_order xa) x y \<noteq> Lt) = le xa y x"
pv: fixes v :: "'a"   and w :: "'a"   and e :: "'a" assumes "vector v" shows "prim_P w v e\<^sup>T * v = bot"