hcju/isabelleps huggingface dataset:
onorm_le_matrix_component_sum: fixes A :: "((real, 'a) vec, 'b) vec" shows "onorm ((*v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
Abs: shows "Abs = (\<lambda>b. SOME b'. A b b')"
II1: shows "II1 \<in> UNpart1 cl dl P"
lzip_mono: fixes xs :: "'c llist \<times> 'd llist" shows "mono_llist (\<lambda>lzip. case xs of (LNil, ys) \<Rightarrow> LNil | (LCons x xs', LNil) \<Rightarrow> LNil | (LCons x xs', LCons y ys') \<Rightarrow> LCons (x, y) (curry lzip xs' ys'))"
expected_value_mix_pmf_distrib: fixes p :: "'a pmf"   and q :: "'a pmf"   and a :: "real"   and f :: "'a \<Rightarrow> real" assumes "finite (set_pmf p)"   and "finite (set_pmf q)"   and "a \<in> {0<..<1}" shows "measure_pmf.expectation (mix_pmf a p q) f = a * measure_pmf.expectation p f + (1 - a) * measure_pmf.expectation q f"
that: fixes t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t'"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t" shows "thesis"
col_space_QR_decomposition: fixes A :: "((real, 'b) vec, 'a) vec" shows "col_space A = col_space (fst (QR_decomposition A))"
Lxy_snoc: fixes xs :: "'c list"   and x :: "'c"   and S :: "'c set" shows "Lxy (xs @ [x]) S = (if x \<in> S then Lxy xs S @ [x] else Lxy xs S)"
presWls_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'f, 'sort, 'g, 'h, 'c, 'i, 'j) Recursion.model_scheme" shows "presWls h MOD \<equiv> \<forall>s X. wls s X \<longrightarrow> gWls MOD s (h X)"
alphabet\<^sub>F_insortTree: fixes t :: "'a tree"   and ts :: "'a tree list" shows "alphabet\<^sub>F (insortTree t ts) = alphabet t \<union> alphabet\<^sub>F ts"
vec_to_iarray_least_non_zero_position_of_vector_from_index'': fixes j :: "'e"   and i :: "'f"   and A :: "(('d, 'e) vec, 'f) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat j, vec_to_iarray (row i A))" shows "least_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (mod_type_class.to_nat j) = mod_type_class.to_nat (LEAST n. A $ i $ n \<noteq> (0::'d) \<and> j \<le> n)"
dvd_dict: shows "(dvd) \<equiv> dvd.dvd (*)"
X_def: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.X = {x. (\<forall>i. 0 $h i \<le> x $h i) \<and> x \<noteq> 0}"
inf_ccSUP: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and a :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_distrib_lattice Inf Sup inf less_eq less sup bot top"   and "countable B" shows "inf a (Sup (f ` B)) = Sup ((\<lambda>b. inf a (f b)) ` B)"
eq1: shows "coeff (reduce_fps_poly F) (degree F) = (1::'a)"
substT_substT: fixes i :: "nat"   and j :: "nat"   and T :: "POPLmark.type"   and V :: "POPLmark.type"   and U :: "POPLmark.type" assumes "i \<le> j" shows "T[Suc j \<mapsto>\<^sub>\<tau> V]\<^sub>\<tau>[i \<mapsto>\<^sub>\<tau> U[j - i \<mapsto>\<^sub>\<tau> V]\<^sub>\<tau>]\<^sub>\<tau> = T[i \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau>[j \<mapsto>\<^sub>\<tau> V]\<^sub>\<tau>"
the_least_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "\<mu> f = (THE x. is_least_fixpoint f x)"
closure_contains_Sup: fixes S :: "real set" assumes "S \<noteq> {}"   and "bdd_above S" shows "Sup S \<in> closure S"
NSdifferentiable_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
select_convs_3: fixes id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "nhid \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = nhid"
lunit'_naturality: fixes f :: "'d" assumes "D.arr f" shows "(\<I>\<^sub>D \<otimes>\<^sub>D f) \<cdot>\<^sub>D D.lunit' (D.dom f) = D.lunit' (D.cod f) \<cdot>\<^sub>D f"
relH_array: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap"   and r :: "'b array" assumes "relH as h h'"   and "addr_of_array r \<in> as" shows "Array.get h r = Array.get h' r"
intros_3: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and R :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), R:*cast(m), \<sigma>', s') \<in> S"   and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j" shows "((\<sigma>, s), \<tau>, \<sigma>', s') \<in> ocnet_sos S"
tendsto_bot: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open a) \<bottom>"
i: fixes i :: "nat"   and n :: "nat" shows "e (i, n) * indicator {0} i = e (i, n) * indicator V (i, n)"
base_RedBlack_paths: assumes "fst (root (red prb)) = init (black prb)"   and "edges (red prb) = {}"   and "subs prb = {}"   and "confs prb (root (red prb)) = init_conf prb"   and "marked prb = (\<lambda>rv. False)"   and "strengthenings prb = (\<lambda>rv \<sigma>. True)" shows "RedBlack_paths prb = subpaths_from (black prb) (init (black prb))"
S2: shows "c_assoc_value ls (c_pair (c_fst (c_snd (c_fst key))) (c_snd key)) = univ_for_pr (c_pair (c_fst (c_snd (c_fst key))) (c_snd key))"
ivl_subset_existence_ivl: fixes t :: "real"   and t0 :: "real"   and x0 :: "'c" assumes "t - t0 \<in> existence_ivl0 x0" shows "{t0..t} \<subseteq> (+) t0 ` existence_ivl0 x0"
hyps: assumes "mod_type_class.to_nat i < k"   and "\<not> is_zero_row i A"   and "k \<le> nrows A" shows "Hermite_of_upt_row_i A k ass res $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range ass"
stream_space_singleI: fixes X :: "'c \<Rightarrow> 'a stream"   and r :: "'a stream" assumes "\<exists>U. U \<in> sets (stream_space borel) \<and> U \<inter> range X = {r}" shows "stream_space_single X r \<in> sets (stream_space borel) \<and> stream_space_single X r \<inter> range X = {r}"
abs_eq: fixes x :: "unit \<Rightarrow> 'a" shows "delay x = Abs_lazy (x ())"
R: fixes P :: "assn"   and s :: "nat \<times> nat"   and bi :: "nat \<times> nat \<Rightarrow> bool Heap"   and f :: "nat \<times> nat \<Rightarrow> (nat \<times> nat) Heap" assumes "wf (measure (\<lambda>(l, h). h - l))"   and "P \<Longrightarrow>\<^sub>A (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (h < n \<longrightarrow> p \<le> snd (xs ! h))))"   and "\<And>s. <(case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (h < n \<longrightarrow> p \<le> snd (xs ! h)))) * true> bi s <\<lambda>r. (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (h < n \<longrightarrow> p \<le> snd (xs ! h)))) * \<up> (r = (case s of (l, h) \<Rightarrow> l < h))>\<^sub>t"   and "\<And>s. case s of (l, h) \<Rightarrow> l < h \<Longrightarrow> <(case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (h < n \<longrightarrow> p \<le> snd (xs ! h)))) * true> f s <\<lambda>r. (case r of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (h < n \<longrightarrow> p \<le> snd (xs ! h)))) * \<up> ((r, s) \<in> measure (\<lambda>(l, h). h - l))>\<^sub>t"   and "\<And>s. \<not> (case s of (l, h) \<Rightarrow> l < h) \<Longrightarrow> (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (h < n \<longrightarrow> p \<le> snd (xs ! h)))) \<Longrightarrow>\<^sub>A (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (l < n \<longrightarrow> p \<le> snd (xs ! l))))" shows "<P> heap_WHILET bi f s <\<lambda>r. case r of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> n \<and> (\<forall>j<l. snd (xs ! j) < p) \<and> (l < n \<longrightarrow> p \<le> snd (xs ! l)))>\<^sub>t"
OAlist_tc_prod_ord_alt: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('b, 'c) oalist_tc"   and ys :: "('b, 'd) oalist_tc" shows "OAlist_tc_prod_ord P xs ys = (\<forall>k\<in>fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys). P k (OAlist_tc_lookup xs k) (OAlist_tc_lookup ys k))"
r_neq_def: shows "r_neq \<equiv> Cn 2 r_not [r_eq]"
sgn_bernoulli: fixes n :: "nat" shows "sgn (bernoulli n) = (if n = 0 then 1 else if n = 1 then - 1 else if odd n then 0 else (- 1) ^ Suc (n div 2))"
min_in_range: fixes k :: "nat"   and R :: "'a set" assumes "0 < k" shows "min_arg R k \<in> {1..k}"
zero_less_dist_iff: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open" shows "(0 < dist x y) = (x \<noteq> y)"
elims_2: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "LeftDerivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = ab__" and "xa = []" and "xb = b__" and "ab__ = b__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. LeftDerives1 a (fst d) (snd d) x \<and> LeftDerivation x D b"
lim_decreasing_cl: fixes f :: "nat \<Rightarrow> 'b" assumes "\<And>n m. m \<le> n \<Longrightarrow> f n \<le> f m" obtains l :: "'b" where "f \<longlonglongrightarrow> l"
Gets_imp_knows_Spy_parts: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> bankerb_gets" shows "X \<in> parts (knows Spy evs)"
mem_set_list_relE1: fixes xs :: "'d list"   and ys :: "'e list"   and R :: "('d \<times> 'e) set"   and x :: "'d" assumes "(xs, ys) \<in> \<langle>R\<rangle>list_rel"   and "x \<in> set xs" obtains y :: "'e" where "y \<in> set ys" and "(x, y) \<in> R"
A_in_sets: fixes i :: "nat"   and j :: "nat" shows "A i \<inter> Q j \<in> sets M"
simp_filter: fixes x :: "'c Filters.filter" shows "ord_class.filter (Filters.filter.Rep_filter x)"
splitAt_dist_ram: fixes vs :: "'c list"   and a :: "'c list"   and ram :: "'c"   and b :: "'c list" assumes "distinct vs"   and "vs = a @ ram # b" shows "(a, b) = splitAt ram vs"
chain_mono: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist" assumes "\<forall>x y. R x y \<longrightarrow> R' x y"   and "chain R xs" shows "chain R' xs"
nth_list_update: fixes i :: "nat"   and xs :: "'g list"   and x :: "'g"   and j :: "nat" assumes "i < length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
unsigned_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero 1 = one"
foldl_assoc: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" shows "foldl (+) (x + y) zs = x + foldl (+) y zs"
beta: fixes T :: "Lambda_Example.type"   and s :: "dB"   and t :: "dB" shows "Abs T s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"
set_drop_auxlist: fixes x :: "nat \<times> event_data option list set \<times> event_data option list set" assumes "x \<in> set (drop (length done) auxlist)" shows "\<not> check_before I nt x"
norm_assertion_simps_2: fixes a :: "assn" shows "a * emp = a"
negation_implies: fixes y :: "'a"   and x :: "'a" shows "- (y - x) = - - x \<squnion> - y"
infinite_solutions_no_homogeneous_imp_dim_solution_set_not_zero_imp: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "infinite (solution_set A b)" shows "0 < vec.dim (solution_set A 0)"
convex_hull_explicit: fixes p :: "'b set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
nonzero_H_rcoset_reps_def: shows "nonzero_H_rcoset_reps \<equiv> SOME hs. Supgroup.is_rcoset_replist G ((0::'g) # hs)"
AF_DF: fixes p :: "nat"   and q :: "nat"   and k :: "nat" assumes "p < q" shows "AF k p \<lless> DF k q"
Domain_Id_on: fixes A :: "'b set" shows "Domain (Id_on A) = A"
degree_def: fixes v :: "'a"   and g :: "('a, 'b) graph" shows "degree v g \<equiv> card {e \<in> edges g. fst e = v}"
doctor_optimal_match_optimal: fixes ds :: "'b set"   and Y :: "'a set"   and x :: "'a"   and X :: "'a set" assumes "TruePref_tax.doctor_optimal_match ds Y"   and "x \<in> X"   and "TruePref_tax.stable_on ds X" shows "\<exists>y\<in>Y. (x, y) \<in> Pd'_tax (Xd x)"
vsubsts_def: fixes n :: "nat"   and vs' :: "nat list"   and vs :: "nat list" shows "vsubsts n vs' vs = map (\<lambda>v. if v < n - length vs' then v else if v < n then vs' ! (n - v - 1) + (n - length vs') else v - length vs') vs"
div_diff1_eq: fixes b :: "nat"   and a :: "nat"   and m :: "nat" shows "(b - a) div m = b div m - a div m - (m + a mod m - Suc (b mod m)) div m"
gh: fixes x :: "'c" assumes "x \<in> A" shows "g x = h x"
assms_2: shows "b \<leftrightarrow>\<^sup>* c"
selsort_eq_empty_iff: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "(linorder_list0.selsort le xs = []) = (xs = [])"
inv: shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"
K_def: fixes C :: "acom"   and LVQ :: "char list set"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "K C LVQ Q \<equiv> \<forall>l s1 s2. s1 = s2 on qdeps C LVQ \<longrightarrow> pre C Q l s1 = pre C Q l s2"
del_min': fixes h :: "'b heap" assumes "is_root' h" shows "hom (del_min' h) = Pairing_Heap_Tree.del_min (hom h)"
Cons_in_shuffles_iff: fixes z :: "'b"   and zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" shows "(z # zs \<in> shuffles xs ys) = (xs \<noteq> [] \<and> hd xs = z \<and> zs \<in> shuffles (tl xs) ys \<or> ys \<noteq> [] \<and> hd ys = z \<and> zs \<in> shuffles xs (tl ys))"
map_ann_hoare_def: fixes Ts :: "('a ann_com option \<times> 'a set) list" shows "[\<turnstile>] Ts \<equiv> \<forall>i<length Ts. \<exists>c q. Ts ! i = (Some c, q) \<and> \<turnstile> c  q"
simps_1: fixes l :: "nat"   and u :: "nat" shows "asubst\<^sub>2 l u (Less 0 0) = FalseF"
stack_is_tree_path: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s"   and "stack s \<noteq> []" shows "\<exists>v0\<in>g_V0 G. path (tree_edges s) v0 (rev (tl (stack s))) (hd (stack s))"
fps_zero_nth: fixes n :: "nat" shows "0 $ n = (0::'b)"
prod_ballI: fixes AxB :: "('a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. (a, b) \<in> AxB \<Longrightarrow> P a b" shows "\<forall>(a, b)\<in>AxB. P a b"
effective_redundancy_criterion_axioms_def: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" shows "effective_redundancy_criterion_axioms \<Gamma> Rf Ri \<equiv> \<forall>\<gamma> N. \<gamma> \<in> \<Gamma> \<longrightarrow> concl_of \<gamma> \<in> N \<union> Rf N \<longrightarrow> \<gamma> \<in> Ri N"
NotE: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> \<^bold>\<not> F"   and "\<Gamma> \<turnstile> F" shows "\<Gamma> \<turnstile> \<bottom>"
euclidean_all_zero_iff: fixes x :: "'a" shows "(\<forall>u\<in>Basis. x \<bullet> u = 0) = (x = (0::'a))"
bigo_abs_powr_iff: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "0 < p" shows "((\<lambda>x. \<bar>f x\<bar> powr p) \<in> O[F](\<lambda>x. \<bar>g x\<bar> powr p)) = (f \<in> O[F](g))"
I: assumes "finite (\<Union>p\<in>S. (\<lambda>r. ((r, p), the (v_hist r p))) ` {0..<r})" shows "finite vs"
\<Psi>_def: fixes n :: "nat" shows "\<Psi> n = (\<forall>\<^sub>1 \<not> (\<forall>\<^sub>1 \<not> \<not> x\<^sub>0 + 1 < x\<^sub>0 \<or> foldr (\<lambda>i \<phi> n. \<not> x\<^sub>n \<in> X\<^bsub>i\<^esub> \<or> \<phi> n) [0..<n] (\<lambda>m. x\<^sub>m \<in> X\<^bsub>n\<^esub>) 0) \<or> foldr (\<lambda>i \<phi> n. \<not> (\<forall>\<^sub>1 \<not> \<not> x\<^sub>n + 1 < x\<^sub>0 \<or> x\<^sub>0 \<in> X\<^bsub>i\<^esub>) \<or> \<phi> n) [0..<n] (\<lambda>na. \<forall>\<^sub>1 \<not> \<not> x\<^sub>na + 1 < x\<^sub>0 \<or> x\<^sub>0 \<in> X\<^bsub>n\<^esub>) 0)"
assms_2: shows "\<S>' = update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"
uniqueness_of_induced_hom: fixes T :: "'a freeword set permutation \<Rightarrow> 'b" assumes "GroupHom (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) T"   and "\<forall>s\<in>S. T (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {Abs_freeletter s})) = f s" shows "T = F"
nvC_Fax: fixes c :: "(('fsym, 'tp) efsym, 'b) lit list" assumes "c \<in> Fax" shows "nvC c = {}"
row_mat_of_rows_list: fixes nc :: "nat"   and rs :: "'b list list"   and i :: "nat" assumes "list_all (\<lambda>r. length r = nc) rs"   and "i < length rs" shows "row (mat_of_rows_list nc rs) i = vec_of_list (rs ! i)"
indspaceD_supp': fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set"   and x :: "('a, 'b) aezfun" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V"   and "x \<notin> FH" shows "f x = (0::'c)"
permutation_inverse_works_1: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "inv' p \<circ> p = id"
reduced_normif: fixes env :: "('d, bool) mapping"   and t :: "'d ifex"   and t1 :: "'d ifex"   and t2 :: "'d ifex" shows "reduced (normif env t t1 t2) (Mapping.keys env)"
Cons_eq_appendI: fixes x :: "'b"   and xs1 :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
BrnFT_notIn_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n" shows "BrnFT cl dl \<notin> UNpart1 cl dl P"
VSagree_def: fixes \<nu> :: "(real, 'd) vec"   and \<nu>' :: "(real, 'd) vec"   and V :: "'d set" shows "VSagree \<nu> \<nu>' V = (\<forall>i\<in>V. \<nu> $ i = \<nu>' $ i)"
M_M: fixes x :: "int" shows "M (M x) = M x"
compatible_def: fixes x :: "'a"   and y :: "'a" shows "compatible x y = (\<exists>z. {x, y} <<| z)"
\<phi>_in_hhom: shows "\<guillemotleft>\<phi> : src f \<rightarrow> trg r\<guillemotright>"
osum_minus_Id: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Total r"   and "\<not> r \<subseteq> Id"   and "Total r'"   and "\<not> r' \<subseteq> Id" shows "r +o r' - Id \<subseteq> (r - Id) +o (r' - Id)"
App_1: fixes b :: "name"   and ba :: "Lambda_Z.term" shows "subst b ba\<^sup>\<bullet> t\<^sub>1\<^sup>\<bullet> \<rightarrow>\<^sub>\<beta>\<^sup>* (subst b ba t\<^sub>1)\<^sup>\<bullet>"
takeWhile_dropWhile_id: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "takeWhile P xs @ dropWhile P xs = xs"
Some_eq_map_of_iff: fixes xys :: "('c \<times> 'd) list"   and y :: "'d"   and x :: "'c" assumes "distinct (map fst xys)" shows "(Some y = table_of xys x) = ((x, y) \<in> set xys)"
INF_top_conv_2: 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 B :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(top = Inf (B ` A)) = (\<forall>x\<in>A. B x = top)"
suffix_zero: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "\<sigma> |\<^sub>s 0 = \<sigma>"
uniform_complement_block_size: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and bl :: "'b set" assumes "block_design point_set block_collection u_block_size"   and "bl \<in># incidence_system.complement_blocks point_set block_collection" shows "int (card bl) = int (card point_set) - u_block_size"
adjoint_consts: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real"   and f :: "char" shows "Consts (USubst.adjoint \<sigma> I \<omega>) f = term_sem I (case SConst \<sigma> f of None \<Rightarrow> Const f | Some r \<Rightarrow> r) \<omega>"
aless_le_not_le: fixes w :: "ant"   and z :: "ant" shows "(w < z) = (w \<le> z \<and> \<not> z \<le> w)"
perp_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong B A D C"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) A.map (J_C.Map d) (J_C.Map x)"
nxt_not_possible: fixes t :: "'b ttree"   and x :: "'b" assumes "\<not> possible t x" shows "nxt t x = TTree.empty"
id: shows "prod_list qs = map_poly complex_of_real (prod_list rs)"
f2: shows "rk {A', B', A, B, P} \<le> 3"
inj_on_DirProd_carrier: fixes G :: "('a, 'b) monoid_scheme"   and Gs :: "('a, 'c) monoid_scheme list" shows "inj_on (\<lambda>(g, gs). g # gs) (carrier (G \<times>\<times> DirProd_list Gs))"
f_Max\<^sub>0_correct_1: fixes m :: "nat"   and T :: "nat \<Rightarrow> nat" shows "\<forall>x\<in>{1..m}. T x \<le> f_Max\<^sub>0 T m"
lunit'_simps_5: fixes f :: "'a" assumes "ide f" shows "cod \<l>\<^sup>-\<^sup>1[f] = trg f \<star> f"
valid_path_equiv_valid_chain_list: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "chain_subdiv_path \<gamma> one_chain"   and "boundary_chain one_chain"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>" shows "valid_path \<gamma>"
zero_eq_add_iff_both_eq_0: fixes x :: "'c multiset"   and y :: "'c multiset" shows "({#} = x + y) = (x = {#} \<and> y = {#})"
cos_exp_eq: fixes z :: "complex" shows "cos z = (exp (\<i> * z) + exp (- (\<i> * z))) / 2"
represents_imp_degree: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "Polynomial.degree p \<noteq> 0"
self_finance_updated_ind: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market"   and asset :: "'b"   and v :: "real"   and n :: "nat"   and w :: "'c" assumes "portfolio pf"   and "\<forall>n w. prices Mkt asset n w \<noteq> 0" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = cls_val_process Mkt pf (Suc n) w + prices Mkt asset (Suc n) w / prices Mkt asset n w * (val_process Mkt (self_finance Mkt v pf asset) n w - val_process Mkt pf n w)"