hcju/isabelleps huggingface dataset:
elims: fixes x :: "complex mat"   and xa :: "complex mat"   and xb :: "complex mat \<Rightarrow> complex mat"   and xc :: "nat"   and xd :: "complex mat"   and y :: "complex mat" assumes "denote_while_n_iter x xa xb xc xd = y" obtains M0 :: "complex mat" and M1 :: "complex mat" and DS :: "complex mat \<Rightarrow> complex mat" and \<rho> :: "complex mat" where "x = M0__" and "xa = M1__" and "xb = DS__" and "xc = 0" and "xd = \<rho>'__" and "y = \<rho>'__"   | M0 :: "complex mat" and M1 :: "complex mat" and DS :: "complex mat \<Rightarrow> complex mat" and n :: "nat" and \<rho> :: "complex mat" where "x = M0" and "xa = M1" and "xb = DS" and "xc = Suc n" and "xd = \<rho>" and "y = denote_while_n_iter M0 M1 DS n (DS (M1 * \<rho> * adjoint M1))"
2: shows "\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)"
simps_4: fixes f1 :: "'b"   and f2 :: "'b" shows "(case sort_dir.Right of sort_dir.Left \<Rightarrow> f1 | sort_dir.Right \<Rightarrow> f2) = f2"
Union_mono: fixes A :: "'c set set"   and B :: "'c set set" assumes "A \<subseteq> B" shows "\<Union> A \<subseteq> \<Union> B"
FOREACHc_refine_rcg: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> R (f' x' \<sigma>')" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>C S' c' f' \<sigma>0')"
aff_dim_convex_Int_open: fixes S :: "'c set"   and T :: "'c set" assumes "convex S"   and "open T"   and "S \<inter> T \<noteq> {}" shows "aff_dim (S \<inter> T) = aff_dim S"
elabE_3: fixes te :: "ExVar \<Rightarrow> Ty option"   and x :: "ExVar"   and e :: "Ex"   and ty :: "Ty" assumes "te \<turnstile> fn x => e ===> ty" obtains ty1 :: "Ty" and ty2 :: "Ty" where "ty = ty1 -> ty2" and "te(x \<mapsto> ty1) \<turnstile> e ===> ty2"
B: fixes x :: "nat"   and i :: "nat" shows "(x \<in> merge_t i) = (r x \<in> merge\<^sub>R i)"
A_FactGroup_defs_1: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "G A_Mod H = add_monoid G Mod H"
bangPres: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Rel' :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel'> S"   and "Rel \<subseteq> Rel'" shows "!P \<leadsto><bangRel Rel'> !Q"
lc_factor_def: shows "lc_factor = (\<Prod>q\<in>P. l ^ Polynomial.degree q div Polynomial.lead_coeff q)"
Trans_msg: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "isTrans ev" shows "msgs c i = msgs c' i"
cblinfun_apply_inject: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(cblinfun_apply x = cblinfun_apply y) = (x = y)"
ord0_def: fixes t :: "'a tree23" shows "ord0 t = ord' None t None"
update_by_fun_raw_eq_update_by_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv xs" shows "update_by_fun_raw k f xs = update_by_raw (k, f (lookup_raw xs k)) xs"
cadjoint_def: fixes G :: "'b \<Rightarrow> 'a" shows "G\<^sup>\<dagger> = (SOME F. is_cadjoint F G)"
hyps: fixes A :: "(('a, 'n) vec, 'm) vec"   and P :: "(('a, 'm) vec, 'm) vec" assumes "a \<noteq> i"   and "n < nrows A"   and "mod_type_class.to_nat a \<le> n" shows "fst (row_add_iterate_PA (P, A) n i j) $ a $ b = row_add P a i (- A $ a $ j) $ a $ b"
intro: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "connection G"   and "galois_connection_axioms G" shows "galois_connection G"
uvz: fixes K :: "'M set" assumes "K \<in> \<D>" shows "K = cbox (uf K) (vf K) \<and> box (uf K) (vf K) \<noteq> {} \<and> zf K \<in> S \<and> zf K \<in> cbox (uf K) (vf K) \<and> cbox (uf K) (vf K) \<subseteq> ball (zf K) (R (zf K))"
irrefl_lex_prod: fixes R :: "('a \<times> 'a) set"   and S :: "('c \<times> 'c) set" assumes "irrefl R"   and "irrefl S" shows "irrefl (R <*lex*> S)"
subst_eq_if_eq_vars: fixes \<theta> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "\<And>v. Var v \<cdot> \<theta> = Var v \<cdot> \<sigma>" shows "\<theta> = \<sigma>"
disc_1: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and T :: "('b, 'a) process_term" assumes "isACT sys T" shows "isAction (solution sys T)"
idem: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g" assumes "semilattice f" shows "f a a = a"
\<b>_2: fixes X :: "'x set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" assumes "finite X"   and "(h, U) \<in> set ps"   and "i \<le> card U" shows "poly_deg h < \<b> ps i"
matrix_to_iarray_rank: fixes A :: "(('d, 'e) vec, 'f) vec" shows "rank A = rank_iarray (matrix_to_iarray A)"
Cons_2: shows "\<forall>a\<in>set M. case a of (a, b) \<Rightarrow> SecurityInvariant_complies_formal_def a b"
get_parent_parent_in_heap: fixes h :: "(_) heap"   and child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent" shows "parent |\<in>| object_ptr_kinds h"
from_mregex: shows "from_mregex mr \<phi>s = r"
zip_replicate: fixes i :: "nat"   and x :: "'a"   and j :: "nat"   and y :: "'b" shows "zip (replicate i x) (replicate j y) = replicate (min i j) (x, y)"
simps_2: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "removeLeaf (T v (T va vb vc) E) = (fst (removeLeaf (T va vb vc)), T v (snd (removeLeaf (T va vb vc))) E)"
normalized_n_primitive_if_no_primitive: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool" assumes "normalized_nnf_match m"   and "\<not> has_disc disc m" shows "normalized_n_primitive (disc, sel) f m"
vpa_induct: fixes xs :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and ys :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and P :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list \<Rightarrow> ('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list \<Rightarrow> bool" assumes "CFGExit.valid_path_aux xs ys"   and "\<And>cs. P cs []"   and "\<And>cs a as. \<lbrakk>intra_kind (knd a); CFGExit.valid_path_aux cs as; P cs as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q r p fs. \<lbrakk>knd a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; CFGExit.valid_path_aux (a # cs) as; P (a # cs) as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q p f. \<lbrakk>knd a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; cs = []; CFGExit.valid_path_aux [] as; P [] as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q p f c' cs'. \<lbrakk>knd a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; cs = c' # cs'; CFGExit.valid_path_aux cs' as; a \<in> lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind c'; P cs' as\<rbrakk> \<Longrightarrow> P cs (a # as)" shows "P xs ys"
under_ofilter: fixes a :: "'a" shows "local.ofilter (local.under a)"
simps_2: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and opss :: "'a strips_operator list list" shows "execute_parallel_plan s (ops # opss) = (if are_all_operators_applicable s ops \<and> are_all_operator_effects_consistent ops then execute_parallel_plan (execute_parallel_operator s ops) opss else s)"
Gouezel_Karlsson_Kingman: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u"   and "- \<infinity> < subcocycle_avg_ereal u" shows "AE x in M. \<exists>delta. (\<forall>l. 0 < delta l) \<and> delta \<longlonglongrightarrow> 0 \<and> infinite {n. \<forall>l\<in>{1..n}. - delta l * real l < u n x - u (n - l) ((T ^^ l) x) - real l * subcocycle_lim u x}"
GotoElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and pc :: "int" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (goto pc)" shows "n = Suc 0 \<and> t = (os, S, h) \<and> ll = pc"
slice_extend_set: fixes A :: "'a set"   and y :: "'b" shows "slice (extend_set h A) y = A"
SD_inefficient_support_subset: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and p' :: "'b pmf"   and p :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "\<not> SD_efficient R p'"   and "set_pmf p' \<subseteq> set_pmf p"   and "p \<in> lotteries_on alts" shows "\<not> SD_efficient R p"
nextElem_append: fixes y :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and d :: "'a" assumes "y \<notin> set xs" shows "nextElem (xs @ ys) d y = nextElem ys d y"
cResB: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x"   and "y \<sharp> C"   and "\<And>C. F C P a x P'" shows "F C (<\<nu>y>P) a x (<\<nu>y>P')"
X4_pow: fixes X :: "X4 set" shows "X \<in> set ` set (subseqs [Xd1, Xd1', Xd2, Xd2'])"
cod_tensor: fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c"   and d :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow>\<^sub>C d\<guillemotright>" shows "C.cod (f \<otimes>\<^sub>C g) = b \<otimes>\<^sub>C d"
bind_eq_return_spmf: fixes p :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf"   and x :: "'d" shows "(p \<bind> f = return_spmf x) = ((\<forall>y\<in>set_spmf p. f y = return_spmf x) \<and> lossless_spmf p)"
induct: fixes P :: "'c Tree \<Rightarrow> bool"   and a0 :: "'c Tree" assumes "P E"   and "\<And>v. P (T v E E)"   and "\<And>v va vb vc. P (T va vb vc) \<Longrightarrow> P (T v E (T va vb vc))"   and "\<And>v va vb vc. P (T va vb vc) \<Longrightarrow> P (T v (T va vb vc) E)"   and "\<And>v va vb vc vd ve vf. \<lbrakk>P (T vd ve vf); P (T va vb vc)\<rbrakk> \<Longrightarrow> P (T v (T va vb vc) (T vd ve vf))"   and "\<And>v vd ve vf va vb vc. \<lbrakk>P (T va vb vc); P (T vd ve vf)\<rbrakk> \<Longrightarrow> P (T v (T vd ve vf) (T va vb vc))" shows "P a0"
ide_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "partial_magma C" shows "partial_magma.ide C a \<equiv> C a a \<noteq> partial_magma.null C \<and> (\<forall>f. (C f a \<noteq> partial_magma.null C \<longrightarrow> C f a = f) \<and> (C a f \<noteq> partial_magma.null C \<longrightarrow> C a f = f))"
assms: shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P'"
ewp_super_id1: fixes x :: "'a" assumes "(0::'a) \<noteq> (1::'a)"   and "(1::'a) \<le> x" shows "ewp x"
compxE2_size_convs: fixes n :: "nat"   and e :: "'a expr1"   and d :: "nat" assumes "n \<noteq> 0" shows "compxE2 e n d = shift n (compxE2 e 0 d)"
that: fixes g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "uniform_limit S g f sequentially"   and "\<And>n. polynomial_function (g n)" shows "thesis"
prod_pOp_mem: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme"   and X :: "'h \<Rightarrow> 'i"   and Y :: "'h \<Rightarrow> 'i" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A" shows "prod_pOp I A X Y \<in> carr_prodag I A"
h_def: shows "h = (\<lambda>k x. (\<Sum>n = 1..x. ln (real n) ^ k / real n) - ln (real x) ^ (k + 1) / real (k + 1) - stieltjes_gamma k)"
1: shows "! (x \<squnion> y) \<le> ! x"
kmp2_def: fixes s :: "'b list"   and t :: "'b list" shows "kmp2 s t \<equiv> ASSERT (s \<noteq> []) \<bind> (\<lambda>_. let i = 0; j = 0; pos = None in compute_butlast_\<ff>s s \<bind> (\<lambda>\<ff>s. WHILE\<^sub>T\<^bsup>I_outer s t\<^esup> (\<lambda>(i, j, pos). i + length s \<le> length t \<and> pos = None) (\<lambda>(i, j, pos). ASSERT (i + length s \<le> length t \<and> pos = None) \<bind> (\<lambda>_. WHILE\<^sub>T\<^bsup>I_in_na s t i\<^esup> (\<lambda>(j, pos). t ! (i + j) = s ! j \<and> pos = None) (\<lambda>(j, pos). let j = j + 1 in if j = length s then RETURN (j, Some i) else RETURN (j, None)) (j, pos) \<bind> (\<lambda>(j, pos). if pos = None then ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let i = i + (j - \<ff>s ! j + 1); j = max 0 (\<ff>s ! j - 1) in RETURN (i, j, None)) else RETURN (i, j, Some i)))) (i, j, pos) \<bind> (\<lambda>(uu_, uu_, pos). RETURN pos)))"
nths_upt_eq_take: fixes l :: "'b list"   and n :: "nat" shows "nths l {..<n} = take n l"
span_def: fixes S :: "'c set" shows "w.span S = {w.lincomb a A |a A. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R}"
pmat_extension_def: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list"   and m :: "'a mat" assumes "partial_state2 vars1 vars2" shows "partial_state2.pmat_extension dims vars1 vars2 m = partial_state2.ptensor_mat dims vars1 vars2 m (1\<^sub>m (partial_state2.d2 dims vars2))"
AE_T_max_qsend_time: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and e :: "real" assumes "AE bT in T cf. qsend (cf ## bT) < \<infinity>"   and "0 < e" shows "\<exists>N. T.prob cf {bT \<in> space (T cf). \<not> discrCf ((cf ## bT) !! N)} < e"
ket_vec_index: fixes i :: "nat"   and v :: "complex Matrix.vec" assumes "i < dim_vec v" shows "|v\<rangle> $$ (i, 0) = v $ i"
Reds_preserves_iconf: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es',s',b'\<rangle>"   and "iconfs (shp s) es" shows "iconfs (shp s') es'"
eq_Min_iff: fixes A :: "'c Word.word set"   and m :: "'c Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(m = signed.Min A) = (m \<in> A \<and> (\<forall>a\<in>A. m \<le>s a))"
induct: fixes P :: "'b set \<Rightarrow> bool"   and a0 :: "'b set" assumes "\<And>S. (\<And>x2. \<lbrakk>S \<noteq> {}; enum_set_get_one enum_class.enum S = Some x2\<rbrakk> \<Longrightarrow> P (S - {x2})) \<Longrightarrow> P S" shows "P a0"
one_power2: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.semiring_1 one times plus zero" shows "power.power one times one 2 = one"
span_Int: fixes A :: "'c set"   and B :: "'c set" shows "ideal (ideal A \<inter> ideal B) = ideal A \<inter> ideal B"
clsub2: shows "closed_segment (a - complex_of_real d) (a - complex_of_real kde) \<subseteq> closed_segment (a - complex_of_real d) (a + complex_of_real e)"
basis_criterion: fixes A :: "'a mod_ring poly set" assumes "finite A"   and "A \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})" shows "Berlekamp_subspace.basis A = (\<forall>v. v \<in> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}) \<longrightarrow> (\<exists>!a. a \<in> A \<rightarrow>\<^sub>E carrier class_ring \<and> Berlekamp_subspace.lincomb a A = v))"
induct: fixes P4 :: "(cell \<times> interface) list \<Rightarrow> bool"   and P1 :: "sdiagram \<Rightarrow> bool"   and P2 :: "cell \<Rightarrow> bool"   and P3 :: "cell \<times> interface \<Rightarrow> bool"   and cell :: "cell" assumes "\<And>x. P4 x \<Longrightarrow> P1 (SDiagram x)"   and "\<And>x. P2 (Filler x)"   and "\<And>x1a x2 x3. P2 (Basic x1a x2 x3)"   and "\<And>x1a x2. P1 x2 \<Longrightarrow> P2 (Exists_sdia x1a x2)"   and "\<And>x1a x2 x3 x4. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P2 (Choose_sdia x1a x2 x3 x4)"   and "\<And>x1a x2 x3. P1 x2 \<Longrightarrow> P2 (Loop_sdia x1a x2 x3)"   and "\<And>x1a x2. P2 x1a \<Longrightarrow> P3 (x1a, x2)"   and "P4 []"   and "\<And>x1a x2. \<lbrakk>P3 x1a; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1a # x2)" shows "P2 cell"
stoch_vec_def: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "stoch_vec v = (sum (($h) v) UNIV = (1::'a))"
not_nofail_iff: fixes S :: "'a nres" shows "(\<not> nofail S) = (S = FAIL)"
simps_3: fixes v :: "'b"   and va :: "'b" shows "removeShadowRules1_alternative_rev [DenyAllFromTo v va] = [DenyAllFromTo v va]"
R_lin_independentD_all_scalars: fixes rs :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M"   and "length rs \<le> length ms"   and "R_lin_independent ms"   and "rs \<bullet>\<cdot> ms = (0::'m)" shows "set rs \<subseteq> 0"
simps_5: fixes I :: "('e, 'f, 'd) interp"   and r :: "real"   and v :: "(real, 'd) vec" shows "frechet I (Const r) v = (\<lambda>v'. 0)"
DirProds_finite_imp_finite_groups: fixes G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme"   and I :: "'d set"   and i :: "'d" assumes "finite_group (DirProds G I)"   and "finite I"   and "i \<in> I" shows "finite_group (G i)"
lin_indpt: shows "gs.lin_indpt (set (map of_int_hom.vec_hom fs))"
convergent_imp_bounded: fixes S :: "nat \<Rightarrow> 'c"   and l :: "'c" assumes "S \<longlonglongrightarrow> l" shows "bounded (range S)"
simps_3: fixes x :: "'ubx2" shows "box_operand (OpUbx2 x) = OpDyn (box_ubx2 x)"
coprime_int_iff: fixes m :: "nat"   and n :: "nat" shows "coprime (int m) (int n) = coprime m n"
index_zero_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat" assumes "i < nr"   and "j < nc" shows "0\<^sub>m nr nc $$ (i, j) = (0::'c)"
simps_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and u :: "'a local.term" assumes "monoidal_language C" shows "monoidal_language.Diag C (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u) = (partial_magma.arr C f \<and> monoidal_language.Diag C u \<and> u \<noteq> \<^bold>\<I>)"
sub_which1: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<subseteq> B \<or> B \<subseteq> A"   and "x \<in> A"   and "x \<notin> B" shows "B \<subseteq> A"
plane_ofs_def: fixes sctn :: "real list sctn" shows "plane_ofs sctn = {x. inner_lv_rel x (normal sctn) = pstn sctn}"
uniform_limitD: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set"   and l :: "'b \<Rightarrow> 'c"   and F :: "'a filter"   and e :: "real" assumes "uniform_limit S f l F"   and "0 < e" shows "\<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e"
rel_mono_strong: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b list_R1"   and y :: "'c list_R1"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "rel_list_R1 R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_list_R1 x; yb \<in> set_list_R1 y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_list_R1 Ra x y"
le_by_leofI: fixes m' :: "'b nres"   and m :: "'b nres" assumes "nofail m' \<Longrightarrow> nofail m"   and "m \<le>\<^sub>n m'" shows "m \<le> m'"
zz_z: shows "exi zz (cnj (subst \<tau> (Var zz) out) (subst \<sigma> (Var zz) inp)) = exi z (cnj (subst \<tau> (Var z) out) (subst \<sigma> (Var z) inp))"
assms_2: shows "P \<turnstile> \<langle>e,s\<^sub>1,icheck P C e\<rangle> \<rightarrow>* \<langle>e\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>"
chamber_in_subcomplex: fixes Y :: "'a set set"   and C :: "'a set" assumes "ChamberSubcomplex Y"   and "C \<in> Y"   and "chamber C" shows "SimplicialComplex.maxsimp Y C"
replacefacesAt2_nth1': fixes k :: "nat"   and ns :: "nat list"   and F :: "face list list"   and oldf :: "face"   and newfs :: "face list" assumes "k \<in> set ns"   and "k < |F|"   and "distinct ns" shows "replacefacesAt2 ns oldf newfs F ! k = replace oldf newfs (F ! k)"
RFc: shows "RF\<^sup>\<circ> (TER g) = RF\<^sup>\<circ> (TER f)"
list_of_eucl_def: fixes x :: "'e" shows "list_of_eucl x = map ((\<bullet>) x) Basis_list"
iEx_subset: fixes A :: "nat set"   and B :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<diamond> t A. P t" shows "\<diamond> t B. P t"
pz_def: shows "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}"
ipresIGWlsAbs_def: fixes hA :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'h, 'q) model_scheme"   and MOD' :: "('r, 's, 'l, 'm, 't, 'u, 'v, 'i, 'w) model_scheme" shows "ipresIGWlsAbs hA MOD MOD' \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD' (us, s) (hA A)"
MkArr_Map: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f" shows "J_C.MkArr (J_C.Dom f) (J_C.Cod f) (J_C.Map f) = f"
rec_newleft_def: shows "rec_newleft = (let g0 = Cn 3 rec_newleft0 [recf.id 3 0, recf.id 3 1]; g1 = Cn 3 rec_newleft2 [recf.id 3 0, recf.id 3 1]; g2 = Cn 3 rec_newleft3 [recf.id 3 0, recf.id 3 1]; g3 = recf.id 3 0; r0 = Cn 3 rec_disj [Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 0) [recf.id 3 0]], Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 1) [recf.id 3 0]]]; r1 = Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 2) [recf.id 3 0]]; r2 = Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 3) [recf.id 3 0]]; r3 = Cn 3 rec_less [Cn 3 (constn 3) [recf.id 3 0], recf.id 3 2]; gs = [g0, g1, g2, g3]; rs = [r0, r1, r2, r3] in rec_embranch (zip gs rs))"
flow_network_axioms_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" shows "flow_network_axioms \<Delta> g \<equiv> flow \<Delta> g \<and> value_flow \<Delta> g \<noteq> \<top> \<and> \<^bold>V\<^bsub>\<Delta>\<^esub> - {source \<Delta>, sink \<Delta>} \<noteq> {}"
type_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile> t : T"   and "\<turnstile> (\<Delta> @ B # \<Gamma>) ok" shows "\<Delta> @ B # \<Gamma> \<turnstile> t : T"
mem_UnitsI: fixes u :: "'a \<Rightarrow> 'a" assumes "invertible u"   and "u \<in> S \<rightarrow>\<^sub>E S" shows "u \<in> Units"
Deduct_with_False_Disj_Rename_axioms: shows "Deduct_with_False_Disj_Rename var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"
bisim1Sync11: fixes P :: "'b prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, Suc (Suc (length (compE2 e1))), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
simps: fixes A_pos :: "nat set list"   and L_pos :: "'b option list set list"   and A_neg :: "nat set list"   and L_neg :: "'b option list set list" shows "dominate_False A_pos L_pos A_neg L_neg = (case find_sub_False A_pos A_neg of None \<Rightarrow> None | Some (pos_split, neg_split) \<Rightarrow> Some ((pos_split, neg_split), proj_list_3 L_pos pos_split, proj_list_3 L_neg neg_split))"