hcju/isabelleps huggingface dataset:
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck_1: fixes X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and G :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G - set X"
left_commute: fixes b :: "'e"   and a :: "'e"   and c :: "'e" shows "b + (a + c) = a + (b + c)"
narrow_rep_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "narrow_rep p1 p2 = (if is_empty_rep p1 \<or> is_empty_rep p2 then empty_rep else let (l1, h1) = p1; (l2, h2) = p2 in (if l1 = -\<infinity> then l2 else l1, if h1 = \<infinity> then h2 else h1))"
pm_of_poly_prod: fixes x :: "'a"   and f :: "'c \<Rightarrow> 'b poly"   and I :: "'c set" shows "pm_of_poly x (prod f I) = (\<Prod>i\<in>I. pm_of_poly x (f i))"
division_ofD_6: fixes s :: "'b set set"   and i :: "'b set" assumes "s division_of i" shows "\<Union> s = i"
Conj_def: fixes A :: "fm"   and B :: "fm" shows "A AND B \<equiv> Neg (A IMP Neg B)"
P1_cong: fixes tms :: "trm list"   and P :: "trm \<Rightarrow> fmla"   and H :: "fmla set"   and x :: "trm"   and x' :: "trm" assumes "\<And>i t x. atom i \<sharp> tms \<Longrightarrow> (P t)(i::=x) = P (subst i x t)"   and "H \<turnstile> x EQ x'" shows "H \<turnstile> P x IFF P x'"
Mapping_lookup_fold_upd_set_max: fixes xs :: "('a \<times> 'b) list"   and as :: "'c"   and Z :: "'b \<Rightarrow> 'a \<Rightarrow> 'c set"   and m :: "('c, 'a) mapping" assumes "{(t, X). (t, X) \<in> set xs \<and> as \<in> Z X t} \<noteq> {}"   and "sorted (map fst xs)" shows "Mapping.lookup (fold (\<lambda>(t, X) m. upd_set m (\<lambda>_. t) (Z X t)) xs m) as = Some (Max (fst ` {(t, X). (t, X) \<in> set xs \<and> as \<in> Z X t}))"
circle_right_bot_edges_neq: shows "circle_right_edge \<noteq> circle_bot_edge"
select_convs_4: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.nxt \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = nxt"
drlex_pm_plus_monotone: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "drlex_pm s t" shows "drlex_pm (s + u) (t + u)"
simps_1: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and a :: "rat array"   and fl :: "rat"   and fr :: "rat" shows "downI' d 0 p a fl fr = return ()"
le_less: fixes x :: "'b list"   and y :: "'b list" shows "suffix x y = (strict_suffix x y \<or> x = y)"
Ugp_const_gHom: fixes G :: "('c, 'd) Group_scheme"   and E :: "('e, 'f) Group_scheme" assumes "Ugp G"   and "Ugp E" shows "(\<lambda>x\<in>carrier G. \<one>\<^bsub>E\<^esub>) \<in> gHom G E"
linear_irreducible_int: fixes p :: "int poly" assumes "degree p = 1"   and "is_unit (content p)" shows "irreducible p"
valuate_add: fixes p1 :: "linear_poly"   and p2 :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "p1 + p2 \<lbrace> v \<rbrace> = p1 \<lbrace> v \<rbrace> + p2 \<lbrace> v \<rbrace>"
inducts: fixes A :: "'b \<Rightarrow> bool"   and x :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "finlstsp A x"   and "P LNil"   and "\<And>l a. \<lbrakk>finlstsp A l; P l; A a\<rbrakk> \<Longrightarrow> P (a ## l)" shows "P x"
minim_isMinim: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "isMinim B (minim B)"
map_raw_id: fixes xs :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv xs" shows "map_raw id xs = xs"
propa_dom_invariant_auxi: fixes a :: "nat list list"   and b :: "nat list" assumes "wf_dom a b"   and "b \<noteq> []" shows "a ! hd b \<noteq> rev [0..<n] \<and> Sorted_Less.sorted (rev (hd b # a ! hd b)) \<and> set (hd b # a ! hd b) \<subseteq> set nodes \<and> hd b # a ! hd b \<in> A \<and> (\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). q < length a \<and> \<tau> \<in> A)"
lm088: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" shows "Outside' X R \<subseteq> R"
conjugate_square_greater_0_vec: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "((0::'c) < inner_prod v v) = (v \<noteq> 0\<^sub>v n)"
isomorphic_symmetric: fixes f :: "'b"   and g :: "'b" assumes "f \<cong>\<^sub>D g" shows "g \<cong>\<^sub>D f"
carrier_is_subalgebra: fixes K :: "'a list set" assumes "K \<subseteq> carrier (K [X])" shows "subalgebra K (carrier (K [X])) (K [X])"
lincomb_closed: fixes rs :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "rs \<bullet>\<cdot> ms \<in> M"
mk_chain_edges: fixes Rules :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and sel :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and L :: "'d set"   and G :: "nat \<times> ('d \<times> nat \<times> nat) set"   and i :: "nat" assumes "valid_selector Rules sel"   and "\<Union> ((edges \<circ> snd) ` Rules) \<subseteq> L \<times> UNIV"   and "edges (graph_of G) \<subseteq> L \<times> UNIV" shows "edges (graph_of (mk_chain sel Rules G i)) \<subseteq> L \<times> UNIV"
simps_49: fixes x71 :: "key"   and x72 :: "msg"   and x61 :: "msg"   and x62 :: "msg" shows "Crypt x71 x72 \<noteq> \<lbrace>x61, x62\<rbrace>"
absorb2: fixes b :: "nat"   and a :: "nat" assumes "b dvd a" shows "gcd a b = b"
wp_while_inv: fixes p :: "'a"   and i :: "'a"   and r :: "'a"   and q :: "'a"   and x :: "'a" assumes "d p \<le> d i"   and "d i \<cdot> ad r \<le> d q"   and "d i \<cdot> d r \<le> wp x i" shows "d p \<le> wp (while r inv i do x od) q"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>nat z. P z (TVar nat)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1\<rightarrow>ty2)" shows "P z ty"
Gamma_asymp_equiv: shows "Gamma \<sim>[at_top] (\<lambda>x. sqrt (2 * pi / x) * (x / exp 1) powr x)"
upright_alt: fixes a :: "'b zmultiset" shows "upright a = (\<forall>t. 0 < zcount a t \<longrightarrow> supported_strong a t)"
map_eq_Cons_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and y :: "'a"   and ys :: "'a list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
invertible_mat_four_block_mat_lower_right: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "invertible_mat A" shows "invertible_mat (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 n) (0\<^sub>m n 1) A)"
poly_ge_def: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" shows "p \<ge>p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> eval_poly \<alpha> q \<le> eval_poly \<alpha> p)"
mI_vals_LB: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "j \<le> n" shows "\<nu>\<^bsub>K P j\<^esub> ` I \<subseteq> LBset (ant 0)"
basis_def: fixes A :: "'a mod_ring poly set" shows "Berlekamp_subspace.basis A = (Berlekamp_subspace.lin_indpt A \<and> Berlekamp_subspace.gen_set A \<and> A \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}))"
union_ring_is_monoid: assumes "C \<noteq> {}" shows "Group.comm_monoid (union_ring C)"
list_of_dlist_empty: shows "list_of_dlist DList_Set.empty = []"
indexed_Ints_vec_UNIV: shows "\<int>\<^sub>v = indexed_Ints_vec UNIV"
G'_fields_3: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and ecnv :: "('a, 'b) graph_rec_scheme \<Rightarrow> 'd" assumes "g_rename_precond G f" shows "g_E (fr_rename_ext ecnv f G) = rename_E f (g_E G)"
face_of_imp_convex: fixes T :: "'a set"   and S :: "'a set" assumes "T face_of S" shows "convex T"
assms: fixes mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff p mon = MPoly_Type.coeff q mon"
ub_ln2_def: fixes prec :: "nat" shows "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 in float_plus_up prec (Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1))) (float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))"
assms_2: shows "\<forall>s. P s \<and> \<not> bval b s \<longrightarrow> Q s"
filter_id_conv: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
r_find_k_def: shows "r_find_k \<equiv> let k = Cn 2 r_pdec1 [recf.Id 2 0]; r = Cn 2 r_result1 [Cn 2 r_pdec2 [recf.Id 2 0], Cn 2 r_s [Cn 2 r_append_zeros [recf.Id 2 1, k]], Cn 2 r_add [Cn 2 r_length [recf.Id 2 1], k]] in Cn 1 r_pdec1 [Mn 1 (Cn 2 r_eq [r, r_constn 1 1])]"
simps: fixes uu :: "'b extT" shows "uu\<^sup>\<star> = extT.Top"
Cons_1: assumes "\<stileturn> G" shows "\<stileturn> map (\<lambda>p. vars_for_consts p cs) G"
many_strongly_connected_iff_12_eq: fixes x :: "'a" assumes "is_p_fun x" shows "many_strongly_connected x = (x\<^sup>\<star> ; x ; x\<^sup>T = x\<^sup>+)"
ctxtp_ctxt_eq: shows "ctxtp = (\<lambda>x. x \<in> ctxt)"
min_int_poly_real_def: shows "min_int_poly_real = min_int_poly"
left_add_twice: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_numeral one times plus" shows "plus a (plus a b) = plus (times (numeral.numeral one plus (num.Bit0 num.One)) a) b"
row_rank_eq_col_rank: fixes A :: "(('b, 'c) vec, 'd) vec" shows "row_rank A = col_rank A"
pte: shows "pivot_tableau_eq t1 eq t2 x = (t1', eq', t2')"
fls_normalized_product_of_inverses_1: fixes f :: "'a fls"   and g :: "'a fls" assumes "f * g = 1" shows "fls_base_factor f * fls_base_factor g = fls_X ^ nat (- (fls_subdegree f + fls_subdegree g))"
map_ugraph_preserves_sub: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set"   and f :: "nat \<Rightarrow> nat" assumes "subgraph G\<^sub>1 G\<^sub>2" shows "subgraph (map_ugraph f G\<^sub>1) (map_ugraph f G\<^sub>2)"
cut_ge_subset_mono: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "A \<subseteq> B" shows "A \<down>\<ge> t \<subseteq> B \<down>\<ge> t"
statEqvt'': fixes p :: "(name \<times> name) list"   and M :: "'a"   and N :: "'a" shows "p \<bullet> M \<leftrightarrow> N = p \<bullet> M \<leftrightarrow> p \<bullet> N"
vec_vec_Tensor_simpl: fixes i :: "nat"   and y :: "'a list"   and z :: "'a"   and zs :: "'a list" shows "i < length y \<longrightarrow> vec_vec_Tensor (z # zs) y ! i = local.times z y ! i"
bitlen_le_iff_power: fixes x :: "int"   and w :: "int" shows "(bitlen x \<le> w) = (0 \<le> w \<and> x < 2 ^ nat w)"
pseudo_mod_hom: fixes p :: "'a poly"   and q :: "'a poly" shows "pseudo_mod (map_poly to_fract p) (map_poly to_fract q) = map_poly to_fract (pseudo_mod p q)"
NSDERIV_add_minus: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and Da :: "'a"   and g :: "'a \<Rightarrow> 'a"   and Db :: "'a" assumes "NSDERIV f x :> Da"   and "NSDERIV g x :> Db" shows "NSDERIV (\<lambda>x. f x + - g x) x :> Da + - Db"
dim_substandard: fixes d :: "'b set" assumes "d \<subseteq> Basis" shows "dim {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = card d"
simps_2: fixes f :: "'d \<Rightarrow> 'c option"   and x :: "'d" shows "omap_option f (Some x) = f x \<bind> (\<lambda>x. Some (Some x))"
subst_trm_fresh_tyvar: fixes X :: "tyvrs"   and t :: "trm"   and u :: "trm"   and x :: "vrs" assumes "X \<sharp> t"   and "X \<sharp> u" shows "X \<sharp> t[x \<mapsto> u]"
convergentD: fixes X :: "nat \<Rightarrow> 'b" assumes "convergent X" shows "\<exists>L. X \<longlonglongrightarrow> L"
tendsto_at_iff_tendsto_nhds: fixes g :: "'b \<Rightarrow> 'c"   and l :: "'b" shows "g \<midarrow>l\<rightarrow> g l = (g \<longlongrightarrow> g l) (nhds l)"
1: assumes "\<lbrakk>M; D; A @ B\<rbrakk>\<^sub>s \<I>" shows "\<lbrakk>M; D; A\<rbrakk>\<^sub>s \<I>"
nub_body'_nub_body''_eq: shows "nub_body' = nub_body''"
max_v_def: shows "local.max_v = (1 / norm1 (fst local.max_v_ev)) *\<^sub>R fst local.max_v_ev"
LIMSEQ_SEQ_conv2: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and l :: "'d" assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l" shows "f \<midarrow>a\<rightarrow> l"
rep_number: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<B> rep x = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
free_Abelian_group_universal: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int"   and S :: "'c set" assumes "f ` S \<subseteq> carrier (free_Abelian_group S)" obtains h :: "('c \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int" where "h \<in> hom (free_Abelian_group S) (free_Abelian_group S)" and "\<And>x. x \<in> S \<Longrightarrow> h (frag_of x) = f x"
assms: shows "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"
Im_I_max: shows "I m \<subseteq> I (max n m)"
minus_add: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "uminus (plus a b) = plus (uminus b) (uminus a)"
YM4_analz_knows_Spy: fixes A :: "agent"   and NB :: "msg"   and Y :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Gets A \<lbrace>NB, Crypt (shrK A) Y, X\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "X \<in> analz (knows Spy evs)"
Gcd_greatest: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and a :: "'a" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times a b" shows "dvd.dvd times a (Gcd A)"
prems_5: shows "AllowPortFromTo c d po \<in> set (ys @ [y])"
grd_dgr: fixes D :: "'d \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and i :: "'d" shows "grd (step D) i = \<Squnion>{P. \<exists>j. P = grd (D (i, j))}"
P_def: shows "P = \<Prod>(min_int_poly ` X)"
finite_K: shows "finite {h. h \<le> k \<and> P h}"
perm_bool_def: fixes pi :: "('d \<times> 'd) list"   and b :: "bool" shows "pi \<bullet> b = b"
gamma2_x_const: shows "\<forall>x. gamma2 x \<bullet> i = b"
preserves_reflects_arr: fixes f :: "'a" shows "arr (R f) = arr f"
simps: fixes e :: "nat"   and M :: "hmultiset multiset" shows "local.decode e (HMSet M) = (\<Sum>m\<in>#M. base ^ local.decode 0 m) div base ^ e"
greaterThan_subset_iff: fixes x :: "'c"   and y :: "'c" shows "({x<..} \<subseteq> {y<..}) = (y \<le> x)"
I_def: fixes n :: "nat" shows "I n = integral {0..4} (\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x))"
Abs_Assertion_inverse: fixes y :: "'c" assumes "y \<in> assertion" shows "Rep_Assertion (Abs_Assertion y) = y"
Quotient_abs_rep: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "Quotient R Abs Rep T" shows "Abs (Rep a) = a"
lift_map_drop_map_eq: fixes i :: "nat"   and s :: "(nat \<Rightarrow> 'c) \<times> 'd" shows "lift_map i (drop_map i s) = s"
add_independentS_Cons_conv_dirsum_right: fixes A :: "'a set"   and As :: "'a set list" shows "add_independentS (A # As) = (add_independentS [A, \<Oplus>B\<leftarrow>As. B] \<and> add_independentS As)"
assms_3: fixes p :: "real" shows "(\<lambda>x. g x powr p) \<in> o[F](f)"
exhaust: fixes y :: "'a tree" obtains "y = \<langle>\<rangle>"   | x21 :: "'a tree" and x22 :: "'a" and x23 :: "'a tree" where "y = \<langle>x21, x22, x23\<rangle>"
finite_prime_divisors: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "finite {p. normalization_semidom.prime zero times one normalize p \<and> dvd.dvd times p x}"
xor_def: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y = x \<^bold>\<sqinter> \<sim> y \<^bold>\<squnion> \<sim> x \<^bold>\<sqinter> y"
matchC_C_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis" shows "PL_Indis.matchC_C aval tval cval indis theta c d \<equiv> \<forall>s t. (s, t) \<in> indis \<longrightarrow> (\<exists>P F. PL_Indis.mC_C aval tval cval indis theta c d s t P F)"
comp_in_homI': fixes f :: "'d"   and g :: "'d"   and a :: "'d"   and c :: "'d" assumes "D.arr f"   and "D.arr g"   and "D.dom f = a"   and "D.cod g = c"   and "D.dom g = D.cod f" shows "\<guillemotleft>g \<cdot>\<^sub>D f : a \<rightarrow>\<^sub>D c\<guillemotright>"
stage: fixes bs :: "real list"   and p :: "fm"   and d :: "int"   and e :: "MIR.num"   and c :: "int" shows "Ifm bs (stage p d (e, c)) = (\<exists>j\<in>{1..c * d}. Ifm bs (\<sigma> p c (Add e (C j))))"
diff_diff_add: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a - b - c = a - (b + c)"
k_invariantAD: fixes a :: "'a"   and ec :: "'rep"   and A :: "('ma, 'mt) AlgState" assumes "KBP.k_isNode a ec"   and "KBP.k_memb ec A"   and "KBP.k_invariant a A" shows "\<exists>acts. lookup aOps (aActs A) ec = Some acts \<and> set acts = set (simAction a ec)"
f: fixes a :: "'a" assumes "a \<in> Field r" shows "f a \<in> Field s \<and> f ` r.underS a \<subseteq> s.underS (f a)"
interior_of_subtopology_subsets: fixes T :: "'c set"   and U :: "'c set"   and X :: "'c topology"   and S :: "'c set" assumes "T \<subseteq> U" shows "T \<inter> subtopology X U interior_of S \<subseteq> subtopology X T interior_of S"
assms: fixes x :: "'x" assumes "x \<in> indets p" shows "poly_deg (f x) = 0"
is_sup_upper: fixes x :: "'a"   and y :: "'a"   and s :: "'a" assumes "is_sup x y s" obtains "x \<sqsubseteq> s" and "y \<sqsubseteq> s"
dim_eq: fixes X :: "'a set" assumes "X \<subseteq> S" shows "S.dim X = dim X"
prems_1: shows "\<forall>a\<in>{(i, t, s)} \<union> set (D @ [di]). case a of (j, p) \<Rightarrow> \<forall>a\<in>{(i, t, s)} \<union> set (D @ [di]). case a of (k, q) \<Rightarrow> (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"
is_pole_divide: fixes z :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "isCont f z"   and "filterlim g (at (0::'c)) (at z)"   and "f z \<noteq> (0::'c)" shows "is_pole (\<lambda>z. f z / g z) z"
shift_bounds_cl_nat_ivl: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..n + k} = (\<Sum>i = m..n. g (i + k))"
Hintikka_Extend: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set" assumes "infinite UNIV"   and "finite A"   and "maximal A S"   and "consistent A S"   and "saturated S" shows "Hintikka A S"
WhileFalse: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c :: "com"   and cs :: "com list" assumes "\<not> b s" shows "((WHILE b DO c) # cs, s) \<rightarrow> (cs, s)"
frac_from_carrier_local: fixes X :: "('a \<times> 'a) set" assumes "X \<in> carrier_local_ring_at" obtains r :: "'a" and s :: "'a" where "r \<in> R" and "s \<in> R" and "s \<notin> I" and "X = cq.frac r s"
normal_invI: fixes N :: "'a set" assumes "subgroup N G"   and "\<And>x h. \<lbrakk>x \<in> carrier G; h \<in> N\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> inv x \<in> N" shows "N \<lhd> G"
agsum_add1m: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "a \<in> carrier A"   and "n < 0"   and "m < 0" shows "(n + m)\<triangleright>a\<^bsub>A\<^esub> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> m\<triangleright>a\<^bsub>A\<^esub>"
atLeastatMost_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "class.order less_eq less"   and "less b a" shows "ord.atLeastAtMost less_eq a b = {}"
locally_path_connected: fixes S :: "'b set" shows "locally path_connected S = (\<forall>V x. openin (top_of_set S) V \<and> x \<in> V \<longrightarrow> (\<exists>U. openin (top_of_set S) U \<and> path_connected U \<and> x \<in> U \<and> U \<subseteq> V))"
exec_iff_execn: 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" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t = (\<exists>n. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t)"
gram_schmidt_fs_Rn_axioms: shows "gram_schmidt_fs_Rn n fs"
ent_disjI2': fixes A :: "assn"   and C :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>A C" shows "A \<Longrightarrow>\<^sub>A B \<or>\<^sub>A C"
extremum: fixes a :: "'b list" shows "\<epsilon> \<le>p a"
update_inf_same: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(x[y\<longmapsto>z]) \<sqinter> y = z\<^sup>T \<sqinter> y"
B_Un_2: shows "B_Int \<subseteq> B_Un"
scale_closed: fixes x :: "'m"   and a :: "'cr1" assumes "x \<in> U\<^sub>M" shows "a *s\<^sub>M x \<in> U\<^sub>M"
a_translate_eq: fixes c :: "'a"   and a :: "'a" assumes "c \<in> carrier R"   and "a \<in> carrier R" shows "a_translate c a = c \<oplus> a"
name_id: fixes a :: "name"   and x :: "'a" shows "[(a, a)] \<bullet> x = x"
less_max_iff_disj: fixes z :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(z \<prec> ordered_powerprod_lin.max x y) = (z \<prec> x \<or> z \<prec> y)"
ctermsSI: fixes p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and pn :: "'c" assumes "p \<in> sterms \<Gamma> (\<Gamma> pn)" shows "p \<in> cterms \<Gamma>"
data_cong: fixes tas :: "nat \<times> 'a option list" assumes "tas \<in> ts_tuple_rel (set (linearize data_prev) \<union> set (linearize data_in))" shows "valid_tuple tuple_since' tas = valid_tuple tuple_since tas"
Union_upper: fixes B :: "'b set"   and A :: "'b set set" assumes "B \<in> A" shows "order.greater_eq (\<Union> A) B"
simps_2: fixes x2 :: "jump"   and y2 :: "jump" shows "(Jump x2 = Jump y2) = (x2 = y2)"
\<S>\<^sub>G_imp_\<A>_refl: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r" assumes "\<S>\<^sub>G s"   and "r \<in> dom s" shows "\<A> r r s"