hcju/isabelleps huggingface dataset:
equivu_imp_le: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_utrans t u" shows "le_utrans t u"
scaleR_fds_def: fixes c :: "real"   and f :: "'a fds" shows "c *\<^sub>R f = fds (\<lambda>n. c *\<^sub>R fds_nth f n)"
last_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "length xs = length ys" shows "last (zip xs ys) = (last xs, last ys)"
calc_ideal_points_on_circline: fixes H :: "circline" assumes "is_poincare_line H" shows "\<forall>z\<in>calc_ideal_points H. z \<in> circline_set H"
c: shows "length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))))) = roundup (length (nat_to_bv (p * q)) - Suc 0) 8 * 8 - 168"
normal_self: shows "carrier (relative_homology_group (int n) (nsphere n) (equator n)) \<lhd> relative_homology_group (int n) (nsphere n) (equator n)"
SecurityInvariant_preliminaries_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "SecurityInvariant_preliminaries sinvar \<equiv> (\<forall>G nP. wf_graph G \<longrightarrow> \<not> sinvar G nP \<longrightarrow> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<noteq> {}) \<and> (\<forall>N E E' nP. wf_graph \<lparr>nodes = N, edges = E\<rparr> \<longrightarrow> E' \<subseteq> E \<longrightarrow> sinvar \<lparr>nodes = N, edges = E\<rparr> nP \<longrightarrow> sinvar \<lparr>nodes = N, edges = E'\<rparr> nP) \<and> (\<forall>G ff nP f'. wf_graph G \<longrightarrow> SecurityInvariant_withOffendingFlows.is_offending_flows sinvar ff G nP \<longrightarrow> SecurityInvariant_withOffendingFlows.is_offending_flows sinvar (ff \<union> f') G nP)"
antisym_conv2: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subseteq># y" shows "(\<not> x \<subset># y) = (x = y)"
eventually_gt_at_top: fixes c :: "'a" shows "eventually ((<) c) at_top"
g: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) = - (1::'rows)"
r_inv_ex: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"
\<gamma>': shows "ladder_\<gamma> (\<delta> @ \<alpha>) D L (Suc index) = \<delta> @ \<gamma>' \<and> LeftDerivationIntro (ladder_\<alpha> \<alpha> D' L' (Suc index)) (ladder_i L (Suc index) - length \<delta>) (snd e) (ladder_ix L (Suc index)) (derivation_shift E (length \<delta>) 0) (ladder_j L (Suc index) - length \<delta>) \<gamma>'"
bidel_arc_head_isolated_axioms_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_head_isolated_axioms G M a \<equiv> head G a \<in> pre_digraph.isolated_verts (bidel_arc.H G M a)"
roolx: fixes f :: "'a" assumes "f \<in> Hom A B" shows "f \<in> Ar"
arity_sym\<^sub>h_lt_\<omega>_if_\<delta>\<^sub>h_gt_0: fixes f :: "'s" assumes "0 < \<delta>\<^sub>h" shows "arity_sym\<^sub>h f < \<omega>"
prv_all_imp_gen: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<notin> Fvars \<phi>"   and "bprv (imp \<phi> \<chi>)" shows "bprv (imp \<phi> (all x \<chi>))"
nodes_semi_auxi: shows "let A = (rev \<circ> sorted_list_of_set) ` Pow (set nodes); r = nodes_le; f = inter_sorted_rev in semilat (A, r, f)"
that: fixes s :: "('fun, 'var) Term.term"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "s \<in> M"   and "\<Gamma> t = \<Gamma> s"   and "mgu t s = Some \<delta>"   and "inj_on \<delta> (fv t)"   and "\<delta> ` fv t \<subseteq> range Var" shows "thesis"
one_strongly_connected_iff_7_aux: fixes x :: "'a" shows "x\<^sup>+ \<le> x ; 1 ; x"
ctr_transfer_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(R ===> llist_all2 R ===> llist_all2 R) LCons LCons"
same_empty_unreach: fixes Q :: "'a set"   and a :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q" shows "\<emptyset> Q a = {}"
mat_col_eq: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list" assumes "mat nr nc m1"   and "mat nr nc m2" shows "(m1 = m2) = (\<forall>i<nc. col m1 i = col m2 i)"
hyp4: shows "fst (snd (snd (foldl echelon_form_of_column_k_det_iarrays (1::'a, matrix_to_iarray A, 0, bezout) [0..<Suc k]))) = snd (snd (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k]))"
is_aligned_mask: fixes w :: "'b word"   and n :: "nat" shows "is_aligned w n = (w AND mask n = 0)"
is_test_def: fixes x :: "'a" shows "is_test x \<equiv> x \<le> 1'"
mod_div_trivial: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (modulo a b) b = zero"
while_rel1_def: shows "while_rel1 = measure (\<lambda>x. trivCond (x = [])) <*lex*> measure (\<lambda>x. trivCond (x = (0::'b))) <*lex*> measure (\<lambda>x. trivCond (\<not> x))"
lists_succ_Cons_pf: fixes xs :: "'b list"   and xss :: "'b list list" shows "lists_succ (xs # xss) = list_succ xs \<circ> lists_succ xss"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d list" shows "map g (map f v) = map (g \<circ> f) v"
set_intros_3: fixes x1 :: "'a" shows "x1 \<in> vars_term (Var x1)"
set_iterator_rev_linord_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'c set" assumes "class.linorder less_eq less" shows "linorder.set_iterator_rev_linord less_eq it S0 \<equiv> set_iterator_genord it S0 (\<lambda>x y. less_eq y x)"
\<sigma>: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1"
det_nz_iff_inj_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "(det (matrix f) \<noteq> (0::'a)) = inj f"
normalize_poly_normalize_poly_p: fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> unsorted_poly_rel" shows "PAC_Polynomials_Operations.normalize_poly p \<le> \<Down> sorted_poly_rel (SPEC (normalize_poly_p\<^sup>*\<^sup>* p'))"
dec_interp_def: fixes n :: "nat"   and FO :: "nat set"   and w :: "('a \<times> bool list) list" shows "dec_interp n FO w \<equiv> map (\<lambda>i. if i \<in> FO then Inl (the_elem (positions_in_row w i)) else Inr (positions_in_row w i)) [0..<n]"
gcsort_sort_form: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and t :: "nat \<times> nat list \<times> 'c list" shows "find ((<) (Suc 0)) (fst (snd (gcsort_aux index key p t))) = None"
simps_2: fixes n :: "nat"   and R :: "('b \<times> 'b) set" shows "R ^^ Suc n = R ^^ n O R"
frontier_of_subset_topspace: fixes X :: "'b topology"   and S :: "'b set" shows "X frontier_of S \<subseteq> topspace X"
corec: fixes g1 :: "'a \<Rightarrow> real"   and g2 :: "'a \<Rightarrow> real"   and a :: "'a" shows "corec_complex g1 g2 a = Complex (g1 a) (g2 a)"
lookup_fresh: fixes x :: "atom"   and e :: "name list"   and n :: "nat"   and y :: "name" shows "x \<sharp> lookup e n y = (y \<in> set e \<or> x \<noteq> atom y)"
\<psi>_def: shows "\<psi> \<equiv> PP (encF N (Var xx))"
mat_match: shows "M_mat_R (get_signs_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) (get_subsets_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) = get_matrix_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))"
termi_c1': shows "\<Gamma>\<turnstile>c1' \<down> Normal s"
tab2: shows "table (args_n args) (args_R args) rel2"
Im_divide_of_real: fixes z :: "complex"   and r :: "real" shows "Im (z / complex_of_real r) = Im z / r"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom" shows "time (C\<^sub>1;; C\<^sub>2) = (\<lambda>s. time C\<^sub>1 s + preT C\<^sub>1 (time C\<^sub>2) s)"
mlex_eq: fixes f :: "'a \<Rightarrow> nat"   and R :: "('a \<times> 'a) set" shows "f <*mlex*> R = {(x, y). f x < f y \<or> f x \<le> f y \<and> (x, y) \<in> R}"
p2s_neg_hom: fixes P :: "'b \<Rightarrow> bool" shows "p2s (- P) = - p2s P"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "trg\<^sub>C \<mu> = trg\<^sub>C f"
fset_eqvt: fixes p :: "perm"   and S :: "'a fset" shows "p \<bullet> fset S = fset (p \<bullet> S)"
hcmod_i: fixes x :: "real star"   and y :: "real star" shows "hcmod (HComplex x y) = (*f* sqrt) (x\<^sup>2 + y\<^sup>2)"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "Left.in_hom f a b"   and "Left.in_hom g b c" shows "Left.seq g f"
0: shows "sp F xs (map Map.empty \<Sigma>) = Ok res"
induct_step: shows "\<lbrace>invar \<and> b\<rbrace> C \<lbrace>invar\<rbrace>\<^sub>u"
lift_Comp_def: fixes f :: "int \<Rightarrow> int \<Rightarrow> bool"   and g :: "real \<Rightarrow> real \<Rightarrow> bool" shows "lift_Comp f g \<equiv> map_int_pair (\<lambda>a b. BoolVal (f a b)) (map_real_pair (\<lambda>a b. BoolVal (g a b)) (\<lambda>_. FALSE))"
set_eq_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A = B) = (\<forall>x. s2p A x = s2p B x)"
UN_ball_bex_simps_2: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> (B ` A). P x) = (\<forall>a\<in>A. \<forall>x\<in>B a. P x)"
is_singleton_altdef: fixes A :: "'b set" shows "is_singleton A = (card A = 1)"
s: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> s) = (\<exists>k u v. meets k p \<and> meets p u \<and> meets u v \<and> meets k q \<and> meets q v)"
iFROM_eq_conv: fixes n :: "nat"   and n' :: "nat" shows "([n\<dots>] = [n'\<dots>]) = (n = n')"
arith_extra_simps_13: fixes a :: "'a" shows "a - (0::'a) = a"
lmeasurable_negligible_symdiff: fixes S :: "'c set"   and T :: "'c set" assumes "S \<in> lmeasurable"   and "negligible (S - T \<union> (T - S))" shows "T \<in> lmeasurable"
v11_vinsert: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "v11 r"   and "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<notin>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "v11 (vinsert \<langle>a, b\<rangle> r)"
search_converg_2: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "eval (total_cons.r_search s) [b] \<down>" shows "e_length b \<le> the (eval (total_cons.r_search s) [b])"
simps_8: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and i :: "nat" shows "\<lbrakk> K\<^sub>1 kills K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> i\<^esup> = {\<rho>. \<forall>n\<ge>i. hamlet (Rep_run \<rho> n K\<^sub>1) \<longrightarrow> (\<forall>m\<ge>n. \<not> hamlet (Rep_run \<rho> m K\<^sub>2))}"
assms_2: shows "A \<inter> B \<subseteq> {\<one>}"
bounds_compare_contradictory_2: fixes c :: "'c"   and b :: "'c option" assumes "c \<le>\<^sub>u\<^sub>b b"   and "c >\<^sub>u\<^sub>b b" shows "False"
fls_lr_inverse_regpart_2: fixes f :: "'a fls"   and y :: "'a" assumes "fls_subdegree f = 0" shows "fls_regpart (fls_right_inverse f y) = fps_right_inverse (fls_regpart f) y"
NormalAction: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x' :: "'x1"   and m' :: "'m1" assumes "r1.r_syntax t x m ta x' m'" shows "r1.init_fin t ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
counit_simps_1: shows "S.seq (S.inv (UP.unit (trg f))) (S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g))"
update_convs_1: fixes verts' :: "'c set \<Rightarrow> 'c set"   and verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d" shows "verts_update verts' \<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr> = \<lparr>verts = verts' verts, arcs = arcs, \<dots> = more\<rparr>"
rcond_false_trel: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and b :: "'b upred"   and P :: "'b hrel"   and Q :: "'b hrel" assumes "\<sigma> \<dagger> b = false" shows "(\<sigma>, P \<triangleleft> b \<triangleright>\<^sub>r Q) \<rightarrow>\<^sub>u (\<sigma>, Q)"
dvd_mult_div_cancel: fixes a :: "'a"   and b :: "'a" assumes "a dvd b" shows "a * (b div a) = b"
assms: shows "\<forall>i\<in>set_tree y. 0 < i"
Cd_Above_dX: fixes X :: "'a set"   and d :: "'b" assumes "ManiPref'.dX X d \<subseteq> Field (override_on Pd Pds' ds' d)" shows "ManiPref.Cd d X = Above (override_on Pd Pds' ds' d) (ManiPref'.dX X d) \<inter> X"
assms_1: fixes x :: "'a" assumes "x \<in># A" shows "R x y"
preserves_trg: fixes x :: "'c" shows "E (trg\<^sub>C x) = trg (E x)"
axiom: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "A \<in> axioms \<Theta>"   and "wf_inst \<Theta> \<rho>" shows "\<Theta>,\<Gamma> \<turnstile> tsubst A \<rho>"
isNSContD: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and y :: "'c star" assumes "isNSCont f a"   and "y \<approx> star_of a" shows "(*f* f) y \<approx> star_of (f a)"
assms_10: shows "r\<^sup>\<infinity> = r\<^sup>\<star>"
pq_def: shows "pq = embed_pmf (\<lambda>(x, y). case x of None \<Rightarrow> case y of None \<Rightarrow> pmf q None | Some y' \<Rightarrow> spmf q y' - spmf p y' | Some x' \<Rightarrow> case y of None \<Rightarrow> 0 | Some y' \<Rightarrow> if x' = y' then spmf p x' else 0)"
ereal_divide_one: fixes x :: "ereal" shows "x / 1 = x"
domainI: fixes R :: "('a, 'b) ring_scheme" assumes "cring R"   and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<And>a b. \<lbrakk>a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>; a \<in> carrier R; b \<in> carrier R\<rbrakk> \<Longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>" shows "domain R"
filter_insort_key_pos: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "sorted (map f xs)"   and "P x" shows "filter P (insort_key f x xs) = insort_key f x (filter P xs)"
tmbound_I: fixes bs :: "'b list"   and t :: "tm"   and n :: "nat"   and vs :: "'b list"   and x :: "'b" assumes "tmboundslt (length bs) t"   and "tmbound n t"   and "n \<le> length bs" shows "Itm vs (bs[n := x]) t = Itm vs bs t"
term_ok_beta_norm: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and t' :: "Core.term" assumes "term_ok thy t"   and "beta_norm t = Some t'" shows "term_ok thy t'"
mustT_MtransT: fixes c :: "('test, 'atom) com"   and s :: "'state" assumes "mustT c s" shows "\<exists>s'. (c, s) \<rightarrow>*t s'"
trivial_FGModuleHom: fixes smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>r. r \<in> FG \<Longrightarrow> smult' r (0::'d) = (0::'d)" shows "GRepHom smult' 0"
interval_inner_leI_2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b"   and i :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "x \<in> ord.atLeastAtMost less_eq a b"   and "less_eq zero i" shows "inner x i \<le> inner b i"
rel_mono: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_prod R1 R2 \<le> rel_prod R1a R2a"
Rep_quot: fixes x :: "'d quot" shows "Rep_quot x \<in> quot"
absE: fixes implem :: "chan \<Rightarrow> msg"   and X :: "chan"   and H :: "msg set" assumes "semivalid_implem implem"   and "X \<in> semivalid_implem.abs implem H" obtains x :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "X = Chan x A B M" and "M \<in> payload" and "implem X \<in> H"
not_inf_Abrupt: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<not> \<Gamma>\<turnstile> (c, Abrupt s) \<rightarrow> \<dots>(\<infinity>)"
Gram_Schmidt_def: fixes xs :: "('a, 'b) vec list" shows "Gram_Schmidt xs = foldr Gram_Schmidt_step xs []"
phiUsesE: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> phiUses g n" obtains n' :: "'node" and v' :: "'val" and vs :: "'val list" where "n' \<in> set (old.successors g n)" and "(n, v) \<in> set (zip (old.predecessors g n') vs)" and "phis g (n', v') = Some vs"
Acyclic_Graph_axioms: shows "Acyclic_Graph task.nodes task.inPorts task.outPorts vertices nodeOf task_edges task.hyps"
card_of_underS: fixes r :: "('d \<times> 'd) set"   and a :: "'d" assumes "Card_order r"   and "a \<in> Field r" shows "|underS r a| <o r"
dgrad_p_setD: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'a" assumes "p \<in> punit.dgrad_p_set d m"   and "v \<in> keys p" shows "d (local.punit.pp_of_term v) \<le> m"
replacement: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "small X" shows "small (f ` X)"
no_cs_change_if_no_event: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "isRecv ev \<longrightarrow> getId ev \<noteq> i"   and "regular_event ev" shows "cs c i = cs c' i"
R27_4: shows "pmf (sds R27) d = 1 / 2"
path_from_toI: fixes G :: "('e, 'f) Graph_scheme"   and xs :: "'e list"   and v :: "'e"   and w :: "'e" assumes "Graph G"   and "Graph.walk G xs \<and> distinct xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
ANDSepaux: fixes x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'c) Combinators"   and z :: "('b set set, 'c) Combinators list" assumes "allNetsDistinct (x # y # z)" shows "allNetsDistinct (x \<oplus> y # z)"
carrier_update_1: fixes S :: "'a set" shows "carrier (R\<lparr>carrier := S\<rparr>) = S"
wadjust_goon_left_moving_O_Oc_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_on_left_moving_O m rs (c, Oc # list)"   and "hd c = Oc" shows "wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
assms_4: fixes a' :: "'b" assumes "a' \<in> A"   and "a' \<noteq> a" shows "f a' = (0::'a)"
on_circline_cmat_cvec_unit: fixes z1 :: "complex"   and z2 :: "complex" shows "on_circline_cmat_cvec unit_circle_cmat (z1, z2) = (z1 * cnj z1 = z2 * cnj z2)"
vreal_zero: shows "0\<^sub>\<real> = []\<^sub>\<circ>"
execn_call_Normal_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =n\<Rightarrow> t" obtains bdy :: "('e, 'd, 'f) com" and i :: "nat" and t' :: "'e" where "\<Gamma> p = Some bdy__" and "\<Gamma>\<turnstile> \<langle>bdy__,Normal (init s)\<rangle> =i__\<Rightarrow> Normal t'__" and "\<Gamma>\<turnstile> \<langle>c s t'__,Normal (return s t'__)\<rangle> =Suc i__\<Rightarrow> t" and "n = Suc i__"   | bdy :: "('e, 'd, 'f) com" and i :: "nat" and t' :: "'e" where "\<Gamma> p = Some bdya__" and "\<Gamma>\<turnstile> \<langle>bdya__,Normal (init s)\<rangle> =ia__\<Rightarrow> Abrupt t'" and "n = Suc ia__" and "t = Abrupt (return s t')"   | bdy :: "('e, 'd, 'f) com" and i :: "nat" and f :: "'f" where "\<Gamma> p = Some bdyb__" and "\<Gamma>\<turnstile> \<langle>bdyb__,Normal (init s)\<rangle> =ib__\<Rightarrow> Fault f" and "n = Suc ib__" and "t = Fault f"   | bdy :: "('e, 'd, 'f) com" and i :: "nat" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =ic__\<Rightarrow> Stuck" and "n = Suc ic__" and "t = Stuck"   | i :: "nat" where "\<Gamma> p = None" and "n = Suc i" and "t = Stuck"
insert_ops_appendD: fixes xs :: "('b \<times> 'b option) list"   and ys :: "('b \<times> 'b option) list" assumes "insert_ops (xs @ ys)" shows "insert_ops xs"
assms_1: shows "u\<^sub>1 @ [a] @ u\<^sub>2 = v\<^sub>1 @ [a] @ v\<^sub>2"
ran_restrict2_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and S :: "'d decision set" shows "p \<triangleright>2 S \<equiv> \<lambda>x. if the (p x) \<in> S then p x else \<bottom>"
power_Suc0_right: fixes a :: "'a" shows "power.power 1' (;) a (Suc zero_class.zero) = a"
null_part_def: fixes M :: "'b measure"   and A :: "'b set" shows "null_part M A = snd (Eps (split_completion M A))"
wf_dickson_less_p: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfP (dickson_less_p d m)"
aNDSetsEqaux: fixes a :: "('b, 'c) Combinators list"   and b :: "('b, 'c) Combinators list" assumes "singleCombinators a"   and "singleCombinators b"   and "set a = set b" shows "set (net_list_aux a) = set (net_list_aux b)"
const: fixes R :: "('c \<Rightarrow> real) set"   and S :: "'c set"   and c :: "real" assumes "function_ring_on R S" shows "(\<lambda>_. c) \<in> R"
assms_1: fixes v :: "'a" assumes "phi v" shows "chi v"