hcju/isabelleps huggingface dataset:
simps_8: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 kills K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. hamlet (Rep_run \<rho> n K\<^sub>1) \<longrightarrow> (\<forall>m\<ge>n. \<not> hamlet (Rep_run \<rho> m K\<^sub>2))}"
\<I>_2: shows "subst_domain \<I> = subst_domain \<theta>"
loc_defs_27: shows "mut_m.hs_load_pending \<equiv> ''hs_load_pending''"
open_segment_translation_eq: fixes d :: "'b"   and x :: "'b"   and a :: "'b"   and b :: "'b" shows "(d + x \<in> open_segment (d + a) (d + b)) = (x \<in> open_segment a b)"
dagger_subdist: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<dagger> \<le> (x + y)\<^sup>\<dagger>"
card_le_Suc_iff: fixes n :: "nat"   and A :: "'b set" shows "(Suc n \<le> card A) = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
add1: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' 0 a b"   and "ccw' 0 c b" shows "ccw' 0 (a + c) b"
modif: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)"
id_eq_2: fixes v :: "i"   and x :: "'a"   and y :: "'a" shows "[x \<^bold>= y \<^bold>\<rightarrow> y \<^bold>= x in v]"
coeff_monom: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "coeff (UP R) (monom (UP R) a m) n = (if m = n then a else \<zero>\<^bsub>R\<^esub>)"
i_Exec_Stream_Previous_i_Exec_Stream_Init: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c n = (i_Exec_Comp_Stream trans_fun input c)\<^bsup>\<leftarrow> c\<^esup> n"
\<Lambda>_PBD_axioms: shows "\<Lambda>_PBD \<V> \<B> \<Lambda> {\<k>}"
domain_very_strict: fixes x :: "'a" shows "(domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x = \<bottom>) = (x = \<bottom>)"
assms_3: shows "has_least_prefixpoint (\<lambda>x. \<mu> f x)"
lmap_lappend_distrib: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist"   and ys :: "'d llist" shows "lmap f (lappend xs ys) = lappend (lmap f xs) (lmap f ys)"
semigroup_axioms: shows "semigroup (\<^bold>\<squnion>)"
sum_list_fun_apply: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and xs :: "'c list"   and y :: "'b" shows "sum_list (map f xs) y = (\<Sum>x\<leftarrow>xs. f x y)"
run_limit_not_empty: fixes \<phi> :: "'a ltl" shows "limit (run\<^sub>t (delta \<Sigma>) (initial \<phi>) w) \<noteq> {}"
equivalence_implies_bisimilarity: fixes P :: "'state"   and Q :: "'state" assumes "S_transform.logically_equivalent P Q" shows "P \<sim>\<cdot>\<^sub>S Q"
wf_term_imp_term_ok': fixes \<Sigma> :: "(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)"   and t :: "Core.term" assumes "wf_term \<Sigma> t" shows "term_ok' \<Sigma> t"
elims_1: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'd"   and y :: "bool" assumes "mat_diagonal x = y" obtains A :: "'a" and B :: "'b" and C :: "'c" and D :: "'d" where "x = (A, B, C, D)" and "y = (B = (0::'b) \<and> C = (0::'c))"
spr_jview_tInit_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envObs :: "'b \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'b"   and t :: "'f Trace"   and obs :: "'e" assumes "Environment jkbp" shows "(Environment.spr_jview envObs a t = tInit obs) = (\<exists>s. t = tInit s \<and> envObs a s = obs)"
rel_pmf_return_pmfI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "P x y" shows "rel_pmf P (return_pmf x) (return_pmf y)"
full_poly_input_assn_def: shows "full_poly_input_assn = hr_comp (hr_comp polys_assn_input (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel\<rangle>fmap_rel)) polys_rel"
balance_right_inv2_with_inv1: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "inv2 lt"   and "inv2 rt"   and "bheight lt = bheight rt + 1"   and "inv1 lt" shows "inv2 (balance_right lt k v rt) \<and> bheight (balance_right lt k v rt) = bheight lt"
less_le_trans: fixes y :: "'d"   and x :: "'d"   and z :: "'d" assumes "y < x"   and "z \<le> y" shows "z < x"
onllD: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and P :: "('i \<times> 'h) \<times> 'j \<times> 'i \<times> 'h \<Rightarrow> bool"   and \<xi> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp"   and a :: "'j"   and \<xi>' :: "'i"   and p' :: "('f, 'g, 'e, 'h) seqp" assumes "onll \<Gamma> P ((\<xi>, p), a, \<xi>', p')" shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
rref_suc_if_all_rows_not_zero: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k"   and "\<forall>n. \<not> is_zero_row_upt_k n k A" shows "reduced_row_echelon_form_upt_k A (Suc k)"
image_closure_subset: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on (closure S) f"   and "closed T"   and "f ` S \<subseteq> T" shows "f ` closure S \<subseteq> T"
length_prefix_ex2: fixes i :: "nat"   and xs :: "'a list"   and j :: "nat" assumes "i < length xs"   and "j < length xs"   and "i < j" shows "\<exists>ys zs vs. xs = ys @ xs ! i # zs @ xs ! j # vs \<and> length ys = i \<and> length zs = j - i - 1"
s'_in01: shows "s' \<inter> {0..1} \<subseteq> {0..1}"
affine_independent_span_eq: fixes S :: "'a set" assumes "\<not> affine_dependent S"   and "card S = Suc DIM('a)" shows "affine hull S = UNIV"
lb1: shows "\<forall>x. poly p x = 0 \<longrightarrow> lb1 < x"
i'_top: shows "i' < (i + 1) * 2 ^ (l' - l)"
completion_distr_eq: fixes M :: "'b measure"   and X :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "complete_measure M"   and "X \<in> M \<rightarrow>\<^sub>M N"   and "null_sets (distr M N X) = null_sets N" shows "completion (distr M N X) = distr M (completion N) X"
distrib_sup_le: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<sqinter> z) \<le> x \<squnion> y \<sqinter> x \<squnion> z"
mat_Ph_split: shows "mat_Ph = 2 \<cdot>\<^sub>m pre - 1\<^sub>m N"
f''_within: fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "(f' has_derivative ($) (f'' tx)) (at tx within T \<times> X)"
preference_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" shows "preference carrier relation \<equiv> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> y \<in> carrier) \<and> preorder_on carrier relation"
asym_lex_prod: fixes R :: "('a \<times> 'a) set"   and S :: "('c \<times> 'c) set" assumes "asym R"   and "asym S" shows "asym (R <*lex*> S)"
not_alw: fixes \<phi> :: "'d stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> alw \<phi> xs) = ev (\<lambda>xs. \<not> \<phi> xs)"
"8_4": shows "f T5 \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t (((T1 @ T2) @ T3) @ T4) \<union> X"
col_rank_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "col_rank A = vec.dim (col_space A)"
domsMT_notND_DD3: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "x \<in> dom (Cp (DenyAllFromTo a b))"   and "x \<in> dom (Cp (DenyAllFromTo c d))" shows "\<not> netsDistinct a c"
function_zero_closed: fixes R :: "('c, 'd) ring_scheme"   and S :: "'e set" assumes "ring_functions R" shows "\<zero>\<^bsub>function_ring S R\<^esub> \<in> carrier (function_ring S R)"
has_subprob_density_imp_subprob_space': fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f"   and "is_subprob_density N f" shows "subprob_space M"
r_ltz: shows "poly [:c, b, a:] r < 0"
root_2_2: fixes x :: "real_alg_2"   and n :: "nat" assumes "invariant_2 x" shows "invariant_2 (root_2 n x)"
zero_a_B: shows "is_zero_row_upt_k a (Suc k) B"
bdd_aboveI2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and M :: "'a" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "preorder.bdd_above less_eq (f ` A)"
gf: fixes x :: "'a" assumes "x \<in> T - S" shows "g x = f x"
affine_hull_2: fixes a :: "'b"   and b :: "'b" shows "affine hull {a, b} = {u *\<^sub>R a + v *\<^sub>R b |u v. u + v = 1}"
coeffs_monom: fixes a :: "'a"   and n :: "nat" shows "coeffs (monom a n) = (if a = (0::'a) then [] else replicate n (0::'a) @ [a])"
hyp1: shows "fst (echelon_form_of_upt_k_det bezout (1::'a, A) k) = fst (echelon_form_of_upt_k_det_iarrays (1::'a, matrix_to_iarray A) k bezout)"
has_bochner_integral_null: fixes N :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "N \<in> null_sets lebesgue" shows "has_bochner_integral (lebesgue_on N) f (0::'b)"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "f x = f' x"
magQ_scaleQ: fixes x :: "'c"   and y :: "'c['d, 'e]" shows "\<lbrakk>x *\<^sub>Q y\<rbrakk>\<^sub>Q = x \<cdot> \<lbrakk>y\<rbrakk>\<^sub>Q"
max_add_distrib_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus x (ord.max less_eq y z) = ord.max less_eq (plus x y) (plus x z)"
flows_edges: shows "flows_fix \<T> \<subseteq> edges G"
csn: shows "cs\<^bsup>\<pi> @\<^bsup>l\<^esup> \<pi>l\<^esup> (l + n) = cs\<^bsup>\<pi>'\<^esup> n'"
linear_split_append: fixes xs :: "('b \<times> 'c) list"   and p :: "'c"   and ys :: "('b \<times> 'c) list"   and ls :: "('b \<times> 'c) list"   and rs :: "('b \<times> 'c) list" assumes "linear_split_help xs p ys = (ls, rs)" shows "ls @ rs = ys @ xs"
SeqSubstFormP_Eq: fixes s :: "name"   and k :: "name"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, x, y, x', y', v, i)"   and "atom k \<sharp> (x, y, x', y', v, i)" shows "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i (Q_Eq x y) (Q_Eq x' y') (Var s) (Var k)))"
split: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and option :: "'h option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
inputIneqTrans: fixes a :: "name"   and x :: "name"   and P :: "pi"   and b :: "subject"   and y :: "name"   and P' :: "pi" assumes "a<x>.P \<longmapsto> b\<guillemotleft>y\<guillemotright> \<prec> P'"   and "a \<sharp> b" shows "False"
list_all2_split: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> \<exists>z. Q x z \<and> R z y" shows "\<exists>zs. list_all2 Q xs zs \<and> list_all2 R zs ys"
vsv_axioms: shows "vsv (\<BB>\<lparr>Cod\<rparr>)"
indRelLSTPO_impl_SRel_and_TRel_reflect_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelLSTPO SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
cproper_interval_int_def: shows "cproper_interval = proper_interval"
finite_imp_compact_convex_hull: fixes S :: "'b set" assumes "finite S" shows "compact (convex hull S)"
scaleR_Pair: fixes r :: "real"   and a :: "'a"   and b :: "'b" shows "r *\<^sub>R (a, b) = (r *\<^sub>R a, r *\<^sub>R b)"
card_odd: fixes n :: "nat" shows "card {a \<in> {a. odd a}. a < 2 * n} = n"
nth_drop: fixes n :: "nat"   and xs :: "'d list"   and i :: "nat" assumes "n \<le> length xs" shows "drop n xs ! i = xs ! (n + i)"
app: fixes env :: "(name, name) fmap"   and n1 :: "nterm"   and n2 :: "nterm"   and m1 :: "nterm"   and m2 :: "nterm" assumes "alpha_equiv env n1 n2"   and "alpha_equiv env m1 m2" shows "alpha_equiv env (n1 $\<^sub>n m1) (n2 $\<^sub>n m2)"
B: assumes "\<lbrakk>M; map Send T\<rbrakk>\<^sub>d \<theta>"   and "t \<in> set T" shows "\<lbrakk>M; [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>d \<theta>"
rep_eq: fixes x :: "rat" shows "Rep_mini_alg (ma_of_rat x) = (x, 0, 0)"
sign_changes_uminus: fixes xs :: "'a list" shows "sign_changes (map uminus xs) = sign_changes xs"
AE_support_countable: assumes "events = UNIV" shows "(AE x in M. prob {x} \<noteq> 0) = (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
distinct_1: fixes x1 :: "nat"   and x21 :: "'a"   and x22 :: "'a form list" shows "At x1 \<noteq> Compound x21 x22"
Suc: shows "inverse (f ^ n) \<langle>$\<rangle> a = (f ^ (order f a - n mod order f a)) \<langle>$\<rangle> a"
c_diff_is_pr: shows "c_diff \<in> PrimRec2"
update_eq: fixes xs :: "'c list"   and opt :: "('a, 'b, 'c) env option"   and env :: "('a, 'b, 'c) env" shows "update xs opt env = (case xs of [] \<Rightarrow> case opt of None \<Rightarrow> env | Some e \<Rightarrow> e | x # xs \<Rightarrow> case env of Val a \<Rightarrow> Val a | Env b es \<Rightarrow> case xs of [] \<Rightarrow> Env b (es(x := opt)) | y # ys \<Rightarrow> Env b (es(x := case es x of None \<Rightarrow> None | Some e \<Rightarrow> Some (update (y # ys) opt e))))"
span_closed: fixes U :: "rat vec set"   and v :: "rat vec" assumes "U \<subseteq> Rn"   and "v \<in> gs.span U" shows "v \<in> Rn"
simps_2: fixes S :: "nat set"   and n :: "nat" shows "pick S (Suc n) = (LEAST a. a \<in> S \<and> pick S n < a)"
strict_mono_less_eq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "strict_mono f" shows "(f x \<le> f y) = (x \<le> y)"
rel_vset_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c set"   and ys :: "'d set" shows "rel_vset R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> (x \<in> xs) = (y \<in> ys)"
module_ow_axioms: shows "module_ow U\<^sub>M\<^sub>_\<^sub>1 (+\<^sub>M\<^sub>_\<^sub>1) 0\<^sub>M\<^sub>_\<^sub>1 (-\<^sub>M\<^sub>_\<^sub>1) uminus\<^sub>M\<^sub>_\<^sub>1 (*s\<^sub>M\<^sub>_\<^sub>1)"
rep_eq: fixes x :: "'a"   and xa :: "'a"   and xb :: "'a scene" shows "x \<oplus>\<^sub>S xa on xb = Rep_scene xb x xa"
nderiv_zero: fixes f :: "real \<Rightarrow> 'a"   and X :: "real set" shows "nderiv 0 f X = f"
irreducible_reachable_imp_not_unifiable: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, E') \<in> unif\<^sup>!"   and "E' \<noteq> {#}" shows "\<not> unifiable (set_mset E)"
cases: fixes a :: "letter list list" assumes "good a" obtains w :: "letter list" and ws :: "letter list list" where "a = w__ # ws__" and "L w__ ws__"   | ws :: "letter list list" and w :: "letter list" where "a = w # ws" and "good ws"
1: shows "\<forall>m<length t + length u. \<forall>x xa. m = length x + length xa \<longrightarrow> (\<forall>xb xc. \<exists>u1 s1. xb setinterleaves ((x, xa), A) \<and> xc < x \<longrightarrow> u1 \<le> xa \<and> s1 < xb \<and> s1 setinterleaves ((xc, u1), A))"
ex_NleastTr: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "ndec_seq A a n \<le> a - n"
mult_poly_p'_mult_poly_spec: fixes p :: "(char list multiset \<times> int) multiset"   and p' :: "int mpoly"   and q :: "(char list multiset \<times> int) multiset"   and q' :: "int mpoly" assumes "(p, p') \<in> mset_poly_rel"   and "(q, q') \<in> mset_poly_rel" shows "mult_poly_p' p q \<le> \<Down> mset_poly_rel (mult_poly_spec p' q')"
analytic_on_Un: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and T :: "complex set" shows "(f analytic_on S \<union> T) = (f analytic_on S \<and> f analytic_on T)"
F: shows "1 + nat (s ''x'') = 2 ^ k"
A: shows "OpposedThinChamberComplexFoldings X f g A"
ip: shows "ipath (mkTree (sdrop n rs) s') (sdrop m steps)"
quot_exhaust: fixes A :: "'b quot" shows "\<exists>a. A = \<lfloor>a\<rfloor>"
split_completion_def: fixes M :: "'b measure"   and A :: "'b set"   and p :: "'b set \<times> 'b set" shows "split_completion M A p = (if A \<in> sets M then p = (A, {}) else \<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets M)"
closest_point_exists_2: fixes S :: "'b set"   and a :: "'b" assumes "closed S"   and "S \<noteq> {}" shows "\<forall>y\<in>S. dist a (closest_point S a) \<le> dist a y"
Not\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "p \<bullet> Not\<^sub>\<alpha> t\<^sub>\<alpha> = Not\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
lcm_least: 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 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"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b c" shows "dvd.dvd times (lcm a b) c"
assms_1: shows "f \<cdot> g = k \<cdot> l"
perm_app: fixes pi :: "(name \<times> name) list"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "pi \<bullet> f x = (pi \<bullet> f) (pi \<bullet> x)"
simplex_in_max: fixes y :: "'a set" assumes "y \<in> X" shows "\<exists>x. domain.chamber x \<and> order.greater_eq x y"
trace_reject_no_possible_steps_atomic: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and a :: "String.literal \<times> value list"   and t :: "(String.literal \<times> value list) list" assumes "possible_steps e s r (fst a) (snd a) = {||}" shows "\<not> recognises_execution e s r (a # t)"
is_shortest_path_map_def: fixes v0 :: "'V"   and res :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option" shows "is_shortest_path_map v0 res \<equiv> \<forall>v\<in>V. case res v of None \<Rightarrow> \<nexists>p. is_path v0 p v | Some p \<Rightarrow> is_path v0 p v \<and> (\<forall>p'. is_path v0 p' v \<longrightarrow> path_weight p \<le> path_weight p')"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Dom (t \<^bold>\<cdot> u) = Dom u"
discr_imp_Sbis: fixes c :: "('test, 'atom, 'choice) cmd" assumes "proper c"   and "discr c" shows "c \<approx>s c"
at_lst_asgd_ge: fixes x :: "'b"   and y :: "'b"   and B :: "'b set"   and Pr :: "('b \<times> 'b) set" assumes "x \<in> at_least_as_good y B Pr" shows "x \<succeq>[Pr] y"
card_cb: shows "int (card (c - b)) + aff_dim S = int DIM('a)"
simps_3: fixes n :: "nat"   and m :: "nat"   and y :: "nat"   and x :: "nat" assumes "n \<noteq> 0"   and "even n" shows "mod_exp_nat.efficient_power m y x n = mod_exp_nat.efficient_power m y (x * x mod m) (n div 2)"
suff_appE: fixes r :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "r \<in> A\<^sup>\<star>"   and "t \<in> suff A r" obtains s :: "'b llist" where "s \<in> A\<^sup>\<infinity>" and "t = r @@ s"
index_mat_multrow_mat_2: fixes n :: "nat"   and k :: "nat"   and a :: "'a" shows "dim_row (multrow_mat n k a) = n"
tMT: shows "(\<Union> (range (relations M)))\<^sup>* `` {t} \<subseteq> T"
f6: fixes a :: "'a"   and aa :: "'a" shows "ad (ad (ad a + aa)) = ad (ad aa \<cdot> a)"
elementary_divisor_ring_class_def: shows "OFCLASS('b, elementary_divisor_ring_class) \<equiv> (OFCLASS('b, comm_ring_1_class) &&& class.elementary_divisor_ring TYPE('b))"
sp_it_is_the_strongest_post: fixes p :: "'b upred"   and C :: "'b hrel"   and Q :: "'b upred" assumes "`p sp C \<Rightarrow> Q`" shows "\<lbrace>p\<rbrace> C \<lbrace>Q\<rbrace>\<^sub>u"
has_integral_restrict_Int: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and T :: "'c set" shows "((\<lambda>x. if x \<in> S then f x else (0::'d)) has_integral i) T = (f has_integral i) (S \<inter> T)"
inv_stereographic_on_sphere: fixes X :: "real"   and z :: "complex"   and Y :: "real"   and Z :: "real" assumes "X = Re (2 * z / (1 + z * cnj z))"   and "Y = Im (2 * z / (1 + z * cnj z))"   and "Z = ((cmod z)\<^sup>2 - 1) / (1 + (cmod z)\<^sup>2)" shows "X * X + Y * Y + Z * Z = 1"
Fun_ide: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.ide S a" shows "set_category_given_img.Fun S img a = (\<lambda>x\<in>set_category_data.set S img a. x)"
abs_fun_eq1: fixes a :: "'a"   and x :: "'b"   and y :: "'b" shows "([a].x = [a].y) = (x = y)"
spr_sim_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "('b, 'e, 'f) BEState list"   and envAction :: "('b, 'e, 'f) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> ('b, 'e, 'f) BEState"   and envVal :: "('b, 'e, 'f) BEState \<Rightarrow> 'c \<Rightarrow> bool"   and agents :: "'b odlist"   and envObsC :: "'e \<Rightarrow> 'h" assumes "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC" shows "FiniteDetBroadcastEnvironment.spr_sim jkbp envInit envAction envTrans envVal envObsC \<equiv> \<lambda>t. \<lparr>sprFst = tFirst t, sprLst = tLast t, sprCRel = FiniteDetBroadcastEnvironment.tObsC_abs jkbp envInit envAction envTrans envVal envObsC t\<rparr>"
mls: shows "measure_space \<Omega> ls (outer_measure M f)"
union_le_mono2: fixes B :: "'b multiset"   and D :: "'b multiset"   and C :: "'b multiset" assumes "B < D" shows "C + B < C + D"
run_new_array: fixes n :: "nat"   and x :: "'b"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'b array" assumes "run (Array.new n x) \<sigma> \<sigma>' r"   and "\<not> is_exn \<sigma>" obtains "\<sigma>' = Some (snd (Array.alloc (replicate n x) (the_state \<sigma>)))" and "r = fst (Array.alloc (replicate n x) (the_state \<sigma>))" and "Array.get (the_state \<sigma>') r = replicate n x"
0: shows "nat n = l0 * nat p ^ k \<and> \<not> nat p dvd l0"
poly_eqI: fixes p :: "'b poly"   and q :: "'b poly" assumes "\<And>n. poly.coeff p n = poly.coeff q n" shows "p = q"
exit_validFrom: fixes vl :: "'a list"   and step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b"   and \<phi> :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and f :: "('b, 'c, 'd) trans \<Rightarrow> 'a"   and T :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and s :: "'b"   and tr :: "('b, 'c, 'd) trans list" assumes "vl \<noteq> []"   and "BD_Security.exit step \<phi> f T s (hd vl)"   and "IO_Automaton.validFrom step s tr"   and "BD_Security.V \<phi> f tr = vl"   and "never T tr" shows "False"
trms_of_deg_leq_0: fixes f :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP INTEG)" shows "INTEG.trms_of_deg_leq 0 f = monom (UP INTEG) (f 0) 0"
assms: shows "(s, (d, \<alpha>, t) # xs) \<in> conv ars"
state_from_induceES_trace: fixes \<beta> :: "'e list"   and \<alpha> :: "'e list" assumes "\<beta> @ \<alpha> \<in> Tr\<^bsub>(induceES SES)\<^esub>" shows "\<exists>s\<in>S\<^bsub>SES\<^esub>. s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s \<and> enabled SES s \<alpha> \<and> reachable SES s"