hcju/isabelleps huggingface dataset:
mod_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo zero a = zero"
below_up_def: shows "(\<sqsubseteq>) \<equiv> \<lambda>x y. case x of Ibottom \<Rightarrow> True | Iup a \<Rightarrow> case y of Ibottom \<Rightarrow> False | Iup b \<Rightarrow> a \<sqsubseteq> b"
cong_minus_minus_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (uminus b) (uminus c) a = unique_euclidean_semiring.cong modulo b c a"
integral_add: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x + g x = integral\<^sup>L M f + integral\<^sup>L M g"
OrdLeast_le: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "Ord k"   and "P k" shows "(LEAST n. Ord n \<and> P n) \<le> k"
override_on_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "f ++\<^bsub>A\<^esub> g = (\<lambda>a. if a \<in> A then g a else f a)"
inf_insertI: fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>i. is_glb L i (insert x A) \<Longrightarrow> P i"   and "is_glb L a A"   and "x \<in> carrier L"   and "A \<subseteq> carrier L" shows "P (\<Sqinter>insert x A)"
reachable_mk_symmetric_eq: fixes u :: "'a"   and v :: "'a" assumes "symmetric (with_proj (mk_symmetric G))" shows "(u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric (with_proj (mk_symmetric G)))\<^esub> v) = (u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v)"
edges_sym_def: fixes g :: "graph" shows "edges_sym g \<equiv> \<forall>a b. (a, b) \<in> \<E> g \<longrightarrow> (b, a) \<in> \<E> g"
derived_set_subset: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "(X derived_set_of S \<subseteq> S) = closedin X S"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> C.CC.arr f" shows "T\<^sub>C f = C.null"
smallest_ccdi_sets: fixes M :: "'b measure" shows "smallest_ccdi_sets (space M) (sets M) \<subseteq> Pow (space M)"
get_disconnected_document_ptr_in_heap: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> ok get_disconnected_document node_ptr" shows "node_ptr |\<in>| node_ptr_kinds h"
card_fun_def: fixes a :: "nat"   and b :: "nat" shows "card_fun a b = (if a \<noteq> 0 \<and> b \<noteq> 0 \<or> b = 1 then b ^ a else 0)"
THROW_rule: fixes \<Phi> :: "'c \<Rightarrow> bool"   and e :: "'c"   and \<Psi> :: "'d \<Rightarrow> bool" assumes "\<Phi> e" shows "THROW e \<le> ESPEC \<Phi> \<Psi>"
fraction_def: shows "fraction \<equiv> map_fun Rep_float id (\<lambda>(_, uu_, y). unat y)"
dist_nz: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "(x \<noteq> y) = (0 < dist x y)"
ma_floor: fixes r :: "mini_alg" shows "\<lfloor>real_of r\<rfloor> = ma_floor r"
dominates_def: fixes g :: "'a"   and n :: "'node"   and m :: "'node" shows "gE2.dominates g n m \<equiv> m \<in> set (\<alpha>n2 g) \<and> (\<forall>ns. gE2.path2 g (Entry2 g) ns m \<longrightarrow> n \<in> set ns)"
refine: fixes n :: "nat"   and Xi :: "real list \<times> real list"   and X :: "'b set"   and sctni :: "real list sctn"   and sctn :: "'b sctn" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> lvivl_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(nres_of (restrict_to_halfspace_impl n sctni Xi), restrict_to_halfspace sctn X) \<in> \<langle>lvivl_rel\<rangle>nres_rel"
in_set_enumerate_eq: fixes p :: "nat \<times> 'a"   and n :: "nat"   and xs :: "'a list" shows "(p \<in> set (List.enumerate n xs)) = (n \<le> fst p \<and> fst p < length xs + n \<and> xs ! (fst p - n) = snd p)"
idealI: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set" assumes "ring R"   and "subgroup I (add_monoid R)"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>R\<^esub> a \<in> I"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> x \<in> I" shows "ideal I R"
pull_up_senior_older_seniors: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "senior x n = senior y n" shows "older_seniors x n = older_seniors y n"
front_tickFree_append: fixes s :: "'c event list"   and t :: "'c event list" assumes "tickFree s"   and "front_tickFree t" shows "front_tickFree (s @ t)"
length_getVars: fixes \<sigma>l :: "'tp list" shows "length (GE.getVars \<sigma>l) = length \<sigma>l"
saturate_def: shows "saturate \<equiv> \<lambda>CC. {x. saturatep (\<lambda>x. x \<in> CC) x}"
simps_1: fixes A :: "'c llist set" shows "lnull (lSup A) = (\<forall>x\<in>A. lnull x)"
polys_1: fixes qa :: "int poly"   and ria :: "root_info" assumes "(qa, ria) \<in> set polys" shows "root_info_cond ria qa"
Rat_induct: fixes P :: "rat \<Rightarrow> bool"   and q :: "rat" assumes "\<And>a b. \<lbrakk>0 < b; coprime a b\<rbrakk> \<Longrightarrow> P (Rat.Fract a b)" shows "P q"
extend_indets_subst_def: fixes x :: "'b" shows "extend_indets_subst x = monomial (1::'c) (monomial 1 (Some x))"
abs_eq: shows "top = Abs_stone_phi_pair stone_phi.pairs_top"
Scons_inject2: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" assumes "Scons M N = Scons M' N'" shows "N = N'"
cinsertE: fixes a :: "'f"   and b :: "'f"   and A :: "'f cset" assumes "cin a (cinsert b A)" obtains "a = b"   | "cin a A"
Suc_0: shows "moebius_mu (Suc 0) = (1::'a)"
intros_1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "Subobjs P C Cs"
simps_2: fixes x :: "'d"   and xs :: "'d list" shows "rotate1 (x # xs) = xs @ [x]"
from_list_correct: fixes xs :: "('fun \<times> ('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef) list"   and k :: "'fun" shows "map_of xs k = Finca_get (Sinca.Fenv.from_list xs) k"
f_add: fixes m1 :: "'c"   and m2 :: "'c" assumes "m1 \<in> carrier M"   and "m2 \<in> carrier M" shows "f (m1 \<oplus>\<^bsub>M\<^esub> m2) = f m1 \<oplus>\<^bsub>N\<^esub> f m2"
except_plus: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" shows "except (p + q) S = except p S + except q S"
diff_numeral_simps_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n) = uminus (numeral.numeral one plus (m + n))"
B2a: shows "\<^bold>\<forall>x. cod (dom x) \<cong> dom x"
v_2: shows "vec\<^sub>h v \<noteq> 0\<^sub>v n"
neighbourhood_base_at_mono: fixes x :: "'b"   and P :: "'b set \<Rightarrow> bool"   and X :: "'b topology"   and Q :: "'b set \<Rightarrow> bool" assumes "neighbourhood_base_at x P X"   and "\<And>S. \<lbrakk>P S; x \<in> S\<rbrakk> \<Longrightarrow> Q S" shows "neighbourhood_base_at x Q X"
return_time_exivl: fixes P :: "'a set"   and x :: "'a" assumes "Y.returns_to P x"   and "closed P" shows "Y.return_time P x \<in> Y.existence_ivl0 x"
asset_iii_used: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "(Asset n, \<lbrace>Num 4, PubKey A\<rbrace>) \<in> s \<longrightarrow> PriKey A \<in> used s"
usubst_condr: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and b :: "'d upred"   and P :: "('b, 'd) uexpr"   and Q :: "('b, 'd) uexpr" shows "\<sigma> \<dagger> (P \<triangleleft> b \<triangleright> Q) = \<sigma> \<dagger> P \<triangleleft> \<sigma> \<dagger> b \<triangleright> \<sigma> \<dagger> Q"
leg1_commutes: shows "HfHgh.cod.leg1 \<cdot> HfHgh.chine = HfHgh.dom.leg1"
product_category_axioms: shows "product_category (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2)"
obsv_round_def: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and r_decisions :: "process \<Rightarrow> val option"   and Os :: "process set" shows "obsv_round r S v r_decisions Os = {(s, s'). r = next_round s \<and> (S \<noteq> {} \<longrightarrow> obs_safe r s v) \<and> d_guard r_decisions (const_map v S) \<and> (S \<in> Quorum \<longrightarrow> Os = UNIV) \<and> (Os \<noteq> {} \<longrightarrow> S \<noteq> {}) \<and> s' = s\<lparr>next_round := Suc r, votes := (votes s)(r := const_map v S), decisions := decisions s ++ r_decisions, obsv_state.obs := (obsv_state.obs s)(r := const_map v Os)\<rparr>}"
ceiling_unique: fixes z :: "int"   and x :: "'b" assumes "of_int z - (1::'b) < x"   and "x \<le> of_int z" shows "\<lceil>x\<rceil> = z"
CAND_OrR2_elim: fixes a :: "coname"   and b :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:OrR2 <b>.M a \<in> \<parallel><B>\<parallel>"   and "<a>:OrR2 <b>.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)" shows "\<exists>B1 B2. B = B1 OR B2 \<and> <b>:M \<in> \<parallel><B2>\<parallel>"
UP_smult_assoc1: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "p \<in> carrier (UP R)" shows "a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>UP R\<^esub> p = a \<odot>\<^bsub>UP R\<^esub> (b \<odot>\<^bsub>UP R\<^esub> p)"
dvdnumcoeff_trans: fixes g :: "int"   and g' :: "int"   and t :: "MIR.num" assumes "g dvd g'"   and "dvdnumcoeff t g'" shows "dvdnumcoeff t g"
convex_on_def: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" shows "convex_on S f = (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
P_mod_pre: fixes I :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal S I"   and "pol_coeff S (Suc n, f)"   and "P_mod R S X I (polyn_expr R X (Suc n) (Suc n, f))" shows "P_mod R S X I (polyn_expr R X n (n, f))"
NoOutMessageLoss: fixes cfg1 :: "('p, 'v, 's) configuration"   and cfg2 :: "('p, 'v, 's) configuration"   and v :: "bool" assumes "reachable cfg1 cfg2" shows "msgs cfg1 <\<bottom>, outM v> \<le> msgs cfg2 <\<bottom>, outM v>"
True: shows "finite (A \<union> B) \<and> A \<noteq> {} \<and> B \<noteq> {}"
concat_eq_append_conv: fixes xss :: "'a list list"   and ys :: "'a list"   and zs :: "'a list" shows "(concat xss = ys @ zs) = (if xss = [] then ys = [] \<and> zs = [] else \<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2)"
iteration_unfold_eq: fixes x :: "'a" shows "x\<^sup>\<infinity> \<cdot> x = x \<cdot> x\<^sup>\<infinity>"
of_nat_0: shows "\<guillemotleft>0\<guillemotright>\<^sub>\<nat> = \<zero>"
enabled_force_enabled: fixes K :: "'c \<Rightarrow> 'c pmf"   and x :: "'c"   and \<omega> :: "'c stream" shows "MC_syntax.enabled K x (MC_syntax.force_enabled K x \<omega>)"
abel_semigroup_axioms: shows "abel_semigroup (\<union>#)"
OrdP_IN_SUCC_D: fixes H :: "fm set"   and x :: "tm"   and y :: "tm" assumes "H \<turnstile> OrdP x"   and "H \<turnstile> SUCC y IN SUCC x" shows "H \<turnstile> y IN x"
foldl2_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b" shows "foldl2 f xs a = foldl (\<lambda>a b. f b a) a xs"
as_3: shows "S \<subseteq> e1 \<union> e2"
vanishes_add: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.vanishes X"   and "Real.vanishes Y" shows "Real.vanishes (\<lambda>n. X n + Y n)"
irreducible\<^sub>dD_2: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "irreducible\<^sub>d p"   and "degree q < degree p"   and "degree r < degree p" shows "p \<noteq> q * r"
le_length_mono: fixes s :: "'b list"   and t :: "'b list" assumes "s \<le> t" shows "length s \<le> length t"
scale_left_distrib: fixes x :: "'b"   and a :: "'a"   and b :: "'a" assumes "x \<in> U\<^sub>M" shows "s (a + b) x = s a x + s b x"
disj_elim: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "(P \<or> Q) x" obtains "P x"   | "Q x"
assms_2: shows "transforms_to_unique_keys (\<alpha>1 m) f"
mute_preserves: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "mute mf q k l (r, A) = (r', A')"   and "i < dim_row A"   and "j < dim_col A"   and "l < dim_row A"   and "k < dim_row A"   and "i \<noteq> k" shows "A' $$ (i, j) = A $$ (i, j)"
mult_left_mono_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times c a) (times c b)"
ith_def: fixes B :: "('b \<Rightarrow> 'a) set"   and i :: "'b" shows "ith B i = {f i |f. f \<in> B}"
summable_bounded_partials: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F x0 in sequentially. \<forall>a\<ge>x0. \<forall>b>a. norm (sum f {a<..b}) \<le> g a"   and "g \<longlonglongrightarrow> 0" shows "summable f"
t_def: shows "t = \<lfloor>real u1 * x\<rfloor> - \<lfloor>real u2 * x\<rfloor>"
ord_strict_p_asymmetric: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<prec>\<^sub>p q" shows "\<not> q \<prec>\<^sub>p p"
holdsIstate_def: fixes istate :: "'a"   and \<phi> :: "'a \<Rightarrow> bool" shows "IO_Automaton.holdsIstate istate \<phi> \<equiv> \<phi> istate"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "a *k (b *k x) = a * b *k x"
eq_inf_concat_start: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a list"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind" shows "traces.eq_inf ind (w @- w\<^sub>1) (w @- w\<^sub>2) = traces.eq_inf ind w\<^sub>1 w\<^sub>2"
sp_spec_subj_obj_def: fixes a :: "'a"   and x :: "obj_id_t"   and m :: "mode_t" shows "sp_spec_subj_obj a x m \<equiv> configuration_subj_obj a x m \<or> (\<exists>y. x = PAGE y \<and> m = READ \<and> configuration_subj_obj a x WRITE)"
module_pair_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" shows "module_pair s1 s2 \<equiv> module s1 \<and> module s2"
axioms_3: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and M' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'" shows "Group_Theory.monoid M' composition' unit'"
eq_diff_left_iff: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m \<le> k"   and "n \<le> k" shows "(k - m = k - n) = (m = n)"
obs_some_kth_obs: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" assumes "obs \<sigma> k \<noteq> None" obtains i :: "nat" where "is_kth_obs (path \<sigma>) k i"
semiring_normalization_rules_11: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times one a = a"
mono_Sync_sym: assumes "Q \<sqsubseteq> Q'" shows "(P \<lbrakk>A\<rbrakk> Q) \<sqsubseteq> (P \<lbrakk>A\<rbrakk> Q')"
gc_deriv: shows "chain (\<leadsto>GC) (lmap lclss_of_state Sts)"
prerun_valid_def: fixes sys :: "('e, 'f, 'g, 'h, 'i, 'j) pre_system_scheme"   and \<phi> :: "('e, 'f, 'g, 'h, 'i) system_state \<Rightarrow> bool" shows "(sys \<Turnstile>\<^bsub>pre\<^esub> \<phi>) = (\<forall>\<sigma>. prerun sys \<sigma> \<longrightarrow> (\<box>\<lceil>\<phi>\<rceil>) \<sigma>)"
isLub_isUb: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isLub R S x" shows "isUb R S x"
hom_int_pow: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c"   and n :: "int" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (x [^]\<^bsub>G\<^esub> n) = h x [^]\<^bsub>H\<^esub> n"
map_restrict_dom: fixes f :: "'c \<Rightarrow> 'd option" shows "f |` dom f = f"
init_AND_def: fixes X :: "'g \<times> 'a"   and W :: "'h \<times> 'b" shows "init_AND X W = (let (x0, x1) = X; (w0, w1) = W in init0 x0 w0 \<bind> (\<lambda>(r0, a0). init1 x1 w1 \<bind> (\<lambda>(r1, a1). return_spmf ((r0, r1), a0, a1))))"
SN_on_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "SN_on r A = (\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r))"
sccs_altdef2: shows "c.sccs = {H. c.max_subgraph strongly_connected H}"
assms_1: fixes x :: "'a" shows "(f x \<in> L) = (x \<in> L)"
arg_less_sf_imp_c_sum_less_arg: fixes u :: "nat"   and x :: "nat" assumes "u < sf x" shows "c_sum u < x"
homogeneousD_poly_deg: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p"   and "t \<in> keys p" shows "deg_pm t = poly_deg p"
fold_val_elem\<^sub>S\<^sub>e\<^sub>t: fixes \<tau> :: "'d state \<times> 'd state"   and f :: "'g \<Rightarrow> 'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and p :: "'g"   and s_set :: "'f list" assumes "\<tau> \<Turnstile> \<upsilon> foldl UML_Set.OclIncluding Set{} (map (f p) s_set)" shows "list_all (\<lambda>x. \<tau> \<Turnstile> \<upsilon> f p x) s_set"
Inf_lower: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "Inf A \<le> x"
rewrite_iter_fast_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (rewrite_iter_fast \<phi>) \<subseteq> atoms_ltln \<phi>"
t6: shows "typ_of (Abs \<tau>' s $ Fv x \<tau>') = typ_of (Abs \<tau>' t $ Fv x \<tau>')"
fi_rule: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and Ps :: "assn"   and F :: "assn" assumes "<P> c <Q>"   and "Ps \<Longrightarrow>\<^sub>A P * F" shows "<Ps> c <\<lambda>x. Q x * F>"
while_option_rule: fixes P :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "while_option b c s = Some t"   and "P s" shows "P t"
h3c: shows "\<forall>x. List.member (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) x \<longrightarrow> x < length subsets"
below_lub: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a"   and x :: "'a"   and i :: "nat" assumes "class.cpo below"   and "po.chain below S"   and "below x (S i)" shows "below x (po.lub below (range S))"
poly_div_hom: fixes p :: "int poly"   and q :: "int poly" shows "of_int_poly (poly_div p q) = poly_div (of_int_poly p) (of_int_poly q)"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e multiset" shows "g `# f `# v = (g \<circ> f) `# v"
Q: shows "dist qa qb \<le> max (5 * deltaG TYPE('a)) (4 * exp (1 / 2 * ln 2) * lambda * (b - a) * exp (- (D - K - C / 2) * ln 2 / (5 * delta)))"
span_neg: fixes U :: "'c set"   and v :: "'c" assumes "U \<subseteq> carrier V"   and "v \<in> span U" shows "\<ominus>\<^bsub>V\<^esub> v \<in> span U"
asset_pwd_genuine: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey \<inter> (bad_pwd \<union> bad_prikey)"   and "(Asset n, Crypt (SesK SK) (Num 0)) \<in> s" shows "(Owner n, Crypt (SesK SK) (Pwd n)) \<in> s"
le_supE: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and x :: "'d" assumes "class.semilattice_sup sup less_eq less"   and "less_eq (sup a b) x" obtains "less_eq a x" and "less_eq b x"
2: assumes "q'' g1 \<odot> g1 \<noteq> 0" shows "lt (q'' g1 \<odot> g1) \<prec>\<^sub>t v"
exhaust: fixes y :: "init_state" obtains "y = Done"   | "y = Processing"   | "y = Prepared"   | "y = Error"
assms_2: shows "A\<^sub>F \<sharp>* \<phi>"
lowerPowers\<^sub>f_def: shows "lowerPowers\<^sub>f \<equiv> map_fun id (map_fun id (map_fun fmlookup Abs_fmap)) (\<lambda>b i p x. if x \<in> {b..<b + i} then None else p (dec_above b i x))"
eexSTUT: fixes F :: "(state \<Rightarrow> 'b) \<Rightarrow> (nat \<Rightarrow> state) \<Rightarrow> bool"   and x :: "state \<Rightarrow> 'b" assumes "STUTINV F x" shows "STUTINV (\<exists>\<exists> x. F x)"
Hermite_of_row_i_def: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" shows "Hermite_of_row_i ass res A i = (if is_zero_row i A then A else let j = LEAST n. A $ i $ n \<noteq> (0::'a); Aij = A $ i $ j; A' = mult_row A i (ass Aij div Aij) in Hermite_reduce_above A' (mod_type_class.to_nat i) i j res)"
partial_span: fixes i :: "nat" assumes "i \<le> length (map of_int_hom.vec_hom fs)" shows "gs.span (gs1.gso ` {0..<i}) = gs.span (set (take i (map of_int_hom.vec_hom fs)))"
impl_adjoin: fixes P :: "'b upred"   and Q :: "'b upred"   and R :: "'b upred" shows "((P \<Rightarrow> Q) \<and> R) = ((P \<and> R \<Rightarrow> Q \<and> R) \<and> R)"
23: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z \<squnion> x \<sqinter> (y \<squnion> z) = x \<sqinter> (y \<squnion> z)"
x_in_trapG: shows "(1, 0) \<in> trapG"
vproductI: fixes f :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "vsv f"   and "\<D>\<^sub>\<circ> f = I"   and "\<forall>i\<in>elts I. f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i" shows "f \<in>\<^sub>\<circ> vproduct I A"
zero_index: fixes i :: "'c" shows "0 $h i = (0::'b)"
dist_les_8: shows "ValFF \<notsqsubseteq> ValTT"
euclidean_size_of_nat: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (semiring_1.of_nat one plus zero n) = n"
empty_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_set A {} {}"
singletonD: fixes b :: "'b"   and a :: "'b" assumes "(b::'b) \<in> {a::'b}" shows "(b::'b) = (a::'b)"