hcju/isabelleps huggingface dataset:
LFOci_refine: fixes li :: "'c list"   and l :: "'d set"   and A :: "('c \<times> 'd) set"   and R :: "('f \<times> 'e) set"   and ci :: "'f \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and fi :: "'c \<Rightarrow> 'f \<Rightarrow> 'f nres"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e nres"   and s0i :: "'f"   and s0 :: "'e"   and I :: "'d set \<Rightarrow> 'e \<Rightarrow> bool" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>s si. (si, s) \<in> R \<Longrightarrow> ci si = c s"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li ci fi s0i \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>I\<^esup> l c f s0)"
strip_all_vars_incr_bv: fixes inc :: "nat"   and lev :: "nat"   and t :: "Core.term" shows "strip_all_vars (incr_bv inc lev t) = strip_all_vars t"
MakeNT: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTransP \<eta>" shows "NatTrans (MakeNT \<eta>)"
H: shows "map (inv q) (filter (\<lambda>x. x \<in> range q) xs') \<in> divergences Q"
pfun_override_dist_comp: fixes f :: "('c, 'b) pfun"   and g :: "('c, 'b) pfun"   and h :: "('a, 'c) pfun" shows "f + g \<circ>\<^sub>p h = (f \<circ>\<^sub>p h) + (g \<circ>\<^sub>p h)"
graph_empty_e: fixes v :: "'f set" shows "graph (LG {} v)"
vrestriction_vconst_on_nex: fixes c :: "V"   and B :: "V"   and A :: "V" assumes "c \<notin>\<^sub>\<circ> B" shows "vconst_on A c \<restriction>\<^sub>\<circ> B = 0"
Subst_updates_cls: fixes X :: "'b list set"   and xrhs :: "'b trm set"   and rhs :: "'b trm set" assumes "X \<notin> rhss xrhs" shows "rhss (Subst rhs X xrhs) = rhss rhs \<union> rhss xrhs - {X}"
a2_map'_def: shows "a2_map' = Mapping.update (tp - len + 1) mc a2_map"
UN_le_eq_Un0: fixes M :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "\<Union> (M ` {..n}) = \<Union> (M ` {1..n}) \<union> M 0"
VPi_D: fixes f :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V"   and x :: "V" assumes "f \<in> elts (VPi A B)"   and "x \<in> elts A" shows "app f x \<in> elts (B x)"
length_Keys_to_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "length (Keys_to_list ps) = card (Keys (set ps))"
StrongPostdomination_axioms: fixes prog :: "cmd" shows "StrongPostdomination sourcenode targetnode kind (valid_edge prog) (_Entry_) (_Exit_)"
hfprod_def: fixes RR :: "('g \<Rightarrow> 'i \<Rightarrow> assn) \<times> ('g \<Rightarrow> 'i \<Rightarrow> assn)"   and SS :: "('h \<Rightarrow> 'j \<Rightarrow> assn) \<times> ('h \<Rightarrow> 'j \<Rightarrow> assn)" shows "RR *\<^sub>a SS \<equiv> (fst RR \<times>\<^sub>a fst SS, snd RR \<times>\<^sub>a snd SS)"
map_bind_pmf: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'c pmf"   and g :: "'c \<Rightarrow> 'b pmf" shows "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
set_eq_sym: fixes S :: "('d, 'e) eq_object_scheme"   and A :: "'d set"   and B :: "'d set" assumes "equivalence S"   and "A \<subseteq> carrier S"   and "B \<subseteq> carrier S"   and "A {.=}\<^bsub>S\<^esub> B" shows "B {.=}\<^bsub>S\<^esub> A"
Ex_status_iff: fixes P :: "status \<Rightarrow> bool" shows "(\<exists>a. P a) = (P SUCCESS \<or> P FOUND \<or> P FAILED)"
ennreal_leI: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "ennreal x \<le> ennreal y"
zs'_def: shows "zs' = snd (poly_pfd_simple p cs)"
matches_rule_iptables_bigstep: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and m' :: "'c match_expr"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p = matches \<gamma> m' p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m' a], s\<rangle> \<Rightarrow> t"
X: fixes H' :: "('a, 'b) pre_digraph"   and H :: "('a, 'b) pre_digraph" assumes "strongly_connected H'"   and "induced_subgraph H' G"   and "max_subgraph strongly_connected H"   and "verts H \<subseteq> verts H'" shows "H = H'"
wf_mlex: fixes R :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> nat" assumes "wf R" shows "wf (f <*mlex*> R)"
ltl_const_equiv_or_const_1: fixes \<phi>\<^sub>1 :: "'a ltln"   and \<phi>\<^sub>2 :: "'a ltln" shows "\<phi>\<^sub>1 or\<^sub>n \<phi>\<^sub>2 \<sim>\<^sub>C true\<^sub>n = (\<phi>\<^sub>1 \<sim>\<^sub>C true\<^sub>n \<or> \<phi>\<^sub>2 \<sim>\<^sub>C true\<^sub>n)"
homeomorphic_maps_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "homeomorphic_maps X Y f g = (homeomorphic_map X Y f \<and> homeomorphic_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y))"
assms_1: shows "(\<lambda>x. ereal (real (u x))) \<longlonglongrightarrow> \<infinity>"
msg_sender_def: fixes m :: "msg" shows "msg_sender m \<equiv> case m of Rreq x xa xb xc xd xe ipc \<Rightarrow> ipc | Rrep x xa xb xc ipc \<Rightarrow> ipc | Rerr x ipc \<Rightarrow> ipc | Pkt x xa ipc \<Rightarrow> ipc"
le_enn2realI: fixes x :: "real"   and y :: "ennreal" assumes "ennreal x \<le> y"   and "y = \<top> \<Longrightarrow> x \<le> 0" shows "x \<le> enn2real y"
is_Prim_def: fixes "term" :: "'b local.term" shows "is_Prim term = (\<exists>x2. term = \<^bold>\<langle>x2\<^bold>\<rangle>)"
one_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" shows "\<one>\<^bsub>subgroup_generated G S\<^esub> = \<one>\<^bsub>G\<^esub>"
iffs: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'a"   and break_status' :: "bool"   and return_status' :: "bool"   and more' :: "'a" shows "(\<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = \<lparr>break_status = break_status', return_status = return_status', \<dots> = more'\<rparr>) = (break_status = break_status' \<and> return_status = return_status' \<and> more = more')"
normalize_quot_normalize_left: fixes x :: "'b"   and y :: "'b" shows "normalize_quot (normalize x, y) = (fst (normalize_quot (x, y)) div unit_factor x, snd (normalize_quot (x, y)))"
swap0_swap0: fixes n :: "nat"   and i :: "nat"   and x :: "nat \<Rightarrow>\<^sub>0 'b" shows "swap0 n i (swap0 n i x) = x"
simps_7: fixes V :: "'f"   and A :: "'f set option" shows "\<D> (Var V) A = (V \<in>\<in> A)"
reads_consistent_flush_other': fixes sb :: "'b memref list"   and xs :: "'c memref list"   and RO :: "nat set"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<inter> (outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<union> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b xs \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) xs) \<union> (outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) xs) - RO) \<union> (\<O> \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) xs))) = {}"   and "reads_consistent False \<O> m xs"   and "read_only_reads \<O> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) xs) \<subseteq> RO" shows "reads_consistent False \<O> (flush sb m) xs"
inj_idom_hom_axioms: shows "inj_idom_hom of_int"
rel_envirD: fixes p :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f" assumes "rel_envir p f g" shows "p (f x) (g x)"
degree_eq_length_coeffs: fixes p :: "'b poly" shows "degree p = length (coeffs p) - 1"
mult_numeral_1_right: fixes a :: "'a" shows "a * Numeral1 = a"
non_redundant_def: fixes S :: "'at Literal set set" shows "non_redundant S = (\<forall>C. C \<in> S \<longrightarrow> \<not> strictly_redundant C S)"
elims_2: fixes x :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and xa :: "'b match_expr list"   and xb :: "action"   and xc :: "'c" assumes "match_list x xa xb xc" obtains \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)" and m :: "'b match_expr" and ms :: "'b match_expr list" and a :: "action" and p :: "'c" where "x = \<gamma>" and "xa = m # ms" and "xb = a" and "xc = p" and "if matches \<gamma> m a p then True else match_list \<gamma> ms a p"
Abs_ordinal_oStrictLimit: fixes f :: "nat \<Rightarrow> ord0" shows "Abs_ordinal (ord0rel `` {ord0_Lim f}) = oStrictLimit (\<lambda>n. Abs_ordinal (ord0rel `` {f n}))"
sortedP_Cons: fixes x :: "'a"   and xs :: "'a list" assumes "sortedP (x # xs)" obtains "\<forall>xa\<in>set xs. le x xa" and "sortedP xs"
rel_pred_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_pred R = R ===> (=)"
emeasure_ge_1_iff: fixes A :: "'a set" shows "(1 \<le> emeasure M A) = (emeasure M A = 1)"
strict_iff_order: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b \<subset># a) = (b \<subseteq># a \<and> a \<noteq> b)"
assms: shows "vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t)"
fi_f: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "g_rename_precond G f"   and "x \<in> g_V G" shows "g_rename_precond.fi G f (f x) = x"
weakly_reduced_def: fixes n :: "nat"   and fs :: "'a vec list"   and \<alpha> :: "'a"   and k :: "nat" shows "gram_schmidt_fs.weakly_reduced n fs \<alpha> k = (\<forall>i. Suc i < k \<longrightarrow> \<parallel>gram_schmidt_fs.gso n fs i\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n fs (Suc i)\<parallel>\<^sup>2)"
inj_tnam': fixes x :: "tnam'"   and y :: "tnam'" shows "(tnam' x = tnam' y) = (x = y)"
h1: shows "\<not> t mod cycleLength (nC i) mem schedule (nC i)"
fun_upd_same: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a" shows "(f(x := y)) x = y"
mat_det_12_mm_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det_12 (A *\<^sub>m\<^sub>m M) (B *\<^sub>m\<^sub>m M) = mat_det M * mat_det_12 A B"
gen_step'_refine: fixes s :: "'s" assumes "gen_rwof s"   and "gen_cond s" shows "gen_step' s \<le> gen_step s"
prems2: assumes "\<S>\<bullet>None \<tturnstile>_ i vs;cs_es : ts'"   and "\<And>k lholed. \<not> Lfilled k lholed [$Return] cs_es"   and "\<And>i k lholed. Lfilled k lholed [$Br i] cs_es \<Longrightarrow> i < k"   and "cs_es \<noteq> [Trap]"   and "\<not> const_list cs_es"   and "store_typing s \<S>" shows "\<exists>a s' vs' cs_es'. \<lparr>s;vs;cs_es\<rparr> \<leadsto>_ i \<lparr>s';vs';cs_es'\<rparr>"
n_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.n p q = card (GAUSS.E q (int p))"
OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>real\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "X->oclAsType\<^sub>R\<^sub>e\<^sub>a\<^sub>l(Boolean) = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>\<lceil>\<lceil>X \<tau>\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor> else invalid \<tau>)"
simps: fixes t :: "nat"   and a1 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and a2 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" shows "alstep t a1 a2 = (\<exists>c n ts c' n' s'. a1 = (c, n, ts) \<and> a2 = (c', n', ts(t := s')) \<and> t < N \<and> astep_sng (c, n, ts t) (c', n', s'))"
Arg_eq_pi: fixes z :: "complex" shows "(Arg z = pi) = (Re z < 0 \<and> Im z = 0)"
onode_deliver: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and i :: "nat"   and R :: "nat set" assumes "((\<sigma>, s), deliver d, \<sigma>', s') \<in> S"   and "\<forall>j. j \<noteq> i \<longrightarrow> \<sigma>' j = \<sigma> j" shows "((\<sigma>, NodeS i s R), i:deliver(d), \<sigma>', NodeS i s' R) \<in> onode_sos S"
assms_1: shows "a \<in>\<^sub>\<circ> ZFC_in_HOL.succ n"
ZObis_matchC_ZO: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01 t" shows "matchC_ZO ZObis s t"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "zero_hom hom"   and "zero_hom_0_axioms hom" shows "zero_hom_0 hom"
affine_hull_UNIV: fixes S :: "'b set" assumes "aff_dim S = int DIM('b)" shows "affine hull S = UNIV"
InvariantConsistentAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantConsistent (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "getQ state \<noteq> []"   and "\<not> getConflictFlag state" shows "let state' = applyUnitPropagate state in InvariantConsistent (getM state')"
continuous_on: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
domainCharacteristicFunction: fixes Y :: "'c set"   and X :: "'c set" shows "Domain (X <|| Y) = X \<union> Y"
sees_method_mono2: fixes C' :: "char list"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'b"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D'" shows "P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
nzero: shows "\<forall>x. c < x \<and> x \<le> d \<longrightarrow> poly p x \<noteq> 0"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "order.greater_eq b a" shows "order.greater b a"
put_put: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'b"   and u :: "'b" assumes "mwb_lens x" shows "put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
Gcd_complex_def: shows "Gcd = normalization_euclidean_semiring_class.Gcd"
distinct_17: fixes x31 :: "'b pltl"   and x32 :: "'b pltl"   and x51 :: "'b pltl"   and x52 :: "'b pltl" shows "x31 implies\<^sub>p x32 \<noteq> x51 U\<^sub>p x52"
D_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "D m i j k \<equiv> Min {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
F: fixes i :: "nat" shows "F i \<in> sets (M i)"
ivl_disj_un_14: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{l..<u} \<union> {u..} = {l..}"
normalize_prod_mset_primes: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a multiset" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "\<And>p. p \<in># A \<Longrightarrow> normalization_semidom.prime zero times one normalize p" shows "normalize (comm_monoid_mult.prod_mset times one A) = comm_monoid_mult.prod_mset times one A"
elims: fixes x :: "real poly"   and xa :: "real poly list \<times> rat mat \<times> (nat list \<times> nat list) list \<times> rat list list"   and xb :: "real poly list \<times> rat mat \<times> (nat list \<times> nat list) list \<times> rat list list"   and y :: "real poly list \<times> rat mat \<times> (nat list \<times> nat list) list \<times> rat list list" assumes "combine_systems_R x xa xb = y" obtains p :: "real poly" and qs1 :: "real poly list" and m1 :: "rat mat" and sub1 :: "(nat list \<times> nat list) list" and sgn1 :: "rat list list" and qs2 :: "real poly list" and m2 :: "rat mat" and sub2 :: "(nat list \<times> nat list) list" and sgn2 :: "rat list list" where "x = p" and "xa = (qs1, m1, sub1, sgn1)" and "xb = (qs2, m2, sub2, sgn2)" and "y = smash_systems_R p qs1 qs2 sub1 sub2 sgn1 sgn2 m1 m2"
list_decomp_1: fixes l :: "'a list" assumes "length l = 1" shows "\<exists>a. l = [a]"
vfsequence_vremove_vcons_vfsequence: fixes xs' :: "V"   and x :: "V" assumes "xs = xs' #\<^sub>\<circ> x" shows "vfsequence xs'"
WhileBodyThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and c :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
E_def: shows "E \<equiv> \<lambda>s. 1 + 5 * Discrete.log (nat (s ''r'' - s ''l''))"
assms_3: fixes n :: "nat" shows "f n \<in> space\<^sub>N (\<LL> (ennreal p) M)"
simps_13: fixes x2 :: "'d" shows "Stuck \<noteq> Abrupt x2"
const_1_bigo_n_ln_n: shows "(\<lambda>n. 1) \<in> O(\<lambda>n. real n * ln (real n))"
order_top_def: fixes X :: "'b set" shows "order_top X = topo (order_base X)"
less_divide_eq_1_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less one (divide b a) = less b a"
abs_eq: shows "\<infinity>\<^sub>h = abs_complex_homo \<infinity>\<^sub>h\<^sub>c"
\<psi>_minus_\<theta>_bigo: shows "(\<lambda>x. primes_psi x - primes_theta x) \<in> O(\<lambda>x. ln x * sqrt x)"
le_imp_less_or_eq: fixes x :: "'a list"   and y :: "'a list" assumes "prefix x y" shows "strict_prefix x y \<or> x = y"
step: fixes y :: "'a"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "y \<in> orbit f x" shows "f y \<in> orbit f x"
plus_times_less_one_lemma: fixes i :: "real"   and u :: "real" assumes "i < 1"   and "u < 1" shows "i + u - i * u < 1"
size_list_conv_sum_list: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and f :: "'e \<Rightarrow> nat"   and xs :: "'e list" assumes "class.monoid_add plus zero" shows "size_list f xs = sum_list (map f xs) + length xs"
simps_2: fixes v :: "'b" shows "degreen (v)\<^sub>p = (\<lambda>m. 0)"
rep_eq: fixes x :: "ennreal"   and xa :: "ennreal" shows "(x < xa) = (enn2ereal x < enn2ereal xa)"
simps: fixes a1 :: "'e"   and a2 :: "'e"   and r :: "('e \<times> 'e) set" shows "((a1, a2) \<in> r\<^sup>*) = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>* \<and> (b, c) \<in> r))"
c_row: shows "c_row \<le> 2 * n + (2 * n + 1 + 2 * i) * i"
valof_eq: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "\<not> IEEE.is_zero x \<or> \<not> IEEE.is_zero y" shows "(valof x = valof y) = (x = y)"
simps_4: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
conv1: shows "plossless_gpv \<I>' (run_converter conv a)"
Deliver_added_ids: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e :: "'b" assumes "xs prefix of j"   and "i \<in> set (added_ids xs e)" shows "Deliver (i, Create i e) \<in> set xs \<or> (\<exists>mo. Deliver (i, Expunge e mo i) \<in> set xs)"
pm_of_poly_sum: fixes x :: "'a"   and f :: "'c \<Rightarrow> 'b poly"   and I :: "'c set" shows "pm_of_poly x (sum f I) = (\<Sum>i\<in>I. pm_of_poly x (f i))"
mult_closed_power: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly"   and m' :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x [^]\<^bsub>R\<^esub> CARD('a) ^ m' = x"   and "y [^]\<^bsub>R\<^esub> CARD('a) ^ m' = y" shows "(x \<otimes>\<^bsub>R\<^esub> y) [^]\<^bsub>R\<^esub> CARD('a) ^ m' = x \<otimes>\<^bsub>R\<^esub> y"
terminates_implies_exec: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<exists>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
closest_point_unique: fixes S :: "'b set"   and x :: "'b"   and a :: "'b" assumes "convex S"   and "closed S"   and "x \<in> S"   and "\<forall>z\<in>S. dist a x \<le> dist a z" shows "x = closest_point S a"
sc_r_distr: fixes a :: "'b"   and m :: "'a"   and n :: "'a" assumes "a \<in> carrier R"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "a \<cdot>\<^sub>s (m \<plusminus> n) = a \<cdot>\<^sub>s m \<plusminus> a \<cdot>\<^sub>s n"
Number_LtK: fixes X :: "nat"   and X' :: "ltkey" shows "Number X \<noteq> LtK X'"
nextl_snoc: fixes q :: "hf"   and xs :: "'a list"   and x :: "'a" shows "ST.nextl q (xs @ [x]) = dfa.nxt \<lparr>dfa.states = {\<langle>q1, q2\<rangle> |q1 q2. q1 \<in> dfa.states MS \<and> q2 \<in> dfa.states MT}, init = \<langle>dfa.init MS, dfa.init MT\<rangle>, final = {\<langle>q1, q2\<rangle> |q1 q2. q1 \<in> dfa.final MS \<and> q2 \<in> dfa.final MT}, nxt = \<lambda>\<langle>qs, qt\<rangle> x. \<langle>dfa.nxt MS qs x, dfa.nxt MT qt x\<rangle>\<rparr> (ST.nextl q xs) x"
word_split: fixes v :: "nat \<Rightarrow> 'b"   and k :: "nat" obtains v\<^sub>1 :: "'b list" and v\<^sub>2 :: "nat \<Rightarrow> 'b" where "v = v\<^sub>1 \<frown> v\<^sub>2" and "length v\<^sub>1 = k"
gL_rel_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>L y"   and "y \<preceq>\<^sub>L z" shows "x \<preceq>\<^sub>L z"
swap_apply_2: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "Fun.swap a b f b = f a"
assms: shows "Var x \<rightarrow>\<^sub>\<beta> t"
coset_eq: shows "\<forall>x\<in>carrier G. B #> x = x <# B"
pt_eqvt_fun1b: fixes f :: "'b \<Rightarrow> 'c" assumes "\<forall>pi. pi \<bullet> f = f" shows "supp f = {}"
finite_total_preorder_on_iff: fixes A :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "finite_total_preorder_on A R = (total_preorder_on A R \<and> finite A)"