hcju/isabelleps huggingface dataset:
l2_def: shows "l2 = snd (case z' of (t, x) \<Rightarrow> (t mod \<bar>k\<bar>, int x mod \<bar>k\<bar>))"
sgas_set: shows "set sgas = consistent_sign_vectors ftrs UNIV"
bernoulli_stream_space: fixes N :: "bool stream measure"   and p :: "real" assumes "N = bernoulli_stream p" shows "space N = streams UNIV"
irbd_inj: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and \<psi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>"   and "Sup_pres \<psi>"   and "bd\<^sup>-\<^sub>\<R> \<phi> = bd\<^sup>-\<^sub>\<R> \<psi>" shows "\<phi> = \<psi>"
trancl_mono: fixes p :: "'a \<times> 'a"   and r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "p \<in> r\<^sup>+"   and "r \<subseteq> s" shows "p \<in> s\<^sup>+"
conjugate_conjr1: fixes y :: "'a" shows "conjugate ((\<rhd>) y) = (\<cdot>) y"
fps_inverse_0_iff: fixes f :: "'a fps" shows "(fps_nth (inverse f) 0 = (0::'a)) = (fps_nth f 0 = (0::'a))"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and cmp_val :: "cmp_val" shows "P (case cmp_val of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = ((cmp_val = LT \<longrightarrow> P f1) \<and> (cmp_val = EQ \<longrightarrow> P f2) \<and> (cmp_val = GT \<longrightarrow> P f3))"
ang_vec_c_bounded: fixes z1 :: "complex"   and z2 :: "complex" shows "0 \<le> \<angle>c z1 z2 \<and> \<angle>c z1 z2 \<le> pi"
ccpo_Sup_upper: fixes Sup :: "'f set \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'f set"   and x :: "'f" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "x \<in> A" shows "less_eq x (Sup A)"
reflpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
WEAK_EX_def: fixes I :: "'b \<Rightarrow> bool"   and X :: "'b set"   and P :: "'b \<Rightarrow> bool nres" shows "WEAK_EX\<^bsup>I\<^esup> X P = WHILE\<^bsup>\<lambda>(Y, b). Y \<subseteq> X \<and> (b \<longrightarrow> (\<exists>x\<in>X. I x))\<^esup> (\<lambda>(X, b). \<not> b \<and> X \<noteq> {}) (\<lambda>(X, b). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(x, X'). P x \<bind> (\<lambda>b'. RETURN (X', b' \<or> b))))) (X, False) \<bind> (\<lambda>(uu_, b). RETURN b)"
tuple_simps_1: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and h :: "('a, 'a set) Maps.arr"   and k :: "('a, 'a set) Maps.arr" assumes "Maps.commutative_square f g h k" shows "Maps.arr (Maps.tuple h f g k)"
frc: shows "FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"
assms: shows "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vdomain A"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VxV.seq g f = (seq (fst g) (fst f) \<and> seq (snd g) (snd f))"
to_fun_monom_plus: fixes a :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp"   and "g \<in> carrier (UP Zp)"   and "c \<in> carrier Zp" shows "(monom (UP Zp) a n \<oplus>\<^bsub>UP Zp\<^esub> g) \<bullet> c = a \<otimes> c [^] n \<oplus> g \<bullet> c"
C\<^sub>4: shows "\<^bold>\<forall>x. \<^bold>\<not> (\<^bold>\<forall>xa. \<^bold>\<not> (((\<^bold>\<forall>x. xa \<cdot> x \<^bold>= x \<^bold>\<leftarrow> E (xa \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> xa \<^bold>= x \<^bold>\<leftarrow> E (x \<cdot> xa))) \<^bold>\<and> E (xa \<cdot> x)))"
assms_2: fixes u :: "'a" assumes "u \<in> carrier L"   and "u \<sqsubseteq> f u" shows "u \<sqsubseteq> x"
pq_order_unique_subgrp: fixes q :: "nat"   and p :: "nat" assumes "finite (carrier G)"   and "order G = q * p"   and "prime p"   and "prime q"   and "p < q" shows "\<exists>!Q. Q \<in> subgroups_of_size q"
c_manifolds_order_le: fixes l :: "enat" assumes "l \<le> k" shows "c_manifolds l charts1 charts2"
sorted_sort: fixes cmp :: "'b comparator"   and xs :: "'b list" shows "Sorting_Algorithms.sorted cmp (Sorting_Algorithms.sort cmp xs)"
integrable_product_swap: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) f" shows "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f (y, x))"
cases: fixes x :: "ty" obtains C :: "String.literal" where "x = Class C"   | T :: "ty" where "x = T\<lfloor>\<rceil>"   | "x = Void"   | "x = Boolean"   | "x = Integer"   | "x = NT"
borel_comp: fixes A :: "'c set" assumes "A \<in> sets borel" shows "- A \<in> sets borel"
hyps_1: fixes j :: "nat" assumes "x = j - i"   and "ssorted s"   and "i \<le> j" shows "s !! i \<le> s !! j"
weight_lub_spmf: fixes Y :: "'c spmf set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "weight_spmf (lub_spmf Y) = \<Squnion> (weight_spmf ` Y)"
y_prop: shows "r < y' \<and> (\<forall>x\<in>{r<..y'}. 0 < poly [:c, b, a:] x)"
Infinitesimal_mult_disj: fixes x :: "'a star"   and y :: "'a star" assumes "x * y \<in> Infinitesimal" shows "x \<in> Infinitesimal \<or> y \<in> Infinitesimal"
safe_distance_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" shows "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<equiv> (0 \<le> v\<^sub>e \<and> 0 \<le> v\<^sub>o) \<and> a\<^sub>e < 0 \<and> a\<^sub>o < 0 \<and> s\<^sub>e < s\<^sub>o"
Red_F_of_subset_F: fixes N :: "'f set"   and N' :: "'f set" assumes "N \<subseteq> N'" shows "local.Red_F_\<G> N \<subseteq> local.Red_F_\<G> N'"
StartBallot_HInv3_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "StartBallot s s' p"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
g': shows "g' = of_int_poly_i ff_ops (Mp g)"
fairness_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "fairness P = (stable P \<and> absolute_liveness P)"
looseD_wave: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "loose \<Gamma>"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "f = (\<lambda>_. 0)"
boundSummandDest: fixes x :: "name"   and y :: "name"   and P' :: "pi"   and P :: "pi" assumes "<\<nu>x>x{y}.P' \<in> summands P" shows "False"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>F 'e" shows "(x)\<^sub>F = (\<lambda>xa. if xa \<in> dom (fmlookup x) then the (fmlookup x xa) else undefined)"
mset_ran_insert: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'a" assumes "finite r"   and "i \<notin> r" shows "mset_ran a (Set.insert i r) = add_mset (a i) (mset_ran a r)"
filter_empty_ivls_impl_def: fixes le :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and ivls :: "('b \<times> 'c) list" shows "filter_empty_ivls_impl le ivls = filter (\<lambda>(i, s). le i s) ivls"
Minimal_Truth_Soundness_HBL1iff_Compl_Pf_axioms: shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"
simps: fixes status :: "status"   and x :: "'x2" shows "r2.init_fin_final (status, x) = (status = Finished \<and> final2 x)"
pref_mod_list': fixes u :: "'b list"   and ws :: "'b list list" assumes "u \<le>np concat ws" obtains j :: "nat" and r :: "'b list" where "j < \<^bold>|ws\<^bold>|" and "r \<le>np ws ! j" and "concat (take j ws) \<cdot> r = u"
PV: shows "partn_lst_VWF (\<omega> \<up> (1 + ord_of_nat (n - 1) * ord_of_nat (k - 1))) [ord_of_nat (2 ^ (k - 1)), \<omega> \<up> (1 + ord_of_nat (n - 1))] 2"
pax4: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "|~ \<box>[P]_v = (_actrans P v \<and> \<circle>\<box>[P]_v)"
is_extensional: fixes f :: "'a" assumes "\<not> \<sigma>.A.arr f" shows "\<sigma> f = \<sigma>.B.null"
axiom_iffimp2_def: fixes p :: "fol fm"   and q :: "fol fm" shows "axiom_iffimp2 p q \<equiv> Thm (Imp (Iff p q) (Imp q p))"
prime_factorization_uminus: fixes x :: "'b" shows "prime_factorization (- x) = prime_factorization x"
quotients_length: shows "length normal\<GG>butlast.quotients + 1 = length (take n \<GG>)"
less': shows "P \<turnstile> \<tau> \<le>' \<tau>'"
poly_pos: fixes p :: "real poly" shows "(\<forall>x. 0 < poly p x) = (poly_inf p = 1 \<and> (\<forall>x. poly p x \<noteq> 0))"
Seq_MtransT_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.MtransT tval aval (c1, s) s'" shows "PL.MtransC tval aval (c1 ;; c2, s) (c2, s')"
Gromov_extension_isometry_inv_2: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry f" shows "bij (Gromov_extension f)"
scast_down_range: fixes w :: "'a Word.word" assumes "sint w \<in> sints LENGTH('b)" shows "sint w = sint (SCAST('a \<rightarrow> 'b) w)"
\<gamma>_st_def: fixes \<gamma> :: "'c \<Rightarrow> 'b set"   and F :: "'c st" shows "\<gamma>_st \<gamma> F = {f. \<forall>x. f x \<in> \<gamma> (fun F x)}"
bind_get: fixes f :: "'s \<Rightarrow> 'm"   and g :: "'a option \<Rightarrow> 'm" shows "bind (get f) g = get (\<lambda>s. bind (f s) g)"
cmp_simps_1: fixes \<mu>\<nu> :: "'a \<times> 'a" assumes "S.VV.arr \<mu>\<nu>" shows "B.arr (cmp \<mu>\<nu>)"
less_imp_triv: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y" shows "(less y x \<longrightarrow> P) = True"
assms: shows "f \<bullet> ns n \<noteq> \<zero>"
\<sigma>W1_2: fixes v :: "'a" assumes "v \<in> W1" shows "winning_strategy p** \<sigma>W1 v"
change_respecting_doesnt_modify': fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and Y :: "'Var set"   and x :: "'Var" assumes "(cms, mem) \<leadsto> (cms', mem')"   and "\<forall>f. dom f = Y \<longrightarrow> (cms, subst f mem) \<leadsto> (cms', subst f mem')"   and "x \<in> Y" shows "mem x = mem' x"
fact_psi_bound_odd: fixes k :: "nat"   and n :: "nat" assumes "odd k" shows "ln (fact n) - 2 * ln (fact (n div 2)) \<le> (\<Sum>d = 1..k. (- 1) ^ (d + 1) * psi (n div d))"
BinOpRedsThrow2: fixes e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and r :: "nat \<times> char list list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Throw r,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw r,s\<^sub>2\<rangle>"
disj_one_right: fixes x :: "'c" shows "x OR - (1::'c) = - (1::'c)"
idom: fixes R :: "('e, 'f) Ring_scheme"   and a :: "'e"   and b :: "'e" assumes "Idomain R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>" shows "a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>"
map_filter_simps_1: fixes f :: "'d \<Rightarrow> 'c option"   and x :: "'d"   and xs :: "'d list" shows "List.map_filter f (x # xs) = (case f x of None \<Rightarrow> List.map_filter f xs | Some y \<Rightarrow> y # List.map_filter f xs)"
pair_by_index_inj: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "pair_by_index x1 y1 = pair_by_index x2 y2" shows "x1 = x2 \<and> y1 = y2"
l3_def: shows "l3 \<equiv> \<lparr>init = l3_init, trans = l3_trans, obs = id\<rparr>"
vector_to_cblinfun_adj_apply: fixes \<psi> :: "'b"   and \<phi> :: "'b" shows "vector_to_cblinfun \<psi>* *\<^sub>V \<phi> = of_complex (\<psi> \<bullet>\<^sub>C \<phi>)"
elims: fixes x :: "'a arctic_delta"   and xa :: "'a arctic_delta"   and y :: "'a arctic_delta" assumes "x * xa = y" obtains y :: "'a arctic_delta" where "xa__ = MinInfty_delta" and "xa = yb__" and "ya__ = MinInfty_delta"   | v :: "'a" where "xa__ = Num_arc_delta v" and "xa = MinInfty_delta" and "ya__ = MinInfty_delta"   | x :: "'a" and y :: "'a" where "xa__ = Num_arc_delta x" and "xa = Num_arc_delta y" and "ya__ = Num_arc_delta (x + y)"
fixing_ide_gives_functor_1: fixes a1 :: "'a" assumes "ide a1" shows "functor (\<cdot>) (\<cdot>) (\<lambda>f2. T (a1, f2))"
closed_map_iff_upper_hemicontinuous_preimage: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and T :: "'a set" assumes "f ` S \<subseteq> T" shows "(\<forall>U. closedin (top_of_set S) U \<longrightarrow> closedin (top_of_set T) (f ` U)) = (\<forall>U. openin (top_of_set S) U \<longrightarrow> openin (top_of_set T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> U})"
mult_coprime: fixes a :: "nat"   and b :: "nat" assumes "coprime a b" shows "g (a * b) = g a * g b"
check_poly_strict_mono_discrete_def: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "('c monom \<times> 'b) list"   and v :: "'c" shows "check_poly_strict_mono_discrete gt p v \<equiv> check_poly_gt gt (poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'b), PVar v] else PVar w)) p) p"
kuhn_lemma: fixes p :: "nat"   and n :: "nat"   and label :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" assumes "0 < p"   and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. label x i = 0 \<or> label x i = 1)"   and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow> label x i = 0)"   and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow> label x i = 1)" obtains q :: "nat \<Rightarrow> nat" where "\<forall>i<n. q i < p" and "\<forall>i<n. \<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and> (\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and> label r i \<noteq> label s i"
abs_eq: fixes xa :: "nat \<Rightarrow>\<^sub>0 nat"   and x :: "'a" shows "monom xa x = MPoly (Poly_Mapping.single xa x)"
step_4: shows "P,E,hp s,shp s \<turnstile> e : T"
ffold_empty: fixes fa :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "comp_fun_commute fa" shows "ffold f z {||} = z"
until_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(P \<U> Q) = (\<lambda>\<sigma>. \<exists>i. Q (\<sigma> |\<^sub>s i) \<and> (\<forall>k<i. P (\<sigma> |\<^sub>s k)))"
cancel_left_a: fixes b :: "'a"   and a :: "'a" shows "b l\<rightarrow> a * b = a"
simps_6: fixes x8 :: "('a, 'b) form"   and y8 :: "('a, 'b) form" shows "(Forall x8 = Forall y8) = (x8 = y8)"
Lim_ident_at: fixes x :: "'b"   and s :: "'b set" assumes "at x within s \<noteq> bot" shows "netlimit (at x within s) = x"
assms_1: shows "is_loc_connective_on A ord (\<rightarrow>)"
eval_to_evaln: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and Y' :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y', s')"   and "type_ok G t s"   and "wf_prog G" shows "\<exists>n. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y', s')"
cases: fixes x :: "'a exp list" obtains e :: "'a exp" and es :: "'a exp list" where "x = e # es"   | "x = []"
is_normal: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and canonical_basis :: "'b list"   and x :: "'b" assumes "class.onb_enum scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner canonical_basis"   and "x \<in> set canonical_basis" shows "norm x = 1"
map_indets_plus: fixes f :: "'e \<Rightarrow> 'c"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "map_indets f (p + q) = map_indets f p + map_indets f q"
ls_update_map_of_neq: fixes x :: "'a"   and k :: "'a"   and v :: "'b"   and ls :: "('a \<times> 'b) list" assumes "x \<noteq> k" shows "map_of (fst (ls_update k v ls)) x = map_of ls x"
tab_def: shows "r\<^sub>0s\<^sub>1.tab \<equiv> r\<^sub>0s\<^sub>1.T0.trnr\<^sub>\<eta> p\<^sub>1 (rep (r\<^sub>0\<^sup>* \<star> s\<^sub>1))"
assms_2: shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
fcomp_vinsert_right: fixes r :: "V"   and a :: "V"   and b :: "V"   and s :: "V" shows "r \<circ>\<^sub>\<bullet> vinsert [a, b]\<^sub>\<circ> s = r \<circ>\<^sub>\<bullet> set {[a, b]\<^sub>\<circ>} \<union>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> s"
map_1: fixes f :: "'c \<Rightarrow> 'b" shows "map_option f None = None"
wf_ineq_append': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list"   and X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t V (S @ \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S')"
non_empty_elem_in: fixes i :: "nat_int" shows "(i \<noteq> bot) = (\<exists>n. n \<^bold>\<in> i)"
assms: shows "wf_fmla_atom (ty_term Q constT) a"
ANR_imp_locally_path_connected: fixes S :: "'a set" assumes "ANR S" shows "locally path_connected S"
eventually_conjE: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. P x \<and> Q x" obtains "eventually P F" and "eventually Q F"
simps_2: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" shows "equal_mod_timpls TI (Fun f T) (Fun g S) = ((f = g \<or> is_Abs f \<and> is_Abs g \<and> ((the_Abs f, the_Abs g) \<in> set TI \<or> (the_Abs g, the_Abs f) \<in> set TI \<or> (\<exists>ti\<in>set TI. (the_Abs f, snd ti) \<in> set TI \<and> (the_Abs g, snd ti) \<in> set TI))) \<and> list_all2 (equal_mod_timpls TI) T S)"
elims: fixes x :: "Monitor.args"   and xa :: "nat"   and xb :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and y :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping" assumes "shift_end x xa xb = y" obtains args :: "Monitor.args" and nt :: "nat" and t :: "nat" and gc :: "nat" and maskL :: "bool list" and maskR :: "bool list" and data_prev :: "(nat \<times> 'a option list set) queue" and data_in :: "(nat \<times> 'a option list set) queue" and tuple_in :: "('a option list, nat) mapping" and tuple_since :: "('a option list, nat) mapping" where "x = args" and "xa = nt" and "xb = (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)" and "y = (let I = args_ivl args; data_prev' = dropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_prev; (data_in, discard) = takedropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_in; tuple_in = fold (\<lambda>(t, X) tuple_in. Mapping.filter (filter_cond X tuple_in t) tuple_in) discard tuple_in in (t, gc, maskL, maskR, data_prev', data_in, tuple_in, tuple_since))"
of_bool_half_eq_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "bool" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (zero_neq_one.of_bool one zero b) (numeral.numeral one plus (num.Bit0 num.One)) = zero"
Gamma_series_LIMSEQ: fixes z :: "'a" shows "Gamma_series z \<longlonglongrightarrow> Gamma z"
nice_ind_inv_is_inductive_invariant: shows "inductive_invariant (\<lambda>s. \<S>\<^sub>G s \<and> \<A>\<^sub>G s)"
prems_2: shows "subst \<rho> s \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
sgn_less: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (sgn a) zero = less a zero"
cases: fixes r :: "'b span_data" obtains Leg0 :: "'b" and Leg1 :: "'b" where "r = \<lparr>Leg0 = Leg0, Leg1 = Leg1\<rparr>"
cos_c_: fixes z1 :: "complex"   and z2 :: "complex" shows "cos (\<angle>c z1 z2) = cos (\<angle> z1 z2)"
runit_char_1: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>B.runit f : L f \<rightarrow>\<^sub>B f\<guillemotright>"
congruent_triangle_sas_7: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "dist a1 b1 = dist b2 a2"
ex_cm_imp_conds_2: fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "\<exists>c'. next_change_multiplicity' c c' loc t n" shows "n \<noteq> 0"
DRLEX_def: shows "DRLEX = Abs_nat_term_order (deg_comp (pot_comp (\<lambda>x y. lex_comp y x)))"
prod_list_remove1: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "prod_list (remove1 x xs) * x = prod_list xs"
m_comm: fixes x :: "int"   and y :: "int" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y = y \<oplus> x"
li_le_dim_2: fixes A :: "'c set" assumes "fin_dim"   and "A \<subseteq> carrier V"   and "lin_indpt A" shows "card A \<le> dim"
s1: shows "t = p + s1 *\<^sub>R (q - p)"
transfer: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (list_all2 R) (list_R1.rel_list_R1 R) list_R1.list_to_list_R1 list_R1.list_to_list_R1"
a0_2: shows "0 < PZ.prob {x. a0 < \<bar>x\<bar>}"
AsCatchExpr2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com"   and a :: "'e set"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com" assumes "assertionsR \<Gamma> \<Theta> Q A a2 c2 a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnComp a1 a2) (Catch c1 c2) a"
absorb_iff2: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (sup_class.sup a b = b)"