hcju/isabelleps huggingface dataset:
span_extend_basis: fixes B :: "'b set" assumes "local.independent B" shows "local.span (local.extend_basis B) = UNIV"
fb: shows "b = \<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr>"
induct: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'b list" assumes "\<And>Pa. P Pa [] []"   and "\<And>Pa a as b bs. P Pa as bs \<Longrightarrow> P Pa (a # as) (b # bs)"   and "\<And>Pa v va. P Pa (v # va) []"   and "\<And>Pa v va. P Pa [] (v # va)" shows "P a0 a1 a2"
simps_7: fixes vb :: "'b"   and vc :: "'b"   and vd :: "'c" shows "separate [AllowPortFromTo vb vc vd] = [AllowPortFromTo vb vc vd]"
O: shows "k < offs_next (offs (enum xs index key n mi ma) 0) (length xs) xs index key mi ma (Max {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> k \<and> 0 < offs_num n xs index key mi ma i})"
backflows_def: fixes E :: "('a \<times> 'a) set" shows "backflows E \<equiv> {(r, s). (s, r) \<in> E}"
filter_inf_absorb_sup: fixes F :: "'a set"   and G :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G" shows "filter_sup F G \<inter> G = G"
dg_is_arrE: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" obtains "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a" and "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
get_parent_pure: fixes ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (Shadow_DOM.get_parent ptr) h"
c6_1: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "f x \<le> 1 / 2 * (c6 * f_approx x)"
val_Zp_of_a_inv: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "val_Zp a = val_Zp (\<ominus> a)"
smash_systems_R_def: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list"   and subsets1 :: "(nat list \<times> nat list) list"   and subsets2 :: "(nat list \<times> nat list) list"   and signs1 :: "rat list list"   and signs2 :: "rat list list"   and mat1 :: "rat mat"   and mat2 :: "rat mat" shows "smash_systems_R p qs1 qs2 subsets1 subsets2 signs1 signs2 mat1 mat2 = (qs1 @ qs2, kronecker_product mat1 mat2, subsets_smash_R (length qs1) subsets1 subsets2, signs_smash signs1 signs2)"
sorted_sort_key: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f (linorder.sort_key less_eq f xs))"
ahm_lookup_def: fixes k :: "'b"   and hm :: "('b, 'a) ArrayHashMap_Impl.hashmap" shows "ArrayHashMap_Impl.ahm_lookup k hm = ArrayHashMap_Impl.ahm_\<alpha> hm k"
qPsubst_preserves_qFresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and zs :: "'varSort"   and z :: "'var" assumes "qGood X"   and "qGoodEnv rho"   and "qFresh zs z X"   and "qFreshEnv zs z rho" shows "qFresh zs z X #[[rho]]"
effect_success_conv: fixes c :: "unit Heap"   and h :: "heap"   and I :: "heap \<Rightarrow> bool" shows "(\<exists>h'. effect c h h' () \<and> I h') = (success c h \<and> I (snd (the (execute c h))))"
currentD_OUT_eq_0: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f" shows "(d_OUT f x = 0) = (\<forall>y. f (x, y) = 0)"
Quotient3_abs_rep: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and a :: "'f" assumes "Quotient3 R Abs Rep" shows "Abs (Rep a) = a"
DynCom: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "'b \<Rightarrow> ('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile> (DynCom c, Normal s) \<rightarrow> (c s, Normal s)"
empty_setinterleaving: fixes t :: "'a event list"   and A :: "'a event set"   and u :: "'a event list" assumes "[] setinterleaves ((t, u), A)" shows "t = []"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case Msg x2 of NoMsg \<Rightarrow> f1 | Msg x \<Rightarrow> f2 x) = f2 x2"
v_def: shows "v \<equiv> glb {x. (restrict f intY1 x, x) \<in> induced intY1 r \<and> x \<in> intY1} \<lparr>pset = intY1, order = induced intY1 r\<rparr>"
simps_1: fixes xs :: "'f"   and x :: "'g"   and xs' :: "'f"   and x' :: "'g" shows "(qVar xs x #= qVar xs' x') = (xs = xs' \<and> x = x')"
f'_alt: shows "f' = augment_edge f (u, v) \<Delta>"
eq_\<I>_converter_reflI: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv"
ActualImplE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]" shows "[\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi> in v]"
cases: fixes p :: "'a \<times> 'b"   and a :: "'b" assumes "sndsp p a" obtains "a = snd p"
IntD1: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
hm_size_def: fixes ht :: "('a, 'b) hashtable" shows "hm_size ht \<equiv> return (the_size ht)"
assms_3: shows "selected_edge h j g \<le> - forest_components f"
UnitsI_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<otimes> b = \<one>" shows "b \<in> Units G"
trelRS: shows "rel_weakly_reflects_barbs TRel TWB"
measurable_ident_sets: fixes M :: "'a measure"   and M' :: "'a measure" assumes "sets M = sets M'" shows "(\<lambda>x. x) \<in> M \<rightarrow>\<^sub>M M'"
sources_sinks: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and u :: "'a"   and xs :: "'b list" shows "sources I D u xs = sinks (I\<inverse>) D u (rev xs)"
deg': shows "degree (\<Prod>i\<leftarrow>[0..<n]. [:- cis (real i * 2 * pi / real n), 1:]) = n"
AllE: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and x :: "'a" assumes "[forall \<phi> in v]" shows "[\<phi> x in v]"
ltake_lfind_lset: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" assumes "x \<in> lset (ltake (enat (lfind P lst)) lst)" shows "\<not> P x"
sees_method_fun: 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 M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list"   and mthd' :: "ty list \<times> ty \<times> char list list \<times> expr" assumes "P \<turnstile> C has least M = mthd via Cs"   and "P \<turnstile> C has least M = mthd' via Cs" shows "mthd = mthd'"
comp_null_1: fixes f :: "'d" shows "D.null \<cdot>\<^sub>D f = D.null"
width_def: fixes G :: "('a, 'b) Graph_scheme"   and T :: "('c, 'd) Graph_scheme"   and bag :: "'c \<Rightarrow> 'a set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.width T bag \<equiv> if V\<^bsub>T\<^esub> = {} then 0 else TreeDecomposition.max_bag_card T bag - 1"
FunctorMapsTo: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ dom\<^bsub>CatDom F\<^esub> f to F @@ cod\<^bsub>CatDom F\<^esub> f"
ide_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "ide (src \<mu>)"
finsum_cong': fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> rat"   and g :: "'a \<Rightarrow> rat" assumes "A = B"   and "True"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "sum f A = sum g B"
wf_until_aux_Nil: fixes \<phi>' :: "Formula.formula"   and \<psi>' :: "Formula.formula"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and I :: "\<I>"   and n :: "nat"   and b :: "bool" assumes "fv \<phi>' \<subseteq> fv \<psi>'" shows "wf_until_aux \<sigma> V R (init_args I n (fv \<phi>') (fv \<psi>') b) \<phi>' \<psi>' (init_muaux (init_args I n (fv \<phi>') (fv \<psi>') b)) 0"
card_Diff1_less: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) < card A"
ncrit_spec_def: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" shows "ncrit_spec crit = (\<forall>d m data gs bs hs ps B q_in_bs p q. dickson_grading d \<longrightarrow> set gs \<union> set bs \<union> set hs \<subseteq> B \<longrightarrow> fst ` B \<subseteq> dgrad_p_set d m \<longrightarrow> snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs) \<longrightarrow> unique_idx (gs @ bs @ hs) data \<longrightarrow> is_Groebner_basis (fst ` set gs) \<longrightarrow> (q_in_bs \<longrightarrow> q \<in> set gs \<union> set bs) \<longrightarrow> (\<forall>p' q'. (p', q') \<in>\<^sub>p snd ` set ps \<longrightarrow> fst p' \<noteq> 0 \<longrightarrow> fst q' \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')) \<longrightarrow> (\<forall>p' q'. p' \<in> set gs \<union> set bs \<longrightarrow> q' \<in> set gs \<union> set bs \<longrightarrow> fst p' \<noteq> 0 \<longrightarrow> fst q' \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')) \<longrightarrow> p \<in> set hs \<longrightarrow> q \<in> set gs \<union> set bs \<union> set hs \<longrightarrow> fst p \<noteq> 0 \<longrightarrow> fst q \<noteq> 0 \<longrightarrow> crit data gs bs hs q_in_bs ps p q \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q))"
mtf2_forward_effect4: fixes q :: "'b"   and xs :: "'b list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "i < index xs q - n" shows "index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < index xs q - n"
pick_inverse: fixes A :: "'d quot" shows "\<lfloor>pick A\<rfloor> = A"
alg2: fixes x :: "complex" assumes "x \<in> X" shows "algebraic (\<beta> x)"
of_nat_index: fixes n :: "nat"   and i :: "'b" shows "of_nat n $ i = of_nat n"
valEF_E: fixes M :: "'a kripke"   and f :: "'a set"   and x :: "'a" assumes "M \<turnstile> EF f"   and "x \<in> init M" shows "x \<in> EF f"
set1_G_map: fixes l1 :: "'p \<Rightarrow> 'w"   and l2 :: "'q \<Rightarrow> 'x" shows "set1_G \<circ> mapl_G l1 l2 = (`) l1 \<circ> set1_G"
emeasure_sigma: fixes \<Omega> :: "'d set"   and A :: "'d set set" shows "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'k list" shows "take n (drop m xs) = drop m (take (n + m) xs)"
y: shows "(y solves_ode f) {t0--t2} X"
map_poly_cong: fixes p :: "'a poly"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = g x" shows "map_poly f p = map_poly g p"
ideal_prod_commute: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "I \<cdot> J = J \<cdot> I"
rel_pmf_of_set_bij: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bij_betw f A B"   and "A \<noteq> {}"   and "finite A"   and "B \<noteq> {}"   and "finite B"   and "\<And>x. x \<in> A \<Longrightarrow> R x (f x)" shows "rel_pmf R (pmf_of_set A) (pmf_of_set B)"
algebraic_div: fixes x :: "'b"   and y :: "'b" assumes "algebraic x"   and "algebraic y" shows "algebraic (x / y)"
pre_iso_test4: shows "\<forall>g\<in>set Quad. pre_iso_test g"
sim: shows "weak_reduction_simulation ((indRelLTPO TRel)\<inverse>) (STCal Source Target)"
continuous_on_empty: fixes f :: "'b \<Rightarrow> 'c" shows "continuous_on {} f"
circ_decompose_11: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> = (x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> * x\<^sup>\<circ>"
eventually_nhds_ball: fixes d :: "real"   and z :: "'b" assumes "0 < d" shows "\<forall>\<^sub>F x in nhds z. x \<in> ball z d"
well_valued0_update_rightmost: fixes t :: "'b tree"   and as :: "'b list"   and x :: "'b" assumes "well_shaped t"   and "well_valued0 as t"   and "length as = r t" shows "well_valued0 (butlast as @ [last as + x]) (update_rightmost (\<lambda>a. a + x) t)"
simps_1: 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 uu :: "variable"   and T :: "typ" shows "term_ok' \<Sigma> (Fv uu T) = typ_ok_sig \<Sigma> T"
pdevs_val_perm: fixes xs :: "'a list"   and ys :: "'a list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "xs <~~> ys"   and "e \<in> funcset UNIV I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> funcset UNIV I" and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
dX_singular: fixes X :: "'a set"   and x :: "'a"   and d :: "'b" assumes "inj_on Xd X"   and "x \<in> X"   and "d = Xd x" shows "Singleton_for_d.dX X d = {x}"
A: shows "\<forall>xs A. (\<forall>X\<in>A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X\<in>A. X) \<in> failures P"
T'_def: shows "T' = (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R fs k (real_of_int a))"
allUses_def_ass: fixes v :: "'e"   and g :: "'a"   and n :: "'b" assumes "v \<in> allUses g n"   and "n \<in> set (\<alpha>n g)" shows "defAss g n v"
sin_theta_div_2: shows "sin (\<theta> / 2) = sqrt (real M / real N)"
a_in_rcs: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G" shows "a \<in> H \<bullet>\<^bsub>G\<^esub> a"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "a \<cong>\<^sub>B a'" obtains f :: "'a" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f"
elims_2: fixes atom_ordering :: "('a \<times> 'a) set"   and x :: "'a Literal" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.positive x" obtains A :: "'a" where "x = Pos A"
pq_eq2: shows "int p * int q = 2 * ((int p * int q - 1) div 2) + 1"
valid_sops_stmt_sb: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)"   and s' :: "stmt"   and t' :: "nat" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)"   and "(s', t') \<in> prog_configs sb" shows "valid_sops_stmt t' s'"
comp_converter_id_left: fixes conv :: "('i, 'j, 'k, 'l) converter" shows "1\<^sub>C \<odot> conv = conv"
pd: shows "ipd (\<pi> j) pd\<rightarrow> \<pi> m"
Suc0_mod: fixes m :: "nat" assumes "m \<noteq> Suc 0" shows "Suc 0 mod m = Suc 0"
connected_space_subtopology: fixes X :: "'c topology"   and S :: "'c set" assumes "connectedin X S" shows "connected_space (subtopology X S)"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "local.merge (x # xs) (y # ys) = (if compare cmp x y = Greater then y # local.merge (x # xs) ys else x # local.merge xs (y # ys))"
Not_Ord_hpair: fixes x :: "hf"   and y :: "hf" shows "\<not> Ord \<langle>x, y\<rangle>"
1: shows "\<exists>x y z. f c * c = x * c + y * c + z * c"
C6: shows "rel_weakly_respects_barb_set Rel (STCalWB SWB TWB) {success}"
not_outside_2: fixes x :: "'alt"   and y :: "'alt" assumes "Pareto R x y" shows "y \<in> alts"
minus_exp_eq_not_mask: fixes n :: "nat" shows "- ((2::'a) ^ n) = NOT (mask n)"
ratfps_nth_bigo_square_free_factorization': fixes q :: "complex poly"   and b :: "complex"   and cs :: "(complex poly \<times> nat) list"   and R :: "real"   and k :: "nat"   and p :: "complex poly" assumes "square_free_factorization q (b, cs)"   and "q \<noteq> 0"   and "0 < R"   and "list_all (\<lambda>cl. proots_ball_card (fst cl) 0 (1 / R) = 0) cs"   and "list_all (\<lambda>cl. proots_sphere_card (fst cl) 0 (1 / R) = 0) (filter (\<lambda>cl. k < snd cl) cs)" shows "($) (fps_of_poly p / fps_of_poly q) \<in> O(\<lambda>n. of_nat n ^ k * complex_of_real R ^ n)"
simps_6: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c \<Rightarrow> 'c" shows "rec_nat f1 f2 0 = f1"
isLimOrd_def: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "isLimOrd r \<equiv> \<not> isSuccOrd r"
runStateT_zeroU: fixes s :: "'b" shows "runStateT\<cdot>zeroU\<cdot>s = mzero"
comp_def: fixes g :: "'a"   and f :: "'a" shows "g \<cdot>\<^sub>S f = (if left b f \<and> left b g \<and> cod f = local.dom g then g \<cdot> f else null)"
inv_tensor: fixes f :: "'a"   and g :: "'a" assumes "local.iso f"   and "local.iso g" shows "local.inv (f \<otimes>\<^sub>1 g) = local.inv f \<otimes>\<^sub>1 local.inv g"
pre_4: fixes S :: "'xa set"   and \<sigma> :: "'\<sigma>" assumes "S \<subseteq> S0"   and "S \<noteq> {}"   and "\<not> c \<sigma>"   and "I S \<sigma>"   and "\<forall>x\<in>S. \<forall>y\<in>S0 - S. R y x" shows "P \<sigma>"
zdiv_mult_self: fixes m :: "int"   and a :: "int"   and n :: "int" assumes "m \<noteq> 0" shows "(a + m * n) div m = a div m + n"
subtensor_combine_dims: fixes As :: "'b tensor list"   and ds :: "nat list" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "dims (subtensor_combine ds As) = length As # ds"
divide_strict_left_mono: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less zero c"   and "less zero (times a b)" shows "less (divide c a) (divide c b)"
vimage_def: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'c set" shows "f -` B \<equiv> _Collect x (f x \<in> B)"
no_multi_arcs: fixes e1 :: "'b"   and e2 :: "'b" assumes "e1 \<in> arcs G"   and "e2 \<in> arcs G"   and "arc_to_ends G e1 = arc_to_ends G e2" shows "e1 = e2"
l_inv_ex: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
mono_setup_loc_def: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "mono_setup_loc le \<equiv> \<forall>x. le x x"
rank_invar_rank_skew: fixes q :: "('c, 'd) SkewBinomialTree list" assumes "rank_invar q" shows "rank_skew_invar q"
basis_proj_def: shows "basis_proj = (SOME B. countable B \<and> topological_basis B)"
u: shows "(\<lambda>x. ereal (aa x / bb x)) \<longlonglongrightarrow> \<infinity>"
hom_mult: fixes Ra :: "('e, 'f) ring_scheme"   and Sa :: "('g, 'h) ring_scheme"   and ha :: "'e \<Rightarrow> 'g"   and h :: "'i \<Rightarrow> 'j"   and R :: "('i, 'k) ring_scheme"   and S :: "('j, 'l) ring_scheme"   and x :: "'i"   and y :: "'i" assumes "ring_hom_ring Ra Sa ha"   and "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
count_mset_lal_ge: fixes x :: "'a"   and n :: "nat"   and xs :: "('a \<times> nat) list" assumes "(x, n) \<in> set xs" shows "n \<le> count (mset (ass_list_to_single_list xs)) x"
simps_5: fixes uu :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uw :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom" shows "strip ({uu/uv/uw} CONSEQ C) = strip C"
cases: fixes a1 :: "'a"   and a2 :: "'a list"   and a3 :: "'a" assumes "trace a1 a2 a3" obtains final :: "'a" where "a1 = final__" and "a2 = []" and "a3 = final__"   | conf' :: "'a" and T :: "'a list" and final :: "'a" and conf :: "'a" where "a1 = conf" and "a2 = conf' # T" and "a3 = final" and "trace conf' T final" and "conf \<Rightarrow> conf'"
add_nonneg_nonneg: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"
finsertI1: fixes a :: "'b"   and B :: "'b fset" shows "a |\<in>| finsert a B"
sublist_Nil_left: fixes ys :: "'c list" shows "sublist [] ys"
le_log_of_power: fixes b :: "real"   and n :: "nat"   and m :: "real" assumes "b ^ n \<le> m"   and "1 < b" shows "real n \<le> log b m"
fps_shift_fps_const: fixes n :: "nat"   and c :: "'a" shows "fps_shift n (fps_const c) = (if n = 0 then fps_const c else 0)"
IH: fixes i :: "nat" assumes "sorted xs"   and "i < length xs" shows "filter_mset ((<) (xs ! i)) (mset xs) \<subseteq># mset (drop (Suc i) xs)"
holds_mono: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c stream"   and Q :: "'c \<Rightarrow> bool" assumes "holds P xs"   and "\<And>x. P x \<Longrightarrow> Q x" shows "holds Q xs"
left_tree_def: fixes s :: "'a"   and t :: "'a" shows "left_tree s t \<equiv> {u \<in> V. s \<in> set (u \<leadsto> t)}"
Ln_approx_linear: fixes z :: "complex" assumes "cmod z < 1" shows "cmod (Ln (1 + z) - z) \<le> (cmod z)\<^sup>2 / (1 - cmod z)"
interleave_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<otimes> ys = Collect (interleave' xs ys)"
bool_to_ternary_simps_1: fixes X :: "bool" shows "(bool_to_ternary X = TernaryTrue) = X"
subst_del_def: fixes \<sigma> :: "'e \<Rightarrow> 'e"   and x :: "'f \<Longrightarrow> 'e" shows "\<sigma> -\<^sub>s x = \<sigma>(x \<mapsto>\<^sub>s &x)"
take_bit_word_eq_self: fixes n :: "nat"   and w :: "'a word" assumes "LENGTH('a) \<le> n" shows "take_bit n w = w"
followed_by_summary: fixes t :: "'t"   and s1 :: "'sum"   and s2 :: "'sum" shows "results_in (results_in t s1) s2 = results_in t (followed_by s1 s2)"
a1: shows "\<psi> \<in> subfrmlsn \<phi>"
eventually_at_left_linorder: fixes b :: "'c"   and a :: "'c" assumes "b < a" shows "\<forall>\<^sub>F x in at_left a. x \<in> {b<..<a}"
domains_def: fixes f :: "'a" shows "C.domains f \<equiv> {a. C.ide a \<and> f \<cdot> a \<noteq> C.null}"
less_oStrictLimitD: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "x < oStrictLimit f" shows "\<exists>n. x \<le> f n"
smallest_prime_beyond_aux_def: fixes k :: "nat"   and n :: "nat" shows "smallest_prime_beyond_aux k n = smallest_prime_beyond n"
igSubstIGVar1_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIGVar1 MOD \<equiv> \<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x"
int_of_real_real: fixes x :: "int" shows "int_of_real (real_of_int x) = x"
HomFtorContra: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "Hom\<^bsub>C\<^esub>[\<midarrow>,X] = Hom\<^bsub>Op C\<^esub>[X,\<midarrow>]"