hcju/isabelleps huggingface dataset:
Schwartz_inequality_abs: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "\<bar>l2product S f g\<bar> \<le> l2norm S f * l2norm S g"
g: shows "\<forall>x\<in>{0..1} - S. g differentiable at x"
out_arcs_in_arcs: fixes x :: "'e"   and G :: "('f, 'e) pre_digraph"   and u :: "'f" assumes "x \<in> out_arcs G u" shows "x \<in> arcs G"
divides_cong_r: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and y' :: "'c" assumes "Group.monoid G"   and "x divides\<^bsub>G\<^esub> y"   and "y \<sim>\<^bsub>G\<^esub> y'"   and "x \<in> carrier G" shows "x divides\<^bsub>G\<^esub> y'"
inorder_delete: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c" assumes "invar t"   and "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (AA_Map.delete x t) = AList_Upd_Del.del_list x (Tree2.inorder t)"
simps: fixes f :: "'b list"   and n :: "nat"   and g :: "'b list"   and m :: "nat" shows "karatsuba_main f n g m = (if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then let ff = poly_of_list f in foldr (\<lambda>a p. smult a ff + pCons (0::'b) p) g 0 else let n2 = n div 2 in if n2 < m then case split_at n2 f of (f0, f1) \<Rightarrow> case split_at n2 g of (g0, g1) \<Rightarrow> let p1 = karatsuba_main f1 (n - n2) g1 (m - n2); p2 = karatsuba_main (coeffs_minus f1 f0) n2 (coeffs_minus g1 g0) n2; p3 = karatsuba_main f0 n2 g0 n2 in monom_mult (n2 + n2) p1 + (monom_mult n2 (p1 - p2 + p3) + p3) else case split_at n2 f of (f0, f1) \<Rightarrow> let p1 = karatsuba_main f1 (n - n2) g m in Let (karatsuba_main f0 n2 g m) ((+) (monom_mult n2 p1)))"
ireturn_rule: fixes \<Phi> :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
mult_minus2_right: fixes x :: "'b" shows "x * - (2::'b) = - (x * (2::'b))"
1: shows "f * fls_deriv R + fls_deriv f * R = 0"
strict_boundedE: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < inf b c" obtains "a < b" and "a < c"
init_tableau_valuated: fixes t :: "(nat \<times> linear_poly) list" shows "\<nabla> (init t)"
CallThrowParams: fixes es :: "char list exp list"   and vs :: "val list"   and e :: "char list exp"   and es' :: "char list exp list"   and v :: "val"   and M :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "es = map Val vs @ throw e # es'" shows "P \<turnstile> \<langle>Val v\<bullet>M(es),s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
pre_subdivFace'_Some1: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and u :: "nat"   and vol :: "nat option list"   and f21 :: "face"   and g' :: "graph" assumes "pre_subdivFace' g f v' v n (Some u # vol)"   and "f \<in> \<F> g"   and "f \<bullet> v = u \<longrightarrow> n \<noteq> 0"   and "\<V> f \<subseteq> \<V> g"   and "f21 = fst (snd (splitFace g v u f [countVertices g..<countVertices g + n]))"   and "g' = snd (snd (splitFace g v u f [countVertices g..<countVertices g + n]))" shows "pre_subdivFace' g' f21 v' u 0 vol"
Q_def: fixes g :: "'a \<Rightarrow> 'b" shows "Q g = P (Abs_poly_mapping g)"
mult_rat: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int" shows "Rat.Fract a b * Rat.Fract c d = Rat.Fract (a * c) (b * d)"
bheight_rbtreeify_g_eq_plog2I: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list"   and t :: "('a, 'b) rbt"   and kvs' :: "('a \<times> 'b) list" assumes "rbtreeify_g n kvs = (t, kvs')"   and "n \<le> Suc (length kvs)" shows "bheight t = RBT_Impl.plog2 n"
cases: fixes a1 :: "('a, 'b) Term.term"   and a2 :: "('a, 'b) Term.term" assumes "a1 \<le>\<cdot> a2" obtains t :: "('a, 'b) Term.term" and s :: "('a, 'b) Term.term" and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" where "a1 = s" and "a2 = t" and "t = s \<cdot> \<sigma>"
mset_factorsD_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and F :: "'b multiset"   and p :: "'b"   and f :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F p"   and "f \<in># F" shows "comm_semiring_1.irreducible times one zero f"
inj_comm_monoid_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_comm_monoid_mult_hom hom \<equiv> comm_monoid_mult_hom hom \<and> inj_monoid_mult_hom hom"
local_2: fixes tvs :: "t list"   and \<C> :: "t_context"   and arb_return :: "t list option"   and arb_labs :: "t list list"   and ts :: "t list"   and ts' :: "t list" assumes "store_typing s \<S>"   and "tvs = map typeof vls"   and "i < length (inst s)"   and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ tvs, label := arb_labs, return := arb_return\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> es' : ts _> ts' \<and> map typeof vls = map typeof vs'"
round_num_denom: fixes num :: "int"   and denom :: "int" shows "round_num_denom num denom = round (rat_of_int num / rat_of_int denom)"
oinvariant_weakenD: fixes A :: "('d \<times> 'e, 'f) automaton"   and S' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<times> 'e \<Rightarrow> bool"   and \<sigma> :: "'d"   and p :: "'e"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "A \<Turnstile> (S', U' \<rightarrow>) P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"   and "\<And>s s'. U s s' \<Longrightarrow> U' s s'" shows "P (\<sigma>, p)"
tss_list_def: shows "tss_list = linearize (takeWhile_queue (\<lambda>t. enat t + right I < enat nt) tss)"
known_ptrs_def: shows "ShadowRootClass.known_ptrs \<equiv> l_known_ptrs\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_known_ptrs ShadowRootClass.known_ptr"
prc_stri_imp_nstri: fixes prc :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and prl :: "'a \<Rightarrow> bool"   and f :: "'a"   and g :: "'a" assumes "precedence prc prl"   and "fst (prc f g)" shows "snd (prc f g)"
proper_uprod_def: shows "proper_uprod = case_uprod Uprod.not_equal"
inj_comm_monoid_mult_hom_axioms: shows "inj_comm_monoid_mult_hom hom"
fml_seq_valid: fixes \<phi> :: "('sf, 'sc, 'sz) formula" assumes "valid \<phi>" shows "seq_valid ([], [\<phi>])"
satisfies_bounded_fut_derivs: fixes \<AA> :: "'i"   and k :: "nat" assumes "#\<^sub>V \<AA> = idx" shows "(\<AA> \<Turnstile>\<^sub>b fut_derivs k \<psi>) = ((SNOC (zero idx) ^^ k) \<AA> \<Turnstile>\<^sub>b \<psi>)"
frechet_derivative_scaleR_fun: fixes y :: "'c \<Rightarrow> 'd"   and a :: "'c"   and x :: "real" assumes "y differentiable at a" shows "frechet_derivative (x *\<^sub>R y) (at a) = x *\<^sub>R frechet_derivative y (at a)"
lemNullLine: fixes lineA :: "'a Line"   and x :: "'a Point" assumes "direction lineA = 0"   and "inLine x lineA" shows "x = basepoint lineA"
isomorphic_implies_ide_1: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>B g" shows "B.ide f"
23: shows "bprv (imp (PP \<langle>\<phi>G\<rangle>) (cnj (PP \<langle>PP \<langle>\<phi>G\<rangle>\<rangle>) (PP \<langle>neg (PP \<langle>\<phi>G\<rangle>)\<rangle>)))"
const_in_iff: fixes c :: "'b"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>_. c) \<in> O[F](f)) = ((\<lambda>_. 1::'b) \<in> O[F](f))"
set_empty_axioms: shows "set_empty rs.\<alpha> rs.invar rs.empty"
take_bin2bl_lem: fixes m :: "nat"   and n :: "nat"   and w :: "int"   and bs :: "bool list" shows "take m (bin_to_bl_aux (m + n) w bs) = take m (bin_to_bl (m + n) w)"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b" assumes "constant_functor A B b" shows "category B"
simps_4: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e2 :: "('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr" shows "(Ite\<^sub>\<E> \<E> e1 e2) [e] = Ite (\<E> [e]) e1 e2"
inverse_equality: fixes u :: "'a"   and v :: "'a" assumes "addA u v = zeroA"   and "addA v u = zeroA"   and "u \<in> A"   and "v \<in> A" shows "maxI.additive.inverse u = v"
lexord_def: fixes r :: "('c \<times> 'c) set" shows "lexord r = {(x, y). \<exists>a v. y = x @ a # v \<or> (\<exists>u a b v w. (a, b) \<in> r \<and> x = u @ a # v \<and> y = u @ b # w)}"
simps_1: fixes P :: "'state"   and tset :: "('b, 'pred, 'act, 'effect) Tree set['b]" shows "FL_valid_Tree P (tConj tset) = Ball (set_bset tset) (FL_valid_Tree P)"
join_bot: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> \<bottom>\<^bsub>L\<^esub> = x"
flowderiv_def: fixes x0 :: "'a"   and t :: "real" shows "flowderiv x0 t = comp12 (Dflow x0 t) (blinfun_scaleR_left (f (flow0 x0 t)))"
min_None_None: fixes xs :: "('c, 'd) bintree option list" shows "(Binomial_Queue.min xs = None) = (xs = [] \<or> set xs = {None})"
WhileInvInt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and A :: "'e set"   and V :: "('e \<times> 'e) set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c P,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnno b P V c (P \<inter> - b),A"
vec_nth_Basis: fixes x :: "(real, 'a) vec"   and i :: "'a" assumes "x \<in> Basis" shows "x $ i = (if x = axis i 1 then 1 else 0)"
le_fininf_suffixE: fixes w\<^sub>1 :: "'action list"   and w :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w" obtains w\<^sub>2 :: "'action stream" where "w =\<^sub>I w\<^sub>1 @- w\<^sub>2"
VV_equivalence: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player" assumes "ParityGame G"   and "v \<in> V\<^bsub>G\<^esub>" shows "(v \<notin> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)) = (v \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>))"
word_of_int_1: shows "word_of_int 1 = 1"
AE_embed_measure': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and P :: "'d \<Rightarrow> bool" assumes "inj_on f (space M)" shows "almost_everywhere (embed_measure M f) P = (AE x in M. P (f x))"
succ_in_diff: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}" shows "(i, succ i) \<in> r \<and> succ i \<noteq> i"
appR: fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "u \<degree> s \<rightarrow>\<^sub>\<eta> u \<degree> t"
assms: shows "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l Q"
strict_bicategory_axioms: shows "strict_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg"
R_cons: fixes p :: "'a"   and p' :: "'a"   and q' :: "'a"   and q :: "'a" assumes "\<tt>\<tt> p \<le> \<tt>\<tt> p'"   and "\<tt>\<tt> q' \<le> \<tt>\<tt> q" shows "Ref p' q' \<le> Ref p q"
flat_ord_top: fixes b :: "'b"   and x :: "'b" shows "flat_ord b b x"
lm093: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" assumes "R \<noteq> {}"   and "Domain R \<inter> X \<noteq> {}" shows "R `` X \<noteq> {}"
simps: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and T :: "nat \<times> nat \<Rightarrow> 'c list"   and i :: "nat"   and k :: "nat"   and j :: "nat" shows "inner G T i k j = (if k < j then match_prods G (T (i, k)) (T (i + k, j - k)) \<cdot> inner G T i (k + 1) j else [])"
lift_transform_simps_2: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and a :: "'a"   and e :: "exp" shows "lift_transform t (up\<cdot>a) e = t a e"
intros_3: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "ce \<sigma> (Assign v e) (\<sigma>(v := e \<sigma>))"
Qs: fixes x :: "'a option list" assumes "wf_tuple n C x"   and "P x" shows "Q x = (list_all2 (\<lambda>A Qi. Qi (restrict A x)) A_pos Q_pos \<and> list_all2 (\<lambda>A Qi. \<not> Qi (restrict A x)) A_neg Q_neg)"
P1: shows "B Midpoint C D \<and> Cong A C A D"
\<Lambda>_PBD_axioms_def: fixes sizes :: "int set" shows "\<Lambda>_PBD_axioms sizes \<equiv> \<forall>k. k \<in> sizes \<longrightarrow> 2 \<le> k"
inf_left_idem: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (x \<cdot> y) = x \<cdot> y"
enat_arith_defs_7: fixes a :: "enat"   and b :: "enat" shows "a mod b \<equiv> case a of enat x \<Rightarrow> case b of enat y \<Rightarrow> enat (x mod y) | \<infinity> \<Rightarrow> a | \<infinity> \<Rightarrow> \<infinity>"
hcomplex_of_hypreal_i: fixes r :: "real star" shows "hcomplex_of_hypreal r * iii = HComplex 0 r"
measure_PiM_emb: fixes J :: "nat set"   and X :: "nat \<Rightarrow> 'a set" assumes "J \<subseteq> UNIV"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> M.events" shows "P.prob (emb UNIV J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. M.prob (X i))"
compose_conv': fixes xs :: "('c \<times> 'e) list"   and ys :: "('e \<times> 'd) list" shows "map_of (AList.compose xs ys) = map_of ys \<circ>\<^sub>m map_of xs"
in_hom_char: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] b\<guillemotright> = (Cop_S.arr f \<and> Cop_S.ide a \<and> Cop_S.ide b \<and> Cop_S.Dom f = Cop_S.Dom a \<and> Cop_S.Cod f = Cop_S.Dom b)"
psi_converg_imp_prefix': fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and x :: "nat" assumes "total_cons s"   and "eval (total_cons.r_psi s) [j, x] \<down>" shows "\<exists>t b. eval (total_cons.r_prefixes s) [t, j] \<down>= b \<and> x < e_length b \<and> eval (total_cons.r_psi s) [j, x] \<down>= e_nth b x"
minus_matrix_def: fixes A :: "'b matrix" shows "- A = apply_matrix uminus A"
k_mkAlgAuto_mkAutoSim_act_eq: fixes t :: "'s Trace" assumes "t \<in> jkbpC" shows "set \<circ> actJP k_mkAlgAuto t = set \<circ> actJP mkAutoSim t"
input2state_some: fixes i :: "nat"   and ia :: "value list"   and x :: "value" assumes "i < length ia"   and "ia ! i = x" shows "input2state ia $ i = Some x"
neq_Nil_conv: fixes xs :: "'c list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = Cons y ys)"
Cases: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A H \<turnstile> B"   and "insert (Neg A) H \<turnstile> B" shows "H \<turnstile> B"
vpra_Cons_Return: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "CFG.valid_path_rev_aux kind get_return_edges cs as" shows "CFG.valid_path_rev_aux kind get_return_edges cs (a # as)"
f_Exec_Stream_expand_aggregate_nth: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and trans_fun :: "'d message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e"   and ag :: "'e list \<Rightarrow> 'e" assumes "0 < k"   and "n < length xs" shows "f_aggregate (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c) k ag ! n = ag (f_Exec_Comp_Stream trans_fun (xs ! n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (xs \<down> n \<odot> k) c))"
top_on_post: fixes C :: "(char list \<Rightarrow> 'av) option acom"   and X :: "char list set" assumes "top_on\<^sub>c C X" shows "top_on\<^sub>o (post C) X"
distinct_19: fixes x3 :: "'b"   and x41 :: "'b rexp"   and x42 :: "'b rexp" shows "Atom x3 \<noteq> Plus x41 x42"
h1: shows "\<forall>(p, i)\<in>set []. p = (0::'a)"
diag_eq_iff: fixes x :: "'c"   and y :: "'c" shows "(diag x = diag y) = (x = y)"
int_pow_neg_int: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> - int n = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> n)"
obj_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "obj (I.map \<mu>)"
Cons_2: shows "(s, t' # ts) \<in> seq ars"
delete_conv': fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "map_of (delete k al) = (map_of al)(k := None)"
assms_3: fixes a :: "'n"   and b :: "'n" shows "Sigma_Algebra.measure lebesgue (f ` cbox a b) = m * Sigma_Algebra.measure lebesgue (cbox a b)"
bernoulli_nat_filtration: fixes p :: "real"   and M :: "bool stream measure"   and N :: "bool stream measure"   and q :: "real" assumes "infinite_coin_toss_space p M"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "0 < p"   and "p < 1" shows "infinite_cts_filtration q N (infinite_coin_toss_space.nat_filtration M)"
mult_less_cancel_left_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less c zero" shows "less (times c a) (times c b) = less b a"
prime_factorization_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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.prime_factorization zero times one normalize zero = {#}"
SAT_def: fixes \<Phi> :: "(('fsym, 'tp) efsym, 'psym) lit list set" shows "TE.SAT \<Phi> \<equiv> \<forall>\<xi>. TE.wtE \<xi> \<longrightarrow> TE.satPB \<xi> \<Phi>"
diff_union_cancelL: fixes N :: "'c multiset"   and M :: "'c multiset" shows "N + M - N = M"
mono2: fixes f :: "'b \<Rightarrow> 'a" assumes "monotone (\<sqsubseteq>) (\<le>) f" shows "monotone (\<sqsubseteq>) (\<le>) (F f)"
simps_2: fixes \<mu> :: "'a" shows "Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>trg \<mu>\<^bold>\<rangle>\<^sub>0"
MC_acc_cfg_onI: fixes cfg :: "'s cfg"   and cfg' :: "'s cfg"   and x :: "'s" assumes "(cfg, cfg') \<in> MC.acc"   and "cfg \<in> cfg_on x" shows "cfg' \<in> cfg_on (state cfg')"
msg_nGet1: fixes nReturn :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and k :: "'b"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "'b \<Rightarrow> Config"   and nStore :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "'b \<Rightarrow> nat \<Rightarrow> nat list" assumes "FlexRayController (nReturn k) recv (nC k) (nStore k) (nSend k) (nGet k)" shows "msg (Suc 0) (nGet k)"
i_append_i_take: fixes n :: "nat"   and xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "(xs \<frown> f) \<Down> n = xs \<down> n @ f \<Down> (n - length xs)"
zmult_div_aux2: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "(a - a mod b) mod b = 0"
exhaust: fixes y :: "('b, 'a) phantom" obtains x :: "'a" where "y = phantom x"
countable_insert_eq: fixes x :: "'b"   and A :: "'b set" shows "countable (insert x A) = countable A"
ipPartitioning_helper: fixes As :: "'b set set"   and ts :: "'b set list" assumes "disjoint As" shows "ipPartition (set ts) (partitioning ts As)"
coeff_keys: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'a mpoly" shows "(m \<in> keys (mapping_of p)) = (MPoly_Type.coeff p m \<noteq> (0::'a))"
Borsukian_continuous_logarithm: fixes S :: "'a set" shows "Borsukian S = (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> - {0} \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (g x))))"
bind_def: fixes f :: "['f]"   and g :: "'f \<Rightarrow> ['e]" shows "f \<bind> g = Heap.Heap (\<lambda>h. case execute f h of None \<Rightarrow> None | Some (x, h') \<Rightarrow> execute (g x) h')"
initial_segment_length_eq: fixes xs :: "'a list"   and ys :: "'a list" assumes "initial_segment xs ys"   and "length xs = length ys" shows "xs = ys"
nat_power_mult_distrib: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.nat_power one times plus zero nat_power"   and "0 < m"   and "0 < n" shows "nat_power (m * n) a = times (nat_power m a) (nat_power n a)"
get_scdom_component_subset_get_dom_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and c :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c" shows "set c \<subseteq> set sc"
pseudo_proj_preimage: fixes g :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and N :: "'a measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_True n -` g -` {g z} = pseudo_proj_True n -` pseudo_proj_True n ` g -` {g z}"
Ide_Nmlize_Ide: fixes t :: "'a E.term" assumes "E.Ide t" shows "E.Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
subsumes_accessible_def: fixes r\<^sub>1 :: "'r"   and r\<^sub>2 :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "\<A> r\<^sub>1 r\<^sub>2 s = (r\<^sub>2 \<in> RID\<^sub>L (the (s r\<^sub>1)) \<longrightarrow> LID\<^sub>S (the (s r\<^sub>2)\<^sub>\<sigma>) \<subseteq> doms (the (s r\<^sub>1)))"
assms_1: fixes a :: "'a"   and b :: "'b" assumes "mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < k \<and> mod_type_class.to_nat b + 1 < k" shows "A $ a $ b dvd A $ (a + (1::'a)) $ (b + (1::'b))"
Deduct_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool" shows "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv \<equiv> Syntax_with_Numerals_and_Connectives var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi \<and> Deduct_axioms var trm fmla Var Fvars subst eql cnj imp all exi prv"
up_below: fixes x :: "'a"   and y :: "'a" shows "(up\<cdot>x \<sqsubseteq> up\<cdot>y) = (x \<sqsubseteq> y)"