hcju/isabelleps huggingface dataset:
fds_ln_completely_multiplicative: fixes f :: "'a fds"   and l :: "'a" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_nth f 1 \<noteq> (0::'a)" shows "fds_ln l f = fds (\<lambda>n. if n = 1 then l else fds_nth f n * mangoldt n /\<^sub>R ln (real n))"
rep_eq: fixes x :: "literal"   and xa :: "literal" shows "explode (x + xa) = explode x @ explode xa"
trace_parallel_plan_strips_head_is_initial_state: fixes I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" shows "trace_parallel_plan_strips I \<pi> ! 0 = I"
I_def: fixes v :: "nat" shows "I v \<equiv> if v < length astI then Some (astI ! v) else None"
matrix_inv_right: fixes M :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "invertible M" shows "M ** matrix_inv M = Finite_Cartesian_Product.mat (1::'b)"
daf: fixes k :: "nat" assumes "k < n" shows "adjoint (f k) * f k \<in> carrier_mat d d"
intro: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real" assumes "\<And>e ts c. 0 < sensors e ts c" shows "sensors sensors"
UP_ring_subring_one: fixes S :: "'a set" assumes "subring S R" shows "\<one>\<^bsub>UP R\<^esub> = \<one>\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub>"
cont_Seq_notFinished: fixes c1 :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat"   and c2 :: "('test, 'atom, 'choice) cmd" assumes "\<not> finished (local.cont c1 s i)" shows "local.cont (c1 ;; c2) s i = local.cont c1 s i ;; c2"
term_trans: fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<leftrightarrow> t"   and "t \<leftrightarrow> u" shows "s \<leftrightarrow> u"
card_set_tree_bst: fixes t :: "'a tree" assumes "bst t" shows "card (set_tree t) = size t"
directional_derivative_linear_on_diff_fun_space: fixes k :: "enat"   and a :: "'c"   and x :: "'c" assumes "k \<noteq> 0" shows "linear_on (manifold_eucl.diff_fun_space k) UNIV (*\<^sub>R) (*\<^sub>R) (directional_derivative k a x)"
AE_pmfI: fixes M :: "'e pmf"   and P :: "'e \<Rightarrow> bool" assumes "\<And>y. y \<in> set_pmf M \<Longrightarrow> P y" shows "almost_everywhere (measure_pmf M) P"
\<R>\<^sub>\<beta>_def: shows "\<R>\<^sub>\<beta> \<equiv> {Regions_Beta.region X I J r |I J r. Regions_Beta.valid_region X k I J r}"
execute_2: shows "ran (fst fP) \<subseteq> {0..2 * card (nodes A)}"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_3: fixes c :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (\<langle>c: t \<doteq> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
loxodromic_isometry_def: fixes f :: "'b \<Rightarrow> 'b" shows "loxodromic_isometry f = (isometry f \<and> \<not> elliptic_isometry f \<and> stable_translation_length f \<noteq> 0)"
hyps_2: fixes thesis :: "bool" assumes "\<And>A\<^sub>P' \<Psi>\<^sub>P' \<Psi>'. \<lbrakk>extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>; \<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'; A\<^sub>P' \<sharp>* P'; A\<^sub>P' \<sharp>* C; distinct A\<^sub>P'\<rbrakk> \<Longrightarrow> thesis"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>P \<sharp>* P" shows "thesis"
oexp_mult_commute: fixes \<alpha> :: "V"   and j :: "nat" assumes "Ord \<alpha>" shows "\<alpha> \<up> ord_of_nat j * \<alpha> = \<alpha> * \<alpha> \<up> ord_of_nat j"
bisimilar_eqvt: fixes P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "P \<sim>\<cdot> Q" shows "(p \<bullet> P) \<sim>\<cdot> (p \<bullet> Q)"
Rep_ell2: fixes x :: "'c ell2" shows "Rep_ell2 x \<in> {x. has_ell2_norm x}"
empty_def: shows "Containers_Userguide.empty \<equiv> Abs_trie trie_empty"
LeftDerivation_breakdown': fixes u :: "symbol list"   and v :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and w :: "symbol list" assumes "LeftDerivation (u @ v) D w" shows "\<exists>n w1 w2. n \<le> length D \<and> w = w1 @ w2 \<and> LeftDerivation u (take n D) w1 \<and> derivation_ge (drop n D) (length w1) \<and> LeftDerivation v (derivation_shift (drop n D) (length w1) 0) w2"
alpha_qOp_iff: fixes X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and delta' :: "'j"   and inp' :: "'f \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qTerm option"   and binp' :: "'g \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qAbs option" assumes "FixVars TYPE('d) TYPE('e)" shows "(X #= qOp delta' inp' binp') = (\<exists>inp binp. X = qOp delta' inp binp \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
props_6: shows "\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>"
opp_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p :: "'c point" assumes "ell_field R"   and "ell_field.on_curve R a b p" shows "ell_field.on_curve R a b (ell_field.opp R p)"
b_e_type_checker_compose: fixes \<C> :: "t_context"   and es :: "b_e list"   and t1s :: "t list"   and t2s :: "t list"   and e :: "b_e"   and t3s :: "t list" assumes "b_e_type_checker \<C> es (t1s _> t2s)"   and "b_e_type_checker \<C> [e] (t2s _> t3s)" shows "b_e_type_checker \<C> (es @ [e]) (t1s _> t3s)"
phi_6: shows "\<phi> -` {0..1} \<subseteq> {0..1}"
akra_bazzi_recursion_axioms: shows "akra_bazzi_recursion x\<^sub>0 x\<^sub>1 k as bs ts"
preserves_dom: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "local.dom (L f) = L (local.dom f)"
mono: fixes x :: "int"   and z :: "int"   and y :: "int" assumes "int_mono x"   and "z < y"   and "0 \<le> x" shows "x * z < x * y"
fmrel_to_trancl: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'f \<rightharpoonup>\<^sub>f 'e"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "reflp r"   and "fmrel r\<^sup>+\<^sup>+ xm ym" shows "(fmrel r)\<^sup>+\<^sup>+ xm ym"
fin_dim_li_fin: fixes A :: "'a mod_ring poly set" assumes "Berlekamp_subspace.fin_dim"   and "Berlekamp_subspace.lin_indpt A"   and "A \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})" shows "finite A"
cwiseext_compat_list: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and xs :: "'b list"   and xs' :: "'b list" assumes "gt y x" shows "cwiseext gt (xs @ y # xs') (xs @ x # xs')"
lfilled_size: fixes j :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and LI :: "e list" assumes "Lfilled j lholed es LI" shows "size_list size es \<le> size_list size LI"
hgf_app: assumes "2\<^sub>\<nat> \<in>\<^sub>\<circ> 3\<^sub>\<nat>" shows "hgf\<lparr>2\<^sub>\<nat>\<rparr> \<in>\<^sub>\<circ> (if 2\<^sub>\<nat> = []\<^sub>\<circ> \<Rightarrow> \<AA> | 2\<^sub>\<nat> = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>)\<lparr>Arr\<rparr>"
STCal_def: fixes Source :: "'a processCalculus"   and Target :: "'b processCalculus" shows "STCal Source Target \<equiv> \<lparr>Reductions = \<lambda>P P'. (\<exists>SP SP'. P = SourceTerm SP \<and> P' = SourceTerm SP' \<and> SP \<longmapsto>Source SP') \<or> (\<exists>TP TP'. P = TargetTerm TP \<and> P' = TargetTerm TP' \<and> TP \<longmapsto>Target TP')\<rparr>"
max_stutter_sampler: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "stutter_sampler (max_stutter_sampler \<sigma>) \<sigma>"
union_mset_add_mset_left: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "add_mset a A + B = add_mset a (A + B)"
C: shows "\<forall>x<a. poly p x \<noteq> 0"
cyclic_group_commute: fixes G :: "('d, 'e) cyclic_group_scheme"   and a :: "'d"   and b :: "'d" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> b = b \<otimes>\<^bsub>G\<^esub> a"
nth_Cons_simp_1: fixes n :: "Integer"   and x :: "'b"   and xs :: "['b]" assumes "eq\<cdot>n\<cdot>0 = TT" shows "(x : xs) !! n = x"
simps_2: shows "sch_to_pr Base_suc = Suc"
Minimal_Solutions_imp_Solutions: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Minimal_Solutions a b" shows "(x, y) \<in> Solutions a b"
fgs_mono: fixes H :: "'a set"   and J :: "'a set"   and K :: "'a set" assumes "free_generator R M H"   and "J \<subseteq> K"   and "K \<subseteq> H" shows "fgs R M J \<subseteq> fgs R M K"
subseteq_guards_Call: fixes c :: "('d, 'e, 'f) com"   and p :: "'e" assumes "c \<subseteq>\<^sub>g Call p" shows "c = Call p"
R: shows "r \<longlonglongrightarrow> R \<and> (\<exists>k. (\<lambda>j. \<theta> j - real_of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>)"
EndPhase1_HInv3: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase1 s s' t"   and "HInv3_inner s p q d" shows "HInv3_inner s' p q d"
poly_mult_const'_1: fixes p :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R" shows "poly_mult [a] p = local.normalize (map ((\<otimes>) a) p)"
distinct_17: fixes x101 :: "stmt"   and x102 :: "stmt" shows "Skip \<noteq> x101 Finally x102"
continuous_on_subset: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'a set" assumes "continuous_on s f"   and "t \<subseteq> s" shows "continuous_on t f"
len_lexext_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "len_lexext gt ys xs" shows "len_lexext gt' ys xs"
Imp_intro: fixes M :: "('b, 'c) kripke"   and s :: "'c"   and p :: "'b fm"   and q :: "'b fm" assumes "M, s \<Turnstile> p \<Longrightarrow> M, s \<Turnstile> q" shows "M, s \<Turnstile> p \<^bold>\<longrightarrow> q"
mem_restrI: fixes y :: "'b list"   and A :: "'b list set"   and n :: "nat"   and V :: "nat set"   and x :: "'b option list" assumes "y \<in> A"   and "length y = n"   and "wf_tuple n V x"   and "\<forall>i\<in>V. x ! i = Some (y ! i)" shows "mem_restr A x"
distrib_3: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
assms_1: shows "p \<in> sig_inv_set' (length fs)"
assms_2: shows "(\<box>(\<lambda>s. enabled s \<longrightarrow> (\<circle>(\<lambda>s. enabled s \<or> taken s)) s)) \<sigma>"
R: shows "(si, l) \<in> \<langle>vrel\<rangle>stack_rel lrel srel"
elims_3: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b stream" assumes "\<not> holds x xa" obtains P :: "'b \<Rightarrow> bool" and xs :: "'b stream" where "x = P" and "xa = xs" and "\<not> P (shd xs)"
set_iterator_rev_linord_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'a set" shows "set_iterator_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>x y. y \<le> x)"
g_def: shows "g \<equiv> extend (fst G'') R f"
weakBisimE_4: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<rhd> Q \<approx> P"
concurrent_store_expunge_independent_technical2: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e1 :: "'b"   and mo2 :: "'a"   and r :: "'a"   and e :: "'b"   and mo :: "'a" assumes "xs prefix of j"   and "(i, Store e1 mo2 i) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e mo r) \<in> set (node_deliver_messages xs)" shows "mo2 \<noteq> r"
iterate: fixes Y :: "'a set"   and p :: "'a" assumes "\<And>q. q \<in> Y \<Longrightarrow> P q"   and "p \<in> f Y" shows "P p"
T: shows "S \<inter> {x. a \<bullet> x = b} face_of S"
ackloop_dom: fixes l :: "nat list" shows "ackloop_dom l"
dgrad_p_set_varnum: fixes X :: "'a set" shows "punit.dgrad_p_set (varnum X) 0 = P[X]"
mult_minus_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ring plus zero minus uminus times" shows "times a (uminus b) = uminus (times a b)"
list_all2_replicate: fixes xs :: "'a list" shows "list_all2 (\<le>) xs xs"
mt_super: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and CDef :: "classDef"   and m :: "nat"   and D :: "nat"   and Bs :: "nat list"   and B :: "nat" assumes "CT C = Some CDef"   and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = None"   and "cSuper CDef = D"   and "mtype(CT,m,D) = Bs \<rightarrow> B" shows "mtype(CT,m,C) = Bs \<rightarrow> B"
put_def: fixes s :: "'d" shows "put s \<equiv> \<lambda>_. (((), s), False)"
ab1_ba1: shows "a * - Q $$ (0, 1) = b * Q $$ (1, 1)"
assms_6: shows "M' = p @ [(bLiteral, False)]"
star_invol: fixes x :: "'a" shows "(x\<^sup>\<star>)\<^sup>\<star> = x\<^sup>\<star>"
eval_G2': fixes s :: "complex" shows "eval_fps G2' (s - 1) = pre_zeta 1 s"
has_derivative_vec_lambda: fixes f :: "real \<Rightarrow> ('c, 'b) vec"   and f' :: "real \<Rightarrow> ('c, 'b) vec"   and x :: "real"   and T :: "real set" assumes "\<forall>i. D (\<lambda>t. f t $ i) \<mapsto> (\<lambda>h. h *\<^sub>R f' x $ i) at x within T" shows "D f \<mapsto> (\<lambda>h. h *\<^sub>R f' x) at x within T"
pred_numeral_simps_2: fixes k :: "num" shows "pred_numeral (num.Bit0 k) = numeral (Num.BitM k)"
r_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<otimes> inv x = \<one>"
inject_2: fixes x31 :: "'b pltl"   and x32 :: "'b pltl"   and y31 :: "'b pltl"   and y32 :: "'b pltl" shows "(x31 implies\<^sub>p x32 = y31 implies\<^sub>p y32) = (x31 = y31 \<and> x32 = y32)"
anycast_def: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a seq_action" shows "anycast P a \<equiv> case a of broadcast m \<Rightarrow> P m | groupcast x m \<Rightarrow> P m | unicast x m \<Rightarrow> P m | _ \<Rightarrow> True"
dc: shows "dim_col (submatrix A I J) = dim_col (submatrix (submatrix A I UNIV) UNIV J)"
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and stmt :: "stmt" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; \<And>x7a. x7a \<in> set x7 \<Longrightarrow> P2 x7a\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)" shows "P3 stmt"
min_dist_suc: fixes v :: "nat"   and v' :: "nat"   and d :: "nat" assumes "connected v v'"   and "min_dist v v' = Suc d" shows "\<exists>v''. connected v v'' \<and> (v'', v') \<in> E \<and> min_dist v v'' = d"
chain_def: fixes A :: "'c set set"   and C :: "'c set set" shows "subset.chain A C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"
retract1_bc: fixes x :: "'Fbc"   and y :: "'Fb" assumes "R (\<pi>_Fbcb x) y" shows "\<exists>z. Ebc x z \<and> y = \<pi>_Fbcb z"
pmaxsimpchain_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "pgallery (xs @ ys)" shows "pgallery xs"
upred_eq_true: fixes p :: "'b upred" shows "(p =\<^sub>u true) = p"
word_test_bit_set_bits: fixes f :: "nat \<Rightarrow> bool"   and n :: "nat" shows "(BITS n. f n) !! n = (n < LENGTH('a) \<and> f n)"
ln_one_minus_pos_lower_bound: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1 / 2" shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
image_mset_empty: fixes f :: "'c \<Rightarrow> 'b" shows "image_mset f {#} = {#}"
Abs_inverse: fixes y :: "atom" assumes "y \<in> {a. sort_of a = s}" shows "Rep (Abs y) = y"
relevant_vars_Entry: fixes V :: "'var"   and S :: "'node set" assumes "V \<in> rv S (_Entry_)"   and "(_High_) \<notin> backward_slice S" shows "V \<in> L"
HInit_HInv5: fixes s :: "state" assumes "HInit s" shows "HInv5 s"
delete_edges_def: fixes G :: "'c graph"   and es :: "('c \<times> 'c) set" shows "delete_edges G es \<equiv> \<lparr>nodes = nodes G, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> (e1, e2) \<notin> es}\<rparr>"
approx_7: shows "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a"
e: shows "S = \<Union> (t ` S) \<inter> closure S"
D: shows "D x = T x - ln (1 + 1 / x)"
int_complexity: shows "mono_matrix_carrier (\<lambda>x y. y < x) 1 nat int_mono"
in_set_member: fixes x :: "'k"   and xs :: "'k list" shows "(x \<in> set xs) = List.member xs x"
OO_eq: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "R OO (=) = R"
fresh_fun_eqvt_app: fixes f :: "'b \<Rightarrow> 'c"   and a :: "atom"   and x :: "'b" assumes "eqvt f"   and "a \<sharp> x" shows "a \<sharp> f x"
sup_mono: fixes a :: "'g multiset"   and c :: "'g multiset"   and b :: "'g multiset"   and d :: "'g multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<union># b \<subseteq># c \<union># d"
three_cycles_incl: fixes n :: "nat" shows "three_cycles n \<subseteq> carrier (alt_group n)"
expand: fixes literal :: "'a literal"   and literal' :: "'a literal" assumes "sign literal = sign literal'"   and "\<lbrakk>sign literal; sign literal'\<rbrakk> \<Longrightarrow> get_pred literal = get_pred literal' \<and> get_terms literal = get_terms literal'"   and "\<lbrakk>\<not> sign literal; \<not> sign literal'\<rbrakk> \<Longrightarrow> get_pred literal = get_pred literal' \<and> get_terms literal = get_terms literal'" shows "literal = literal'"
not_listrel1_Nil: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set" shows "(xs, []) \<notin> listrel1 r"
monad_state_altc_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_altc return bind get put altc" shows "monad_state_altc (return_env return) (bind_env bind) (get_env get) (put_env put) (altc_env altc)"
L_transform_formula_Pred: fixes f :: "'d"   and \<phi> :: "'b" shows "L_transform_formula (FL_Formula.Pred f \<phi>) = Formula.Act\<^sub>\<alpha> (Eff f) (Formula.Pred\<^sub>\<alpha> \<phi>)"
polar_pole: fixes l :: "proj2_line" shows "polar (pole l) = l"
natural_transformation_axioms: shows "natural_transformation C C Adj.FG.map Adj.C.map \<epsilon>"
c_drop_at_Suc1: fixes y :: "nat"   and x :: "nat" shows "c_drop (Suc y) x = c_drop y (c_tl x)"
Pg_minus_def: fixes G :: "Pg" shows "- G = the_elem (\<Union>g\<in>Rep_Pg G. {Abs_Pg (eq_game_rel `` {neg_game g})})"
finite_g: fixes e :: "'v \<times> 'v" shows "g e \<noteq> \<top>"
c: shows "\<Turnstile>\<^sub>p {P'} S {Q'}"
eval_fmla_rename: fixes k' :: "name"   and A :: "fmla"   and e :: "(name, nat) finfun"   and k :: "name"   and x :: "nat" assumes "atom k' \<sharp> A" shows "eval_fmla (finfun_update e k x) A = eval_fmla (finfun_update e k' x) ((k' \<leftrightarrow> k) \<bullet> A)"
map_in_list_rel_conv: fixes l :: "'b list"   and \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "((l, map \<alpha> l) \<in> \<langle>br \<alpha> I\<rangle>list_rel) = (\<forall>x\<in>set l. I x)"
lselect_lnth: fixes i :: "nat"   and w :: "'b llist"   and s :: "nat set" assumes "enat i < llength (lselect s w)" shows "lselect s w ?! i = w ?! nth_least s i"
games_lfp_eq_gfp: shows "games_lfp = games_gfp"
\<psi>_def: shows "\<psi> = sum_upto mangoldt"
asm: shows "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') \<noteq> 0"
1: shows "power.power 1' (;) (x\<^sup>T) (Suc n) = 0"
greatest_cong: fixes x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "x .= x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "is_closed A" shows "greatest L x A = greatest L x' A"
not_empty_imp_ex: fixes A :: "'a set" assumes "A \<noteq> {}" shows "\<exists>x. x \<in> A"
diff: fixes a :: "'a \<Rightarrow>\<^sub>0 int"   and b :: "'a \<Rightarrow>\<^sub>0 int" assumes "P a"   and "P b" shows "P (a - b)"
subst_F_left_commute: fixes x :: "name"   and y :: "name"   and t :: "tm" shows "subst x (F x) (subst y (F y) t) = subst y (F y) (subst x (F x) t)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "(x \<le> xa) = (int_of_integer x \<le> int_of_integer xa)"
a_monoid: fixes G :: "('a, 'b) ring_scheme" assumes "abelian_monoid G" shows "Group.monoid (add_monoid G)"
Well_Shaped_Graph_axioms: shows "Well_Shaped_Graph task.nodes task.inPorts task.outPorts vertices nodeOf task_edges task.hyps"
e_p_below: fixes e :: "'a \<rightarrow> 'b"   and p :: "'b \<rightarrow> 'a"   and y :: "'b" assumes "ep_pair e p" shows "e\<cdot>(p\<cdot>y) \<sqsubseteq> y"
assms: shows "weakly_Pareto_optimal_for_men \<mu>"
closedin_Un: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin U S"   and "closedin U T" shows "closedin U (S \<union> T)"
parallel_subspace_explicit: fixes S :: "'a set"   and a :: "'a"   and L :: "'a set" assumes "affine S"   and "a \<in> S"   and "L \<equiv> {y. \<exists>x\<in>S. - a + x = y}" shows "subspace L \<and> affine_parallel S L"
monoseq_iff: fixes X :: "nat \<Rightarrow> 'b" shows "monoseq X = (incseq X \<or> decseq X)"
2: shows "set_borel_measurable lborel (einterval a b) f"
index_mat_multcol_1: fixes i :: "nat"   and A :: "'d mat"   and j :: "nat"   and k :: "nat"   and a :: "'d" assumes "i < dim_row A"   and "j < dim_col A" shows "multcol k a A $$ (i, j) = (if k = j then a * A $$ (i, j) else A $$ (i, j))"
aform_slp_def: fixes prec :: "nat"   and n :: "nat"   and slp :: "floatarith list"   and xs :: "(real \<times> real pdevs) list" shows "aform_slp prec n slp xs = dRETURN (approx_slp_outer prec n slp xs)"
sp_in_extends_imp1: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v1 :: "'d \<times> nat"   and v2 :: "'d \<times> nat"   and subs' :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and es :: "('d \<times> nat) edge list"   and v :: "'d \<times> nat" assumes "wf_sub_rel_of g subs"   and "SubExt.extends g subs (v1, v2) subs'"   and "SubRel.subpath g v1 es v subs'" shows "es = [] \<or> SubRel.subpath g v2 es v subs'"