hcju/isabelleps huggingface dataset:
pderiv_closed: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)" shows "pderiv p \<in> carrier (UP Zp)"
neg_less_pos: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) a = less zero a"
dlat_axioms_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "dlat_axioms le \<equiv> (\<forall>x y. \<exists>inf. dpo.is_inf le x y inf) \<and> (\<forall>x y. \<exists>sup. dpo.is_sup le x y sup)"
rep_eq: fixes x :: "uint8" shows "nat_of_uint8 x = unat (Rep_uint8 x)"
simps_2: shows "skew \<langle>\<rangle> = \<langle>\<rangle>"
exp1: fixes A :: "nat"   and B :: "nat" shows "exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)"
uminus_eint_def: fixes b :: "eint" shows "- b = (case b of eint m \<Rightarrow> eint (- m) | \<infinity> \<Rightarrow> \<infinity>)"
disc_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "\<not> lnull (x21 % x22)"
subgraph_def2: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" assumes "graph G\<^sub>1"   and "graph G\<^sub>2" shows "subgraph G\<^sub>1 G\<^sub>2 = (vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> edges G\<^sub>1 \<subseteq> edges G\<^sub>2)"
Hilbert_fun_cone_empty: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and z :: "nat" assumes "h \<in> P[X]"   and "h \<noteq> 0"   and "homogeneous h" shows "Hilbert_fun (cone (h, {})) z = (if poly_deg h = z then 1 else 0)"
left_adjoint_extends_to_adjunction: fixes f :: "'a" assumes "is_left_adjoint f" shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f f\<^sup>* (some_unit f) (some_counit f)"
no_root: shows "\<forall>x. a < x \<and> x < b \<longrightarrow> poly p x \<noteq> 0"
transpose_row_code: fixes A :: "(('c, 'd) vec, 'b) vec"   and i :: "'d" shows "($) (transpose_row A i) = (\<lambda>j. A $ j $ i)"
sup_ge2: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<le> y"
BVG_assign_meta: fixes \<theta> :: "trm"   and x :: "variable" assumes "\<And>I \<omega>. term_sem I \<theta> \<omega> = \<omega> x" shows "BVG (x := \<theta>) = {}"
f_VK_1: shows "f \<in> S \<rightarrow> mat_kernel A"
eWlsInp_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and einp :: "'index \<Rightarrow> 'c withERR option" assumes "eWlsInp MOD delta einp" shows "\<exists>inp. igWlsInp MOD delta inp \<and> einp = OKI inp"
fst_dominant_term_uminus: fixes F :: "'a" shows "fst (dominant_term (- F)) = - fst (dominant_term F)"
sin_zero_iff_int2: fixes x :: "real" shows "(sin x = 0) = (\<exists>i. x = real_of_int i * pi)"
disconnected_nodes_eq2_h3: fixes doc_ptr :: "(_) document_ptr" shows "|h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
normal_def: shows "normal = (\<lambda>s. fst s = None)"
\<alpha>_1: shows "D = Dijkstra_Abstract.enat_of_option \<circ> V ++ Q"
max_prob_init: fixes s :: "'b"   and os :: "'a list"   and o :: "'a" shows "max_prob s os = Max {emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'} |as. length as = length os \<and> set as \<subseteq> \<S>}"
P_Cons_3: shows "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>\<^sub>2"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "TE.wtE \<xi>" shows "TE.satPB \<xi> tPB"
simps: fixes a1 :: "'a"   and a2 :: "'a" shows "hb a1 a2 = ((\<exists>m1 i m2. a1 = m1 \<and> a2 = m2 \<and> Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2) \<or> (\<exists>m1 i m2. a1 = m1 \<and> a2 = m2 \<and> Deliver m1 \<sqsubset>\<^sup>i Broadcast m2) \<or> (\<exists>m1 m2 m3. a1 = m1 \<and> a2 = m3 \<and> hb m1 m2 \<and> hb m2 m3))"
intros_2: fixes F :: "'b formula"   and G :: "'b formula" shows "(F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> G \<in> AXH"
map_add_empty: fixes m :: "'b \<Rightarrow> 'c option" shows "m ++ Map.empty = m"
prems: shows "xs \<cdot> [x] \<in> lists {u\<^sub>0, u\<^sub>1}"
Un_mono_approx: fixes A :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and C :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and B :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and D :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set" assumes "A \<lessapprox> C"   and "B \<lessapprox> D" shows "A \<union> B \<lessapprox> C \<union> D"
teq: shows "\<forall>x\<in>H. \<forall>y\<in>H. x \<noteq> y \<longrightarrow> f {x, y} = i"
Lem: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P' :: "'d \<Rightarrow> 'f set"   and c :: "('f, 'e, 'g) com"   and Q' :: "'d \<Rightarrow> 'f set"   and A' :: "'d \<Rightarrow> 'f set"   and P :: "'f set"   and Q :: "'f set"   and A :: "'f set"   and x :: "'h" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c (Q' Z),(A' Z)"   and "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P lem x c Q,A"
bit_not_iff: fixes a :: "'a"   and n :: "nat" shows "bit (NOT a) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> \<not> bit a n)"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.seq g f" obtains "B.VV.arr f" and "B.VV.arr g" and "B.VV.dom g = B.VV.cod f"
homE_2: fixes A :: "'a"   and B :: "'a"   and f :: "'b" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "Cod f = B"
lists_length_eq_Suc_eq_image_Cons: fixes A :: "'a set"   and n :: "nat" shows "{xs. set xs \<subseteq> A \<and> length xs = Suc n} = (\<lambda>(x, xs). x # xs) ` (A \<times> {xs. set xs \<subseteq> A \<and> length xs = n})"
inj_ocircline_set: fixes H :: "ocircline"   and H' :: "ocircline" assumes "ocircline_set H = ocircline_set H'"   and "ocircline_set H \<noteq> {}" shows "H = H' \<or> H = opposite_ocircline H'"
fst_snd_Gauss_Jordan_column_k_det_P_le_nrows: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and n :: "'b"   and k :: "nat" assumes "i \<le> nrows A" shows "fst (snd (Gauss_Jordan_column_k_det_P (n, i, A) k)) \<le> nrows A"
f_approx_def: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and C :: "real"   and x :: "real" assumes "akra_bazzi_real as bs hs k x\<^sub>0 x\<^sub>1 hb e p integrable integral f g C" shows "akra_bazzi_real.f_approx x\<^sub>0 p integral g x = x powr p * (1 + integral (\<lambda>u. g u / u powr (p + 1)) x\<^sub>0 x)"
Cnext_solve_weakly_precedes: fixes K\<^sub>2 :: "clock"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "{\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi>"
g_def: shows "g = (\<lambda>i. Inl ` f i \<union> Inr ` {0..<k - card (f i)})"
inorder_delete: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = del_list x (Tree2.inorder t)"
intruder_synth'_induct: fixes M :: "('fun, 'a) Term.term set"   and t :: "('fun, 'a) Term.term"   and P :: "('fun, 'a) Term.term \<Rightarrow> bool" assumes "intruder_synth' public arity M t"   and "\<And>x. intruder_synth' public arity M (Var x) \<Longrightarrow> P (Var x)"   and "\<And>f T. \<lbrakk>\<And>z. \<lbrakk>z \<in> set T; intruder_synth' public arity M z\<rbrakk> \<Longrightarrow> P z; intruder_synth' public arity M (Fun f T)\<rbrakk> \<Longrightarrow> P (Fun f T)" shows "P t"
simps_2: fixes k :: "nat"   and l :: "'b btree"   and a :: "'b"   and r :: "'b btree" shows "order_up\<^sub>i k (Up\<^sub>i l a r) = (order k l \<and> order k r)"
set_bound_2: fixes n :: "nat" shows "S \<subseteq> snd (A n)"
Im_eq: fixes x :: "'a" assumes "x \<in> A" shows "Im (f x) = Im (g x)"
Goedel_Form_Proofs_Minimal_Truth_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 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 fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and "proof" :: "'d set"   and prfOf :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and encPf :: "'d \<Rightarrow> 'b"   and Pf :: "'c"   and isTrue :: "'c \<Rightarrow> bool" shows "Goedel_Form_Proofs_Minimal_Truth var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv enc S proof prfOf encPf Pf isTrue \<equiv> Goedel_Form_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi fls prv bprv enc S dsj proof prfOf encPf Pf \<and> Minimal_Truth_Soundness var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num bprv isTrue"
constructive_security_obsf_refl: fixes \<eta> :: "nat"   and \<A> :: "('g + 'b, 'h + 'd) distinguisher_obsf" shows "constructive_security_obsf (real \<eta>) (middle \<eta>) (sim_outer \<eta>) (\<I>_real \<eta>) (\<I>_middle \<eta>) (\<I>_common \<eta>) \<A> (advantage \<A> (obsf_resource (sim_outer \<eta> |\<^sub>= 1\<^sub>C \<rhd> middle \<eta>)) (obsf_resource (real \<eta>)))"
is_Send_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "stateful_strand_step.is_Send stateful_strand_step = (\<exists>x1. stateful_strand_step = send\<langle>x1\<rangle>)"
terminates_upt_raw: fixes n :: "nat" shows "terminates (upt_raw n)"
p_antitone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- y \<le> - x"
Fail_parametric': fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'m \<Rightarrow> 'n \<Rightarrow> bool" shows "rel_gpv'' A C R Fail Fail"
wf_1: shows "prefs_from_table_wf agents alts yss"
usubstappp_antimon: fixes V :: "variable set"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<alpha> :: "game" assumes "V \<subseteq> U"   and "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg" shows "snd (usubstappp \<sigma> U \<alpha>) = snd (usubstappp \<sigma> V \<alpha>)"
Mem_Eats_E: fixes u :: "tm"   and t :: "tm"   and H :: "fm set"   and C :: "fm"   and z :: "tm" assumes "insert (u IN t) H \<turnstile> C"   and "insert (u EQ z) H \<turnstile> C" shows "insert (u IN Eats t z) H \<turnstile> C"
product_def: fixes A :: "'c set"   and B :: "'d set" shows "Product_Type.product A B = A \<times> B"
space_def: fixes M :: "'c measure" shows "space M = fst (Rep_measure M)"
assms_1: shows "f \<in> borel_measurable (F (Suc n))"
cmp_def: shows "Tgh.cmp = Tgh_Hgh.chine"
Goedel_I: shows "\<exists>\<phi>. \<not> {} \<turnstile> \<phi> \<and> \<not> {} \<turnstile> Neg \<phi> \<and> eval_fm e0 \<phi>"
H: shows "Ipoly bs (head p) \<noteq> (0::'a)"
wf_disc_sel: shows "wf_disc_sel (disc, sel) C"
\<R>\<^sub>1_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "mds\<^sub>2 = mds" and "\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'''__" and "mds_consistent mds \<Gamma>" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"
f_Exec_Stream_expand_aggregate_nth_eq_i_nth': fixes k :: "nat"   and trans_fun :: "'e message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat"   and input :: "nat \<Rightarrow> 'e message_af"   and c :: "'d"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "f_aggregate (f_Exec_Comp_Stream trans_fun (input \<Down> Suc n \<odot> k) c) k ag ! n = i_aggregate (i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag n"
mset_merge_sort: fixes xs :: "'a list" shows "mset (merge_sort xs) = mset xs"
list_of_dlist_empty: shows "list_of_dlist DList.empty = []"
6: shows "Ind {a} (set (u @ v\<^sub>1))"
k_prop: shows "y < k \<and> 0 < a * k\<^sup>2 + b * k + c"
less_add_same_cancel1: fixes a :: "'m multiset"   and b :: "'m multiset" shows "(a \<subset># a + b) = ({#} \<subset># b)"
cases: fixes x :: "thread" obtains "x = INIT"   | v :: "nat" where "x = WAIT v__"   | v :: "nat" where "x = HOLD va__"   | v :: "nat" where "x = REL v"
pred_def: fixes x :: "'a" shows "pred x = val (pos x - 1)"
f_NF: fixes t :: "dB" assumes "NF t" shows "NF (f t)"
le_imp_diff_is_add: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "a \<le> b" shows "(b - a = c) = (b = c + a)"
select_convs_6: fixes igWls :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and igWlsAbs :: "'b \<times> 'g \<Rightarrow> 'i \<Rightarrow> bool"   and igVar :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and igAbs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'i"   and igOp :: "'h \<Rightarrow> ('e \<Rightarrow> 'd option) \<Rightarrow> ('f \<Rightarrow> 'i option) \<Rightarrow> 'd"   and igFresh :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and igFreshAbs :: "'b \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> bool"   and igSwap :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and igSwapAbs :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> 'i"   and igSubst :: "'b \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and igSubstAbs :: "'b \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'i \<Rightarrow> 'i"   and more :: "'j" shows "igFresh \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igFresh"
val_equiv_axiom1: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v_equiv K v v"
llcp_LCons: fixes x :: "'c"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "llcp (LCons x xs) (LCons y ys) = (if x = y then eSuc (llcp xs ys) else 0)"
conjugate_add_vec: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (v + w) = conjugate v + conjugate w"
trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "??.Sublist.prefix b a"   and "??.Sublist.prefix c b" shows "??.Sublist.prefix c a"
fvi_And: fixes b :: "nat"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.fvi b (MFOTL.And \<phi> \<psi>) = MFOTL.fvi b \<phi> \<union> MFOTL.fvi b \<psi>"
admits_diagonal_imp_admits_triangular: assumes "\<forall>A\<in>carrier_mat 1 2. admits_diagonal_reduction A" shows "\<forall>A. admits_triangular_reduction A"
subst_gt: fixes i :: "nat"   and j :: "nat"   and u :: "db" assumes "i < j" shows "DBVar j<u/i> = DBVar (j - 1)"
cases: fixes x :: "int list" obtains "x = []"   | xa :: "int" where "x = [xa__]"   | xa :: "int" and y :: "int" and xs :: "int list" where "x = xa # y # xs"
abs_eq: fixes xa :: "perm"   and x :: "('a, 'b, 'c, 'd) Tree" shows "xa \<bullet> abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
fields_def: fixes G :: "prog"   and C :: "qtname" shows "DeclConcepts.fields G C = class_rec G C [] (\<lambda>C c. (@) (map (\<lambda>(n, t). ((n, C), t)) (cfields c)))"
filterlim_at_infinity_divide_iff: fixes f :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> (0::'b)" shows "(LIM x F. f x / g x :> at_infinity) = filterlim g (at (0::'b)) F"
cong_exp_iff_simps_2: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and n :: "num"   and q :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 q)) = zero) = (modulo (numeral.numeral one plus n) (numeral.numeral one plus q) = zero)"
freshSets_5: fixes x :: "name"   and xvec :: "name list"   and C :: "'e" shows "(x # xvec) \<sharp>* C = (x \<sharp> C \<and> xvec \<sharp>* C)"
fw_step_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and n :: "nat"   and j :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "less_eq neutral (m 0 0)"   and "i \<le> n"   and "j \<le> n" shows "linordered_ab_monoid_add.fw plus less_eq m n 0 i j i j = ord.min less_eq (m i j) (plus (m i 0) (m 0 j))"
prefixes_not_Nil: fixes xs :: "'c list" shows "prefixes xs \<noteq> []"
lemScaleOrigin: fixes k :: "'a" shows "k ** 0 = 0"
max_fun_diff_alt: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" shows "t.max_fun_diff f g = t.maxim ((rt.SUPP f \<union> rt.SUPP g) \<inter> {a. f a \<noteq> g a})"
B: shows "{0..3} = {0, 1, 2, 3}"
lsd_def: fixes x :: "'b" shows "lsd x = (\<top> \<cdot> x \<le> x)"
lookup_def: fixes x :: "'b" shows "SFP_lookup x = get_state (\<lambda>s. case s x of None \<Rightarrow> fail_state (fail_option return_pmf) | Some y \<Rightarrow> return_state (return_option return_pmf) y)"
m1: assumes "v \<noteq> v'" shows "verticesFrom f21 v' = v' # tl (fst (splitAt v (verticesFrom f v'))) @ v # ws @ u # snd (splitAt u (verticesFrom f v'))"
ncols_iarray_def: fixes A :: "'b iarray iarray" shows "ncols_iarray A = IArray.length (A !! 0)"
sort_conv_fold: fixes xs :: "'a list" shows "sort xs = fold insort xs []"
Ndet_commute: fixes P :: "'b process"   and Q :: "'b process" shows "(P |-| Q) = (Q |-| P)"
content_pos_le: fixes X :: "'a set" shows "0 \<le> content X"
nospoof_def: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and rs :: "'c common_primitive rule list" shows "nospoof TYPE('b) iface ipassmt rs = (\<forall>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface\<rparr>) rs Undecided = Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface))))"
\<A>_split_2: shows "\<A>1 \<inter> \<A>2 = {}"
swaprows_mat_Unit: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and b :: "'d" assumes "k < n"   and "l < n" shows "swaprows_mat n k l \<in> Units (ring_mat TYPE('c) n b)"
inj_zero_hom_axioms: shows "inj_zero_hom hom"
admissible_leI: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "mcont luba orda Sup (\<le>) f"   and "mcont luba orda Sup (\<le>) g" shows "ccpo.admissible luba orda (\<lambda>x. f x \<le> g x)"
\<pi>_conv_\<theta>_integral: assumes "2 \<le> x" shows "((\<lambda>t. \<theta> t / (t * (ln t)\<^sup>2)) has_integral \<pi> x - \<theta> x / ln x) {2..x}"
in_span_insert: fixes a :: "'b"   and b :: "'b"   and S :: "'b set" assumes "a \<in> span (insert b S)"   and "a \<notin> span S" shows "b \<in> span (insert a S)"
Rep_bit_inject: fixes x :: "bit"   and y :: "bit" shows "(Rep_bit x = Rep_bit y) = (x = y)"
image_snd_alist_of: fixes q :: "('d, 'c) pq" shows "the ` priority q ` set |q| = snd ` set (alist_of q)"
neg_closed: fixes G :: "'a set"   and g :: "'a" assumes "Group G"   and "g \<in> G" shows "- g \<in> G"
unitor_coincidence_2: fixes a :: "'d" assumes "D.obj a" shows "\<r>\<^sub>D[a] = \<i>\<^sub>D[a]"
current_plus_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g" shows "current \<Gamma> (plus_current f g)"
wqo_on_imp_wfp_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "wfp_on (strict P) A"
phiDefs_in_allDefs: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> step.phiDefs g n" shows "v \<in> step.allDefs g n"
rg_fo_inv: shows "cf_of (flow_of_cf cf) = cf"
deutsch_transform_def: shows "U\<^sub>f \<equiv> Tensor.mat_of_cols_list 4 (map (map of_nat) [[1 - f 0, f 0, 0, 0], [f 0, 1 - f 0, 0, 0], [0, 0, 1 - f 1, f 1], [0, 0, f 1, 1 - f 1]])"
lossless_iff_pmf_None: fixes p :: "'c spmf" shows "lossless_spmf p = (pmf p None = 0)"
ts_length_hint1: fixes x :: "nat \<Rightarrow> 'd list"   and i :: "nat" assumes "ts x" shows "x i \<noteq> []"
finite_UN_disjointed_eq: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "\<Union> (disjointed A ` {0..<n}) = \<Union> (A ` {0..<n})"
abelian_monoid_def: fixes G :: "('c, 'd) ring_scheme" shows "abelian_monoid G \<equiv> Group.comm_monoid (add_monoid G)"
distinct_of_odd_def: fixes p :: "nat \<Rightarrow> nat" shows "distinct_of_odd p = (\<lambda>i. if index i \<in> bitset (p (oddpart i)) then 1 else 0)"
restrict_shared_empty: fixes A :: "nat set"   and L :: "nat set" shows "Map.empty \<ominus>\<^bsub>A\<^esub> L = Map.empty"
bind_runit: fixes x :: "\<langle>'d\<rangle>\<^sub>\<bottom>" shows "Option.bind x Some = x"
create_unique_indiff_using_distinct_best_worst: fixes l :: "'a pmf"   and b :: "'a pmf"   and w :: "'a pmf" assumes "l \<in> local.\<P>"   and "b \<in> local.best"   and "w \<in> local.worst"   and "b \<succeq>[\<R>] w \<and> (w, b) \<notin> \<R>" shows "\<exists>!\<alpha>. \<alpha> \<in> {0..1} \<and> l \<approx>[\<R>] mix_pmf \<alpha> b w"
sqsubset_is_subset: fixes A :: "'d set"   and B :: "'d set" shows "(A \<sqsubseteq> B) = (A \<subseteq> B)"
F_ctor_o_fold: fixes s :: "('e, 'f) F0 \<Rightarrow> 'f" shows "ctor_fold_F s \<circ> ctor_F = s \<circ> map_pre_F id (ctor_fold_F s)"