hcju/isabelleps huggingface dataset:
last_leq_path: fixes Cs :: "char list list"   and Ds :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,C \<turnstile> Cs \<sqsubset>\<^sup>1 Ds"   and "wf_prog wf_md P" shows "P \<turnstile> last Cs \<prec>\<^sup>1 last Ds"
ldropn_lmap: fixes n :: "nat"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "ldropn n (lmap f xs) = lmap f (ldropn n xs)"
unique_decomposition: fixes Us :: "'a list"   and a :: "'a" assumes "local.independent K Us"   and "a \<in> Span K Us" shows "\<exists>!Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = combine Ks Us"
assms: fixes x :: "'a" shows "\<pi> \<bullet> f x = f (\<pi> \<bullet> x)"
conf_NT: fixes P :: "'b prog"   and h :: "'heap"   and v :: "'addr val" shows "(P,h \<turnstile> v :\<le> NT) = (v = Null)"
U_\<Phi>_def: shows "U_\<Phi> \<equiv> TE_\<Phi>"
induct: fixes P :: "bool \<Rightarrow> (bool \<times> 'b) rexp \<Rightarrow> 'b \<Rightarrow> bool"   and a0 :: "bool"   and a1 :: "(bool \<times> 'b) rexp"   and a2 :: "'b" assumes "\<And>uu_ uv_. P uu_ One uv_"   and "\<And>uw_ ux_. P uw_ Zero ux_"   and "\<And>m uy_ x c. P m (Atom (uy_, x)) c"   and "\<And>m r s c. \<lbrakk>P m r c; P m s c\<rbrakk> \<Longrightarrow> P m (Plus r s) c"   and "\<And>m r s c. \<lbrakk>P m r c; P (final r \<or> m \<and> nullable r) s c\<rbrakk> \<Longrightarrow> P m (Times r s) c"   and "\<And>m r c. P (final r \<or> m) r c \<Longrightarrow> P m (Star r) c" shows "P a0 a1 a2"
gen_dfs_outer_already_vis: fixes v0 :: "'a"   and it :: "'a set"   and V0 :: "'a set"   and V :: "'a set"   and E :: "('a \<times> 'a) set"   and U :: "'a set" assumes "v0 \<in> it"   and "it \<subseteq> V0"   and "v0 \<in> V"   and "gen_dfs_outer E U V0 it V False" shows "gen_dfs_outer E U V0 (it - {v0}) V False"
holomorphic_contract_to_zero: fixes \<xi> :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (cball \<xi> r) f"   and "f holomorphic_on ball \<xi> r"   and "0 < r"   and "\<And>z. cmod (\<xi> - z) = r \<Longrightarrow> cmod (f \<xi>) < cmod (f z)" obtains z :: "complex" where "z \<in> ball \<xi> r" and "f z = 0"
Y: shows "Y = add_aform' p X (ivl_err (real_interval r))"
map_dual_inv_def: fixes f :: "'b dual \<Rightarrow> 'c dual" shows "map_dual_inv f = \<partial>\<^sup>- \<circ> f \<circ> \<partial>"
distinct_49: fixes x4 :: "'a OCL_Types.type"   and x8 :: "'a OCL_Types.type" shows "Collection x4 \<noteq> Sequence x8"
simps_2: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and f :: "bool \<Rightarrow> 'a sorter" shows "eval_sorter R (Query a b f) = eval_sorter R (f ((a, b) \<in> R))"
Insync_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and V :: "nat"   and e :: "'addr expr1" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (sync\<^bsub>V\<^esub> (e) e') h (stk, loc, Suc (Suc (Suc (length (compE2 e) + pc))), xcp) (stk', loc', Suc (Suc (Suc (length (compE2 e) + pc'))), xcp')"
lhs_dot_rewrite: fixes p :: "real poly"   and signs :: "rat list list"   and I :: "nat list \<times> nat list"   and qs :: "real poly list" assumes "p \<noteq> 0" shows "vec_of_list (mtx_row_R signs I) \<bullet> w_vec_R p qs signs = (\<Sum>s\<leftarrow>signs. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})))"
row_echelon_form_imp_1_or_0_row: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "row_echelon_form A" shows "A = 1\<^sub>m n \<or> 0 < n \<and> row A (n - 1) = 0\<^sub>v n"
timpls_transformable_to_inv_Var_1: fixes TI :: "('e set \<times> 'e set) list"   and x :: "'h"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to TI (Var x) t" shows "t = Var x"
face_of_slice: fixes F :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "F face_of S"   and "convex T" shows "F \<inter> T face_of S \<inter> T"
bit_eq_rec: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "(a = b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b \<and> divide a (numeral.numeral one plus (num.Bit0 num.One)) = divide b (numeral.numeral one plus (num.Bit0 num.One)))"
discrL_update: fixes cl :: "('test, 'atom, 'choice) cmd list"   and c' :: "('test, 'atom, 'choice) cmd"   and n :: "nat" assumes "discrL cl"   and "discr c'" shows "discrL (cl[n := c'])"
diff_delta_arctan_upper_44_def: shows "diff_delta_arctan_upper_44 \<equiv> \<lambda>x. - 16384 / ((35 + 1260 * x\<^sup>2 + 6930 * x ^ 4 + 12012 * x ^ 6 + 6435 * x ^ 8)\<^sup>2 * (1 + x\<^sup>2))"
pchoose_def: fixes r :: "real"   and m :: "'m"   and m' :: "'m" shows "m \<lhd> r \<rhd> m' = sample (map_pmf (\<lambda>b. if b then \<^bold>1 else \<^bold>2) (bernoulli_pmf r)) (\<lambda>x. if x = \<^bold>1 then m else m')"
n_3: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
match_list_normalize_match: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" shows "match_list \<gamma> [m] a p = match_list \<gamma> (normalize_match m) a p"
simps_8: fixes v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators"   and vb :: "('b, 'c) Combinators"   and vc :: "('b, 'c) Combinators list" shows "removeShadowRules1_alternative_rev (v \<oplus> va # vb # vc) = v \<oplus> va # removeShadowRules1_alternative_rev (vb # vc)"
assms_1: shows "\<forall>\<^sub>F n in net. norm (f n) \<le> norm (g n)"
dynamically_complete_Liminf: fixes B :: "'f \<times> 'l"   and Ns :: "('f \<times> 'l) set llist" assumes "B \<in> Bot_FL"   and "chain (\<rhd>L) Ns"   and "empty_ord.fair Ns"   and "entails_\<G> (lhd Ns) {B}" shows "\<exists>B'\<in>Bot_FL. B' \<in> Liminf_llist Ns"
history_consistent_hd_prog: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p' :: "'a"   and xs :: "'a memref list"   and p :: "'a" assumes "history_consistent \<theta> p' xs" shows "history_consistent \<theta> (hd_prog p xs) xs"
mono_aux: fixes z :: "'b"   and x :: "'b" shows "mono (\<lambda>y. sup z (x \<cdot> y))"
simps_6: fixes x1 :: "'d" shows "Stuck \<noteq> Normal x1"
Max_ge: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "x \<in> A" shows "x \<preceq> linordered_monoid.Max A"
dghm_proj_ArrMap_app: fixes x :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "x \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = x\<lparr>i\<rparr>"
distinct_17: fixes x61 :: "nat"   and x62 :: "recf" shows "S \<noteq> Mn x61 x62"
space_stock_measure: fixes t :: "pdf_type" shows "space (stock_measure t) = type_universe t"
mojmir_to_rabin_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" shows "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F \<equiv> mojmir \<Sigma> \<delta> q\<^sub>0 w F"
assms_1: shows "n -as\<rightarrow>\<^sub>\<iota>* n'"
pmf_pmf_of_st_vec: fixes x :: "'a st_vec"   and i :: "'a" shows "pmf (pmf_of_st_vec x) i = st_vec x $h i"
cop_nos__ts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D"   and "\<not> Col C A B"   and "\<not> Col D A B"   and "\<not> A B OS C D" shows "A B TS C D"
union_disjoint: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
lunstream_lmap_trans: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> ('e, 'd) step"   and s :: "'d" shows "lunstream (lmap_trans f g) s = lmap f (lunstream g s)"
11: shows "top * e \<le> top * r\<^sup>T * prim_W w v e\<^sup>\<star> * e"
trie_odlist_update_def: shows "trie_odlist_update = (\<lambda>k. trie_update (toList k))"
event_can_go_back_if_no_sender_trace: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and c' :: "('d, 'e, 'f) configuration"   and ev' :: "('d, 'e, 'f) event" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv c t c'"   and "list_all (\<lambda>ev. occurs_on ev \<noteq> occurs_on ev') t"   and "distributed_system.can_occur channel trans send recv ev' c'"   and "\<not> isRecvMarker ev'"   and "list_all (\<lambda>ev. \<not> isSend ev) t" shows "distributed_system.can_occur channel trans send recv ev' c"
M: fixes X :: "msg"   and Y :: "msg" assumes "P X"   and "P Y" shows "P (MPair X Y)"
poly_lift_hom_comm_taylor_expansion_cf: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and a :: "'a"   and i :: "nat" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier (UP R)"   and "a \<in> carrier R" shows "\<phi> (taylor_expansion R a p i) = taylor_expansion S (\<phi> a) (poly_lift_hom R S \<phi> p) i"
communicating_def: shows "KN.communicating = (SIGMA x:UNIV. set_pmf (KN.Kp x))\<^sup>* \<inter> ((SIGMA x:UNIV. set_pmf (KN.Kp x))\<^sup>*)\<inverse>"
acute2__sams: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Acute A B C"   and "Acute D E F" shows "SAMS A B C D E F"
i_def: shows "i = s\<^bsub>01\<^esub> (f \<triangleright> n\<^sub>0)"
prems: shows "\<forall>x\<le>Suc n. 0 < x \<longrightarrow> s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)"
risky_asset_martingale_if: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "q = (1 + r - d) / (u - d)" shows "martingale N G (discounted_value r (prices Mkt stk))"
param_op_nres_ASSERT_bnd: fixes \<Phi>' :: "bool"   and \<Phi> :: "bool"   and m :: "'a nres"   and m' :: "'b nres"   and R :: "('a \<times> 'b) set" assumes "\<Phi>' \<Longrightarrow> \<Phi>"   and "\<lbrakk>\<Phi>'; \<Phi>\<rbrakk> \<Longrightarrow> (m, m') \<in> \<langle>R\<rangle>nres_rel" shows "(op_nres_ASSERT_bnd \<Phi> m, op_nres_ASSERT_bnd \<Phi>' m') \<in> \<langle>R\<rangle>nres_rel"
terminal_path_cons: fixes v\<^sub>1 :: "'vertex"   and p\<^sub>1 :: "('form, 'var) out_port"   and v\<^sub>2 :: "'vertex"   and p\<^sub>2 :: "('form, 'var) antecedent"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> edges"   and "terminal_path v\<^sub>2 v' pth"   and "hyps (nodeOf v\<^sub>1) p\<^sub>1 = None" shows "terminal_path v\<^sub>1 v' (((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) # pth)"
Rep_less_n: fixes x :: "'b bit0" shows "Rep_bit0 x < int CARD('b bit0)"
big_by_small: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "iconf (shp s) e"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>" shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',False\<rangle>"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "inf (inf a b) c = inf a (inf b c)"
topological_tendstoD: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and l :: "'b"   and F :: "'d filter"   and S :: "'b set" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
lunit_simps_5: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "cod \<l>[f] = f"
roofed_idem: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set" shows "roofed_gen G B (roofed_gen G B S) = roofed_gen G B S"
nth_off_MTF: fixes n :: "nat"   and s :: "'a list \<times> unit" assumes "n < length rs" shows "off2 MTF s rs ! n = (length (fst s) - 1, [])"
null_option_def: shows "null \<equiv> \<lfloor>\<bottom>\<rfloor>"
inj_on_strict_subset: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and A :: "'a set" assumes "inj_on f B"   and "A \<subset> B" shows "f ` A \<subset> f ` B"
cut: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
null_sets_outer_le: fixes S :: "'b set"   and M :: "'b measure" shows "(S \<in> null_sets (completion M)) = (\<forall>e>0. \<exists>T\<in>fmeasurable (completion M). S \<subseteq> T \<and> Sigma_Algebra.measure (completion M) T \<le> e)"
ipresFresh_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'b, 'g, 'h) model_scheme" shows "ipresFresh h MOD \<equiv> \<forall>ys y s X. wls s X \<longrightarrow> fresh ys y X \<longrightarrow> igFresh MOD ys y (h X)"
simps_5: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and A :: "'b ptrm"   and B :: "'b ptrm" shows "ptrm_infer_type \<Gamma> (PPair A B) = (case (ptrm_infer_type \<Gamma> A, ptrm_infer_type \<Gamma> B) of (None, b) \<Rightarrow> None | (Some \<tau>, None) \<Rightarrow> None | (Some \<tau>, Some \<sigma>) \<Rightarrow> Some (TPair \<tau> \<sigma>))"
add_frac_num: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero" shows "plus (divide x y) z = divide (plus x (times z y)) y"
IH: shows "nths (nths xs A) (- ind_in_set A ` B) = nths xs (A - B)"
regular_closed_inf: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top"   and "x = uminus (uminus x)"   and "y = uminus (uminus y)" shows "inf x y = uminus (uminus (inf x y))"
encode12_lt: fixes i :: "nat"   and j :: "nat" assumes "i < d1"   and "j < d2" shows "encode12 (i, j) < d"
last_in_set: fixes as :: "'d list" assumes "as \<noteq> []" shows "last as \<in> set as"
eval_e1: shows "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<rightarrow> s1"
EphK_Pair: fixes X :: "ephkey"   and X' :: "msg"   and Y' :: "msg" shows "EphK X \<noteq> \<langle>X', Y'\<rangle>"
eucl_sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "sup x y = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (sup (inner x i) (inner y i)) i) Basis"
ide_in_hom_1: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>f : src\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright>"
sum_strict_mono: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset" assumes "finite A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subset># g x" shows "comm_monoid_add.sum (+) {#} f A \<subset># comm_monoid_add.sum (+) {#} g A"
convex_onD_Icc: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> real"   and t :: "real" assumes "convex_on {x..y} f"   and "x \<le> y"   and "0 \<le> t"   and "t \<le> 1" shows "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
split_face_empty_ram2_ram1_in_f12: fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face oldF ram1 ram2 []"   and "(f12, f21) = split_face oldF ram1 ram2 []" shows "(ram2, ram1) \<in> \<E> f12"
assms_1: shows "prv (imp \<phi>1 \<phi>2)"
Im_power_real: fixes x :: "complex"   and n :: "nat" assumes "is_real x" shows "is_real (x ^ n)"
HaddP_SUCC2: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "{HaddP x y z} \<turnstile> HaddP x (SUCC y) (SUCC z)"
less_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
fold_simps_1: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a" shows "fold f [] s = s"
design_iso_inter_numbers_imp: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and x :: "int" assumes "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "x \<in> incidence_system.intersection_numbers \<B>" shows "x \<in> incidence_system.intersection_numbers \<B>'"
alls2_Cons_tl_conv: fixes A :: "nat"   and B :: "nat"   and as :: "nat list"   and bs :: "nat list" shows "alls2 A B as bs = ((zeroes (length as), 0), zeroes (length bs), 0) # tl (alls2 A B as bs)"
rank_greater_zero: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "A \<noteq> 0" shows "0 < rank A"
matrix_inv_2: fixes M :: "(('d, 'e) vec, 'f) vec" assumes "invertible M" shows "M ** matrix_inv M = mat (1::'d)"
lfinite_finite_index: fixes xs :: "'b llist" assumes "lfinite xs" shows "finite {n. enat n < llength xs}"
fundamental_theorem_of_calculus_interior_strong: fixes S :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "finite S"   and "a \<le> b"   and "\<And>x. x \<in> {a<..<b} - S \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "continuous_on {a..b} f" shows "(f' has_integral f b - f a) {a..b}"
that: fixes q :: "nat" assumes "w = 2 ^ n * word_of_nat q"   and "q < 2 ^ (LENGTH('a) - n)" shows "thesis"
fresh_map_transform: fixes a :: "atom"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>" assumes "a \<sharp> \<Gamma>" shows "a \<sharp> map_transform trans ae \<Gamma>"
nn_integral_erlang_density: fixes l :: "real"   and k :: "nat"   and a :: "real" assumes "0 < l" shows "\<integral>\<^sup>+x\<in>{..a}. ennreal (erlang_density k l x)\<partial>lborel = ennreal (erlang_CDF k l a)"
seq_char: fixes f :: "'a"   and g :: "'a" shows "V.seq f g = (V.ide f \<and> f = g)"
tensor_mat_positive: fixes m1 :: "complex mat"   and m2 :: "complex mat" assumes "m1 \<in> carrier_mat st.d1 st.d1"   and "m2 \<in> carrier_mat st.d2 st.d2"   and "positive m1"   and "positive m2" shows "positive (st.tensor_mat m1 m2)"
edge_end_tup: fixes x :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort" shows "edge_end x = fst (snd x)"
exhaust: fixes y :: "'b sequent" obtains x1 :: "'b form multiset" and x2 :: "'b form multiset" where "y = ( x1 \<Rightarrow>* x2)"
order_trans: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
finite_top_prod: shows "top = \<Squnion>\<^sub>f\<^sub>i\<^sub>n UNIV"
Image_empty2: fixes R :: "('f \<times> 'e) set" shows "R `` {} = {}"
simps_1: fixes x :: "'a"   and y :: "'a" shows "Some x \<squnion> Some y = Some (x \<squnion> y)"