hcju/isabelleps huggingface dataset:
ring_distribs_4: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ring_ow U plus zero minus uminus times"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "times a (minus b c) = minus (times a b) (times a c)"
simps: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "integrable M f = (\<exists>x. has_bochner_integral M f x)"
tendsto_at_right_sequentially: fixes a :: "'a"   and b :: "'a"   and X :: "'a \<Rightarrow> 'b"   and L :: "'b" assumes "a < b"   and "\<And>S. \<lbrakk>\<And>n. a < S n; \<And>n. S n < b; decseq S; S \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" shows "(X \<longlongrightarrow> L) (at_right a)"
independence_symmetric: fixes a :: "'item"   and b :: "'item" assumes "ind a b" shows "ind b a"
porder_torder_tranclpE: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "porder_on A r"   and "torder_on B s"   and "order_consistent r s"   and "B \<subseteq> A"   and "(\<lambda>a b. r a b \<or> s a b)\<^sup>+\<^sup>+ x y" obtains "r x y"   | u :: "'b" and v :: "'b" where "r x u" and "s u v" and "r v y"
simps_35: fixes x61 :: "msg"   and x62 :: "msg"   and x3 :: "fresh_t" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Nonce x3"
irrefl_lex: fixes r :: "('g \<times> 'g) set" assumes "irrefl r" shows "irrefl (lex r)"
permutes_swap_id: fixes a :: "'c"   and S :: "'c set"   and b :: "'c" assumes "a \<in> S"   and "b \<in> S" shows "Fun.swap a b id permutes S"
dependent_biggerset_general: fixes S :: "'b set" assumes "finite S \<Longrightarrow> dim S < card S" shows "dependent S"
injective_publicKey: fixes b :: "bool"   and A :: "agent"   and c :: "bool"   and A' :: "agent" assumes "publicKey b A = publicKey c A'" shows "b = c \<and> A = A'"
igSwapAbsIPresIGWlsAbs_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapAbsPresGWlsAbs MOD" shows "igSwapAbsIPresIGWlsAbs (fromMOD MOD)"
arctan_le_iff: fixes x :: "real"   and y :: "real" shows "(arctan x \<le> arctan y) = (x \<le> y)"
a_subid_aux2: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_pre_dioid antidomain_op plus times one zero less_eq less" shows "less_eq (times x (antidomain_op y)) x"
induct: fixes P :: "'a tree list \<Rightarrow> bool"   and a0 :: "'a tree list" assumes "P []"   and "\<And>t. P [t]"   and "\<And>t\<^sub>1 t\<^sub>2 ts. P (t\<^sub>2 # ts) \<Longrightarrow> P (t\<^sub>1 # t\<^sub>2 # ts)" shows "P a0"
hcomp_reassoc_1: fixes \<tau> :: "'d"   and \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.arr \<tau>"   and "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<tau> = trg\<^sub>D \<mu>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>" shows "(\<tau> \<star>\<^sub>D \<mu>) \<star>\<^sub>D \<nu> = \<a>\<^sub>D\<^sup>-\<^sup>1[D.cod \<tau>, D.cod \<mu>, D.cod \<nu>] \<cdot>\<^sub>D (\<tau> \<star>\<^sub>D \<mu> \<star>\<^sub>D \<nu>) \<cdot>\<^sub>D \<a>\<^sub>D[D.dom \<tau>, D.dom \<mu>, D.dom \<nu>]"
simps_122: fixes x17 :: "'a exp"   and x3 :: "val" shows "throw x17 \<noteq> Val x3"
integral_diff: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x - g x = integral\<^sup>L M f - integral\<^sup>L M g"
eq_\<I>_gpvD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv" assumes "eq_\<I>_gpv A \<I> gpv gpv'" shows "rel_spmf (eq_\<I>_generat A \<I> (eq_\<I>_gpv A \<I>)) (the_gpv gpv) (the_gpv gpv')"
1: shows "(H * d)\<^sup>\<star> * (H * a * top) \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top)"
quasi_inverts_components_2: fixes a :: "'a" assumes "C.obj a" shows "D.isomorphic (\<tau>\<^sub>0 a \<star>\<^sub>D \<sigma>.map\<^sub>0 a) (G.map\<^sub>0 a)"
tan_Arctan: fixes z :: "complex" assumes "z\<^sup>2 \<noteq> - 1" shows "tan (Arctan z) = z"
inj_signed_lconjaction: fixes s :: "'a" shows "inj (signed_lconjaction s)"
prems: shows "filter is_Write\<^sub>s\<^sub>b (x # xs) = Write\<^sub>s\<^sub>b volatile a sop v A L R W # ys"
special_subgroup_special_coset_subset_ordering_iso: fixes w :: "'w" shows "OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater (genby ` Pow S) ((+o) w)"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>" shows "r \<in>\<^sub>\<circ> Vset \<alpha>"
pentagon': fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "C.ide a"   and "C.ide b"   and "C.ide c"   and "C.ide d" shows "(C.tensor (C.assoc' a b c) d \<cdot> C.assoc' a (C.tensor b c) d) \<cdot> C.tensor a (C.assoc' b c d) = C.assoc' (C.tensor a b) c d \<cdot> C.assoc' a b (C.tensor c d)"
eq: shows "KL_div S a d = KL_div (S \<inter> {i. 0 < a i}) a d + KL_div (S \<inter> {i. 0 = a i}) a d"
F2rel_def: fixes R1 :: "'j \<Rightarrow> 'k \<Rightarrow> bool" shows "F2rel R1 = (BNF_Def.Grp (F2in {(x, y). R1 x y}) (F2map fst))\<inverse>\<inverse> OO BNF_Def.Grp (F2in {(x, y). R1 x y}) (F2map snd)"
relcomppI: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c"   and s :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d" assumes "r a b"   and "s b c" shows "(r OO s) a c"
1: shows "isotone (\<lambda>y. y * x \<squnion> (1::'a))"
POR_arg1_rel_def: shows "POR_arg1_rel \<equiv> \<lambda>i. case i of One \<Rightarrow> ValTT | Two \<Rightarrow> \<bottom> | Three \<Rightarrow> ValFF"
pos_add_strict: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "{#} \<subset># a"   and "b \<subset># c" shows "b \<subset># a + c"
simps_7: fixes x :: "'sz"   and y :: "'sz"   and a :: "('sf, 'sc, 'sz) hp"   and b :: "('sf, 'sc, 'sz) hp" shows "PUrename x y (a ;; b) = (PUrename x y a ;; PUrename x y b)"
insertion_aux_irrelevant_vars: fixes p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<And>m v. \<lbrakk>lookup p m \<noteq> (0::'b); lookup m v \<noteq> 0\<rbrakk> \<Longrightarrow> f v = g v" shows "insertion_aux f p = insertion_aux g p"
node_order_is_total: fixes e1 :: "('a \<times> ('a, 'b) operation) event"   and i :: "nat"   and e2 :: "('a \<times> ('a, 'b) operation) event" assumes "e1 \<in> set (history i)"   and "e2 \<in> set (history i)"   and "e1 \<noteq> e2" shows "e1 \<sqsubset>\<^sup>i e2 \<or> e2 \<sqsubset>\<^sup>i e1"
0: fixes \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "\<chi> \<pi>l" shows "\<psi> \<pi>l \<or> \<phi> \<pi>l \<and> \<chi> (map stl \<pi>l)"
power2_eq_square: fixes a :: "'a" shows "a\<^sup>2 = a \<cdot> a"
openin_components_locally_connected: fixes S :: "'a set"   and c :: "'a set" assumes "locally connected S"   and "c \<in> components S" shows "openin (top_of_set S) c"
stc_hcomplex_of_hypreal: fixes z :: "real star" assumes "z \<in> HFinite" shows "stc (hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)"
abs_eq: fixes xa :: "hmultiset zmultiset"   and x :: "hmultiset zmultiset" shows "(ZHMSet xa < ZHMSet x) = (xa < x)"
bg_impl_rel_eext_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "\<langle>Rm, Rv\<rangle>bg_impl_rel_eext \<equiv> \<langle>Rm, \<langle>Rv\<rangle>fun_set_rel\<rangle>gen_bg_impl_rel_eext"
VDMWhile: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and B :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> c : B"   and "\<forall>s. \<not> evalB b (fst s) \<longrightarrow> A s s"   and "\<forall>s r t. evalB b (fst s) \<longrightarrow> B s r \<longrightarrow> A r t \<longrightarrow> A s t" shows "G \<rhd> While b c : \<lambda>s t. A s t \<and> \<not> evalB b (fst t)"
chamber_facet_is_chamber_facet: fixes C :: "'a set"   and D :: "'a set"   and z :: "'a set" assumes "chamber C"   and "chamber D"   and "z \<lhd> C"   and "order.greater_eq D z" shows "z \<lhd> D"
finite_Invs: fixes xs :: "'b list"   and ys :: "'b list" shows "finite (Inv xs ys)"
fvdiff_plus1: fixes t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" shows "FVDiff (Plus t1 t2) = FVDiff t1 \<union> FVDiff t2"
last_coeff_is_hd: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "coeff (Poly xs) (length xs - 1) = hd (rev xs)"
that: fixes y :: "'c" assumes "s = Var y"   and "\<sigma> y = Var x" shows "thesis"
vid_onI: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "\<langle>a, a\<rangle> \<in>\<^sub>\<circ> vid_on A"
intros_2: fixes ft :: "('c, 'd) flow_entry_match list"   and \<gamma> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'e" assumes "\<forall>fe\<in>set ft. \<gamma> (ofe_fields fe) p = False" shows "guha_table_semantics \<gamma> ft p None"
spec_ops_distinct_fst: fixes ops :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set" assumes "spec_ops ops deps" shows "distinct (map fst ops)"
equivalent_iff_not_distinguished: fixes P :: "'state"   and Q :: "'state" shows "(P =\<cdot> Q) = (\<nexists>x. x distinguishes P from Q)"
bind_return_spmf: fixes x :: "'c spmf" shows "x \<bind> return_spmf = x"
prems_4: shows "InvariantWatchListsUniq (getWatchList state')"
class_wf: fixes P :: "'b prog"   and C :: "char list"   and c :: "'b class"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool" assumes "class P C = \<lfloor>c\<rfloor>"   and "wf_prog wf_md P" shows "wf_cdecl wf_md P (C, c)"
insert_subset_eq_iff: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># B - {#a#})"
less_prod_simps: fixes leq_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and x2 :: "'d"   and y1 :: "'c"   and y2 :: "'d" shows "less_prod leq_a less_a less_b (x1, x2) (y1, y2) = (less_a x1 y1 \<or> leq_a x1 y1 \<and> less_b x2 y2)"
distinct_19: fixes x1 :: "agent"   and x11 :: "tag" shows "cAgent x1 \<noteq> cTag x11"
inf_continuous_def: fixes F :: "'b \<Rightarrow> 'c" shows "inf_continuous F = (\<forall>M. decseq M \<longrightarrow> F (Inf (range M)) = (INF i. F (M i)))"
set_sum_list_element_Cons: fixes x :: "'a"   and A :: "'a set"   and As :: "'a set list" assumes "x \<in> (\<Sum>X\<leftarrow>A # As. X)" shows "\<exists>a as. a \<in> A \<and> as \<in> listset As \<and> x = a + (\<Sum>b\<leftarrow>as. b)"
unitary_def: fixes A :: "'a mat" shows "unitary A = (A \<in> carrier_mat (dim_row A) (dim_row A) \<and> inverts_mat A (adjoint A))"
rem_implicit_pres_ops_lookup_op_Some_2': fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and n :: "char list"   and p :: "(nat \<times> nat) list"   and e :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and c :: "nat" assumes "ast_problem.lookup_operator (rem_implicit_pres_ops prob) name = Some (n, p, e, c)" shows "\<exists>op'. ast_problem.lookup_operator prob name = Some op' \<and> (n, p, e, c) = rem_implicit_pres op'"
simps_2: fixes c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt" shows "inv1 (Branch c lt k v rt) = (inv1 lt \<and> inv1 rt \<and> (c = color.B \<or> color_of lt = color.B \<and> color_of rt = color.B))"
le_add_same_cancel2: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and a :: "'r"   and b :: "'r" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
map_filter_simps_2: fixes f :: "'d \<Rightarrow> 'c option" shows "List.map_filter f [] = []"
substitutable_preferences_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B a b. \<lbrakk>B \<subseteq> A; a \<in> B; b \<in> B; a \<noteq> b; a \<in> f B\<rbrakk> \<Longrightarrow> a \<in> f (B - {b})" shows "substitutable_preferences_on A f"
consequence_relation_axioms: shows "consequence_relation Bot_G (\<Turnstile>G)"
ErrorT_def: shows "ErrorT \<equiv> \<Lambda> app. errorT_abs\<cdot>app"
CX4h_range: fixes h :: "H1"   and X :: "X4 set" shows "CX4h h X \<subseteq> {x \<in> X. H = h}"
sig_red_sing_regI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F p q" shows "sig_red (\<preceq>\<^sub>t) top_tail F p q"
fM_rel_invD: fixes x :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state" assumes "fM_rel_inv x" shows "(\<forall>xa. sys_ghost_hs_in_sync xa x, sys_ghost_hs_phase x, gc_fM x, sys_fM x, filter is_mw_fM (sys_mem_store_buffers gc x), sys_mem_lock x = Some gc) \<in> fM_rel"
matchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto><Rel> Q"   and "Rel \<subseteq> Rel'" shows "[a\<frown>b]P \<leadsto><Rel'> [a\<frown>b]Q"
that: fixes d :: "real" assumes "0 < d"   and "\<And>T. \<lbrakk>T \<in> lmeasurable; \<And>y. y \<in> T \<Longrightarrow> \<exists>x. x \<in> S \<and> dist x y < d\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure lebesgue T < Sigma_Algebra.measure lebesgue S + e" shows "thesis"
deadlock2_imp_\<tau>s_deadlock1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r2.deadlock s2" shows "\<exists>s1'. \<tau>mRed1 s1 s1' \<and> r1.deadlock s1' \<and> s1' \<approx>m s2"
ArrayCpy_retv_def: fixes x :: "char list"   and rv :: "char list" shows "ArrayCpy_retv x rv \<equiv> x[] ::= rv"
tiebreakingGivesSingleton: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "card (argmax (sum (randomBidsAlg N \<Omega> b r)) (maximalStrictAllocations N (set \<Omega>) b)) = 1"
assms_3: shows "net_ips s2 = net_tree_ips n2"
SourcesA91_L1: shows "Sources level1 sA91 = {sA6, sA71, sA81, sA91}"
summable': fixes z :: "real" assumes "norm z < K" shows "summable (\<lambda>n. diffs ((gchoose) a) n * z ^ n)"
intros_2: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and es :: "(String.literal \<times> value list) list" assumes "\<forall>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es"   and "\<forall>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. \<exists>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es" shows "executionally_equivalent e1 s1 r1 e2 s2 r2 ((l, i) # es)"
row_uminus: fixes i :: "nat"   and A :: "'a mat" assumes "i < dim_row A" shows "row (- A) i = - row A i"
act_quant_nec: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and x :: "'a" shows "[(\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x) \<^bold>\<equiv> (\<^bold>\<forall>z. \<^bold>\<A>\<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x) in v]"
rel_chain_append: fixes l :: "'b list"   and R :: "('b \<times> 'b) set"   and l' :: "'b list" assumes "rel_chain l R"   and "rel_chain l' R"   and "(last l, hd l') \<in> R" shows "rel_chain (l @ l') R"
select_convs_2: fixes mbal :: "nat"   and bal :: "nat"   and inp :: "InputsOrNi"   and more :: "'a" shows "bal \<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr> = bal"
sconst_case: fixes I :: "('b, 'c, 'a) interp"   and r :: "real" shows "sterm_sem I (Const r) = (\<lambda>v. r)"
non_terminating_refine_DF: fixes P :: "'d process" shows "non_terminating P = DF UNIV \<sqsubseteq>\<^sub>T P"
document_ptr_kinds_eq_h: shows "document_ptr_kinds h = document_ptr_kinds h2"
chine_composite: shows "gf.chine = g.leg0 \<down>\<down> f.leg1"
TryRedsVal: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Val v,s'\<rangle>" shows "P \<turnstile> \<langle>try e catch(C V) e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Val v,s'\<rangle>"
compact_space_subtopology: fixes X :: "'b topology"   and S :: "'b set" assumes "compactin X S" shows "compact_space (subtopology X S)"
exists_algorithm_is_SNF_imp_admits_diagonal_reduction: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n"   and "\<exists>algorithm. is_SNF A (algorithm A)" shows "admits_diagonal_reduction A"
epsclo_nextl: fixes M :: "'b nfa"   and Q :: "hf set"   and xs :: "'b list" assumes "nfa M" shows "nfa.epsclo M (nfa.nextl M Q xs) = nfa.nextl M Q xs"
inv_mult_group: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> (x \<oplus>\<^bsub>F\<^esub> y) = \<ominus>\<^bsub>F\<^esub> y \<oplus>\<^bsub>F\<^esub> \<ominus>\<^bsub>F\<^esub> x"
prios_ins: fixes k :: "'b"   and p :: "'a"   and t :: "('b \<times> 'a) tree" shows "prios (ins k p t) \<subseteq> {p} \<union> prios t"
count_cases: fixes y :: "'a \<Rightarrow> nat" assumes "y \<in> multiset" obtains x :: "'a multiset" where "y = count x"
fst_netgmap_netlift: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and s :: "'f net_state" shows "fst (netgmap sr s) = netlift sr s"
shiftl_zero_size: fixes x :: "'b word"   and n :: "nat" assumes "size x \<le> n" shows "x << n = 0"
inj_on_mult: fixes a :: "'a"   and A :: "'a set" assumes "a \<noteq> (0::'a)" shows "inj_on ((*) a) A"
mem_read_mod_state: fixes asi :: "word8"   and addr :: "word32"   and s :: "'b sparc_state"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "fst (memory_read asi addr s) = fst (memory_read asi addr (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>))"
comp: shows "Some \<circ> \<pi> \<circ>\<^sub>m inv_map (Some \<circ> \<pi>) = Some"
tinres_def: fixes xs :: "cell list"   and ys :: "cell list" shows "tinres xs ys = (\<exists>n. xs = ys @ Bk \<up> n \<or> ys = xs @ Bk \<up> n)"
aadd_minus_le: fixes a :: "ant"   and b :: "ant" assumes "a \<le> 0" shows "a + b \<le> b"
coeffs_of_int_poly: fixes x :: "int poly" shows "coeffs (of_int_poly (Mp x)) = map of_int (coeffs (Mp x))"
is_L_bisimulation_eqvt: fixes R :: "'effect fs_set \<Rightarrow> 'state \<Rightarrow> 'state \<Rightarrow> bool"   and p :: "perm" assumes "is_L_bisimulation R" shows "is_L_bisimulation (p \<bullet> R)"
foldr_node_correct: fixes f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and nd :: "('e, 'f) Node"   and \<sigma> :: "'d" shows "foldr_node f nd \<sigma> = foldr f (nodeToList nd) \<sigma>"
prime_elem_aprimedivisor_nat: fixes d :: "nat" assumes "Suc 0 < d" shows "prime_elem (aprimedivisor d)"
complete\<^sub>N_I: fixes N :: "'c quasinorm" assumes "\<And>u. \<lbrakk>cauchy_in\<^sub>N N u; \<forall>n. u n \<in> space\<^sub>N N\<rbrakk> \<Longrightarrow> \<exists>x\<in>space\<^sub>N N. tendsto_in\<^sub>N N u x" shows "complete\<^sub>N N"
fds_truncate_0: fixes f :: "'b fds" shows "fds_truncate 0 f = 0"
wrap_iso_arcs_def: fixes G :: "('f, 'g) pre_digraph"   and hom :: "('f, 'g, 'h, 'e) digraph_isomorphism"   and f :: "'g \<Rightarrow> 'g" shows "pre_digraph.wrap_iso_arcs G hom f = perm_restrict (iso_arcs hom \<circ> f \<circ> iso_arcs (pre_digraph.inv_iso G hom)) (arcs (app_iso hom G))"
h1: shows "aEval (Less p) (L[var := (Av + Bv * sqrt Cv) / Dv]) = aEval (Less (quadratic_part_1 var a b d (Less p))) (L[var := sqrt Cv])"
ldropn_LNil: fixes n :: "nat" shows "ldropn n LNil = LNil"
homeomorphic_imp_continuous_map: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "homeomorphic_map X Y f" shows "continuous_map X Y f"
betweenI: fixes u :: "real"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "0 \<le> u"   and "u \<le> 1"   and "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" shows "between (a, b) x"
assms: shows "u \<noteq> \<infinity>\<^sub>h"
dest_Num_fa_Some: fixes fa :: "floatarith"   and x :: "float" assumes "dest_Num_fa fa = Some x" obtains "fa = Num x"
restrict_def: fixes f :: "'b \<Rightarrow> 'c"   and U :: "'b set" shows "f \<restriction> U = (\<lambda>n. if n \<in> U then f n else undefined)"
bin_last_BIT: fixes x :: "int"   and b :: "bool" shows "odd (x BIT b) = b"
f_Exec_Stream_Equal_State: fixes localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c1 :: "'d"   and c2 :: "'d"   and xs :: "'f message_af list" assumes "Exec_Equal_State localState trans_fun"   and "localState c1 = localState c2" shows "f_Exec_Comp_Stream trans_fun xs c1 = f_Exec_Comp_Stream trans_fun xs c2"
dist_triangle4: fixes x :: "'c"   and t :: "'c"   and y :: "'c"   and z :: "'c" shows "dist x t \<le> dist x y + dist y z + dist z t"