hcju/isabelleps huggingface dataset:
B_gk: shows "bezout_iterate (interchange_rows A ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<le> n)) (nrows A - Suc 0) ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)) (mod_type_class.from_nat k) bezout $ ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)) $ mod_type_class.from_nat k \<noteq> (0::'a)"
sectionE: fixes f :: "'d" assumes "Dop.section f" obtains g :: "'d" where "Dop.ide (Dop.comp g f)"
divide_less_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and 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 sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide b c) a = (if less zero c then less b (times a c) else if less c zero then less (times a c) b else less zero a)"
frac_eq_0_iff: fixes x :: "'a" shows "(frac x = (0::'a)) = (x \<in> \<int>)"
ss: shows "{b. \<exists>a. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r} \<subseteq> (\<lambda>x. x\<lparr>1\<^sub>\<nat>\<rparr>) ` elts r"
Dom_hcomp: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = trg \<nu>" shows "Dom (\<mu> \<star> \<nu>) = Dom \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom \<nu>"
simps_9: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm"   and q :: "fm" shows "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
qmdecl_def: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "qmdecl sig m = (declclass m, sig, mthd m)"
range_inj_infinite: fixes f :: "nat \<Rightarrow> 'a" assumes "inj f" shows "infinite (range f)"
sameDom_trans: fixes inp :: "'a \<Rightarrow> 'b option"   and inp' :: "'a \<Rightarrow> 'c option"   and inp'' :: "'a \<Rightarrow> 'd option" assumes "sameDom inp inp'"   and "sameDom inp' inp''" shows "sameDom inp inp''"
id_act_prop: fixes forall :: "('a \<Rightarrow> \<o>) \<Rightarrow> \<o>"   and identity :: "'a \<Rightarrow> 'a \<Rightarrow> \<o>"   and v :: "i"   and \<alpha> :: "'a"   and \<beta> :: "'a" assumes "class.id_act forall identity"   and "[\<^bold>\<A>identity \<alpha> \<beta> in v]" shows "[identity \<alpha> \<beta> in v]"
alt_matrix_char: fixes signs :: "rat list list"   and subsets :: "nat list list" shows "alt_matrix_A signs subsets = M_mat signs subsets"
LAssRedsThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
eval_r_s: shows "s = (\<lambda>x. eval r_s [x])"
Left_Coset_memI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H" shows "x \<cdot> h \<in> x \<cdot>| H"
odd_nonzero: fixes z :: "int" shows "1 + z + z \<noteq> 0"
divide_le_eq_1_neg: 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less_eq (divide b a) one = less_eq a b"
isoI: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> hom G H"   and "bij_betw h (carrier G) (carrier H)" shows "h \<in> Group.iso G H"
fneg_def: shows "fneg = (\<lambda>w. max 0 (- f w))"
length_eq_card_rows: fixes A :: "(('a, 'b) vec, 'c) vec" shows "IArray.length (matrix_to_iarray A) = CARD('c)"
loopfree_digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "loopfree_digraph G \<equiv> wf_digraph G \<and> loopfree_digraph_axioms G"
greaterThanAtMost_iff: fixes i :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "(i \<in> ord.greaterThanAtMost less_eq less l u) = (less l i \<and> less_eq i u)"
accepts_Plus_iff: fixes fsm1 :: "'b fsm"   and fsm2 :: "'b fsm"   and xs :: "'b list" shows "accepts (Plus fsm1 fsm2) xs = (accepts fsm1 xs \<or> accepts fsm2 xs)"
stable_regularCard: fixes r :: "('b \<times> 'b) set" assumes "Card_order r"   and "infinite (Field r)"   and "stable r" shows "regularCard r"
SetPC_remove: fixes p :: "'a \<Rightarrow> real"   and x :: "'a"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "p x \<noteq> 0"   and "p x \<noteq> 1"   and "finite (supp p)" shows "SetPC a (\<lambda>_. p) = a x \<^bsub>(\<lambda>_. p x)\<^esub>\<oplus> SetPC a (\<lambda>_. dist_remove p x)"
indices_def: fixes xs :: "('c \<times> ('c, 'd) operation) event list" shows "indices xs \<equiv> List.map_filter (\<lambda>x. case x of Broadcast x \<Rightarrow> Map.empty x | Deliver (i, Insert e n) \<Rightarrow> Some (fst e) | Deliver (i, Delete x) \<Rightarrow> Map.empty x) xs"
distinct_sorted_list_of_set: fixes A :: "'a set" shows "distinct (sorted_list_of_set A)"
fun_upd_twist: fixes a :: "'e"   and c :: "'e"   and m :: "'e \<Rightarrow> 'f"   and b :: "'f"   and d :: "'f" assumes "a \<noteq> c" shows "m(a := b, c := d) = m(c := d, a := b)"
return_bind_spmf: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a spmf" shows "return_spmf x \<bind> f = f x"
imp_igWlsInp: fixes MOD :: "('i, 'j, 'k, 'sort, 'l, 'm, 'n, 'o, 'p) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'n option"   and h :: "'n \<Rightarrow> 'q"   and MOD' :: "('r, 's, 't, 'sort, 'u, 'v, 'q, 'w, 'x) model_scheme" assumes "igWlsInp MOD delta inp"   and "ipresIGWls h MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
simps_2: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and C' :: "char list"   and Cs :: "char list list"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step P h stk loc C M pc (Called (C' # Cs)) frs sh = (None, h, create_init_frame P C' # (stk, loc, C, M, pc, Called Cs) # frs, sh)"
leg1_in_hom_2: shows "\<guillemotleft>s\<^sub>1 : s\<^sub>1 \<Rightarrow> s\<^sub>1\<guillemotright>"
range_cset2seq: fixes A :: "'c cset" assumes "A \<noteq> {}\<^sub>c" shows "range (Sequence.seq.Rep_seq (cset2seq A)) = rcset A"
inf_implies_top: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<le> x \<sqinter> y \<leadsto> x"
constr_constr: fixes ts :: "value list"   and us :: "value list"   and name :: "name" assumes "list_all2 veq_structure ts us" shows "veq_structure (Vconstr name ts) (Vconstr name us)"
prime_elem_imp_irreducible: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.idom times one plus zero minus uminus"   and "comm_semiring_1.prime_elem times one zero p" shows "comm_semiring_1.irreducible times one zero p"
subalgebra_M_M: fixes M :: "'b measure" shows "subalgebra M M"
card_less: shows "int CARD('a) - 1 < int CARD('a)"
obj_at_field_on_heap_imp_valid_ref_1: fixes P :: "'e \<Rightarrow> bool"   and r :: "'e"   and f :: "'f"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state" assumes "obj_at_field_on_heap P r f s" shows "obj_at (\<lambda>s. True) r s"
GroupEnd_inner_dirsum_el_decomp_nth: fixes Gs :: "'a set list"   and n :: "nat" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "n < length Gs" shows "GroupEnd (\<Oplus>G\<leftarrow>Gs. G) (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>Gs. G) then \<Oplus>Gs\<leftarrow> a ! n else (0::'a))"
truncate_error_aform_def: fixes p :: "nat"   and x :: "'a \<times> 'a pdevs" shows "truncate_error_aform p x = (eucl_truncate_down p (fst x) - fst x, trunc_err_pdevs p (snd x))"
CFGExit_axioms: shows "CFGExit sourcenode targetnode kind valid_edge Entry Exit"
real_distribution_axioms_def: fixes M :: "real measure" shows "real_distribution_axioms M \<equiv> sets M = sets borel"
minus_ms_aux_MSLNil_left: fixes ys :: "('a \<times> real) msllist" shows "minus_ms_aux MSLNil ys = uminus_ms_aux ys"
invertible: fixes u :: "'a" assumes "u \<in> G" shows "invertible u"
diff_Pair: fixes a :: "'c"   and b :: "'d"   and c :: "'c"   and d :: "'d" shows "(a, b) - (c, d) = (a - c, b - d)"
of_nat_eq_0_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = zero) = (m = 0)"
nb\<^sub>1: shows "\<forall>op\<in>set \<pi>. op \<in> set (\<Pi>\<^sub>\<O>)"
f3: shows "sqrt (real 2) + - 1 * sqrt ((Ratreal p)\<^sup>2) \<le> 0"
assms: shows "Unifier \<theta> (Fun f (x # X)) (Fun g (y # Y))"
writes_pure: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and S :: "('a, 'b, 'd) prog set"   and h' :: "'a" assumes "pure f h" shows "writes S f h h'"
verit_eq_simplify_12: fixes x3 :: "num" shows "num.One \<noteq> num.Bit1 x3"
SN: shows "SN {(x, y). arc_pos y \<and> x \<succ> y}"
br'_step_invar: fixes \<delta> :: "('d, 'e) ta_rule set"   and \<Sigma> :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<Sigma>' :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" assumes "finite \<delta>"   and "\<Sigma> \<in> br'_invar_add \<delta>"   and "br'_\<alpha> \<Sigma> \<in> br_invar \<delta>"   and "(\<Sigma>, \<Sigma>') \<in> br'_step \<delta>" shows "\<Sigma>' \<in> br'_invar_add \<delta>"
geodesics_nearby: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and H :: "'a set"   and z :: "'a" assumes "geodesic_segment_between G x y"   and "geodesic_segment_between H x y"   and "z \<in> G" shows "infdist z H \<le> 4 * deltaG TYPE('a)"
J_def: shows "J \<equiv> \<lambda>(A, A'). (ok A \<longrightarrow> ok A') \<and> more A = more A'"
transfer: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex" shows "pcr_poly (=) (\<lambda>k. if n < k then 0 else (- 1) ^ (n - k) * power_sum_puzzle_existence.e' f n (n - k)) (power_sum_puzzle_existence.Q' f n)"
interpret_floatarith_FDERIV_floatariths_eucl_of_env: fixes xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list"   and ds :: "nat list" assumes "isFDERIV DIM('a) xs fas vs"   and "freshs_floatariths fas ds"   and "length ds = DIM('a)"   and "\<And>i. i \<in> set xs \<Longrightarrow> i < length vs"   and "distinct xs"   and "\<And>i. i \<in> set ds \<Longrightarrow> i < length vs"   and "distinct ds" shows "((\<lambda>x. einterpret fas (list_updates xs (list_of_eucl x) vs)) has_derivative (\<lambda>d. einterpret (FDERIV_floatariths fas xs (map Var ds)) (list_updates ds (list_of_eucl d) vs))) (at (eucl_of_env xs vs))"
h1: shows "A \<rightarrow>\<^sub>\<S> B"
simps_4: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and m :: "'b"   and x :: "nat" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs (Actual_in (m, x)) = (\<exists>a Q r p fs ins outs. valid_edge a \<and> kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<and> sourcenode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length ins)"
job_lower_bound_machine: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and x :: "nat" assumes "lb T A j"   and "x \<in> {1..j}" shows "\<exists>y\<in>{1..m}. t x \<le> T y"
dghm_HomCod: shows "\<HH>\<lparr>HomCod\<rparr> = \<CC>"
exec_move_LAss: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and V :: "nat"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (V:=e) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
B: fixes y :: "'b process" assumes "cont g" shows "cont (\<lambda>x. y \<lbrakk>A\<rbrakk> g x)"
vbrelation_Limit_in_VsetI: fixes r :: "V"   and \<alpha> :: "V" assumes "vbrelation r"   and "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>"
wx'_ex_prop: fixes X :: "'b program set" shows "ex_prop {F. \<forall>G. F ok G \<longrightarrow> F \<squnion> G \<in> X}"
exp_S: fixes x :: "real" shows "exp (S x) = (\<Prod>p\<in>Pr x. 1 - 1 / real p)"
keys_is_none_rep: fixes k :: "'c"   and m :: "('c, 'd) mapping" shows "(k \<in> Mapping.keys m) = (\<not> Option.is_none (Mapping.lookup m k))"
65615: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - ((x \<squnion> top) \<sqinter> y) = - y \<sqinter> - x"
iterateoi_def: fixes olist_it :: "'f \<Rightarrow> (('d \<times> 'e) list \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> ('d \<times> 'e) list \<Rightarrow> ('d \<times> 'e) list) \<Rightarrow> ('d \<times> 'e) list \<Rightarrow> ('d \<times> 'e) list"   and S :: "'f" shows "poly_map_iterateoi_defs.iterateoi olist_it S \<equiv> it_to_it (olist_it S)"
hyps: fixes s :: "'c" assumes "v \<in> V"   and "s \<in> vertex_subtree v"   and "t \<in> vertex_subtree v"   and "s \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> t" shows "set xs \<subseteq> vertex_subtree v"
that: fixes V :: "'a set" assumes "openin (top_of_set U) V"   and "S \<union> T retract_of V" shows "thesis"
upd_cs_empty_split: fixes cs :: "'edge list"   and as :: "'edge list" assumes "upd_cs cs as = []"   and "cs \<noteq> []"   and "as \<noteq> []" obtains xs :: "'edge list" and ys :: "'edge list" where "as = xs @ ys" and "xs \<noteq> []" and "upd_cs cs xs = []" and "\<forall>xs' ys'. xs = xs' @ ys' \<and> ys' \<noteq> [] \<longrightarrow> upd_cs cs xs' \<noteq> []" and "upd_cs [] ys = []"
div_Integer_def: shows "div = (\<Lambda> (MkI\<cdot>x) (MkI\<cdot>y). MkI\<cdot>(divide x y))"
coprime_diff_one_right_nat: fixes n :: "nat" assumes "0 < n" shows "coprime n (n - 1)"
minarc_min: fixes y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.arc y \<and> y \<otimes> x \<noteq> mbot \<longrightarrow> sum\<^sub>M (minarc\<^sub>M x \<otimes> x) \<preceq> sum\<^sub>M (y \<otimes> x)"
mul_instr_sub1_privilege: fixes s' :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (mul_instr_sub1 instr_name result s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
qGoodBinp_iff_goodBinp_asBinp: fixes qbinp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs option" shows "goodBinp (asBinp qbinp) = qGoodBinp qbinp"
algebra_iff_Un: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<Omega> - a \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M))"
sign_comp_atom: fixes l\<^sub>1 :: "'a literal"   and l\<^sub>2 :: "'a literal" shows "(sign l\<^sub>1 \<noteq> sign l\<^sub>2 \<and> get_atom l\<^sub>1 = get_atom l\<^sub>2) = (l\<^sub>2 = l\<^sub>1\<^sup>c)"
prime_dvd_mult_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "dvd.dvd times p (times a b) = (dvd.dvd times p a \<or> dvd.dvd times p b)"
fin_f: fixes x :: "'a" assumes "x \<in> A" shows "f x \<noteq> \<top>"
mult_left_commute: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> E" shows "a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
map_zip_is_bisum: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and l1 :: "'b list" shows "bisum f l1 = sum_list (map2 f ((0::'b) # l1) (l1 @ [0::'b]))"
hd_in_set: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "hd xs \<in> set xs"
trans_join: fixes r :: "('b \<times> 'b) set" shows "trans r = (\<forall>(x, y1)\<in>r. \<forall>(y2, z)\<in>r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
length_concat: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and xss :: "'d list list" assumes "class.monoid_add plus zero" shows "length (concat xss) = sum_list (map length xss)"
finite_supp_wf_Pred: fixes \<phi> :: "'b"   and x :: "('a, 'b, 'c) formula" shows "finite (supp (binsert (Pred \<phi>) (bsingleton x)))"
2: shows "moebius_circline (moebius_rotation (- pi / 2)) H \<noteq> x_axis"
inverse_le_imp_le_neg: 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq (inverse a) (inverse b)"   and "less b zero" shows "less_eq b a"
divide_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f / g = (\<lambda>x. f x / g x)"
intros_1: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<in> \<Gamma>" shows "\<Gamma> \<turnstile>\<^sub>H F"
coeff_in_ring: fixes a :: "'g \<Rightarrow> 'a"   and A :: "'g set"   and x :: "'g" assumes "a \<in> A \<rightarrow> carrier K"   and "x \<in> A" shows "a x \<in> carrier K"
joinable_components_eq: fixes t :: "'c set"   and s :: "'c set"   and c1 :: "'c set"   and c2 :: "'c set" assumes "connected t \<and> t \<subseteq> s \<and> c1 \<in> components s \<and> c2 \<in> components s \<and> c1 \<inter> t \<noteq> {} \<and> c2 \<inter> t \<noteq> {}" shows "c1 = c2"
rep_eq: shows "lookup 0 = (\<lambda>k. 0::'c)"
discr_MtransC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr c"   and "(c, s) \<rightarrow>*c (c', s')" shows "discr c'"
isoI: fixes f :: "'a2"   and g :: "'a2" assumes "C2.inverse_arrows f g" shows "C2.iso f"
fmadd_assoc: fixes m :: "'a \<rightharpoonup>\<^sub>f 'b"   and n :: "'a \<rightharpoonup>\<^sub>f 'b"   and p :: "'a \<rightharpoonup>\<^sub>f 'b" shows "m ++\<^sub>f (n ++\<^sub>f p) = m ++\<^sub>f n ++\<^sub>f p"
mrw_change_prefix: fixes \<E> :: "('a \<times> ('b, 'a) obs_event action) llist set"   and P :: "'c prog"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and r :: "nat"   and w :: "nat"   and n :: "enat" assumes "executions_aux \<E> P"   and "E' \<in> \<E>"   and "P,E \<turnstile> r \<leadsto>mrw w"   and "thread_start_actions_ok E'"   and "ltake n E [\<approx>] ltake n E'"   and "enat r < n"   and "enat w < n" shows "P,E' \<turnstile> r \<leadsto>mrw w"
eq: fixes J :: "'i set"   and F :: "'i \<Rightarrow> 'a set" assumes "finite J"   and "J \<subseteq> I"   and "\<And>j. j \<in> J \<Longrightarrow> F j \<in> M.events j" shows "emeasure M' (emb I J (Pi\<^sub>E J F)) = (\<Prod>j\<in>J. emeasure (M j) (F j))"
distinct_remdups'_strong: fixes xs :: "'c list"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" assumes "\<forall>x\<in>set xs. \<forall>y\<in>set xs. g x = g y \<longrightarrow> f x = f y" shows "distinct (map g (remdups' f xs))"
Domain_Un_eq: fixes A :: "('d \<times> 'e) set"   and B :: "('d \<times> 'e) set" shows "Domain (A \<union> B) = Domain A \<union> Domain B"
le_neq_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
inj: shows "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
next_state_invariant: shows "\<forall>s execs. invariant s \<longrightarrow> invariant (int.next_state s execs)"
lappend_code_2: fixes xa :: "'a"   and x :: "'a llist"   and ys :: "'a llist" shows "lappend (LCons xa x) ys = LCons xa (lappend x ys)"
assms_2: fixes i :: "nat" shows "p i * i \<le> n"
prems: shows "y \<in> set (butlast (x # xs))"
encode_config: fixes ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "encode_config (ss, rv) = prod_encode (Partial_Recursive.list_encode (map encode_frame ss), encode_option rv)"
underS_empty: fixes a :: "'c"   and r :: "('c \<times> 'c) set" assumes "a \<notin> Field r" shows "underS r a = {}"
389: fixes x :: "'a"   and y :: "'a" shows "- - (x \<sqinter> y) \<sqinter> - x = bot"
maxsplit_eq: fixes P :: "'c list \<Rightarrow> bool" shows "\<forall>xs ps qs. (maxsplit P ([], xs) [] xs = (ps, qs)) = (xs = ps @ qs \<and> is_maxpref P ps xs)"
se_star_append_one: fixes c1 :: "('d, 'e) conf"   and ls :: "('d, 'e) label list"   and l :: "('d, 'e) label"   and c2 :: "('d, 'e) conf" shows "se_star c1 (ls @ [l]) c2 = (\<exists>c. se_star c1 ls c \<and> se c l c2)"
union_single_eq_diff_zmset: fixes x :: "'b"   and M :: "'b zmultiset"   and N :: "'b zmultiset" assumes "add_zmset x M = N" shows "M = N - {#x#}\<^sub>z"
dependent_proj2_abs: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec"   and i :: "real"   and j :: "real" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "i \<noteq> 0 \<or> j \<noteq> 0"   and "i *\<^sub>R p + j *\<^sub>R q = 0" shows "proj2_abs p = proj2_abs q"
Abs_ordinal_le: fixes x :: "ord0"   and y :: "ord0" shows "(Abs_ordinal (ord0rel `` {x}) \<le> Abs_ordinal (ord0rel `` {y})) = (x \<le> y)"
defined_offending: fixes N :: "'v set" assumes "wf_graph \<lparr>nodes = N, edges = {}\<rparr>" shows "c_sinvar m \<lparr>nodes = N, edges = {}\<rparr>"
abs_eq: fixes x :: "cmsg" shows "atomic (Ab x) = catomic x"
assms: shows "f x \<le> \<Down>\<^sub>E E R (f' x')"
scale_prod: fixes x :: "'a"   and a :: "'b"   and b :: "'c" shows "scale x (a, b) = (s1 x a, s2 x b)"
simps_2: fixes xs :: "'b list" shows "chop 0 xs = []"