hcju/isabelleps huggingface dataset:
simps_3: fixes uu :: "le_rel" shows "uu + Lt_Rel = Lt_Rel"
reachable_subterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('f \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and \<xi> :: "'f"   and p :: "('f, 'g, 'e, 'h) seqp"   and I :: "'g seq_action \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "automaton.trans A = seqp_sos \<Gamma>"   and "(\<xi>, p) \<in> reachable A I" shows "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
BasicRule: fixes p :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and q :: "'b set" assumes "p \<subseteq> \<lbrace>\<acute>f \<in> q\<rbrace>" shows "\<turnstile> p Basic f q"
open_A: shows "open {z. \<bar>Re z\<bar> < 1}"
rreq_dip_in_vD_dip_eq_ip: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:16..PRreq-:17} \<longrightarrow> dip \<xi> \<in> vD (rt \<xi>)) \<and> (l \<in> {PRreq-:6, PRreq-:7} \<longrightarrow> dip \<xi> = ip \<xi>) \<and> (l \<in> {PRreq-:15..PRreq-:17} \<longrightarrow> dip \<xi> \<noteq> ip \<xi>))"
accepts_epsilon: fixes w :: "'b list" shows "accepts epsilon w = (w = [])"
simps: fixes n :: "int" shows "the_eint (eint n) = n"
poly_to_mpoly_finite: fixes p :: "'b poly"   and v :: "nat" shows "finite {m. (poly.coeff p (lookup m v) when keys m \<subseteq> {v}) \<noteq> (0::'b)}"
offs_mono: fixes i :: "nat"   and j :: "nat"   and ns :: "nat list"   and k :: "nat" assumes "i \<le> j"   and "j < length ns" shows "offs ns k ! i \<le> offs ns k ! j"
simps_4: 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 a :: "nat"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step P h stk loc C M pc (Throwing [] a) frs sh = (\<lfloor>a\<rfloor>, h, (stk, loc, C, M, pc, No_ics) # frs, sh)"
simps_3: fixes x21 :: "('b btree \<times> 'b) list"   and x22 :: "'b btree" shows "Node x21 x22 \<noteq> Leaf"
fresh_tvar_trm: fixes X :: "W.tvar"   and t :: "trm" shows "X \<sharp> t"
rel_induct_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b" shows "rel_induct P I D u \<equiv> rel_induct_aux P I D `` {u}"
ideD_1: fixes a :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.ide a" shows "B.VVV.arr a"
finite: shows "finite (snd ` {(e1, e2). (e1, e2) \<in> (edges G)\<^sup>+})"
LstSeq_app: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "LstSeq s k y" shows "app s k = y"
bij_betw_imp_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "inj_on f A"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(Iup x2 = Iup y2) = (x2 = y2)"
prv_imp_tru: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp \<phi> tru)"
reflection: shows "rel_reflects_binary_pred (indRelRSTPO SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
order_iff: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a = sup a b)"
3: shows "ltake (enat k) w ?! i = w ?! i"
vconn_implies_hpar_1: fixes \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "src \<mu> = src f"
welltyped_constant0: fixes T :: "Unary_PCF.type" shows "constant0 T ::: T"
hQ0: shows "reciprocal_poly p P \<circ>\<^sub>p [:1, 1:] \<noteq> 0"
zcount_single: fixes b :: "'b"   and a :: "'b" shows "zcount {#b#}\<^sub>z a = (if b = a then 1 else 0)"
hyp_ind: shows "\<forall>y. (y, x) \<in> trm_ord \<longrightarrow> trm_rep (trm_rep y S) S = trm_rep y S"
chamber_distance_def: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplex X" shows "ChamberComplex.chamber_distance X C D = (if C = D then 0 else Suc (length (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D]))))"
integrable_add: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x + g x)"
new_array'_def: fixes v :: "'a" shows "new_array' v \<equiv> new_array v \<circ> nat_of_integer"
jordan_nf_block_size_order_bound: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list"   and n :: "nat"   and a :: "'b" assumes "jordan_nf A n_as"   and "(n, a) \<in> set n_as" shows "n \<le> Polynomial.order a (char_poly A)"
integrable_subinterval_real: fixes f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "f integrable_on {a..b}"   and "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
path_component_mem_1: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "path_component S x y" shows "x \<in> S"
exec_step_input_StepC2D: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and C' :: "char list"   and Cs :: "char list list" assumes "exec_step_input P C M pc ics = StepC2 C' Cs" shows "ics = Called (C' # Cs)"
pair_def: shows "pair \<equiv> \<lambda>f. (f False, f True)"
osp__ncop2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OSP Bet A B C P Q" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C Q"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "reflect (Br a t1 t2) = Br a (reflect t2) (reflect t1)"
assms_2: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(g has_real_derivative g' x) (at x within {a..b})"
hd0_imp_R_lin_dependent: fixes ms :: "'m list" shows "\<not> R_lin_independent ((0::'m) # ms)"
fun_eq_iff: fixes f :: "'q \<Rightarrow> 'r"   and g :: "'q \<Rightarrow> 'r" shows "(f = g) = (\<forall>x. f x = g x)"
gt_ex: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
consistent_eq_rank_ge_code: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "consistent A b = (let GJ_P = Gauss_Jordan_PA A; P_mult_b = fst GJ_P *v b in Let (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ_P) \<noteq> 0) + 1) ((\<le>) (if \<exists>a. P_mult_b $ a \<noteq> (0::'d) then mod_type_class.to_nat (GREATEST a. P_mult_b $ a \<noteq> (0::'d)) + 1 else 0)))"
Riemann_lebesgue_sin_half: fixes f :: "real \<Rightarrow> real" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>n. LINT x|lebesgue_on {- pi..pi}. sin ((real n + 1 / 2) * x) * f x) \<longlonglongrightarrow> 0"
assms_2: fixes P :: "ccs"   and Q :: "ccs" assumes "(P, Q) \<in> X" shows "P \<leadsto>\<^sup>^<X> Q"
X_def: shows "X = [:0::'b, 1::'b:]"
Non_Strict_stable1: shows "Non_Strict.stable {(M1, W2), (M2, W1), (M3, W3)}"
BinOp_\<tau>execI2: 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 e :: "'addr expr1"   and bop :: "bop"   and v :: "'addr val" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e \<guillemotleft>bop\<guillemotright> e') h (stk @ [v], loc, length (compE2 e) + pc, xcp) (stk' @ [v], loc', length (compE2 e) + pc', xcp')"
Inters_None: fixes r :: "'c rexp" shows "(Zero \<in> toplevel_inters r) = (Inters r = None)"
wt_sym_gt_0: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and wt_sym :: "'a \<Rightarrow> nat"   and \<epsilon> :: "nat"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and extf :: "'a \<Rightarrow> (('a, 'b) tm \<Rightarrow> ('a, 'b) tm \<Rightarrow> bool) \<Rightarrow> ('a, 'b) tm list \<Rightarrow> ('a, 'b) tm list \<Rightarrow> bool"   and f :: "'a" assumes "kbo_basic_basis gt_sym wt_sym \<epsilon> ground_heads_var extf" shows "0 < wt_sym f"
plus: fixes p1 :: "perm"   and p2 :: "perm" assumes "P p1"   and "P p2"   and "supp p1 \<subseteq> S"   and "supp p2 \<subseteq> S" shows "P (p1 + p2)"
strassen_mat_mult: fixes A :: "'b mat"   and B :: "'b mat" assumes "dim_col A = dim_row B" shows "strassen_mat_mult A B = A * B"
det_rw: fixes f :: "nat \<Rightarrow> nat" assumes "(\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)" shows "Determinant.det (mat\<^sub>r n n (\<lambda>i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i))) = (\<Prod>i = 0..<n. B $$ (f i, i)) * Determinant.det (mat\<^sub>r n n (\<lambda>i. col A (f i)))"
\<alpha>_dim_1: shows "\<alpha> \<in> carrier_vec N"
integrable: fixes A :: "real set" assumes "A \<in> sets lborel"   and "A \<subseteq> {r..}" shows "complex_set_integrable lborel A (g s)"
Stone_Weierstrass_basic: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "continuous_on S f"   and "0 < e" shows "\<exists>g\<in>Collect P. \<forall>x\<in>S. \<bar>f x - g x\<bar> < e"
total: fixes b :: "'a"   and a :: "'a" assumes "\<not> R b a"   and "a \<in> set (x # xs)"   and "b \<in> set (x # xs)" shows "R a b"
finfun_const_apply: fixes b :: "'a"   and a :: "'b" shows "finfun_apply (finfun_const b) a = b"
count_terminals_append: fixes a :: "symbol list"   and b :: "symbol list" shows "count_terminals (a @ b) = count_terminals a + count_terminals b"
v11_vlrestriction_vrange: fixes s :: "V"   and r :: "V" assumes "v11 s"   and "v11 (r \<restriction>\<^sup>l\<^sub>\<circ> \<R>\<^sub>\<circ> s)" shows "v11 (r \<circ>\<^sub>\<circ> s)"
is_natural_1: fixes f :: "'d" assumes "D.arr f" shows "F f \<cdot>\<^sub>C F (D.dom f) = F f"
jpTraces_step_inv: fixes t :: "'s Trace"   and s :: "'s"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol" assumes "t \<leadsto> s \<in> SPR.jpTraces jp" shows "t \<in> SPR.jpTraces jp \<and> (\<exists>eact\<in>set (envAction (tLast t)). \<exists>aact. (\<forall>a. aact a \<in> set (SPR.actJP jp t a)) \<and> s = envTrans eact aact (tLast t))"
partition_onE: fixes A :: "'a set"   and P :: "'a set set" assumes "partition_on A P" obtains "\<And>p. p \<in> P \<Longrightarrow> p \<noteq> {}" and "\<Union> P = A" and "\<And>p p'. \<lbrakk>p \<in> P; p' \<in> P; p \<noteq> p'\<rbrakk> \<Longrightarrow> p \<inter> p' = {}"
b_span: fixes a :: "'a"   and b :: "'b" assumes "a \<in> basisA'" shows "\<exists>G\<in>cspan {F a b |b. b \<in> basisB}. \<forall>a'\<in>basisA'. G *\<^sub>V a' = (if a' = a then b else (0::'b))"
length_greater_0_conv: fixes xs :: "'a list" shows "(0 < length xs) = (xs \<noteq> [])"
tm_expand_apps: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "head s = head t"   and "args s = args t" shows "s = t"
is_scc_connected': fixes E :: "('a \<times> 'a) set"   and scc :: "'a set"   and x :: "'a"   and y :: "'a" assumes "is_scc E scc"   and "x \<in> scc"   and "y \<in> scc" shows "(x, y) \<in> (Restr E scc)\<^sup>*"
src_cod: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src (cod \<mu>) = src \<mu>"
simps: fixes p :: "'a poly"   and q :: "'a poly" shows "gcd_poly_code_aux p q = (if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))"
congBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s Q" shows "P \<sim> Q"
assms_2: shows "ref \<notin> set (interp_ins pre)"
assms_16: shows "InvariantEquivalentZL (getF state) (getM state) F0'"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> less_option x xa" obtains ao :: "'a option" and bo :: "'a option" where "x = ao" and "xa = bo" and "\<not> (less_eq_option ao bo \<and> ao \<noteq> bo)"
pred_set: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool" shows "pred_xstate P1 P2 = (\<lambda>x. Ball (set1_xstate x) P1 \<and> Ball (set2_xstate x) P2)"
uniformly_convergent_Cauchy: fixes X :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" assumes "uniformly_convergent_on X f" shows "uniformly_Cauchy_on X f"
star_plus: fixes y :: "'a" shows "top \<sqinter> y = y \<sqinter> top"
HStartBallot_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "p \<noteq> q"   and "HInv4b s q" shows "HInv4b s' q"
mult_cvec_def: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" shows "z *\<^sub>v w = (let (z1, z2) = z; (w1, w2) = w in if z1 = 0 \<and> w2 = 0 \<or> w1 = 0 \<and> z2 = 0 then (1, 1) else (z1 * w1, z2 * w2))"
simps_1: fixes l :: "'c list" shows "list_diff l [] = Some l"
rel_T_neg_distr_cond_def: fixes Co1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Co1' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and Contra1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra2 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and uu :: "('n \<times> 'o \<times> 'p \<times> 'q) itself" shows "rel_T_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1'. rel_T (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_T L1 Co1 Co2 Contra1 Contra2 OO rel_T L1' Co1' Co2' Contra1' Contra2')"
prod_tOp_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" shows "prod_tOp I A = (\<lambda>f\<in>carr_prodag I A. \<lambda>g\<in>carr_prodag I A. \<lambda>x\<in>I. f x \<cdot>\<^sub>r\<^bsub>A x\<^esub> g x)"
ass: shows "coprime a b \<and> a\<^sup>2 + 3 * b\<^sup>2 = w ^ 3 \<and> odd w"
a_asymptotics: shows "(\<lambda>x. T x - x * ln x) \<in> O(\<lambda>x. x)"
ccompare: fixes ccompare :: "('a \<Rightarrow> 'a \<Rightarrow> order) option"   and comp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ccompare = Some comp" shows "comparator comp"
Units_l_inv: fixes x :: "'a" assumes "x \<in> Units G" shows "inv x \<otimes> x = \<one>"
PSQ_P'S'Q': shows "(P, S, Q) = (Mod_Type_Connect.to_hma\<^sub>m P', Mod_Type_Connect.to_hma\<^sub>m S', Mod_Type_Connect.to_hma\<^sub>m Q')"
any_has_not_element_type: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "element_type \<tau> \<sigma>"   and "\<tau> \<le> OclAny[?] \<or> \<tau> \<le> Tuple fmempty" shows "False"
simps_3: fixes v :: "bool"   and va :: "bool list" shows "path (v # va) Leaf = False"
pdg_dg_prod_Obj_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "dg_prod I \<AA>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
val_in_type_universe: fixes v :: "val" shows "v \<in> type_universe (val_type v)"
distinct_take_cols_id: fixes inds :: "nat list"   and n :: "nat" assumes "distinct inds" shows "distinct (cols (take_cols (1\<^sub>m n) inds))"
gs_sq: shows "\<bar>fs.gs.gso i $ j\<bar>\<^sup>2 \<le> rat_of_nat N"
\<dd>_foldr_Times: fixes xs :: "'b language list"   and s :: "'b language"   and a :: "'b" shows "\<dd> (foldr Times xs s) a = (let n = length (takeWhile \<oo> xs) in PLUS (map (\<lambda>zs. TIMES (\<dd> (hd zs) a # tl zs)) (take (Suc n) (tails (xs @ [s])))))"
nat_le_linear: fixes m :: "nat"   and n :: "nat" shows "(m::nat) \<le> (n::nat) \<or> n \<le> m"
supp_prod: fixes x :: "'e"   and y :: "'f" shows "supp (x, y) = supp x \<union> supp y"
span_explicit': fixes b :: "('a, 'b) vec set" shows "vec.span b = {\<Sum>v | f v \<noteq> (0::'a). f v *s v |f. finite {v. f v \<noteq> (0::'a)} \<and> (\<forall>v. f v \<noteq> (0::'a) \<longrightarrow> v \<in> b)}"
clock_numbering'_2: shows "\<forall>c. c \<notin> X \<longrightarrow> n < v c"
C_eqLemmas_id_22: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "distinct p"   and "noDenyAll1 p"   and "all_in_list p l"   and "singleCombinators p" shows "NetsCollected (qsort p l)"
dist_triangle4: fixes x :: "'a"   and t :: "'a"   and y :: "'a"   and z :: "'a" shows "dist x t \<le> dist x y + dist y z + dist z t"
0: shows "f2 * (real l0 * min e c) \<le> c * (f2 * real l0)"
scale_scale_spmf: fixes r :: "real"   and r' :: "real"   and p :: "'a spmf" shows "scale_spmf r (scale_spmf r' p) = scale_spmf (r * max 0 (min (inverse (weight_spmf p)) r')) p"
IH: fixes y :: "nat"   and j' :: "int"   and i :: "int"   and i' :: "int"   and j :: "int" assumes "y < l"   and "y = nat (j' - i)"   and "i \<le> i'"   and "i' \<le> j"   and "j \<le> j'" shows "c i j + c i' j' \<le> c i j' + c i' j"
add_less_imp_less_right: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" assumes "a + c \<subset># b + c" shows "a \<subset># b"
decompose_prod_root_unity: fixes p :: "complex poly"   and ks :: "nat list"   and f :: "complex poly"   and ks' :: "nat list"   and g :: "complex poly" assumes "p = prod_root_unity ks * f"   and "decompose_prod_root_unity p = (ks', g)"   and "\<And>x. cmod x = 1 \<Longrightarrow> poly f x \<noteq> 0"   and "p \<noteq> 0" shows "p = prod_root_unity ks' * f \<and> f = g \<and> set ks = set ks'"
open_map: fixes S :: "'a set" assumes "open S" shows "open (f ` S)"
mkMinj_def: fixes i :: "nat"   and M :: "mon" shows "mkMinj i M = (case M of Mc c \<Rightarrow> Mc c | Minj j M \<Rightarrow> Minj (i + j) M | MX nat mon \<Rightarrow> Minj i M)"
zero_in_deg_le_sect: fixes n :: "'b set"   and d :: "nat" shows "0 \<in> deg_le_sect n d"
ordLess_ordIso_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r <o r'"   and "r' =o r''" shows "r <o r''"
num_params_polyneg: fixes p :: "'a poly" shows "num_params (~\<^sub>p p) = num_params p"
length_lv_rel: fixes ys :: "real list"   and y :: "'a" assumes "(ys, y) \<in> lv_rel" shows "length ys = DIM('a)"
assms: shows "simple_linux_router rt fw mlf ifl pi = Some po"
p_imp_p: fixes P :: "'a upred" shows "(P \<Rightarrow> P) = true"
assms_2: fixes w :: "nat"   and x :: "nat"   and y :: "nat"   and z :: "nat" shows "eval f [w, x, y, z] \<down>"
hyps_4: shows "sibling (Node w t\<^sub>1 t\<^sub>2) b \<noteq> b"
f: shows "(\<Sum>i\<in>set_pmf (mix_pmf a p q). a * pmf p i * f i + (1 - a) * pmf q i * f i) = (\<Sum>i\<in>set_pmf (mix_pmf a p q). a * pmf p i * f i) + (\<Sum>i\<in>set_pmf (mix_pmf a p q). (1 - a) * pmf q i * f i)"
angelic_def: fixes Q :: "'a \<Rightarrow> 'b"   and p :: "'b" shows "{: Q :} p = {s. Q s \<sqinter> p \<noteq> \<bottom>}"
greaterThanAtMost_nat_numeral: fixes m :: "nat"   and k :: "num" shows "{m<..numeral k} = (if m < numeral k then insert (numeral k) {m<..pred_numeral k} else {})"
add_commute: fixes a :: "'a"   and b :: "'a" shows "min a b = min b a"
complete_delete: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c" assumes "complete t" shows "complete (Tree23_Map.delete x t)"
e_if_elim: fixes v :: "val"   and e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E (EIf e1 e2 e3) \<rho>" obtains n :: "nat" where "VNat n \<in> E e1 \<rho>" and "n = 0 \<longrightarrow> v \<in> E e3 \<rho>" and "n \<noteq> 0 \<longrightarrow> v \<in> E e2 \<rho>"
lborel_affine: fixes c :: "real"   and t :: "'b" assumes "c \<noteq> 0" shows "lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>x. ennreal (\<bar>c\<bar> ^ DIM('b)))"
Times_Un_distrib1: fixes A :: "'a set"   and B :: "'a set"   and C :: "'b set" shows "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C"
sig_red_singleD5: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'e"   and q :: "'c \<Rightarrow>\<^sub>0 'e"   and f :: "'c \<Rightarrow>\<^sub>0 'e"   and t :: "'d" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail p q f t" shows "ord.is_le_rel ord ord_strict top_tail"
assms_5: shows "(A ===> (=)) inEdges' inEdges2"
ipresIGFreshAbs_def: fixes hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'ae, 'w, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ai, 'aj, 'ad, 'ak, 'x, 'al) model_scheme" shows "ipresIGFreshAbs hA MOD MOD' \<equiv> \<forall>ys y us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igFreshAbs MOD ys y A \<longrightarrow> igFreshAbs MOD' ys y (hA A)"
setsetmapim_restrict1: fixes B :: "'b set set"   and A :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "order.greater_eq A B" shows "restrict1 f (\<Union> A) \<turnstile> B = f \<turnstile> B"
map2_rel_witness_option: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "rel_option A x y" shows "map_option snd (rel_witness_option (x, y)) = y"
remdups_remdups: fixes xs :: "'a list" shows "remdups (remdups xs) = remdups xs"
V_constotal_def: shows "V_constotal \<equiv> V_constotal_1 \<union> V_constotal_2"
sorted_list_of_fset_simps_1: fixes S :: "'e fset" shows "set (sorted_list_of_fset S) = fset S"
key_sets_crypt_2: fixes K :: "key"   and X :: "msg"   and Y :: "msg" shows "insert (InvKey K) ` key_sets X {Y} \<subseteq> key_sets X {Crypt K Y}"
ennsqrt_def: fixes x :: "ennreal" shows "ennsqrt x = ennexp (eln x / 2)"
combine_substs: fixes V1 :: "'b set"   and V2 :: "'b set"   and \<eta>1 :: "('b \<times> 'b trm) list"   and \<eta>2 :: "('b \<times> 'b trm) list" assumes "finite V1"   and "V1 \<inter> V2 = {}"   and "ground_on V1 \<eta>1" shows "\<exists>\<sigma>. coincide_on \<sigma> \<eta>1 V1 \<and> coincide_on \<sigma> \<eta>2 V2"
scene_inter_compl: fixes X :: "'a scene" assumes "idem_scene X" shows "X \<sqinter>\<^sub>S - X = \<bottom>\<^sub>S"
uint_word_ariths_3: fixes a :: "'b word"   and b :: "'b word" shows "uint (a * b) = uint a * uint b mod 2 ^ LENGTH('b)"
not_lnull_lappend_endE: fixes w :: "'b llist" assumes "\<not> lnull w" obtains a :: "'b" and v :: "'b llist" where "w = v $ <a>"
negligible_Union: fixes \<T> :: "'a set set" assumes "finite \<T>"   and "\<And>t. t \<in> \<T> \<Longrightarrow> negligible t" shows "negligible (\<Union> \<T>)"
smooth_on_open_subsetsI: fixes S :: "'c set"   and k :: "enat"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. x \<in> T \<and> open T \<and> k-smooth_on T f" shows "k-smooth_on S f"
add_zero2: fixes x :: "'b" shows "Sep_Eq.add2 (0::'b) x = Some x"
fsubset_finsertI: fixes B :: "'a fset"   and a :: "'a" shows "B |\<subseteq>| finsert a B"
iFROM_iprev: fixes n :: "nat"   and t :: "nat" assumes "n < t" shows "iprev t [n\<dots>] = t - Suc 0"
Abs_atoms_cases: fixes x :: "'a atoms" obtains y :: "'a set" where "x = Abs_atoms y" and "y \<in> range atom_map"
restrict_extensional: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "restrict f A \<in> extensional A"
pos_mod_bound: fixes b :: "'m"   and a :: "'m" assumes "(0::'m) < b" shows "a mod b < b"
res_neg_eq: fixes m :: "int"   and x :: "int" assumes "residues m" shows "\<ominus>\<^bsub>residue_ring m\<^esub> x = - x mod m"
distinct_product: fixes xs :: "'c list"   and ys :: "'d list" assumes "distinct xs"   and "distinct ys" shows "distinct (List.product xs ys)"
jordan_nf_root_char_poly: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and m :: "nat"   and lam :: "'a" assumes "jordan_nf A n_as"   and "(m, lam) \<in> set n_as" shows "poly (char_poly A) lam = (0::'a)"