hcju/isabelleps huggingface dataset:
Aexp_IfThenElse: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and scrut :: "exp"   and e1 :: "exp"   and a :: "Arity"   and e2 :: "exp" assumes "ArityAnalysisSafe Aexp" shows "Aexp scrut\<cdot>0 \<squnion> Aexp e1\<cdot>a \<squnion> Aexp e2\<cdot>a \<sqsubseteq> Aexp (scrut ? e1 : e2)\<cdot>a"
cases: fixes x :: "'c local.term" obtains "x = \<^bold>\<I>"   | f :: "'c" where "x = \<^bold>\<langle>f\<^bold>\<rangle>"   | a :: "'c local.term" and b :: "'c local.term" where "x = a \<^bold>\<otimes> b"   | v :: "'c local.term" and va :: "'c local.term" where "x = v__ \<^bold>\<cdot> va__"   | v :: "'c local.term" where "x = \<^bold>\<l>\<^bold>[vb__\<^bold>]"   | v :: "'c local.term" where "x = \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[vc__\<^bold>]"   | v :: "'c local.term" where "x = \<^bold>\<r>\<^bold>[vd__\<^bold>]"   | v :: "'c local.term" where "x = \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[ve__\<^bold>]"   | v :: "'c local.term" and va :: "'c local.term" and vb :: "'c local.term" where "x = \<^bold>\<a>\<^bold>[vf__, vaa__, vba__\<^bold>]"   | v :: "'c local.term" and va :: "'c local.term" and vb :: "'c local.term" where "x = \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[v, va, vb\<^bold>]"
plus_bit0_def: fixes x :: "'b bit0"   and y :: "'b bit0" shows "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
ex: shows "i \<in> obs_ids \<pi> \<inter> {..<j}"
unx_none_simps_4: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unconst (free name) = None"
I_FR\<^sub>1: fixes \<phi> :: "atom fm"   and xs :: "real list" assumes "nqfree \<phi>" shows "R.I (FR\<^sub>1 \<phi>) xs = (\<exists>x. R.I \<phi> (x # xs))"
assms_4: fixes n :: "nat" shows "Norm (\<LL> p M) (f n) \<le> C"
constant_process_borel_adapted: fixes c :: "'c" shows "borel_adapt_stoch_proc F (\<lambda>n w. c)"
bfs_impl_def: fixes succ :: "nat \<Rightarrow> nat dlist"   and src :: "nat"   and dst :: "nat" shows "bfs_impl succ src dst \<equiv> WHILE (\<lambda>(f, V, C, N, d). f = False \<and> \<not> ls.isEmpty C) (\<lambda>(f, V, C, N, d). RETURN (the (ls.sel C (\<lambda>_. True))) \<bind> (\<lambda>v. let C = ls.delete v C in if v = dst then RETURN (True, hs.empty (), ls.empty (), ls.empty (), d) else FOREACH (ls.\<alpha> (succ v)) (\<lambda>w (V, N). if \<not> hs.memb w V then RETURN (hs.ins w V, ls.ins_dj w N) else RETURN (V, N)) (V, N) \<bind> (\<lambda>(V, N). if ls.isEmpty C then let C = N; N = ls.empty (); d = d + 1 in RETURN (f, V, C, N, d) else RETURN (f, V, C, N, d)))) (False, hs.sng src, ls.sng src, ls.empty (), 0) \<bind> (\<lambda>(f, uu_, uu_, uu_, d). if f then RETURN (Some d) else RETURN None)"
assms_3: shows "x \<noteq> \<infinity>\<^sub>h"
external_actions_committed_def: fixes P :: "'a prog"   and J :: "nat \<Rightarrow> ('b, 'c) justifying_execution" shows "external_actions_committed P J = (\<forall>n. \<forall>a\<in>external_actions (justifying_exec (J n)). \<forall>a'\<in>committed (J n). P,justifying_exec (J n) \<turnstile> a \<le>hb a' \<longrightarrow> a \<in> committed (J n))"
Collect_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "order.greater_eq (Collect Q) (Collect P)"
minarc_arc: fixes x :: "'a" shows "x \<noteq> bot \<longrightarrow> arc (minarc x)"
Max_gr_iff: fixes A :: "'a DBMEntry set"   and x :: "'a DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(x \<prec> linordered_monoid.Max A) = (\<exists>a\<in>A. x \<prec> a)"
topology\<^sub>N_def: fixes N :: "'b quasinorm" shows "topology\<^sub>N N = topology (\<lambda>U. \<forall>x\<in>U. \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U)"
smap_Union_mono: fixes ms1 :: "('a \<Rightarrow> 'b set) list"   and ms2 :: "('a \<Rightarrow> 'b set) list" assumes "list_all2 smap_less ms1 ms2" shows "\<Union>.ms1 \<subseteq>. \<Union>.ms2"
of_nat_mono: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and i :: "nat"   and j :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "i \<le> j" shows "less_eq (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j)"
assms: shows "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) (is @ [i])"
deltafun_plus: fixes b :: "'d"   and c :: "'d"   and a :: "'c" shows "(b + c) \<delta> a = b \<delta> a + c \<delta> a"
span_eq: fixes S :: "'d set"   and T :: "'d set" shows "(cspan S = cspan T) = (S \<subseteq> cspan T \<and> T \<subseteq> cspan S)"
dist_eqs_2: fixes discr :: "bool discr"   and cfun' :: "Value \<rightarrow> Value" shows "B\<cdot>discr \<noteq> Fn\<cdot>cfun'"
step_3_a_inv: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "i < j"   and "j < n"   and "local.inv_upto local.jb A j"   and "local.inv_from local.uppert A j"   and "local.inv_from_bot (\<lambda>A i. local.one_zero A i j) A i"   and "local.ev_block n A" shows "local.inv_from local.uppert (step_3_a i j A) j \<and> local.inv_upto local.jb (step_3_a i j A) j \<and> local.inv_at local.one_zero (step_3_a i j A) j \<and> local.same_diag A (step_3_a i j A)"
simps: fixes a :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set" shows "(a \<in> generate_ring R H) = (a = \<one>\<^bsub>R\<^esub> \<or> (\<exists>h. a = h \<and> h \<in> H) \<or> (\<exists>h. a = \<ominus>\<^bsub>R\<^esub> h \<and> h \<in> generate_ring R H) \<or> (\<exists>h1 h2. a = h1 \<oplus>\<^bsub>R\<^esub> h2 \<and> h1 \<in> generate_ring R H \<and> h2 \<in> generate_ring R H) \<or> (\<exists>h1 h2. a = h1 \<otimes>\<^bsub>R\<^esub> h2 \<and> h1 \<in> generate_ring R H \<and> h2 \<in> generate_ring R H))"
singleton: fixes x :: "'a" shows "\<Prod>\<^sub># {#x#} = x"
delta_3: shows "ennreal (1 - d) \<le> emeasure MI {x \<in> space MI. 1 - d \<le> upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - delta l * real l < vI n x - vI (n - l) ((TI ^^ l) x)}}"
implements_def: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" shows "implements jp \<equiv> \<forall>t\<in>jkbpC. set \<circ> actJP jp t = set \<circ> jAction MC t"
false: shows "\<not> fst a \<subseteq>\<^sub>f s1"
asTerm_equal_iff_alpha: fixes qX :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and qY :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "qGood qX \<or> qGood qY" shows "(asTerm qX = asTerm qY) = (qX #= qY)"
p1: shows "lead_coeff p1 \<in> carrier R - {\<zero>}"
openInjectAuxAuxAux: fixes x :: "name"   and xvec :: "name list" shows "\<exists>y yvec. x # xvec = yvec @ [y] \<and> length xvec = length yvec"
v: shows "vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s)"
enab_beta2: shows "\<turnstile> $pc2 = #b \<longrightarrow> Enabled beta2"
cross_dot_cancel: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" assumes "x \<bullet> y = x \<bullet> z"   and "x \<times> y = x \<times> z"   and "x \<noteq> 0" shows "y = z"
concat_eq_Nil_conv: fixes xss :: "'c list list" shows "(concat xss = []) = (\<forall>xs\<in>set xss. xs = [])"
Liminf_obtain': fixes F :: "'d filter"   and u :: "'d \<Rightarrow> 'c"   and c :: "'c"   and P :: "'d \<Rightarrow> bool" assumes "Liminf F u < c"   and "eventually P F" shows "\<exists>n. P n \<and> u n < c"
set_mset_eq_implies_surj_on: fixes A :: "'c set"   and M :: "'d multiset"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "size M = card A"   and "set_mset M = B"   and "f \<in> functions_of A M" shows "f ` A = B"
Dickson_poly_mapping: fixes V :: "'a set" assumes "finite V" shows "almost_full_on (adds) {x. keys x \<subseteq> V}"
execlass_leq_def: fixes cs :: "('a \<times> 'b) list"   and c1 :: "'a"   and c2 :: "'b" shows "execlass_leq cs c1 c2 = List.member cs (c1, c2)"
lifting_q: fixes q :: "'q" assumes "q \<in> Q" shows "labeled_tiebreaker_lifting Bot_F Inf_F Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_q q) (\<G>_I_q q) (\<lambda>g Cl Cl'. False) Inf_FL"
has_bochner_integral_integrable: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "integrable M f" shows "has_bochner_integral M f (integral\<^sup>L M f)"
vld_def: fixes \<eta> :: "nat" shows "vld \<eta> \<equiv> nlists UNIV \<eta>"
lcm_div_gt_0: fixes a :: "nat"   and b :: "nat" assumes "0 < a"   and "0 < b" shows "0 < lcm a b div a"
not_in_complement_partition_False: fixes A :: "'a set" assumes "A \<in> infinite_complement_partition"   and "infinite UNIV" shows "(- A \<in> infinite_complement_partition) = False"
sorted_wrt_list_set: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list" assumes "linorder_on A R"   and "set xs \<subseteq> A" shows "sorted_wrt_list_of_set R (set xs) = insort_wrt R (remdups xs)"
summands_part_implies_sum_part: fixes x :: "i"   and z :: "i"   and y :: "i" assumes "P x z \<and> P y z" shows "P (x \<oplus> y) z"
simps_4: fixes X :: "'b list"   and F :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>)\<rangle>\<^sub>s\<^sub>t"
foldl_Cons: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a"   and x :: "'b"   and xs :: "'b list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
simps_3: fixes n :: "nat"   and m :: "nat" shows "geod (Suc n) (2 * m) = geod n m"
disc_49: fixes x71 :: "'a expr"   and x72 :: "call_kind"   and x73 :: "'a call_expr" shows "is_Call (Call x71 x72 x73)"
dtor2_o_unfold2: fixes s1 :: "'f \<Rightarrow> ('e, 'f, 'd) F1"   and s2 :: "'d \<Rightarrow> ('e, 'f, 'd) F2" shows "dtor2 \<circ> unfold2 s1 s2 = F2map id (unfold1 s1 s2) (unfold2 s1 s2) \<circ> s2"
directly_attracted_disjoint: fixes p :: "Player"   and W :: "'a set" shows "directly_attracted p W \<inter> W = {}"
notEq: shows "val[p,c] \<noteq> {True, False}"
simps_2: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) BinomialTree list" shows "tree_invar (Node e a (Suc r) ts) = (\<exists>e1 a1 ts1 e2 a2 ts2. tree_invar (Node e1 a1 r ts1) \<and> tree_invar (Node e2 a2 r ts2) \<and> Node e a (Suc r) ts = link (Node e1 a1 r ts1) (Node e2 a2 r ts2))"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs G" shows "edge_rev M a = a"
mult_le_less_imp_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq a b"   and "less c d"   and "less zero a"   and "less_eq zero c" shows "less (times a c) (times b d)"
epred_enat: fixes m :: "nat" shows "epred (enat m) = enat (m - 1)"
Group_RSpan_single: fixes m :: "'m" assumes "m \<in> M" shows "Group (RSpan [m])"
funeq_LeastTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and l :: "nat" assumes "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "l < (LEAST k. NfuncPair_neq_at f g k)" shows "f l = g l"
IdI: fixes a :: "'k" shows "(a, a) \<in> Id"
l_def: shows "l = lcs (punit.lt (rep_list p)) (punit.lt (rep_list q))"
RETURN_rel_conv0: fixes f :: "'c"   and g :: "'d"   and A :: "('c \<times> 'd) set" shows "((RETURN f, RETURN g) \<in> \<langle>A\<rangle>nres_rel) = ((f, g) \<in> A)"
bell11_is_state: shows "state 2 |\<beta>\<^sub>1\<^sub>1\<rangle>"
succ_tran_set: fixes G :: "'b list_graph"   and v :: "'b" shows "set (FiniteListGraph.succ_tran G v) = {e2. (v, e2) \<in> (set (edgesL G))\<^sup>+}"
map\<^sub>0_def: fixes a :: "'c" shows "F.map\<^sub>0 a \<equiv> src\<^sub>D (local.map a)"
sqrt_nat_def: fixes x :: "nat" shows "sqrt_nat x = root_nat 2 x"
simps_2: fixes vT :: "variable \<times> typ"   and n :: "nat"   and T :: "typ"   and t :: "Core.term" shows "bind_fv2 vT n (Abs T t) = Abs T (bind_fv2 vT (n + 1) t)"
star_le_def: shows "(\<le>) \<equiv> *p2* (\<le>)"
simps_2: fixes f :: "'fsym"   and Tl :: "'fsym trm list" shows "tpOf (Fn f Tl) = resOf f"
even_of_nat: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (semiring_1.of_nat one plus zero a) = even a"
decode_instr_result_3: fixes x :: "Exception list + sparc_operation \<times> inst_operand list"   and v1 :: "word32"   and y :: "Exception list + sparc_operation \<times> inst_operand list"   and v2 :: "word32" assumes "x = decode_instruction v1 \<and> y = decode_instruction v2 \<and> v1 = v2" shows "x = y"
assms_1: fixes u :: "'a"   and v :: "'a" assumes "snd (rep_nat_term u) = snd (rep_nat_term v)"   and "fst (rep_nat_term u) = 0" shows "cmp u v \<noteq> Gt"
nat_term_compare_proj_ord: fixes cmp :: "(('a, 'b) pp \<times> nat) nat_term_order" shows "nat_term_compare (proj_ord cmp) = proj_comp cmp"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_monoid_add_hom hom"   and "inj_monoid_add_hom hom" shows "inj_comm_monoid_add_hom hom"
simps_22: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a" shows "(case a\<^sub>2 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3 | a\<^sub>4 \<Rightarrow> f4 | a\<^sub>5 \<Rightarrow> f5) = f2"
mult_four_block_mat: fixes A1 :: "'b mat"   and nr1 :: "nat"   and n1 :: "nat"   and B1 :: "'b mat"   and n2 :: "nat"   and C1 :: "'b mat"   and nr2 :: "nat"   and D1 :: "'b mat"   and A2 :: "'b mat"   and nc1 :: "nat"   and B2 :: "'b mat"   and nc2 :: "nat"   and C2 :: "'b mat"   and D2 :: "'b mat" assumes "A1 \<in> carrier_mat nr1 n1"   and "B1 \<in> carrier_mat nr1 n2"   and "C1 \<in> carrier_mat nr2 n1"   and "D1 \<in> carrier_mat nr2 n2"   and "A2 \<in> carrier_mat n1 nc1"   and "B2 \<in> carrier_mat n1 nc2"   and "C2 \<in> carrier_mat n2 nc1"   and "D2 \<in> carrier_mat n2 nc2" shows "four_block_mat A1 B1 C1 D1 * four_block_mat A2 B2 C2 D2 = four_block_mat (A1 * A2 + B1 * C2) (A1 * B2 + B1 * D2) (C1 * A2 + D1 * C2) (C1 * B2 + D1 * D2)"
zero_le_square: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a a)"
inv_closed: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> x \<in> carrier V"
ex_nonseparating_wall_imp_wall_crossings_not_distinct: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set"   and H :: "'a set set set" assumes "gallery (C # Cs @ [D])"   and "H \<in> set (wall_crossings (C # Cs @ [D]))"   and "H \<noteq> {}"   and "\<not> separated_by H C D" shows "\<not> distinct (wall_crossings (C # Cs @ [D]))"
lsorted_LCons_LCons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a llist" shows "lsorted (LCons x (LCons y xs)) = (x \<le> y \<and> lsorted (LCons y xs))"
case_nat_unfold: fixes x :: "'f"   and f :: "nat \<Rightarrow> 'f"   and n :: "nat" shows "case n of 0 \<Rightarrow> x | Suc x \<Rightarrow> f x \<equiv> if n = 0 then x else f (n - 1)"
d_transitive: fixes x :: "'a" assumes "\<forall>y.  |x\<rangle>  |x\<rangle> y \<le>  |x\<rangle> y" shows "\<forall>y.  |x\<rangle> y =  |x\<^sup>\<star>\<rangle>  |x\<rangle> y"
impossibility_of_trisecting_pi_over_3_lemma: fixes x :: "real" assumes "x \<in> radical_sqrt"   and "x ^ 3 - 3 * x - 1 = 0" shows "False"
I_R_iprod_a: fixes cs :: "real list"   and as :: "atom list"   and xs :: "real list" assumes "length cs = length as"   and "\<forall>(c, a)\<in>set (zip cs as). I\<^sub>R (c *\<^sub>a a) xs" shows "I\<^sub>R (cs \<odot>\<^sub>a as) xs"
complement_ranking: fixes w :: "'c stream"   and A :: "('c, 'd) nba" assumes "w \<in> language (complement A)" obtains f :: "nat \<times> 'd \<Rightarrow> nat" where "ranking A w f"
A: shows "A \<in> carrier_mat n n \<and> isDiagonal_mat A"
asymp_equiv_eventually_pos_iff: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. (0 < f x) = (0 < g x)"
R: fixes x :: "'b"   and \<sigma> :: "'a" shows "RETURN (fi x \<sigma>) \<le> f x \<sigma>"
f12_def: shows "f12 = fst (split_face f v w ws)"
simps_2: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and y :: "'c"   and ys :: "'c list" shows "insort_key f x (y # ys) = (if f x \<le> f y then x # y # ys else y # insort_key f x ys)"
lossless_seed_gen: shows "lossless_spmf seed_gen"
epi_implies_arr: fixes f :: "'a" assumes "epi f" shows "arr f"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> ns_shared"   and "KAB \<notin> range shrK" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
tendsw: shows "\<xi> \<longlonglongrightarrow> w"
v_eq: shows "\<v> = (\<Lambda> * (\<v> - 1) div (\<k> - 1) * (\<k> - 1) + \<Lambda>) div \<Lambda>"
mcont_eadd2: fixes x :: "enat" shows "mcont Sup (\<le>) Sup (\<le>) ((+) x)"
length_runs: fixes xs :: "'b list" shows "length (runs xs) \<le> length xs"
basis_def: shows "basis = {F a b |a b. a \<in> basisA \<and> b \<in> basisB}"
genby_lcoset_refl: fixes a :: "'b"   and S :: "'b set" shows "a \<in> a +o \<langle>S\<rangle>"
nonneg_coeffs: fixes n :: "nat" shows "fds_nth g n \<in> \<real>\<^sub>\<ge>\<^sub>0"
induct: fixes P :: "'a ifex \<Rightarrow> 'a ifex \<Rightarrow> 'a ifex \<Rightarrow> bool"   and a0 :: "'a ifex"   and a1 :: "'a ifex"   and a2 :: "'a ifex" assumes "\<And>i t e. \<lbrakk>\<And>x2. lowest_tops [i, t, e] = Some x2 \<Longrightarrow> P (restrict_top i x2 True) (restrict_top t x2 True) (restrict_top e x2 True); \<And>x2. lowest_tops [i, t, e] = Some x2 \<Longrightarrow> P (restrict_top i x2 False) (restrict_top t x2 False) (restrict_top e x2 False)\<rbrakk> \<Longrightarrow> P i t e" shows "P a0 a1 a2"
equivalence_comp_right_complement: fixes x :: "'a" assumes "equivalence x" shows "- x * x = - x"
Comp_assoc: fixes A :: "'o"   and B :: "'o"   and C :: "'o"   and D :: "'o"   and f :: "'a"   and g :: "'a"   and h :: "'a" assumes "A \<in> Obj"   and "B \<in> Obj"   and "C \<in> Obj"   and "D \<in> Obj"   and "f \<in> Hom A B"   and "g \<in> Hom B C"   and "h \<in> Hom C D" shows "Comp D C A h (Comp C B A g f) = Comp D B A (Comp D C B h g) f"
hyps: fixes v :: "symbol list"   and w :: "symbol list"   and u :: "symbol list" assumes "Derivation v D w"   and "is_sentence u" shows "Derivation (u @ v) (derivation_shift D 0 (length u)) (u @ w)"
tabulation_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" shows "tabulation V H \<a> \<i> src trg r \<rho> f g \<equiv> tabulation_data V H \<a> \<i> src trg r \<rho> f g \<and> tabulation_axioms V H \<a> r \<rho> f g"
SUP_lessD: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b"   and i :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less (Sup (f ` A)) y"   and "i \<in> A" shows "less (f i) y"
assms_3: shows "Complete_Partial_Order.chain (\<le>) Y"
eval\<^sub>G_respectfulness_2: fixes \<phi> :: "'c ltl"   and \<psi> :: "'c ltl"   and S :: "'c ltl set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "eval\<^sub>G S \<phi> \<equiv>\<^sub>P eval\<^sub>G S \<psi>"
monom_mult_monomial: fixes c :: "'c"   and s :: "'a"   and v :: "'t"   and d :: "'c" shows "monom_mult c s (monomial d v) = monomial (c * d) (s \<oplus> v)"
updVal_swapVal: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and x :: "'var"   and sX :: "'b"   and xs :: "'varSort" shows "val (x := sX)_xs ^[y1 \<and> y2]_ys = val ^[y1 \<and> y2]_ys (x @xs[y1 \<and> y2]_ys := sX)_xs"
span_subspace: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B"   and "B \<subseteq> cspan A"   and "csubspace B" shows "cspan A = B"
ts\<^sub>h_def: shows "ts\<^sub>h \<equiv> map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, [], \<D>, \<O>, \<R>)) ts"
holomorphic_on_add: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on s"   and "g holomorphic_on s" shows "(\<lambda>z. f z + g z) holomorphic_on s"
select_convs_3: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "subs \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = subs"
star_mon: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<star> \<le> y\<^sup>\<star>"
squarefree_part_subset: fixes n :: "nat" shows "squarefree_part n \<subseteq> prime_factors n"
assms_8: shows "kKS m \<notin> LocalSecrets P \<union> LocalSecrets Q"
median_def: fixes xs :: "'a list" shows "median xs = select ((length xs - 1) div 2) xs"
ar_r_iff: fixes x :: "'a" shows "ar x = (1::'a) \<sqinter> - (\<top> \<cdot> x)"
has_fps_expansion_exp1: shows "exp has_fps_expansion fps_exp (1::'a)"
m_def: shows "m = (n - 1) div 2 ^ multiplicity 2 (n - 1)"
dense_interval: fixes L :: "real set"   and l :: "real"   and x :: "real"   and P :: "real \<Rightarrow> bool" assumes "finite L"   and "l \<in> L"   and "l < x"   and "P x"   and "\<And>y l. \<lbrakk>\<forall>y\<in>{l<..<x}. y \<notin> L; l < x; l < y; y \<le> x\<rbrakk> \<Longrightarrow> P y" shows "\<exists>l\<in>L. l < x \<and> (\<forall>y\<in>{l<..<x}. y \<notin> L) \<and> (\<forall>y. l < y \<and> y \<le> x \<longrightarrow> P y)"
Step: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e" assumes "UP_cring R" shows "shift (Suc n) f = poly_shift (shift n f)"