hcju/isabelleps huggingface dataset:
map_le_on_disj_left: fixes h' :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h' \<subseteq>\<^sub>m h"   and "h\<^sub>0 \<bottom> h\<^sub>1"   and "h' = h\<^sub>0 ++ h\<^sub>1" shows "h\<^sub>0 \<subseteq>\<^sub>m h"
rep_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and xa :: "'a \<Rightarrow>\<^sub>0 'b"   and xb :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less" shows "linorder.dord_pm x xa xb = linorder.dord_fun (map_fun Abs_poly_mapping id \<circ> x \<circ> Abs_poly_mapping) (lookup xa) (lookup xb)"
rep_eq: fixes x :: "uint8" shows "bit x = bit (Rep_uint8 x)"
find_largest_block_main: fixes block :: "nat \<times> nat"   and blocks :: "(nat \<times> nat) list"   and m_b :: "nat"   and m_e :: "nat" assumes "??.Jordan_Normal_Form_Existence.find_largest_block block blocks = (m_b, m_e)" shows "(m_b, m_e) \<in> insert block (set blocks) \<and> (\<forall>b\<in>insert block (set blocks). snd b - fst b \<le> m_e - m_b)"
continuous_on_imp_absolutely_integrable_on: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a..b} f" shows "norm (integral {a..b} f) \<le> integral {a..b} (\<lambda>x. norm (f x))"
assms_4: shows "i \<subseteq> j \<union> S"
llist_less_le_not_le: fixes s :: "'a llist"   and t :: "'a llist" shows "(s < t) = (s \<le> t \<and> \<not> t \<le> s)"
sorted_list_of_multiset_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_multiset less_eq {#} = []"
those_def: shows "those \<equiv> rec_list (Some []) (\<lambda>x xs xsa. case x of None \<Rightarrow> None | Some y \<Rightarrow> map_option ((#) y) xsa)"
wfactors_mult_single: fixes a :: "'a"   and fb :: "'a list"   and b :: "'a" assumes "irreducible G a"   and "wfactors G fb b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set fb \<subseteq> carrier G" shows "wfactors G (a # fb) (a \<otimes> b)"
abs: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "Abs s \<rightarrow>\<^sub>\<beta> Abs t"
HEndPhase2_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p" shows "blocksOf s' q \<subseteq> blocksOf s q"
subring_props_1: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set" assumes "subfield K L" shows "K \<subseteq> carrier L"
Fubini_integrable: fixes f :: "'a \<times> 'b \<Rightarrow> 'd" assumes "f \<in> borel_measurable (count_space A \<Otimes>\<^sub>M count_space B')"   and "integrable (count_space A) (\<lambda>x. LINT y|count_space B'. norm (f (x, y)))"   and "AE x in count_space A. integrable (count_space B') (\<lambda>y. f (x, y))" shows "integrable (count_space A \<Otimes>\<^sub>M count_space B') f"
exp_cf3_pos: fixes x :: "real" assumes "0 < numer_cf3 x"   and "0 < numer_cf3 (- x)" shows "0 < exp_cf3 x"
map_le_antisym: fixes f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option" assumes "f \<subseteq>\<^sub>m g"   and "g \<subseteq>\<^sub>m f" shows "f = g"
vpeq_s_t: shows "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s t'"
simps_1: fixes y :: "'a"   and arr :: "(nat \<times> 'a) list" shows "addmult_spvec y arr [] = arr"
HLD_Stream: fixes X :: "'a set"   and x :: "'a"   and \<omega> :: "'a stream" shows "HLD X (x ## \<omega>) = (x \<in> X)"
id_Bij: fixes S :: "'c set" shows "(\<lambda>x\<in>S. x) \<in> Bij S"
elims: fixes x :: "wall"   and y :: "int" assumes "codomain_wall x = y" obtains xa :: "brick list" where "x = basic xa__" and "y = codomain_block xa__"   | xa :: "brick list" and ys :: "wall" where "x = xa * ys" and "y = codomain_wall ys"
nonempty_imp_ex_extremal: fixes Q :: "'a set" assumes "Q \<subseteq> X"   and "Q \<noteq> {}" shows "\<exists>z\<in>Q. \<forall>y\<in>Q. \<not> y \<sqsubset> z"
m_expr_1: shows "m1 = m1' * a mod n \<and> m1' \<in> kernel G H f"
append_butlast_last_id: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) node_ptr" assumes "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')" shows "get ptr h = get ptr h'"
prems_2: shows "x \<in> set (take n (y # ys))"
sim_while_cong_aux: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and b' :: "bexp"   and c' :: "com" assumes "\<pi>: (WHILE b DO c, s) \<Rightarrow> t"   and "bval b = bval b'"   and "c \<sim> c'" shows "\<pi>: (WHILE b' DO c', s) \<Rightarrow> t"
cases: fixes a :: "'a list" assumes "local.steps a" obtains x :: "'a" where "a = [xa__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "a = x # y # xs" and "x \<rightarrow> y" and "local.steps (y # xs)"
a_star_OrL: fixes M :: "trm"   and M' :: "trm"   and N :: "trm"   and N' :: "trm"   and x :: "name"   and y :: "name"   and z :: "name" assumes "M \<longrightarrow>\<^sub>a* M'"   and "N \<longrightarrow>\<^sub>a* N'" shows "OrL x.M y.N z \<longrightarrow>\<^sub>a* OrL x.M' y.N' z"
root_hash_F_def: fixes rha :: "'e \<Rightarrow> 'f"   and rhb :: "'g \<Rightarrow> 'h" shows "root_hash_F rha rhb = root_hash_F' \<circ> map_F\<^sub>m rha id rhb id"
left_tree_initial: fixes G :: "('e, 'f) Graph_scheme"   and s :: "'e"   and t :: "'e" assumes "Tree G"   and "s \<in> V\<^bsub>G\<^esub>"   and "t \<in> V\<^bsub>G\<^esub>" shows "s \<in> Tree.left_tree G s t"
proj2_incident_switch: fixes p :: "proj2"   and l :: "proj2_line" assumes "proj2_incident p l" shows "proj2_incident (L2P l) (P2L p)"
row_space_subset_phull: fixes ps :: "('d \<Rightarrow>\<^sub>0 'e) list"   and ts :: "'d list" assumes "Keys (set ps) \<subseteq> set ts" shows "row_to_poly ts ` row_space (polys_to_mat ts ps) \<subseteq> phull (set ps)"
inverse_distributive_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "(b - c) \<cdot> a = b \<cdot> a - c \<cdot> a"
pre_def: shows "pre = (\<lambda>(blues, reds, onstack, u). gen_dfs_pre E reachable onstack blues u \<and> add_inv blues reds onstack)"
apply_cltn2_line_unique: fixes p :: "proj2"   and q :: "proj2"   and l :: "proj2_line"   and C :: "cltn2"   and m :: "proj2_line" assumes "p \<noteq> q"   and "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident (apply_cltn2 p C) m"   and "proj2_incident (apply_cltn2 q C) m" shows "apply_cltn2_line l C = m"
point_in_one_group: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and x :: "'b"   and G1 :: "'b set"   and G2 :: "'b set" assumes "group_design point_set block_collection groups"   and "x \<in> G1"   and "G1 \<in> groups"   and "G2 \<in> groups"   and "G1 \<noteq> G2" shows "x \<notin> G2"
Call: fixes r :: "(state \<times> state) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "wf r"   and "\<forall>s'. {(\<lambda>z s. P z s \<and> (s, s') \<in> r, CALL, Q)} \<turnstile>\<^sub>t {\<lambda>z s. P z s \<and> s = s'} body {Q}" shows "{} \<turnstile>\<^sub>t {P} CALL {Q}"
mult2_alt_emptyL: fixes Y :: "'b multiset"   and b :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "(({#}, Y) \<in> mult2_alt b ns s) = (b \<or> Y \<noteq> {#})"
meet_def: fixes x :: "nat"   and y :: "nat" shows "nat.meet x y = (THE inf. nat.is_inf x y inf)"
derivation_const_eq_zero: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and k :: "enat"   and p :: "'c"   and c :: "real" assumes "X \<in> manifold_eucl.tangent_space k p"   and "p \<in> manifold_eucl.dest.carrier" shows "X (restrict0 manifold_eucl.dest.carrier (\<lambda>x. c)) = 0"
Span_Cons: fixes m :: "'v"   and ms :: "'v list" shows "RSpan (m # ms) = RSpan [m] + RSpan ms"
rel_with_target_impl_transC_TRel_is_strong_reduction_simulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "strong_reduction_simulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "strong_reduction_simulation (TRel\<^sup>+) Target"
imp_def: fixes p :: "fm"   and q :: "fm" shows "imp p q = (if p = F \<or> q = T then T else if p = T then q else if q = F then not p else Imp p q)"
per_not_colp: 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 P :: "'a"   and R :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A \<noteq> P"   and "B \<noteq> R"   and "Tarski_neutral_dimensionless.Per Bet Cong B A P"   and "Tarski_neutral_dimensionless.Per Bet Cong A B R" shows "\<not> Tarski_neutral_dimensionless.Col Bet P A R"
Units_inv_comm: fixes x :: "'a mat"   and y :: "'a mat" assumes "x \<otimes>\<^bsub>ring_mat TYPE('a) n ()\<^esub> y = \<one>\<^bsub>ring_mat TYPE('a) n ()\<^esub>"   and "x \<in> Units (ring_mat TYPE('a) n ())"   and "y \<in> Units (ring_mat TYPE('a) n ())" shows "y \<otimes>\<^bsub>ring_mat TYPE('a) n ()\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n ()\<^esub>"
eval_rewrite: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "p = ring.eval (K [X]) (map poly_of_const p) X"
strict_coboundedI2: fixes b :: "nat"   and c :: "nat"   and a :: "nat" assumes "b dvd c \<and> b \<noteq> c" shows "gcd a b dvd c \<and> gcd a b \<noteq> c"
minarc_below: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "minarc\<^sub>M x \<preceq> \<ominus> \<ominus> x"
hn_ctxt_def: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> assn"   and a :: "'g"   and c :: "'h" shows "hn_ctxt P a c \<equiv> P a c"
cancels_to_1_unfold: fixes x :: "(bool \<times> 'c) list"   and y :: "(bool \<times> 'c) list" assumes "cancels_to_1 x y" obtains xs1 :: "(bool \<times> 'c) list" and x1 :: "bool \<times> 'c" and x2 :: "bool \<times> 'c" and xs2 :: "(bool \<times> 'c) list" where "x = xs1 @ x1 # x2 # xs2" and "y = xs1 @ xs2" and "canceling x1 x2"
wary_num_args_le_arity_head: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and s :: "('a, 'b) tm" assumes "arity.wary arity_sym arity_var s" shows "enat (num_args s) \<le> arity.arity_hd arity_sym arity_var (head s)"
simps_2: fixes e :: "name list"   and t :: "tm"   and u :: "tm" shows "trans_fm e (t EQ u) = DBEq (trans_tm e t) (trans_tm e u)"
subcls1D: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C1D" shows "C \<noteq> Object \<and> (\<exists>c. class G C = Some c \<and> super c = D)"
assms_1: shows "(t, t') \<in> red\<^sup>*"
simp_elI: fixes b :: "'b list"   and G :: "'b list set" assumes "b \<in> G"   and "b \<noteq> \<epsilon>"   and "\<forall>u v. u \<noteq> \<epsilon> \<and> u \<in> \<langle>G\<rangle> \<and> v \<noteq> \<epsilon> \<and> v \<in> \<langle>G\<rangle> \<longrightarrow> u \<cdot> v \<noteq> b" shows " b \<in>B G "
effect_deterministic_1: fixes f :: "'a Heap"   and h :: "heap"   and h' :: "heap"   and a :: "'a"   and h'' :: "heap"   and b :: "'a" assumes "effect f h h' a"   and "effect f h h'' b" shows "a = b"
semiring_norm_160: shows "(1::'a) \<noteq> (0::'a)"
L_def: shows "L \<equiv> \<lambda>A. remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A))"
subsequence_prefix_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "prefix (j - i) (suffix i w) = w [i \<rightarrow> j]"
result: shows "closure ([(init r, init s)], {(post (init r), post (init s))}) = None"
m_comm: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V" shows "x \<oplus>\<^bsub>V\<^esub> y = y \<oplus>\<^bsub>V\<^esub> x"
assms_2: assumes "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>n. A)" shows "P"
total_pair_less: fixes A :: "(nat \<times> nat) set" shows "total_on A pair_less"
Basic: fixes P :: "'d set"   and f :: "'d \<Rightarrow> 'd"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> {s. f s \<in> Q}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Basic f Q,A"
euclidean_vec_componentwise: fixes f :: "('b, 'c) vec \<Rightarrow> 'a" shows "sum f Basis = (\<Sum>a\<in>Basis. \<Sum>b\<in>UNIV. f (axis b a))"
j: shows "i \<le> j \<and> enat j < llength Ns \<and> C \<notin> lnth Ns j"
fold_mset_union: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and s :: "'l"   and M :: "'k multiset"   and N :: "'k multiset" assumes "comp_fun_commute f" shows "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
less_cSupD: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set"   and z :: "'c" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "less z (Sup X)" shows "\<exists>x\<in>X. less z x"
subset_closed_segment: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(closed_segment a b \<subseteq> closed_segment c d) = (a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
ba_bot_unique: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - x) = - (y \<squnion> - y)"
simps_6: fixes v :: "('a, 'b) form"   and va :: "('a, 'b) form" shows "compl (Or v va) = Neg (Or v va)"
arith_simps_43: shows "\<bar>0::'a\<bar> = (0::'a)"
same_level_path_aux_valid_path_aux: 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 cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path_aux kind get_return_edges cs as" shows "CFG.valid_path_aux kind get_return_edges cs as"
assms_3: shows "(r, Expunge e i r) \<in> set (node_deliver_messages xs)"
IH_2: fixes V :: "nat set" assumes "randomfree e2"   and "free_vars e2 \<subseteq> V" shows "(\<lambda>\<sigma>. expr_sem_rf \<sigma> e2) \<in> state_measure V (\<lambda>a. case a of 0 \<Rightarrow> t1 | Suc x \<Rightarrow> \<Gamma> x) \<rightarrow>\<^sub>M stock_measure t2"
ldropT: fixes t :: "'b llist"   and A :: "'b set"   and i :: "nat" assumes "t \<in> A\<^sup>\<infinity>" shows "t \<up> i \<in> A\<^sup>\<infinity>"
Sup_upper2: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and u :: "'b::type"   and A :: "'b::type set"   and v :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(u::'b::type) \<in> (A::'b::type set)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (v::'b::type) (u::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (v::'b::type) ((Sup::'b::type set \<Rightarrow> 'b::type) (A::'b::type set))"
rep_eq: fixes x :: "'a poly" shows "Rep_ratfps (local.ratfps_of_poly x) = to_fract x"
connected_Icc: fixes a :: "'c"   and b :: "'c" shows "connected {a..b}"
wf: shows "wf {(c2, c1). \<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* c1 \<and> \<Gamma>\<turnstile> c1 \<rightarrow> c2}"
minimize_wrtD: fixes c :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "c \<in> set (minimize_wrt P xs)" shows "c \<in> set xs"
e: shows "1151 = 9 * 2 ^ (2 * 4 - 1) - 1"
ide_def: fixes a :: "'a" shows "S.ide a \<equiv> S.comp a a \<noteq> S.null \<and> (\<forall>f. (S.comp f a \<noteq> S.null \<longrightarrow> S.comp f a = f) \<and> (S.comp a f \<noteq> S.null \<longrightarrow> S.comp a f = f))"
minBC: fixes x :: "'a" assumes "x \<in> B"   and "x \<notin> C" shows "bc \<le> x \<and> (x \<le> bc \<longrightarrow> bc = x)"
Ks_1: shows "set Ks \<subseteq> carrier R"
lunstream_lfilter_trans: fixes P :: "'c \<Rightarrow> bool"   and g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lunstream (lfilter_trans P g) s = lfilter P (lunstream g s)"
VWF_iff_Ord_less: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "((\<alpha>, \<beta>) \<in> VWF) = (\<alpha> < \<beta>)"
chop_append: fixes d :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "d dvd length xs" shows "chop d (xs @ ys) = chop d xs @ chop d ys"
simps_7: fixes \<phi> :: "'b ltlr" shows "ltlr_to_ltln (X\<^sub>r \<phi>) = X\<^sub>n ltlr_to_ltln \<phi>"
af_congruent_def: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" shows "af_congruent eq \<equiv> ltl_equivalence eq \<and> af_congruent_axioms eq"
inj_into_WW: fixes k :: "nat" shows "inj (into_WW k)"
embed_measure_count_space: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "embed_measure (count_space A) f = count_space (f ` A)"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>src\<^sub>D f : src\<^sub>D a \<Rightarrow>\<^sub>D src\<^sub>D b\<guillemotright>"
count_mset_set_3: fixes x :: "'c"   and A :: "'c set" assumes "x \<notin> A" shows "count (mset_set A) x = 0"
X: shows "\<forall>x\<ge>X. (\<forall>h\<in>set hs'. \<bar>h x\<bar> \<le> hb * x / ln x powr (1 + e)) \<and> (\<forall>b\<in>set bs. C * x \<le> b * x - hb * x / ln x powr (1 + e)) \<and> (\<forall>b\<in>set bs. akra_bazzi_asymptotics b hb e p x) \<and> (\<forall>u\<in>{C * x..x}. c1 * g' x \<le> g' u) \<and> 0 \<le> g' x"
C_def: shows "C = (\<lambda>N. \<Union>m\<in>{1..N}. Pair m ` totatives m)"
eval_pos_if_dominant_term_pos': fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "trimmed_ms_aux F"   and "is_expansion_aux F f basis"   and "0 < fst (dominant_term_ms_aux F)" shows "\<forall>\<^sub>F x in at_top. 0 < f x"
Invariants_of_foTn: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> real"   and n :: "nat" assumes "fmpt M T"   and "integrable M f" shows "AE x in M. real_cond_exp M (qmpt.Invariants M T) (f \<circ> T ^^ n) x = real_cond_exp M (qmpt.Invariants M T) f x"
simps: fixes a1 :: "'t"   and a2 :: "(status \<times> 'x) \<times> 'm"   and a3 :: "('l, 't, status \<times> 'x, 'm, 'w, 'o action) thread_action"   and a4 :: "(status \<times> 'x) \<times> 'm" shows "a1 \<turnstile> a2 -a3\<rightarrow>i a4 = ((\<exists>t x m ta x' m'. a1 = t \<and> a2 = ((Running, x), m) \<and> a3 = convert_TA_initial (convert_obs_initial ta) \<and> a4 = ((Running, x'), m') \<and> t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>) \<or> (\<exists>t x m. a1 = t \<and> a2 = ((PreStart, x), m) \<and> a3 = \<lbrace>InitialThreadAction\<rbrace> \<and> a4 = ((Running, x), m)) \<or> (\<exists>x t m. a1 = t \<and> a2 = ((Running, x), m) \<and> a3 = \<lbrace>ThreadFinishAction\<rbrace> \<and> a4 = ((Finished, x), m) \<and> final x))"
eval_fds_at_numeral: fixes f :: "'a fds"   and k :: "num" shows "eval_fds f (numeral k) = (\<Sum>n. fds_nth f n / of_nat n ^ numeral k)"
LeadsTo_Diff: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "F \<in> A - B \<longmapsto>w C"   and "F \<in> A \<inter> B \<longmapsto>w C" shows "F \<in> A \<longmapsto>w C"
diff_mono: fixes a :: "complex"   and b :: "complex"   and d :: "complex"   and c :: "complex" assumes "a \<le>\<^sub>\<complex> b"   and "d \<le>\<^sub>\<complex> c" shows "a - c \<le>\<^sub>\<complex> b - d"
2: shows "[a] @ v\<^sub>1 =\<^sub>F [a] @ v\<^sub>2"
pr_comp2_2': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "pr_conv_2_to_1 f \<in> PrimRec1'"   and "pr_conv_2_to_1 g \<in> PrimRec1'"   and "pr_conv_2_to_1 h \<in> PrimRec1'" shows "pr_conv_2_to_1 (\<lambda>x y. f (g x y) (h x y)) \<in> PrimRec1'"
case_eq_if: fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and prod :: "'l \<times> 'm" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
lowner_lub_scale: fixes d :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and c :: "real" assumes "matrix_seq d f"   and "\<forall>n. trace (complex_of_real c \<cdot>\<^sub>m f n) \<le> 1"   and "0 \<le> c" shows "matrix_seq.lowner_lub (\<lambda>n. complex_of_real c \<cdot>\<^sub>m f n) = complex_of_real c \<cdot>\<^sub>m matrix_seq.lowner_lub f"
DBM_val_bounded_len_2'_aux: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and c :: "'c"   and vs :: "nat list" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "v c \<le> n"   and "\<forall>k\<in>set vs. 0 < k \<and> k \<le> n \<and> (\<exists>c. v c = k)" shows "dbm_entry_val u None (Some c) (len m 0 (v c) vs)"
dvd_minus_mod: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and a :: "'e" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times b (minus a (modulo a b))"
rep_eq: fixes x :: "('a, 'b, 'c) Vardi.var" shows "set1_var x = set1_var0 (rep_var x)"
upper: shows "ln (real n) - ln (fact n) / real n \<le> 1"