hcju/isabelleps huggingface dataset:
closure_Some_Inv: fixes as :: "'b list"   and r :: "'b rexp"   and s :: "'b rexp"   and p :: "('b rexp set \<times> 'b rexp set) list \<times> ('b rexp set \<times> 'b rexp set) list" assumes "pEquivalence_Checking.closure as ([({r}, {s})], []) = Some p" shows "\<forall>(R, S)\<in>set (fst p). \<exists>w. R = pderivs w r \<and> S = pderivs w s"
list_le_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<le> ys) = (xs < ys \<or> xs = ys)"
ifex_sat_list_NoneD: fixes i :: "'a ifex"   and ass :: "'a \<Rightarrow> bool" assumes "ifex_sat_list i = None" shows "val_ifex i ass = False"
that: assumes "h' = h"   and "x = get h r" shows "thesis"
Loop: fixes A :: "triple set"   and P :: "assn"   and x :: "vname"   and c :: "stmt" assumes "A \<turnstile> {\<lambda>s. P s \<and> s<x> \<noteq> Null} c {P}" shows "A \<turnstile> {P} While (x) c {\<lambda>s. P s \<and> s<x> = Null}"
square_sym: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "square R S T U" shows "square S R U T"
thetaPar_Sbis: shows "thetaPar \<subseteq> Sbis"
that: fixes D :: "'a literal multiset"   and E :: "'a literal multiset" assumes "D \<in> N"   and "INTERP N \<Turnstile> D"   and "production N D \<noteq> {}"   and "unord_resolve D C E"   and "\<not> INTERP N \<Turnstile> E"   and "E < C" shows "thesis"
subset_2: shows "\<Union> (S ` B) \<subseteq> carrier G"
preserves_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "G (g \<cdot>\<^sub>C f) = G g \<cdot>\<^sub>D G f"
dec_evaluate_not_timeout: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and s' :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_dec False mn env s d (s', r)" shows "r \<noteq> Rerr (Rabort Rtimeout_error)"
poly_of_vars_simps_2: fixes xs :: "char list multiset"   and ys :: "char list multiset" shows "poly_of_vars (xs + ys) = poly_of_vars xs * poly_of_vars ys"
B0c: fixes x :: "i" shows "E x \<^bold>\<leftarrow> E (cod x)"
assms: shows "\<forall>x\<le>n. 0 < x \<longrightarrow> s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)"
simps: fixes os :: "'t iarray"   and s :: "'s"   and t_end :: "'s"   and n :: "nat" shows "viterbi_ix_rec os s t_end n = (if IArray.length os \<le> n then ([], indicator {t_end} s) else fst (argmax snd (map (\<lambda>t. let (xs, v) = viterbi_ix_rec os t t_end (n + 1) in (t # xs, ennreal (pmf (\<O> t) (os !! n) * pmf (\<K> s) t) * v)) state_list)))"
int: fixes b :: "real"   and c :: "real"   and z :: "complex"   and e :: "real" assumes "a \<le> b"   and "c \<le> x"   and "z \<in> U" shows "(\<lambda>t. complex_of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}"
sigma_finite_iff_density_finite': fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable N" shows "sigma_finite_measure (density N f) = (AE x in N. f x \<noteq> \<infinity>)"
indets_sum_subset: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and A :: "'d set" shows "indets (sum f A) \<subseteq> (\<Union>a\<in>A. indets (f a))"
leadsETo_cancel_Diff1: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and B :: "'d set"   and A' :: "'d set"   and B' :: "'d set" assumes "F \<in> A leadsTo[CC] (B \<union> A')"   and "F \<in> (B - A') leadsTo[CC] B'" shows "F \<in> A leadsTo[CC] (B' \<union> A')"
solution_usolves_ode: shows "(solution usolves_ode f from t_max) (cball t_max et) (cball (x t_max) ex)"
split_beta': fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'e" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
maxsimp_preimage: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "C \<in> X"   and "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` C)" shows "SimplicialComplex.maxsimp X C"
representation_sum: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and basis :: "'b set"   and I :: "'c set"   and v :: "'c \<Rightarrow> 'b" assumes "Modules.module scale"   and "\<not> module.dependent scale basis"   and "\<And>i. i \<in> I \<Longrightarrow> v i \<in> module.span scale basis" shows "module.representation scale basis (sum v I) = (\<lambda>b. \<Sum>i\<in>I. module.representation scale basis (v i) b)"
runit_closed: fixes f :: "'a" assumes "Arr f"   and "B.ide f" shows "Arr (B.\<rr> f)"
DSourcesA4_L0: shows "DSources level0 sA4 = {sA3}"
eval_tm_HTuple: fixes e :: "(name, hf) finfun"   and n :: "nat" shows "\<lbrakk>HTuple n\<rbrakk>e = htuple n"
light_edge_is_safe: fixes A :: "('v \<times> 'v) set"   and C :: "'v set"   and u :: "'v"   and v :: "'v" assumes "is_subset_MST w rg A"   and "respects_cut A C"   and "light_edge C u v" shows "is_subset_MST w rg ({(v, u)} \<union> A)"
lin_independent_vs_lincomb0: fixes ms :: "'v list"   and n :: "'v"   and ns :: "'v list"   and rs :: "'f list"   and s :: "'f"   and ss :: "'f list" assumes "set (ms @ n # ns) \<subseteq> V"   and "R_lin_independent (ms @ n # ns)"   and "set (rs @ s # ss) \<subseteq> UNIV"   and "length rs = length ms"   and "(rs @ s # ss) \<bullet>\<cdot> (ms @ n # ns) = (0::'v)" shows "s = (0::'f)"
Ord_odiff_le_odiff: fixes x :: "V"   and y :: "V"   and \<alpha> :: "V" assumes "x \<le> y"   and "Ord x"   and "Ord y" shows "odiff x \<alpha> \<le> odiff y \<alpha>"
distinct_map_default: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "distinct (list.map fst xs)" shows "distinct (list.map fst (AList.map_default k v f xs))"
chain4: shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. \<Squnion>f\<in>Z. f x) ` Y)"
INF_ereal_add_left: fixes I :: "'a set"   and c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "c \<noteq> - \<infinity>"   and "\<And>x. x \<in> I \<Longrightarrow> 0 \<le> f x" shows "(INF i\<in>I. f i + c) = Inf (f ` I) + c"
is_map_to_sorted_listE: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Rk :: "('g \<times> 'f) set"   and Rv :: "('h \<times> 'i) set"   and Rm :: "('g \<times> 'f) set \<Rightarrow> ('h \<times> 'i) set \<Rightarrow> ('j \<times> ('f \<Rightarrow> 'i option)) set"   and tsl :: "'j \<Rightarrow> ('g \<times> 'h) list"   and m :: "'j"   and m' :: "'f \<Rightarrow> 'i option" assumes "is_map_to_sorted_list ordR Rk Rv Rm tsl"   and "(m, m') \<in> \<langle>Rk, Rv\<rangle>Rm" obtains l' :: "('f \<times> 'i) list" where "(tsl m, l') \<in> \<langle>Rk \<times>\<^sub>r Rv\<rangle>list_rel" and "RETURN l' \<le> it_to_sorted_list (key_rel ordR) (map_to_set m')"
length_entries_rbtreeify_f: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> length kvs" shows "length (entries (fst (rbtreeify_f n kvs))) = n"
cases: fixes x :: "('c, 'd, 'varSort, 'var, 'e) qAbs" obtains xs :: "'varSort" and x :: "'var" and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" where "xa__ = qAbs xs x X"
id_iso: fixes G :: "('a, 'b) monoid_scheme" shows "id \<in> Group.iso G G"
blinding_of_tree: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "blinding_of_on UNIV h bo" shows "blinding_of_on UNIV (hash_tree h) (blinding_of_tree h bo)"
simps_2: fixes x :: "char list"   and e :: "aexp"   and P :: "'b" shows "post (x ::= e {P}) = P"
of_real_prod: fixes f :: "'b \<Rightarrow> real"   and s :: "'b set" shows "of_real (prod f s) = (\<Prod>x\<in>s. of_real (f x))"
index_of_c_fst_def: shows "index_of_c_fst = sch_to_nat Base_fst"
AbstTermP_Var_same: fixes v :: "tm"   and t :: "tm" shows "{VarP v, OrdP t} \<turnstile> AbstTermP v t v (Q_Ind t)"
below_halfspaces_def: fixes sctns :: "'c sctn set" shows "below_halfspaces sctns = \<Inter> (below_halfspace ` sctns)"
tendsto_norm_zero_cancel: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" assumes "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> (0::'d)) F"
nonzero_mult_real: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "complex" assumes "A \<noteq> mat_zero"   and "k \<noteq> 0" shows "k *\<^sub>s\<^sub>m A \<noteq> mat_zero"
tj_stack_more_cong: fixes s :: "('a, 'b) tarjan_state_scheme"   and s' :: "('a, 'b) tarjan_state_scheme" assumes "state.more s = state.more s'" shows "tj_stack s = tj_stack s'"
Heap_ord_def: shows "Heap_ord = img_ord execute option.le_fun"
assms_1: shows "g \<longlonglongrightarrow> 0"
ck_unifier_thm: fixes t :: "'b trm"   and s :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences" assumes "ck_unifier t s \<sigma> k" shows "t \<lhd> \<sigma> = s \<lhd> \<sigma>"
call_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "call_instr instr \<equiv> let op_list = snd instr; mem_addr = ucast (get_operand_w30 (op_list ! 0)) << 2 in do curr_win \<leftarrow> get_curr_win (); pc_val \<leftarrow> gets (cpu_reg_val PC); npc_val \<leftarrow> gets (cpu_reg_val nPC); write_reg pc_val curr_win (word_of_int 15); write_cpu npc_val PC; write_cpu (pc_val + mem_addr) nPC; return () od"
in_listsD: fixes xs :: "'a list"   and A :: "'a set" assumes "xs \<in> lists A" shows "\<forall>x\<in>set xs. x \<in> A"
1: fixes n :: "nat" assumes "n < (LEAST n. xs !! n \<in> A)" shows "xs !! n \<notin> A"
is_unit_unit_factor: 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 unit_factor :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor"   and "dvd.dvd times a one" shows "unit_factor a = a"
se_star_succs_states: fixes c :: "('d, 'e) conf"   and ls :: "('d, 'e) label list"   and c1 :: "('d, 'e) conf"   and c2 :: "('d, 'e) conf" assumes "se_star c ls c1"   and "se_star c ls c2" shows "states c1 = states c2"
card_insert_if: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "card (insert x A) = (if x \<in> A then card A else Suc (card A))"
finite_total_preorder_on_map: fixes f :: "'b \<Rightarrow> 'a" assumes "finite (f -` inv f -` carrier)" shows "finite_total_preorder_on (f -` inv f -` carrier) (map_relation f (map_relation (inv f) le))"
bind_def: fixes f :: "'b \<Rightarrow> ('d \<times> 'b) \<times> bool"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) \<times> bool" shows "f >>= g \<equiv> \<lambda>s. let fs = h1 f s; v = h2 g fs in (fst v, snd v \<or> snd fs)"
prems_2: fixes k :: "'b" assumes "k \<in> set ys" shows "filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) (x # xs)"
sorted_wrt_distinct_set_unique: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "antisymp rel"   and "sorted_wrt rel xs"   and "distinct xs"   and "sorted_wrt rel ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
unsigned_neg_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (- numeral n) = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('c) (- numeral n)))"
W: shows "openin (top_of_set (S \<times> T)) W"
rel_intros: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "unit + 'c \<times> 'c list_R1"   and y :: "unit + 'd \<times> 'd list_R1" assumes "rel_sum (=) (rel_prod R (rel_list_R1 R)) x y" shows "rel_list_R1 R (list_R1 x) (list_R1 y)"
arith_extra_simps_6: fixes a :: "'d" shows "a + (0::'d) = a"
r: shows "C \<union> S \<subseteq> ball 0 r"
comm_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and a :: "'a" assumes "Groups.comm_monoid f z" shows "f a z = a"
simps: fixes arss :: "('a \<times> ('b \<times> 'c list) list) list" shows "exe_ars_conds' arss = (alist_conds arss \<and> (\<forall>ars\<in>snd ` set arss. alist_conds ars))"
Invariants_def: shows "I.Invariants = sigma (space M) {A \<in> sets M. Tinv -` A \<inter> space M = A}"
gauss_cnj_of_int: fixes n2 :: "int" shows "gauss_cnj (of_int n2) = of_int n2"
prefix_fininf_extend: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a stream" assumes "u \<le>\<^sub>F\<^sub>I v @- w"   and "length v \<le> length u" shows "v \<le> u"
reachable_adj_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow>\<^sup>* b"   and "b \<rightarrow> c" shows "a \<rightarrow>\<^sup>* c"
normalize_def: shows "Execute.normalize \<equiv> \<lambda>x. pred.Pred ((\<Down>) x)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_trans_axioms ext" shows "ext_trans ext"
order_bal_nonempty_lasttreebal: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and t :: "'c btree" assumes "split split"   and "0 < k"   and "root_order k t"   and "bal t" shows "nonempty_lasttreebal t"
xsclosed: fixes x :: "'a set" assumes "x \<in> S" shows "x closed"
direct_decomp_def: fixes A :: "'a set"   and ss :: "'a set list" shows "direct_decomp A ss = bij_betw sum_list (listset ss) A"
arr_def: fixes f :: "('obj, 'obj) concrete_category.arr" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
hasty_const: fixes c :: "Const"   and t :: "Ty" assumes "c isof t" shows "v_const c hasty t"
in_set_conv_decomp_first: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
insert_3: shows "\<exists>q. (\<forall>b\<in>bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs"
disc_call_option_lgeom: fixes T :: "nat"   and K :: "real"   and w :: "bool stream" shows "discounted_value r (\<lambda>m. call_option T K) T w = det_discount r T * max (lgeom_proc u d init (stake T w) - K) 0"
times_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "x * y = (case (x, y) of (a\<^sub>1, b) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2)"
bex_cong2: fixes I :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "I \<subseteq> A"   and "\<forall>x\<in>I. f x = g x" shows "(\<exists>x\<in>I. P (f x)) = (\<exists>x\<in>I. P (g x))"
cdependent_raw_def: shows "cdependent \<equiv> module.dependent (*\<^sub>C)"
min_product_subseteq: fixes x :: "'b fset"   and A :: "'b fset set"   and B :: "'b fset set" assumes "x \<in> A \<otimes>\<^sub>m B" shows "\<exists>a. a |\<subseteq>| x \<and> a \<in> A"
qVar_alpha_iff: fixes xs :: "'h"   and x :: "'i"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" shows "(qVar xs x #= X') = (X' = qVar xs x)"
enum_n_top: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s" shows "(a = kuhn_simplex.enum base upd n) = (\<forall>a'\<in>s. a' \<le> a)"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "F.A.par f f'"   and "F f = F f'" shows "f = f'"
foldli_eq_nfoldli: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c" shows "RETURN (foldli l c f s) = nfoldli l c (\<lambda>x s. RETURN (f x s)) s"
tendsto_inverse_real: fixes u :: "'b \<Rightarrow> real"   and l :: "real"   and F :: "'b filter" assumes "(u \<longlongrightarrow> l) F"   and "l \<noteq> 0" shows "((\<lambda>x. 1 / u x) \<longlongrightarrow> 1 / l) F"
H_3: shows "head (CN c n p) = head (CN c' n' p')"
stc_mult: fixes x :: "complex star"   and y :: "complex star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "stc (x * y) = stc x * stc y"
simps_3: fixes x :: "'a"   and y :: "'a" shows "Val x + Val y = Val (x + y)"
1: shows "FvarsT (rawpsubstT \<chi> txs) \<subseteq> FvarsT r - insert x (snd ` set txs) \<union> \<Union> {FvarsT ta |ta. \<exists>xa. ta = t \<and> xa = x \<or> (ta, xa) \<in> set txs}"
represents_irr_non_0: fixes p :: "int poly"   and x :: "'a" assumes "irreducible p"   and "p represents x"   and "x \<noteq> (0::'a)" shows "poly p 0 \<noteq> 0"
to_polynomial_def: fixes R :: "('e, 'f) ring_scheme" shows "to_polynomial R = (\<lambda>a. monom (UP R) a 0)"
simps_1: fixes f :: "'d option \<Rightarrow> 'd"   and v :: "'d option"   and ps :: "('c \<times> ('c, 'd) trie) list" shows "update_with_trie [] f (Trie v ps) = Trie (Some (f v)) ps"
cont_case_prodI: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and lubb :: "'d set \<Rightarrow> 'd"   and luba :: "'c set \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "monotone (rel_prod orda ordb) less_eq (\<lambda>(x, y). f x y)"   and "\<And>x. cont lubb ordb Sup less_eq (f x)"   and "\<And>y. cont luba orda Sup less_eq (\<lambda>x. f x y)"   and "class.preorder orda (mk_less orda)"   and "class.preorder ordb (mk_less ordb)" shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup less_eq (\<lambda>(x, y). f x y)"
rank: fixes a :: "V" shows "rank a = set (\<Union>y\<in>elts a. elts (ZFC_in_HOL.succ (rank y)))"
Mem_SUCC_Refl: fixes H :: "fm set"   and k :: "tm" shows "H \<turnstile> k IN SUCC k"
DF_unfold: fixes A :: "'d set" shows "DF A = (\<sqinter>z\<in>A \<rightarrow>  DF A)"
functor_axioms: shows "functor VV.comp (\<bullet>) (\<lambda>\<nu>\<mu>. fst \<nu>\<mu> \<star> snd \<nu>\<mu>)"
wf_def: fixes r :: "('a \<times> 'a) set" shows "Wellfounded.wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
zdvd_reduce: fixes k :: "int"   and n :: "int"   and m :: "int" shows "(k dvd n + k * m) = (k dvd n)"
subst_fv_bound_singleton: fixes t :: "('b, 'a) Term.term"   and v :: "'a"   and t' :: "('b, 'a) Term.term" shows "fv (t \<cdot> Var(v := t')) \<subseteq> fv t \<union> fv t'"
succ_notin_self: fixes i :: "hf" shows "succ i \<^bold>\<notin> i"
r: shows "sqrt (real_of r) = real_of_int (sgn a) * real sq / real_of_int b * sqrt (real fact)"
assms_2: shows "fair_price Mkt \<pi> der matur"
is_process_REP_Mp: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "is_process ({(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}, {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))})"
t_ins_form_aux_all: fixes X :: "'b t_type" shows "t_ins_form_all (t_ins_aux X)"
assms_1: shows "cfg \<in> R_G.cfg_on s"
dfc_equals_dwfc_rel_ipurge: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and I :: "('b \<times> 'b) set" shows "d_future_consistent P D (rel_ipurge P I D) = d_weakly_future_consistent P I D (rel_ipurge P I D)"
fMin_less_Length: fixes x :: "nat"   and m1 :: "nat"   and \<AA> :: "interp"   and k :: "WS1S_Prelim.order" assumes "x |\<in>| m1\<^bsup>\<AA>\<^esup>k" shows "fMin (m1\<^bsup>\<AA>\<^esup>k) < Length \<AA>"
assms_10: shows "(xs @ [y], {}) \<in> seq_comp_failures P Q"
mult_neg_neg: fixes 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less a zero"   and "less b zero" shows "less zero (times a b)"
sup2: fixes B :: "'c"   and k :: "'c"   and b :: "'c"   and a :: "'c" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
PREFER_D: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "PREFER P x" shows "P x"
sum_clauses_2: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "finite S" shows "sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
SN: shows "SN_on (S\<^sup>* O (R \<union> R') O S\<^sup>*) X"
wf_bs_insensitive': fixes x :: "'a"   and bs :: "'a list"   and p :: "poly"   and y :: "'a" shows "wf_bs (x # bs) p = wf_bs (y # bs) p"
inv1: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], rev u @ h) v = ([y, x], rev v @ rev u @ h)"
is_list_UNIV_iff: fixes xs :: "'a list" shows "is_list_UNIV xs = (set xs = UNIV)"
cospan_diagram_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b" shows "cospan_diagram C f0 f1 \<equiv> category C \<and> cospan_diagram_axioms C f0 f1"
intro: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
length_list_of_vec: fixes A :: "'b vec" shows "length (list_of_vec A) = dim_vec A"
set_diff_eq: fixes A :: "'a set"   and B :: "'a set" shows "A - B = {x \<in> A. x \<notin> B}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_add_hom hom" shows "comm_monoid_add_hom hom"
edge_succ_cyclic: fixes v :: "'a" assumes "v \<in> verts (with_proj G)"   and "out_arcs (with_proj G) v \<noteq> {}" shows "cyclic_on (edge_succ M) (out_arcs (with_proj G) v)"
prefixes_are_paths: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and p :: "(symbol \<times> character list) list"   and Doc :: "character list"   and k :: "nat"   and u :: "nat"   and x :: "(symbol \<times> character list) list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "p \<in> LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u"   and "is_prefix x p" shows "x \<in> LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u"
monomial_0D: fixes t :: "'c"   and c :: "'d" assumes "monomial c t = 0" shows "c = (0::'d)"
subpath_def: fixes g :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and es :: "'c edge list"   and v2 :: "'c" shows "Graph.subpath g v1 es v2 \<equiv> Graph.ces v1 es v2 \<and> v1 \<in> Graph.vertices g \<and> set es \<subseteq> edges g"
distinct_3: fixes x3 :: "nat" shows "extN.Bot \<noteq> extN.N x3"
assms_3: fixes C :: "('l, 'x) labeled_graph" assumes "consequence_graph Rs C" shows "maintained (S, G) C"
K_def: shows "K = (\<lambda>n. {x \<in> space M. \<exists>k\<ge>n. \<not> P k x})"
assms_1: fixes t :: "'a set"   and x :: "'a" assumes "openin (top_of_set S) t"   and "x \<in> t" shows "openin (top_of_set S) (path_component_set t x)"