hcju/isabelleps huggingface dataset:
div_abs_eq_div_nat: fixes k :: "int"   and l :: "int" shows "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
game_0_def: fixes key_gen :: "'b spmf"   and "prf" :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and \<A> :: "(bool, 'c, 'd) gpv" shows "Pseudo_Random_Function.prf.game_0 key_gen prf \<A> = key_gen \<bind> (\<lambda>key. exec_gpv (Pseudo_Random_Function.prf.prf_oracle prf key) \<A> () \<bind> (\<lambda>(b, uu_). return_spmf b))"
closure_sound: fixes as :: "'a list"   and r :: "'a rexp"   and s :: "'a rexp"   and R :: "('a rexp \<times> 'a rexp) set" assumes "closure as (r, s) = Some ([], R)"   and "atoms r \<union> atoms s \<subseteq> set as" shows "lang r = lang s"
cyc: shows "cyclic_on (edge_succ M) (out_arcs G (tail G a))"
learn_total_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" shows "learn_total \<psi> U s \<equiv> learn_lim \<psi> U s \<and> (\<forall>f\<in>U. \<forall>n. \<psi> (the (s (f \<triangleright> n))) \<in> \<R>)"
two_composable_arrows_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" shows "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<equiv> (span_bicategory C prj0 prj1 \<and> arrow_of_spans C \<mu>) \<and> arrow_of_spans C \<nu> \<and> two_composable_arrows_of_spans_axioms C \<mu> \<nu>"
simps: fixes u :: "'a Vector"   and v :: "'a Vector" shows "(u mdot v) = tdir u * tdir v - (xdir u * xdir v + ydir u * ydir v + zdir u * zdir v)"
inject: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset"   and y1 :: "'b literal multiset multiset"   and y2 :: "'b literal multiset"   and y3 :: "'b literal multiset" shows "(old_Infer x1 x2 x3 = old_Infer y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
frameResChainFresh'': fixes xvec :: "name list"   and yvec :: "name list"   and F :: "'b frame" assumes "xvec \<sharp>* yvec" shows "xvec \<sharp>* (\<lparr>\<nu>*yvec\<rparr>F) = xvec \<sharp>* F"
P'_def: fixes y :: "'b" shows "P' y = (\<exists>x. P x \<and> y = f x)"
completeness_def: shows "completeness \<equiv> \<forall>h w e. (h, w) \<in> Rel \<longrightarrow> e \<in> challenge_space \<longrightarrow> spmf (completeness_game h w e) True = 1"
y_2: shows "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x *\<^sub>R g x) = (\<Sum>(x, k)\<in>p. (content k * f x) *\<^sub>R y)"
finite_theFT: fixes cl :: "('d, 'e, 'f) cmd list" shows "finite (theFT cl)"
None_notin_image_Some: fixes A :: "'b set" shows "None \<notin> Some ` A"
cases: fixes x :: "nat" obtains "x = 0"   | k :: "nat" where "x = Suc k"
funpow_inj_finite: fixes p :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "inj p"   and "finite {(p ^^ n) x |n. True}" shows "\<exists>n>0. (p ^^ n) x = x"
of_int_eq_numeral_power_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "int"   and x :: "num"   and n :: "nat" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus y = power.power one times (numeral.numeral one plus x) n) = (y = numeral x ^ n)"
conv_times: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<smile> = x\<^sup>\<smile> \<cdot> y\<^sup>\<smile>"
finite_fset_map: fixes F :: "'g \<Rightarrow> 'h option" shows "finite (set_fmap F)"
intro: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>a a' b. prod (a + a') b = prod a b + prod a' b"   and "\<And>a b b'. prod a (b + b') = prod a b + prod a b'"   and "\<And>r a b. prod (r *\<^sub>R a) b = r *\<^sub>R prod a b"   and "\<And>a r b. prod a (r *\<^sub>R b) = r *\<^sub>R prod a b"   and "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" shows "bounded_bilinear prod"
eval_InsInitV: fixes G :: "prog"   and s :: "st"   and c :: "stmt"   and w :: "Term.var"   and v :: "vvar"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>InsInitV c w=\<succ>v\<rightarrow> s' = False"
big_forest_d_U_e: fixes f :: "'a"   and j :: "'a"   and h :: "'a"   and d :: "'a"   and g :: "'a" assumes "forest f"   and "vector j"   and "regular j"   and "forest h"   and "forest_components h \<le> forest_components f"   and "big_forest (forest_components h) d"   and "d * top \<le> - j"   and "forest_components h * j = j"   and "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"   and "selected_edge h j g \<le> - forest_components f"   and "selected_edge h j g \<noteq> bot"   and "j \<noteq> bot" shows "big_forest (forest_components h) (d \<squnion> selected_edge h j g)"
1: shows "(\<Prod>i\<in>I. (f i + g i) ^ m i) = (\<Prod>i\<in>I. f i ^ m i) + c"
rbd_translate: fixes R :: "('c \<times> 'd) set"   and S :: "('c \<times> 'd) set" shows "(bd\<^sub>\<R> R = bd\<^sub>\<R> S) = (R = S)"
intr: shows "(intr, (\<inter>)) \<in> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel"
sem\<^sub>e\<^sub>s\<^sub>t_d_decomp_rm\<^sub>e\<^sub>s\<^sub>t_if_sem\<^sub>e\<^sub>s\<^sub>t_c: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (decomp_rm\<^sub>e\<^sub>s\<^sub>t A)"
elims: fixes x :: "('b \<times> 'c) tree"   and y :: "'b multiset" assumes "mset_tree x = y" obtains "x = \<langle>\<rangle>" and "y = {#}"   | l :: "('b \<times> 'c) tree" and a :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" where "x = \<langle>l, (a, uu_), r\<rangle>" and "y = {#a#} + mset_tree l + mset_tree r"
pivotandupdate_tableau_normalized: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<triangle> (\<T> (pivot_and_update x\<^sub>i x\<^sub>j c s))"
undefined: fixes a :: "'b\<^sub>m"   and b :: "'b\<^sub>m" assumes "rh a \<noteq> rh b"   and "a \<in> A" shows "m a b = None"
scc_root_finished_impl_scc_finished: fixes v :: "'v"   and scc :: "'v set" assumes "v \<in> dom (finished s)"   and "scc_root s v scc" shows "scc \<subseteq> dom (finished s)"
small_antisym: fixes F :: "'a filter"   and H :: "('a \<Rightarrow> real) set"   and h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "landau_function_family F H"   and "h1 \<in> H"   and "h2 \<in> H"   and "h1 \<in> o[F](h2)"   and "h2 \<in> o[F](h1)" shows "False"
map_comp: fixes g :: "'g \<Rightarrow> 'f"   and f :: "'h \<Rightarrow> 'g"   and v :: "'e \<Rightarrow> 'h" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
graph_ext_def: fixes nodes :: "'c set"   and edges :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e" shows "\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons graph_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple nodes edges) more"
akra_bazzi_p_strict_mono: fixes x :: "real"   and y :: "real" assumes "x < y" shows "(\<Sum>i<k. as ! i * bs ! i powr y) < (\<Sum>i<k. as ! i * bs ! i powr x)"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "Ik.satL \<xi> (Pos a) = Ik.satA \<xi> a"
presAtm_compatAtm: fixes atm :: "'atom" assumes "presAtm atm" shows "compatAtm atm"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map ((\<otimes>\<^sub>D) \<I>\<^sub>D) D.\<ll>'"
col_sylvester: fixes j :: "nat"   and p :: "'b poly"   and q :: "'b poly" assumes "j < degree p + degree q" shows "col (sylvester_mat p q) j = vec_of_poly_rev_shifted p (degree q) j @\<^sub>v vec_of_poly_rev_shifted q (degree p) j"
dirichlet_prod_inverse: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b" assumes "f 1 * i = (1::'b)" shows "f \<star> dirichlet_inverse f i = (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
G2_G: fixes k :: "nat" assumes "k < i - 1" shows "gs2.gso k = fs.gs.gso k"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "max b (max a c) = max a (max b c)"
red_init_witness_def: fixes u :: "'a"   and v :: "'a" shows "red_init_witness u v = Some ([u], v)"
fcard_Suc_fminus1: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "Suc (fcard (A |-| {|x|})) = fcard A"
unit_mult_left_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd (1::'a)" shows "(a * b = a * c) = (b = c)"
fds_moebius_inverse_zeta: shows "fds moebius_mu = inverse fds_zeta"
dist_triangle_less_add: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x1 :: "'b"   and y :: "'b"   and e1 :: "real"   and x2 :: "'b"   and e2 :: "real" assumes "class.metric_space dist uniformity open"   and "dist x1 y < e1"   and "dist x2 y < e2" shows "dist x1 x2 < e1 + e2"
eNorm_ge1_triangular_ineq: fixes p :: "real"   and M :: "'a measure"   and x :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> real" assumes "1 \<le> p" shows "eNorm (\<LL> (ennreal p) M) (x + y) \<le> eNorm (\<LL> (ennreal p) M) x + eNorm (\<LL> (ennreal p) M) y"
maxsplit_eq: fixes P :: "'b list \<Rightarrow> bool" shows "\<forall>xs ps qs. (maxsplit P ([], xs) [] xs = (ps, qs)) = (xs = ps @ qs \<and> is_maxpref P ps xs)"
higher_deriv_cmult: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and x :: "complex"   and j :: "nat"   and c :: "complex" assumes "f holomorphic_on A"   and "x \<in> A"   and "open A" shows "(deriv ^^ j) (\<lambda>x. c * f x) x = c * (deriv ^^ j) f x"
Acc_to_mojmir_accept: fixes \<delta> :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0 :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and M_fin :: "('b ltl \<Rightarrow> nat option) \<Rightarrow> (('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> 'b set \<times> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set"   and \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltl"   and \<pi> :: "'b ltl \<Rightarrow> nat option"   and \<phi> :: "'b ltl" assumes "ltl_to_rabin_base \<delta> \<delta>\<^sub>M q\<^sub>0 q\<^sub>0\<^sub>M M_fin \<Sigma> w"   and "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "accepting_pair\<^sub>R (ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma>) (ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M \<phi>) (ltl_to_rabin_base_def.Acc_fin \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma> \<pi> (G \<psi>), ltl_to_rabin_base_def.Acc_inf \<delta>\<^sub>M q\<^sub>0\<^sub>M \<pi> (G \<psi>)) w"   and "the (\<pi> (G \<psi>)) < semi_mojmir_def.max_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>)" shows "mojmir_def.accept \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"
emsapss_decode_help9_def: fixes mHash :: "bit list"   and salt_new :: "bit list"   and H :: "bit list" shows "emsapss_decode_help9 mHash salt_new H = emsapss_decode_help10 (generate_M' mHash salt_new) H"
card_of_ordIso: fixes A :: "'a set"   and B :: "'b set" shows "(\<exists>f. bij_betw f A B) = (|A| =o |B|)"
mult: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" assumes "completely_multiplicative_function f" shows "f (a * b) = f a * f b"
route_tables_fresher: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A (recvmsg rreq_rrep_sn \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). \<forall>dip\<in>kD (rt \<xi>). rt \<xi> \<sqsubseteq>\<^bsub>dip\<^esub> rt \<xi>')"
F: shows "F \<le> lfp (\<lambda>a b. P b \<or> HLD X b \<and> a (stl b))"
neq_dP: fixes p :: "'b ref"   and q :: "'b ref"   and h :: "'b \<Rightarrow> 'b ref"   and Ps :: "'b list" assumes "p \<noteq> q"   and "Path h p Ps q"   and "distinct Ps" shows "\<exists>a Qs. p = Ref a \<and> Ps = a # Qs \<and> a \<notin> set Qs"
Afix_e_to_heap: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and n :: "Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "Afix (delete x \<Gamma>)\<cdot>(\<A>\<^sup>\<bottom>\<^bsub>n\<^esub> e \<squnion> ae) \<sqsubseteq> Afix ((x, e) # delete x \<Gamma>)\<cdot>(esing x\<cdot>n \<squnion> ae)"
exec_step_ind_Throw: fixes stk :: "val list"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "hd stk \<noteq> Null" shows "exec_step_ind (StepI Throw) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>the_Addr (hd stk)\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
count_empty: fixes a :: "'c" shows "count {#} a = 0"
case2: fixes y :: "game" assumes "zin y (left_options x)" shows "\<not> ge_game (zero_game, plus_game x (neg_game y))"
stream_eq_Stream_iff: fixes s :: "'b stream"   and x :: "'b"   and t :: "'b stream" shows "(s = x ## t) = (shd s = x \<and> stl s = t)"
up_smult_closed: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> up R" shows "(\<lambda>i. a \<otimes> p i) \<in> up R"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "(case x1\<^sup>+ of x\<^sup>+ \<Rightarrow> f1 x | x\<inverse> \<Rightarrow> f2 x) = f1 x1"
image_mset_add_mset: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and M :: "'b multiset" shows "f `# add_mset a M = add_mset (f a) (f `# M)"
ord_isom_less: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b = f a \<prec>\<^bsub>E\<^esub> f b"
simps_3: fixes uu :: "typ"   and t :: "Core.term"   and k :: "nat" shows "loose_bvar (Abs uu t) k = loose_bvar t (k + 1)"
ln_lower_9_def: shows "ln_lower_9 \<equiv> \<lambda>x. - ln_upper_9 (inverse x)"
abs_summable_on_def: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "(f abs_summable_on A) = integrable (count_space A) f"
subcs_map: fixes c :: "'c"   and s :: "'c FOL_Fitting.term"   and G :: "('c, 'd) form list" shows "subcs c s G = map (subc c s) G"
cyclerecD: fixes i :: "nat"   and j :: "nat"   and as :: "atom list"   and xs :: "'b list" assumes "cyclerec i j as"   and "xs ! i < xs ! j" shows "\<exists>a\<in>set as. \<not> I\<^sub>d\<^sub>l\<^sub>o a xs"
2: shows "invertible P' \<and> invertible Q' \<and> Smith_normal_form S \<and> S = P' ** D ** Q'"
of_real_of_int_eq: fixes z :: "int" shows "of_real (real_of_int z) = of_int z"
iso_cancel_right: fixes g :: "'d"   and f :: "'d"   and f' :: "'d" assumes "D.iso g"   and "f \<cdot>\<^sub>D g = f' \<cdot>\<^sub>D g"   and "D.seq f g"   and "D.iso g" shows "f = f'"
rep_eq: fixes x :: "'a mod_ring"   and xa :: "'a mod_ring" shows "Rep_mod_ring (x * xa) = Rep_mod_ring x * Rep_mod_ring xa mod int CARD('a)"
cases: fixes x :: "d \<times> d list \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour" obtains d :: "d" and ds :: "d list" and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" and b :: "contour" where "x = (d, ds, ve, b)"
map2_val_compatD2: fixes f :: "('a \<times> 'd) list \<Rightarrow> ('a \<times> 'e) list"   and zs :: "('a \<times> 'd) list" assumes "map2_val_compat f"   and "oalist_inv_raw zs" shows "fst ` set (f zs) \<subseteq> fst ` set zs"
RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m_refines_RPHIL: fixes i :: "nat" assumes "0 < i" shows "RPHIL i \<sqsubseteq> RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m i 0"
matrix_equation_hyp: shows "M_mat signs subsets *\<^sub>v w_vec p qs signs = v_vec p qs subsets"
de_morgan_orthogonal_complement_plus: fixes A :: "'b set"   and B :: "'b set" assumes "(0::'b) \<in> A"   and "(0::'b) \<in> B" shows "orthogonal_complement (A +\<^sub>M B) = orthogonal_complement A \<inter> orthogonal_complement B"
fds_nth_fds_moebius_mu: shows "fds_nth (fds moebius_mu) = moebius_mu"
table_left: shows "table (args_n args) (args_L args) X"
valid_dd: shows "valid_data_dependency ts\<^sub>s\<^sub>b"
less_ord0_LimI: fixes f :: "nat \<Rightarrow> ord0"   and n :: "nat" shows "f n < ord0_Lim f"
valuate_sum: fixes f :: "'b \<Rightarrow> linear_poly"   and A :: "'b set"   and v :: "nat \<Rightarrow> 'a" shows "sum f A \<lbrace> v \<rbrace> = (\<Sum>x\<in>A. f x \<lbrace> v \<rbrace>)"
simps_1: fixes uu :: "'b zmultiset" shows "repeat_zmset 0 uu = {#}\<^sub>z"
paid_A_def: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list" assumes "BIT_Off acts qs init" shows "BIT_Off.paid_A acts init = map (filter (\<lambda>x. Suc x < length init)) (BIT_Off.paid_A' acts)"
assms_3: shows "\<exists>i\<in>agents. p \<prec>[SD (R i)] q"
mono: shows "mono (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> mk_less orda x y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> mk_less orda x y \<and> \<not> mk_less orda y x \<and> p xs ys))"
some_list_def: fixes cs :: "'a list" shows "some_list cs = (SOME xs. perm_restrict f (set cs) = list_succ xs \<and> set xs = set cs \<and> distinct xs \<and> hd xs = hd cs)"
tail_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "tail t = (case viewL t of Some (x, m) \<Rightarrow> m)"
less_if_bot: fixes b :: "alt"   and p :: "alt \<Rightarrow> real"   and x :: "alt" assumes "b \<cdot>< p"   and "x \<noteq> b" shows "p b < p x"
pmaxsimpchainI_maxsimpchain: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimpchain X xs"   and "distinct xs" shows "SimplicialComplex.pmaxsimpchain X xs"
hom_0_iff: fixes x :: "int poly" shows "(of_int_poly x = 0) = (x = 0)"
multiset_partition: fixes M :: "'a multiset"   and P :: "'a \<Rightarrow> bool" shows "M = filter_mset P M + {#x \<in># M. \<not> P x#}"
discrete_evolution_def: fixes incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "real"   and t0 :: "real"   and x :: "'b" shows "discrete_evolution incr t1 t0 x = x + (t1 - t0) *\<^sub>R incr (t1 - t0) t0 x"
Infinitesimal_hyperpow: fixes x :: "real star"   and N :: "nat star" assumes "x \<in> Infinitesimal"   and "0 < N" shows "x pow N \<in> Infinitesimal"
emp_trap_set_err_mode: fixes s :: "'b sparc_state" shows "err_mode_val s = err_mode_val (emp_trap_set s)"
find_closest_set: fixes ps :: "(int \<times> int) list"   and p :: "int \<times> int"   and \<delta> :: "real" assumes "0 < length ps" shows "find_closest p \<delta> ps \<in> set ps"
R_def: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and j :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "subresultant_prs_locale.R n \<delta> G1 G2 j = (if j = n 2 then sdiv_poly (Polynomial.smult (lead_coeff G2 ^ \<delta> 1) G2) (lead_coeff G2) else subresultant j G1 G2)"
Int_commute: fixes A :: "'h set"   and B :: "'h set" shows "A \<inter> B = B \<inter> A"
bound_bad2_event: shows "\<bar>spmf (map_spmf fst game_semi_forgetful) True - spmf (map_spmf fst game_A) True\<bar> \<le> spmf (map_spmf snd game_A) True"
lower_triangular_def: fixes A :: "'c mat" shows "lower_triangular A = (\<forall>i j. i < j \<and> i < dim_row A \<and> j < dim_col A \<longrightarrow> A $$ (i, j) = (0::'c))"
distinct_1: fixes x2 :: "'d" shows "\<bottom> \<noteq> \<lfloor>x2\<rfloor>"
Smith_1xn_aux_S'_works: fixes S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat"   and j :: "nat" assumes "(S', Q') = Smith_1xn_aux i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "S \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n"   and "S = A * Q"   and "i < n"   and "0 < j"   and "j < n"   and "\<forall>j\<in>{i + 1..<n}. S $$ (0, j) = (0::'a)" shows "S' $$ (0, j) = (0::'a)"
ih2: shows "eval (change_int f i) p2 = change_tv f (eval i p2)"
last_conv_nth: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "last xs = xs ! (|xs| - 1)"
map_default_def: fixes k :: "'b"   and v :: "'c"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" shows "Mapping.map_default k v f m = Mapping.map_entry k f (Mapping.default k v m)"
cond_def: fixes p :: "'b ppath set"   and x :: "'b ppath set"   and y :: "'b ppath set" shows "ppath_aka.cond p x y = rdom p ; x \<union> pp_a p ; y"
non_interference_induct_case_sub1: fixes n :: "nat"   and s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> NIA t1 t2)" shows "\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> NIA t1 t2 \<and> NIC t1 t2)"
eval\<^sub>1_final: fixes P :: "nat exp prog"   and e :: "nat exp"   and s :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "final e'"
dist: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j \<le> n" shows "local.x i \<noteq> local.x j"
distinct_33: fixes x41 :: "('c, 'd) Term.term"   and x42 :: "('c, 'd) Term.term"   and x61 :: "poscheckvariant"   and x62 :: "('c, 'd) Term.term"   and x63 :: "('c, 'd) Term.term" shows "insert\<langle>x41,x42\<rangle> \<noteq> \<langle>x61: x62 \<in> x63\<rangle>"
path_split_1: 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 n :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n (as @ a # as') n'" shows "CFG.path sourcenode targetnode valid_edge n as (sourcenode a)"
cmovi: shows "m O ov\<inverse> \<subseteq> s \<union> ov \<union> d"
prj1_ext: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "\<not> (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g)" shows "prj1 f g = partial_magma.null C"
prems_1: shows "term_ok \<Theta> (subst_bvs (map (\<lambda>a. case a of (a, b) \<Rightarrow> Fv a b) vs) (Abs T s))"
exhaust_sel: fixes prod :: "'d \<times> 'e" obtains "prod = (fst prod, snd prod)"
vs: shows "vs \<equiv> getFrN (map snd txs) (Var y # map fst txs) [] (length txs)"
indets_extend_indets: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "indets (extend_indets p) = Some ` indets p"