hcju/isabelleps huggingface dataset:
3: shows "idempotent ((v * v\<^sup>T \<sqinter> g)\<^sup>\<star>)"
asymp_equiv_refl: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "f \<sim>[F] f"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp" assumes "\<And>x. x \<in> domA \<Gamma> \<Longrightarrow> P (the (map_of \<Gamma> x))"   and "P exp" shows "P (Let \<Gamma> exp)"
assms_2: shows "a + punit.lt (rep_list p) = b + punit.lt (rep_list q)"
unique_is_poincare_line_general: fixes u :: "complex_homo"   and v :: "complex_homo"   and l1 :: "circline"   and l2 :: "circline" assumes "u \<noteq> v"   and "u \<noteq> inversion v"   and "is_poincare_line l1"   and "is_poincare_line l2"   and "{u, v} \<subseteq> circline_set l1 \<inter> circline_set l2" shows "l1 = l2"
sorted: fixes sort1 :: "('b list \<Rightarrow> 'b) \<Rightarrow> 'b list list \<Rightarrow> 'b list list"   and f :: "'b list \<Rightarrow> 'b"   and xss :: "'b list list" assumes "Radix_Sort sort1" shows "sorted (map f (sort1 f xss))"
weak_reduction_simulations_impl_bisimulation: fixes Rel :: "('a \<times> 'a) set"   and Cal :: "'a processCalculus" assumes "weak_reduction_simulation Rel Cal"   and "weak_reduction_simulation (Rel\<inverse>) Cal" shows "weak_reduction_bisimulation Rel Cal"
permutations_of_multiset_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "permutations_of_multiset (image_mset f A) = map f ` permutations_of_multiset A"
simps: fixes a1 :: "'g"   and a2 :: "'g"   and r :: "('g \<times> 'g) set" shows "((a1, a2) \<in> r\<^sup>*) = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>* \<and> (b, c) \<in> r))"
MtransC_Ex_parTrace: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state" assumes "cf \<rightarrow>*c cf'" shows "\<exists>cfl. parTrace cfl \<and> hd cfl = cf \<and> last cfl = cf'"
yz: shows "\<forall>w. P w (y \<otimes> z) = (P w y \<and> P w z)"
bin_fst_nemp: shows "u\<^sub>1 \<noteq> \<epsilon>"
ide_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.ide (\<F>C.cod f)"
measurable_split_conv: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and A :: "('b \<times> 'c) measure"   and B :: "'d measure" shows "((\<lambda>(x, y). f x y) \<in> A \<rightarrow>\<^sub>M B) = ((\<lambda>x. f (fst x) (snd x)) \<in> A \<rightarrow>\<^sub>M B)"
minus_moebius_def: fixes A :: "moebius"   and B :: "moebius" shows "A - B = A + - B"
EndPhase0_HInv3_R: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase0 s s' t"   and "HInv3_R s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_R s' p q d"
getParts_samefw: fixes A :: "'b word set"   and rs :: "'b simple_rule list"   and s1 :: "'b word"   and s2 :: "'b word"   and p :: "('b, 'c) simple_packet_scheme" assumes "A \<in> set (map wordinterval_to_set (getParts rs))"   and "s1 \<in> A"   and "s2 \<in> A" shows "simple_fw rs (p\<lparr>p_src := s1\<rparr>) = simple_fw rs (p\<lparr>p_src := s2\<rparr>) \<and> simple_fw rs (p\<lparr>p_dst := s1\<rparr>) = simple_fw rs (p\<lparr>p_dst := s2\<rparr>)"
simps: fixes e1 :: "'c"   and a1 :: "'d"   and r1 :: "nat"   and ts1 :: "('c, 'd) BinomialTree list"   and e2 :: "'c"   and a2 :: "'d"   and r2 :: "nat"   and ts2 :: "('c, 'd) BinomialTree list" shows "link (Node e1 a1 r1 ts1) (Node e2 a2 r2 ts2) = (if a1 \<le> a2 then Node e1 a1 (Suc r1) (Node e2 a2 r2 ts2 # ts1) else Node e2 a2 (Suc r2) (Node e1 a1 r1 ts1 # ts2))"
pos_length_pos: fixes w :: "bit list" assumes "0 < bv_to_int w" shows "0 < length w"
mult_right_sub_dist_sup_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> z \<le> x \<squnion> z"
Case_In1: fixes c :: "('c, 'd) Old_Datatype.node set \<Rightarrow> 'b"   and d :: "('c, 'd) Old_Datatype.node set \<Rightarrow> 'b"   and N :: "('c, 'd) Old_Datatype.node set" shows "Case c d (In1 N) = d N"
strT1_def: shows "strT1 = (\<lambda>(Kl, lab). case lab [] of Inl x \<Rightarrow> F1map id (\<lambda>k1. (BNF_Greatest_Fixpoint.Shift Kl (Inl k1), BNF_Greatest_Fixpoint.shift lab (Inl k1))) (\<lambda>k2. (BNF_Greatest_Fixpoint.Shift Kl (Inr k2), BNF_Greatest_Fixpoint.shift lab (Inr k2))) x | Inr x \<Rightarrow> undefined x)"
join_right: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "dlat le" shows "le y (dlat.join le x y)"
image_eqI: fixes b :: "'ag"   and f :: "'ah \<Rightarrow> 'ag"   and x :: "'ah"   and A :: "'ah set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
mapl_G_def: fixes l1 :: "'h \<Rightarrow> 'o"   and l2 :: "'i \<Rightarrow> 'p" shows "mapl_G l1 l2 = map_G l1 l2 id id id id"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "map \<alpha> S T"   and "surjective_map_axioms \<alpha> S T" shows "surjective_map \<alpha> S T"
revcast_down_us: fixes n :: "nat"   and w :: "'b word" assumes "source_size revcast = target_size revcast + n" shows "revcast w = ucast (w >>> n)"
"1_2": shows "(\<Sum>(r, c)\<leftarrow>C. Le_Constraint Leq_Rel (r *R lec_poly c) (r *R lec_const c)) = Le_Constraint Leq_Rel 0 const"
dgrad_sig_setD_lt: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and d :: "'a \<Rightarrow> nat" assumes "p \<in> dgrad_sig_set' j d"   and "p \<noteq> 0" shows "component_of_term (lt p) < j"
ffd_fbb_dual: fixes f :: "'d \<Rightarrow> 'c set" shows "\<partial> \<circ> fd\<^sub>\<F> f = bb\<^sub>\<F> (op\<^sub>K f) \<circ> \<partial>"
True: shows "b = mod_type_class.from_nat (Suc k)"
intros_1: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration" assumes "distributed_system channel" shows "distributed_system.trace channel trans send recv c [] c"
inf_sup_aci_5: fixes 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 x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (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)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = sup y x"
sets_restrict_space_count_space: fixes A :: "'c set"   and B :: "'c set" shows "sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"
elims: fixes x :: "'b"   and xa :: "'b list"   and y :: "nat" assumes "position x xa = y" obtains a :: "'b" where "x = a__" and "xa = []" and "y = 0"   | a :: "'b" and xb :: "'b" and xs :: "'b list" where "x = a" and "xa = xb # xs" and "y = (if a = xb then 1 else Suc (position a xs))"
asymp_powser_coinduct': fixes P :: "(real \<Rightarrow> real) \<Rightarrow> real msstream \<Rightarrow> bool"   and x1 :: "real \<Rightarrow> real"   and x2 :: "real msstream"   and F :: "real filter"   and h :: "real \<Rightarrow> real" assumes "P x1 x2"   and "\<And>x1 x2. P x1 x2 \<Longrightarrow> \<exists>f cs c. x2 = MSSCons c cs \<and> P f cs \<and> x1 \<in> O[F](\<lambda>_. 1) \<and> (\<forall>\<^sub>F x in F. x1 x = c + f x * h x)" shows "asymp_powser h F x1 x2"
test_bit_def': fixes a :: "'b word"   and x :: "nat" shows "a !! x = bit (uint a) x"
selectlenhelp: fixes l :: "bit list"   and i :: "nat" shows "length (selecthelp l i) = i + 1"
CeqStart: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<notin> dom (Cp (list2FWpolicy a))"   and "a \<noteq> []"   and "b \<noteq> []" shows "Cp (list2FWpolicy (b @ a)) x = Cp (list2FWpolicy b) x"
d_def: fixes n :: "nat"   and fs :: "int vec list"   and k :: "nat" shows "LLL.d n fs k = gram_schmidt.Gramian_determinant n fs k"
infinite: fixes A :: "'a set" assumes "infinite A" shows "list_of A = []"
rep_nat_term_inj: fixes x :: "'b"   and y :: "'b" assumes "rep_nat_term x = rep_nat_term y" shows "x = y"
assms_3: shows "p\<^sup>\<star> ; q \<le> p\<^sup>T\<^sup>\<star> ; p"
subdigraphD_3: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and a :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
add2_zero: fixes x :: "'b" shows "Sep_Eq.add2 x (0::'b) = Some x"
not_in_keys_iff_lookup_eq_zero: fixes k :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd" shows "(k \<notin> Poly_Mapping.keys f) = (poly_mapping.lookup f k = (0::'d))"
beta: fixes x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and M :: "'a trm" shows "App (Fn x T A) M \<rightarrow>\<beta> A[x ::= M]"
Liminf_add_ereal_right: fixes F :: "'b filter"   and c :: "ereal"   and g :: "'b \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "Liminf F (\<lambda>n. g n + c) = Liminf F g + c"
measurable_count_space_extend: fixes A :: "'b set"   and B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "A \<subseteq> B"   and "f \<in> space M \<rightarrow> A"   and "f \<in> M \<rightarrow>\<^sub>M count_space B" shows "f \<in> M \<rightarrow>\<^sub>M count_space A"
le_succE: fixes i :: "V"   and j :: "V" assumes "ZFC_in_HOL.succ i \<le> ZFC_in_HOL.succ j" shows "i \<le> j"
finite_lattice_complete_Sup_insert: fixes x :: "'a"   and A :: "'a set" shows "Sup (insert x A) = sup x (Sup A)"
dual_bounded_lattice: shows "class.bounded_lattice (\<squnion>) (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x) (\<sqinter>) \<top> \<bottom>"
le_infI2: fixes b :: "'c multiset"   and x :: "'c multiset"   and a :: "'c multiset" assumes "b \<subseteq># x" shows "a \<inter># b \<subseteq># x"
l_scc: shows "set l \<subseteq> Collect (is_scc E)"
finite_imp_ex_se_star_succ: fixes c :: "('c, 'd) conf"   and ls :: "('c, 'd) label list" assumes "finite (pred c)"   and "\<forall>e\<in>pred c. finite (Bexp.vars e)"   and "finite_labels ls" shows "\<exists>c'. se_star c ls c'"
imageE: fixes b :: "'k"   and f :: "'l \<Rightarrow> 'k"   and A :: "'l set" assumes "b \<in> f ` A" obtains x :: "'l" where "b = f x" and "x \<in> A"
smallomega_iff_smallo: fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "(g \<in> \<omega>[F](f)) = (f \<in> o[F](g))"
is_none_bind: fixes f :: "'n option"   and g :: "'n \<Rightarrow> 'm option" shows "Option.is_none (Option.bind f g) = (Option.is_none f \<or> Option.is_none (g (the f)))"
IH: assumes "degree p1 = degree p2" shows "q2 ^ (degree p1 - degree (p1 + p2)) * fcompose (p1 + p2) q1 q2 = fcompose p1 q1 q2 + fcompose p2 q1 q2"
simps_9: fixes b :: "bool"   and v :: "Term.var" shows "assigns_if b (Acc v) = assignsV v"
hoare_tvalid_def: fixes P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>t {P} c {Q}) = (\<Turnstile> {P} c {Q} \<and> (\<forall>s. P s \<longrightarrow> c \<down> s))"
proper_eq: fixes ct :: "'s \<Rightarrow> 's pmf" assumes "ct \<in> Pi\<^sub>E S K" shows "proper ct = (S\<^sub>r \<subseteq> directed_towards S2 (SIGMA x:S\<^sub>r. set_pmf (ct x)))"
swap_apply1: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "Fun.swap a b f a = f b"
accept_def: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and F :: "'c set" shows "mojmir_def.accept \<delta> q\<^sub>0 w F = (\<forall>\<^sub>\<infinity>x. mojmir_def.token_succeeds \<delta> q\<^sub>0 w F x)"
iMODb_inext_nth_diff: fixes a :: "nat"   and c :: "nat"   and b :: "nat"   and r :: "nat"   and m :: "nat" assumes "a \<le> c"   and "b \<le> c" shows "([ r, mod m, c ] \<rightarrow> a) - ([ r, mod m, c ] \<rightarrow> b) = (a - b) * m"
WT_determ: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and T :: "ty"   and T' :: "ty" assumes "wf_prog wf_md P"   and "P,E \<turnstile> e :: T"   and "P,E \<turnstile> e :: T'" shows "T = T'"
K_2: fixes n :: "nat" shows "norm (f n) \<le> K"
chine_simps_3: shows "C.cod src.chine = src.cod.apex"
in_lset_conv_lnth: fixes x :: "'b"   and xs :: "'b llist" shows "(x \<in> lset xs) = (\<exists>n. enat n < llength xs \<and> xs $ n = x)"
group_free_Abelian_group: fixes S :: "'b set" shows "Group.group (free_Abelian_group S)"
get_ACS_def: fixes M :: "'a SecurityInvariant_configured list" shows "get_ACS M \<equiv> filter (\<lambda>m. \<not> c_isIFS m) M"
len_bin_to_bl: fixes n :: "nat"   and w :: "int" shows "length (bin_to_bl n w) = n"
consistence_error: fixes j :: "nat" assumes "t (Suc j) \<le> t1" shows "dist (x (t j + stepsize j)) (discrete_evolution incr (t j + stepsize j) (t j) (x (t j))) \<le> B * stepsize j ^ (p + 1)"
sf2: fixes m :: "nat" shows "c m \<in> sfis (mu urev m) M"
hyps: fixes x :: "'a llist"   and m :: "nat" assumes "\<not> lfinite x"   and "lnth x m = y"   and "\<And>i. lnth x i = y \<Longrightarrow> \<exists>m>i. lnth x m = y" shows "y \<in> lset (ldropn n x)"
monoD: fixes f :: "'a word \<Rightarrow> 'b"   and x :: "'a word"   and y :: "'a word" assumes "signed.mono f"   and "x \<le>s y" shows "f x \<le> f y"
assms_3: shows "edges (graph_of G) \<subseteq> L \<times> UNIV"
lzip_llist_of_stream: fixes xs :: "'c stream"   and ys :: "'d stream" shows "lzip (llist_of_stream xs) (llist_of_stream ys) = llist_of_stream (szip xs ys)"
lbutlast_lapp_llast: fixes l :: "'b llist"   and A :: "'b set" assumes "l \<in> A\<^sup>\<clubsuit>" shows "l = lbutlast l @@ LList2.llast l ## LNil"
top_conj_1: fixes x :: "'a"   and P :: "bool" shows "(\<top> x \<and> P) = P"
indicator_times_eq_if_1: fixes S :: "'c set"   and x :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "indicator S x * f x = (if x \<in> S then f x else (0::'b))"
f_eq_fc: fixes j :: "nat"   and n :: "nat" assumes "0 < j"   and "j < 2 ^ n" shows "f (b (real (2 * j - 1) / 2 ^ Suc n)) = f (c (real j / 2 ^ n)) \<and> f (a (real (2 * j + 1) / 2 ^ Suc n)) = f (c (real j / 2 ^ n))"
partial_division_of_tagged_division: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "s tagged_partial_division_of i" shows "snd ` s division_of \<Union> (snd ` s)"
reduced_homology_group_nsphere_aux: fixes p :: "int"   and n :: "nat" shows "if p = int n then reduced_homology_group (int n) (nsphere n) \<cong> integer_group else trivial_group (reduced_homology_group p (nsphere n))"
distrib_imp2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup"   and "\<And>x y z. sup x (inf y z) = inf (sup x y) (sup x z)" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
finite_measure_compl: fixes S :: "'a set" assumes "S \<in> PS.events" shows "PS.prob (space P - S) = PS.prob (space P) - PS.prob S"
step_redexes: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and r :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and r' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and c :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')"   and "r \<in> redexes c" shows "\<exists>c'. \<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s') \<and> r' \<in> redexes c'"
hd_upt: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "hd [i..<j] = i"
simps_3: fixes f :: "Core.term"   and t :: "Core.term"   and lev :: "nat"   and args :: "Core.term list" shows "subst_bvs1 (f $ t) lev args = subst_bvs1 f lev args $ subst_bvs1 t lev args"
wtL_glitOfL: fixes l' :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit"   and l :: "('fsym, 'psym) lit" assumes "l' \<in> glitOfL l" shows "GE.wtL l'"
cut_less_empty: fixes t :: "'b" shows "{} \<down>< t = {}"
len_after_drop: fixes p :: "nat"   and q :: "nat"   and w :: "'b list" assumes "p + q \<le> \<^bold>|w\<^bold>|" shows "q \<le> \<^bold>|drop p w\<^bold>|"
fundantivertex_def: fixes s :: "'a permutation" shows "fundantivertex s \<equiv> THE v. v \<in> C0 - s `\<rightarrow> C0"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "(a1 >\<^sub>t a2) = ((\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt s < wt t) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt t = wt s \<and> head t >\<^sub>h\<^sub>d head s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt t = wt s \<and> head t = head s \<and> (\<forall>f\<in>local.ground_heads (head s). extf f (>\<^sub>t) (args t) (args s))))"
mat_of_col_dim_col_1: fixes A :: "'b mat" shows "(dim_col A = 1) = (A = mat_of_col (col A 0))"
fX: shows "f ` topspace X \<subseteq> topspace Y"
of_real_eq_star_of: shows "of_real = hypreal_of_real"
that: fixes f :: "V" assumes "\<And>x. \<lbrakk>x \<in>\<^sub>\<circ> A; x \<noteq> 0\<rbrakk> \<Longrightarrow> f\<lparr>x\<rparr> \<in>\<^sub>\<circ> x"   and "vsv f" shows "thesis"
shiftr_f_def: fixes a :: "'b array"   and i :: "nat ref" shows "shiftr_f a i = !i \<bind> (\<lambda>i'. (a.(i' - 1)) \<bind> (\<lambda>x. (a.(i') \<leftarrow> x) \<bind> (\<lambda>_. i := i' - 1)))"
eq_refl: fixes x :: "'b list"   and y :: "'b list" assumes "x = y" shows "x \<le>p y"
residuated_sup: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "f (x \<squnion> y) = f x \<squnion> f y"
exhaust: fixes y :: "'b rexp" obtains "y = Zero"   | "y = One"   | x3 :: "'b" where "y = Atom x3"   | x41 :: "'b rexp" and x42 :: "'b rexp" where "y = Plus x41 x42"   | x51 :: "'b rexp" and x52 :: "'b rexp" where "y = Times x51 x52"   | x6 :: "'b rexp" where "y = Star x6"
surjec_surj_to: fixes R :: "('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "surjec\<^bsub>R,S\<^esub> f" shows "surj_to f (carrier R) (carrier S)"
zip_arg_index: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> set (zip xs ys)" obtains i :: "nat" where "xs ! i = x" and "ys ! i = y" and "i < length xs" and "i < length ys"
Kp_def: shows "KN.Kp \<equiv> \<lambda>(a, b). pair_pmf (K a) (K' b)"
simps_6: fixes I :: "interp"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "term_sem I (Times \<theta> \<eta>) = (\<lambda>\<omega>. term_sem I \<theta> \<omega> * term_sem I \<eta> \<omega>)"
irreducible_multD: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.idom times one plus zero minus uminus"   and "comm_semiring_1.irreducible times one zero (times a b)" shows "dvd.dvd times a one \<and> comm_semiring_1.irreducible times one zero b \<or> dvd.dvd times b one \<and> comm_semiring_1.irreducible times one zero a"
bij_project_act: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "bij (project_act (\<lambda>(x, u). h x))"
Agg: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and b :: "nat"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and y :: "nat"   and f :: "Formula.trm"   and g0 :: "bool"   and \<omega> :: "Formula.agg_type \<times> event_data" assumes "wf_mformula \<sigma> j P V (b + n) (lift_envs' b R) \<phi> \<phi>'"   and "y < n"   and "y + b \<notin> fv \<phi>'"   and "{0..<b} \<subseteq> fv \<phi>'"   and "fv_trm f \<subseteq> fv \<phi>'"   and "g0 = (fv \<phi>' \<subseteq> {0..<b})" shows "wf_mformula \<sigma> j P V n R (MAgg g0 y \<omega> b f \<phi>) (formula.Agg y \<omega> b f \<phi>')"
disc_4: fixes x4 :: "'a" shows "\<not> isTrans (Snapshot x4)"
simps: fixes a :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls" shows "(a \<in> valid_lists S G) = (a = [] \<or> (\<exists>xs x. a = x # xs \<and> set xs \<subseteq> ccNeighbors x G \<and> xs \<in> valid_lists S G \<and> x \<in> S))"
carrier_0: fixes x :: "'a" shows "(x \<in> carrier mk_monoid) = (x \<noteq> (0::'a))"
bi_unique_conversep: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_unique R\<inverse>\<inverse> = bi_unique R"
Body1: fixes G :: "'a triple set"   and P :: "pname \<Rightarrow> 'a assn"   and Q :: "pname \<Rightarrow> 'a assn"   and Procs :: "pname set"   and pn :: "pname" assumes "G \<union> (\<lambda>p. {P p}. BODY p .{Q p}) ` Procs||-(\<lambda>p. {P p}. the (body p) .{Q p}) ` Procs"   and "pn \<in> Procs" shows "G|-{P pn}. BODY pn .{Q pn}"
simps_16: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = sip"
rep_eq: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a \<Rightarrow> 'b" shows "Rep_pfun (plambda x xa) = (\<lambda>xb. if x xb then Some (xa xb) else None)"
sylvester_index_mat: fixes i :: "nat"   and p :: "'c poly"   and q :: "'c poly"   and j :: "nat" assumes "i < degree p + degree q"   and "j < degree p + degree q" shows "sylvester_mat p q $$ (i, j) = (if i < degree q then if i \<le> j \<and> j - i \<le> degree p then coeff p (degree p + i - j) else (0::'c) else if i - degree q \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'c))"
y_def: shows "y = x /\<^sub>R \<parallel>x\<parallel>"
composite_simps_3: shows "local.dom (f \<star> src f) = local.dom f \<star> local.dom (src f)"
lim_cnj: fixes f :: "'b \<Rightarrow> complex"   and l :: "complex"   and F :: "'b filter" shows "((\<lambda>x. cnj (f x)) \<longlongrightarrow> cnj l) F = (f \<longlongrightarrow> l) F"
assms_2: shows "\<CC> = \<AA> \<times>\<^sub>D\<^sub>G \<BB>"
map_eq_Cons_D: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and y :: "'c"   and ys :: "'c list" assumes "map f xs = y # ys" shows "\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys"
less_le_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"