hcju/isabelleps huggingface dataset:
td_ext_sbin: shows "td_ext sint word_of_int (sints LENGTH('a)) (signed_take_bit (LENGTH('a) - 1))"
eWls_nchotomy: fixes eX :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and s :: "'sort" shows "(\<exists>X. eX = OK X \<and> igWls MOD s X) \<or> \<not> eWls MOD s eX"
A00_A01: shows "\<not> (A $h 0 $h 0 = (0::'a) \<and> A $h 0 $h 1 \<noteq> (0::'a))"
domainOfGraph: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "runiq (graph X f) \<and> Domain (graph X f) = X"
euler_maclaurin_strong_int: fixes b :: "int" assumes "a \<le> b" shows "(\<Sum>k = a..b. f (real_of_int k)) = F (real_of_int b) + C + (\<Sum>i<2 * N + 1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real_of_int b)) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) b"
empty_le: fixes A :: "'a multiset" shows "{#} \<subseteq># A"
cring_functions_def: fixes R :: "('a, 'b) ring_scheme" shows "cring_functions R \<equiv> ring_functions R \<and> cring R"
length_upto_add_nat: fixes a :: "int"   and n :: "nat" shows "length [a..a + int n] = Suc n"
elims: fixes x :: "nat"   and xa :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and xb :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec"   and y :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec" assumes "Diagonal_to_Smith_row_i_PQ x xa xb = y" obtains i :: "nat" and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" and P :: "(('a, 'b) vec, 'b) vec" and A :: "(('a, 'c) vec, 'b) vec" and Q :: "(('a, 'c) vec, 'c) vec" where "x = i" and "xa = bezout" and "xb = (P, A, Q)" and "y = diagonal_to_Smith_i_PQ [i + 1..<min (nrows A) (ncols A)] i bezout (P, A, Q)"
intro: fixes upd :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and cnst :: "'b \<Rightarrow> 'c" assumes "\<And>a b. upd a b (cnst b) = cnst b"   and "\<And>a a' b b' c. a \<noteq> a' \<Longrightarrow> upd a b (upd a' b' c) = upd a' b' (upd a b c)"   and "\<And>b b' a b''. b \<noteq> b' \<Longrightarrow> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)" shows "finfun_rec_wf_aux cnst upd"
zero_eq_power2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a 2 = zero) = (a = zero)"
cone_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and a :: "'c"   and \<chi> :: "'b \<Rightarrow> 'c" shows "Limit.cone J C D a \<chi> \<equiv> (category C \<and> category J) \<and> diagram J C D \<and> constant_functor J C a \<and> natural_transformation J C (constant_functor.map J C a) D \<chi>"
e: fixes i :: "nat" assumes "i < length A" shows "(A @ [b]) ! i = A ! i"
assms_9: shows "z ^ 3 + a * z\<^sup>2 + b * z + c = 0"
graph_def: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "graph X f = {(x, f x) |x. x \<in> X}"
index_take_if_set: fixes x :: "'a"   and n :: "nat"   and xs :: "'a list" assumes "x \<in> set (take n xs)" shows "index (take n xs) x = index xs x"
lookupd_cmaps: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and a :: "coname"   and x :: "name"   and P :: "trm"   and y :: "name" assumes "\<theta>_c cmaps a to Some (x, P)" shows "lookupd y a \<theta>_c = P[x\<turnstile>n>y]"
rtrancl_while_finite_Some: fixes f :: "'b \<Rightarrow> 'b list"   and x :: "'b"   and p :: "'b \<Rightarrow> bool" assumes "finite ({(x, y). y \<in> set (f x)}\<^sup>* `` {x})" shows "\<exists>y. rtrancl_while p f x = Some y"
HEndPhase1_Inv2a_dblock_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv1 s"   and "Inv2a s"   and "Inv2c_inner s p" shows "Inv2a_innermost s' q (dblock s' q)"
nrows_Diagonal_to_Smith_row_i: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "nrows (Diagonal_to_Smith_row_i A i bezout) = nrows A"
amin_ge1: fixes z :: "ant"   and x :: "ant"   and y :: "ant" assumes "z \<le> x"   and "z \<le> y" shows "z \<le> amin x y"
cm_preserves_inv_imps_work_sum: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "next_change_multiplicity' c0 c1 loc t n"   and "inv_imps_work_sum c0" shows "inv_imps_work_sum c1"
invar_start: shows "\<alpha>s \<Sigma> \<in> brw_invar \<delta>"
infnorm_eq_1_2: fixes x :: "(real, 2) vec" shows "(infnorm x = 1) = (\<bar>x $ 1\<bar> \<le> 1 \<and> \<bar>x $ 2\<bar> \<le> 1 \<and> (x $ 1 = - 1 \<or> x $ 1 = 1 \<or> x $ 2 = - 1 \<or> x $ 2 = 1))"
enumerateable_axioms: shows "enumerateable T (\<prec>)"
that: fixes U :: "'a set" assumes "U \<inter> insert (0::'a) S = {}"   and "pairwise orthogonal (S \<union> U)"   and "span (S \<union> U) = span (S \<union> T)" shows "thesis"
minus_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V" shows "x \<ominus>\<^bsub>V\<^esub> y \<in> carrier V"
ChiPlus_HAStates_Right: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(S, T) \<in> ChiPlus A" shows "T \<in> HAStates A"
isCont_add: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "isCont f a"   and "isCont g a" shows "isCont (\<lambda>x. f x + g x) a"
dvd_lcm2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times b (lcm a b)"
is_increasingI: fixes f :: "nat \<Rightarrow> nat" assumes "\<And>n m. m < n \<Longrightarrow> f m < f n" shows "is_increasing f"
common_denom_def: fixes xs :: "rat list" shows "common_denom xs \<equiv> let nds = map quotient_of xs; denom = lcm_list (map snd nds) in Let (map (\<lambda>(n, d). n * denom div d) nds) (Pair denom)"
eq_\<I>_converter_trans: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and conv' :: "('i, 'j, 'k, 'l) converter"   and conv'' :: "('i, 'j, 'k, 'l) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>,\<I>' \<turnstile>\<^sub>C conv' \<sim> conv''" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv''"
restict_ODNmap_func: fixes D :: "'c Order"   and x :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D" shows "restrict (ODNmap D) (segment ODnods (ordinal_number D)) \<in> segment ODnods (ordinal_number D) \<rightarrow> carrier D"
map_add_assoc: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and m3 :: "'a \<Rightarrow> 'b option" shows "m1 ++ (m2 ++ m3) = m1 ++ m2 ++ m3"
single_non_qFresh_ordLess_var: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort" assumes "qGood X" shows "|{x. \<not> qFresh xs x X}| <o |UNIV|"
inf_SUP: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and a :: "'g"   and f :: "'h \<Rightarrow> 'g"   and B :: "'h set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf a (Sup (f ` B)) = Sup ((\<lambda>b. inf a (f b)) ` B)"
ereal3_cases: fixes x :: "ereal"   and xa :: "ereal"   and xb :: "ereal" obtains r :: "real" and ra :: "real" and rb :: "real" where "x = ereal r__" and "xa = ereal ra__" and "xb = ereal rb"   | r :: "real" and ra :: "real" where "x = ereal rc__" and "xa = ereal raa__" and "xb = \<infinity>"   | r :: "real" and ra :: "real" where "x = ereal rd__" and "xa = ereal rab__" and "xb = - \<infinity>"   | r :: "real" and ra :: "real" where "x = ereal re__" and "xa = \<infinity>" and "xb = ereal rac__"   | r :: "real" where "x = ereal rf__" and "xa = \<infinity>" and "xb = \<infinity>"   | r :: "real" where "x = ereal rg__" and "xa = \<infinity>" and "xb = - \<infinity>"   | r :: "real" and ra :: "real" where "x = ereal rh__" and "xa = - \<infinity>" and "xb = ereal rad__"   | r :: "real" where "x = ereal ri__" and "xa = - \<infinity>" and "xb = \<infinity>"   | r :: "real" where "x = ereal rj__" and "xa = - \<infinity>" and "xb = - \<infinity>"   | r :: "real" and ra :: "real" where "x = \<infinity>" and "xa = ereal rk__" and "xb = ereal rae__"   | r :: "real" where "x = \<infinity>" and "xa = ereal rl__" and "xb = \<infinity>"   | r :: "real" where "x = \<infinity>" and "xa = ereal rm__" and "xb = - \<infinity>"   | r :: "real" where "x = \<infinity>" and "xa = \<infinity>" and "xb = ereal rn__"   | "x = \<infinity>" and "xa = \<infinity>" and "xb = \<infinity>"   | "x = \<infinity>" and "xa = \<infinity>" and "xb = - \<infinity>"   | r :: "real" where "x = \<infinity>" and "xa = - \<infinity>" and "xb = ereal ro__"   | "x = \<infinity>" and "xa = - \<infinity>" and "xb = \<infinity>"   | "x = \<infinity>" and "xa = - \<infinity>" and "xb = - \<infinity>"   | r :: "real" and ra :: "real" where "x = - \<infinity>" and "xa = ereal rp__" and "xb = ereal ra"   | r :: "real" where "x = - \<infinity>" and "xa = ereal rq__" and "xb = \<infinity>"   | r :: "real" where "x = - \<infinity>" and "xa = ereal rr__" and "xb = - \<infinity>"   | r :: "real" where "x = - \<infinity>" and "xa = \<infinity>" and "xb = ereal rs__"   | "x = - \<infinity>" and "xa = \<infinity>" and "xb = \<infinity>"   | "x = - \<infinity>" and "xa = \<infinity>" and "xb = - \<infinity>"   | r :: "real" where "x = - \<infinity>" and "xa = - \<infinity>" and "xb = ereal r"   | "x = - \<infinity>" and "xa = - \<infinity>" and "xb = \<infinity>"   | "x = - \<infinity>" and "xa = - \<infinity>" and "xb = - \<infinity>"
max2_underS'_1: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> underS Bonk_Schramm_extension_unfolded_wo y" shows "max2 Bonk_Schramm_extension_unfolded_wo x y = y"
freess_def: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees = ffUnion \<circ> fset_of_list \<circ> map frees"
interrupt_a_conv: fixes ls :: "'b lock_actions"   and nts :: "('a, 'c, 'd) new_thread_action list"   and js :: "'a conditional_action list"   and ws :: "('a, 'e) wait_set_action list"   and "is" :: "'a interrupt_action list"   and obs :: "'f list" shows "\<lbrace>(ls, nts, js, ws, is, obs)\<rbrace>\<^bsub>i\<^esub> = is"
lookup_single_not_eq: fixes k :: "'a"   and k' :: "'a"   and v :: "'b" assumes "k \<noteq> k'" shows "lookup (monomial v k) k' = (0::'b)"
BinOpThrow2: fixes v\<^sub>1 :: "val"   and bop :: "bop"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> throw e,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
w_simps_6: shows "cod \<omega>.chine = \<omega>.chine"
div2_less_self: fixes n :: "nat" assumes "odd n" shows "n div 2 < n"
closedin_compact_eq: fixes S :: "'b set"   and T :: "'b set" assumes "compact S" shows "closedin (top_of_set S) T = (compact T \<and> T \<subseteq> S)"
eq: shows "(\<Sum>v | f v \<noteq> (0::'a). f v *s v) = (\<Sum>v | g v \<noteq> (0::'a). g v *s v)"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "Xa.map f = B.null"
counit_in_hom_2: shows "\<guillemotleft>\<xi>' : S.UP f \<star>\<^sub>S S.UP g \<Rightarrow>\<^sub>S S.trg (S.UP f)\<guillemotright>"
psimps: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "tak_dom (x, y, z)" shows "tak x y z = (if x \<le> y then y else tak (tak (x - 1) y z) (tak (y - 1) z x) (tak (z - 1) x y))"
indRelRTPO_refl: fixes TRel :: "('procT \<times> 'procT) set" assumes "refl TRel" shows "refl (indRelRTPO TRel)"
rank_nullity_main_2: assumes "V.fin_dim"   and "T ` carrier V = carrier W" shows "W.fin_dim"
delete_def: fixes x :: "'a"   and t :: "('a \<times> 'b) tree" shows "delete x t = (let (l, uu_, r) = split t x in join2 l r)"
minimum_elem_sub: fixes T :: "'a set"   and X :: "'a set"   and a :: "'a" assumes "T \<subseteq> carrier D"   and "X \<subseteq> T" shows "minimum_elem D X a = minimum_elem (Iod D T) X a"
simps: fixes a1 :: "nat \<Rightarrow> 'n"   and a2 :: "nat" shows "((a1, a2) \<in> scop) = (\<exists>\<pi> n. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, n) \<in> scp \<and> \<pi> n \<in> dom local.att)"
tr: shows "((\<sigma>, p'), a, \<sigma>', q) \<in> automaton.trans A"
vector_space_with_axioms_def: fixes plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "vector_space_with_axioms plus\<^sub>V\<^sub>S scale \<equiv> ((\<forall>a x y. scale a (plus\<^sub>V\<^sub>S x y) = plus\<^sub>V\<^sub>S (scale a x) (scale a y)) \<and> (\<forall>a b x. scale (a + b) x = plus\<^sub>V\<^sub>S (scale a x) (scale b x))) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'e) x = x)"
gt: fixes i :: "nat" shows "delta_gt \<delta> (f i) (f (Suc i))"
arr_dom_iff_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C" shows "partial_magma.arr C (partial_magma.dom C f) = partial_magma.arr C f"
xs_def: shows "xs = list_neq (map (\<lambda>p. sgn (poly p x)) ps) 0"
simps_1: fixes n :: "nat"   and t\<^sub>1 :: "term"   and t\<^sub>2 :: "term" shows "wellformed' n (t\<^sub>1 $ t\<^sub>2) = (wellformed' n t\<^sub>1 \<and> wellformed' n t\<^sub>2)"
set_inter_compl_diff: fixes A :: "'a set"   and B :: "'a set" shows "- A \<inter> B = B - A"
Rf_mono_iia: fixes f :: "'d set \<Rightarrow> 'd set" shows "mono (Rf f) = iia f"
trms_of_deg_leq_id: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "f \<in> carrier P" shows "f \<ominus>\<^bsub>P\<^esub> trms_of_deg_leq k f = shift (Suc k) f \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
ax_impossible: fixes G :: "prog"   and A :: "'b triple set"   and t :: "Term.term"   and Q :: "'b assn" shows "G,A\<turnstile>{\<lambda>Y s Z. False} t\<succ> {Q}"
bij_betw_partitions_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "bij_betw (partitions_of A B) ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // range_permutation A B) {P. partition_on A P \<and> card P \<le> card B \<and> (\<forall>X\<in>P. card X = 1)}"
swapidseq_ext_of_cycles: fixes cs :: "'a list" assumes "cycle cs" shows "swapidseq_ext (set cs) (length cs - 1) (cycle_of_list cs)"
f_fold_merge: fixes f :: "'d \<Rightarrow> 'e"   and merge :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d list \<Rightarrow> 'f"   and xs :: "'d list"   and x :: "'d" assumes "groupsort f merge g"   and "\<And>y. y \<in> set xs \<Longrightarrow> f y = f x" shows "f (fold merge xs x) = f x"
split_list: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> list.set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
assoc\<^sub>S_def: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" shows "\<a>\<^sub>S[a, b, c] \<equiv> a \<otimes>\<^sub>S b \<otimes>\<^sub>S c"
top_synlr_def: shows "top \<equiv> mksynlr UNIV"
zmult_zless_mono2: fixes i :: "Quotient_Int.int"   and j :: "Quotient_Int.int"   and k :: "Quotient_Int.int" assumes "i < j"   and "0 < k" shows "k * i < k * j"
simps_4: fixes 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> NT) = (P,E,h \<turnstile> e : NT)"
A_SPAWN: fixes s :: "'a list"   and ce :: "'a list multiset"   and p :: "'b"   and s' :: "'a list" assumes "((s, ce), LSpawn p, s', {#[entry fg p]#} + ce) \<in> trss fg"   and "c = {#s#} + ce"   and "c' = {#s'#} + {#[entry fg p]#} + ce"   and "e = LSpawn p" shows "P"
A: shows "p \<ominus>\<^bsub>P\<^esub> q = p \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> q"
alg_dvd_0_right: fixes x :: "'b" shows "x alg_dvd (0::'b)"
cast_t: shows "cast\<cdot>t = EMB('a) oo d oo PRJ('a)"
G_def: shows "G = (\<lambda>m. if m = 0 then ln_Gamma else Polygamma (m - 1))"
fundfoldpairs_induced_autormorph_ntrivial: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "induced_automorph f g \<noteq> id"
add_block_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection" shows "incidence_system.add_block block_collection b \<equiv> block_collection + {#b#}"
cases: fixes a1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "a1 \<Rightarrow> a2" obtains \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>__, App e__ x__, S__)" and "a2 = (\<Gamma>__, e__, Arg x__ # S__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and y :: "Vars.var" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>'__, Lam [y]. ea__, Arg xa__ # Sa__)" and "a2 = (\<Gamma>'__, ea__[y::=xa__], Sa__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and x :: "Vars.var" and e :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>''__, Var xb__, Sb__)" and "a2 = (delete xb__ \<Gamma>''__, eb__, Upd xb__ # Sb__)" and "map_of \<Gamma>''__ xb__ = Some eb__"   | x :: "Vars.var" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>'''__, ec__, Upd x # Sc__)" and "a2 = ((x, ec__) # \<Gamma>'''__, ec__, Sc__)" and "x \<notin> domA \<Gamma>'''__" and "isVal ec__"   | \<Delta> :: "(Vars.var \<times> exp) list" and \<Gamma> :: "(Vars.var \<times> exp) list" and S :: "stack_elem list" and e :: "exp" where "a1 = (\<Gamma>''''__, Let \<Delta> e, Sd__)" and "a2 = (\<Delta> @ \<Gamma>''''__, e, Sd__)" and "atom ` domA \<Delta> \<sharp>* \<Gamma>''''__" and "atom ` domA \<Delta> \<sharp>* Sd__"   | \<Gamma> :: "(Vars.var \<times> exp) list" and scrut :: "exp" and e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>'''''__, scrut ? e1__ : e2__, Se__)" and "a2 = (\<Gamma>'''''__, scrut, Alts e1__ e2__ # Se__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and b :: "bool" and e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>, Bool b, Alts e1 e2 # S)" and "a2 = (\<Gamma>, if b then e1 else e2, S)"
inv_tr_card_min: fixes r :: "nat set" assumes "inv_translation r" shows "2 \<le> card r"
inj_monoid_mult_hom_axioms: shows "inj_monoid_mult_hom (inv hom)"
crsp_step_dec_b_suc: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat"   and la :: "cell list"   and ra :: "cell list"   and e :: "nat" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "inv_locate_a (as, lm) (n, la, ra) ires"   and "abc_fetch as ap = Some (Dec n e)"   and "0 < abc_lm_v lm n" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some (Dec n e))) (steps (start_of ly as + 2 * n, la, ra) (ci (layout_of ap) (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp) ires"
prerun_valid_induct: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and I :: "('g, 'h, 'i, 'j, 'k) system_state \<Rightarrow> bool" assumes "\<And>\<sigma>. prerun sys \<sigma> \<Longrightarrow> \<lceil>I\<rceil> \<sigma>"   and "\<And>\<sigma>. prerun sys \<sigma> \<Longrightarrow> (\<box>(\<lambda>s. \<lceil>I\<rceil> s \<longrightarrow> (\<circle>\<lceil>I\<rceil>) s)) \<sigma>" shows "sys \<Turnstile>\<^bsub>pre\<^esub> I"
norm_axis_eq: fixes i :: "'c"   and k :: "'b" shows "\<parallel>axis i k\<parallel> = \<parallel>k\<parallel>"
simps_2: fixes uu :: "'c match_expr"   and rs :: "'c rule list" shows "rm_LogEmpty (Rule uu Empty # rs) = rm_LogEmpty rs"
dg_dghm_id_is_iso_dghm: shows "dghm_id \<AA> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<AA>"
succ_not_in: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "(succ r i, i) \<notin> r"
simps: fixes A :: "('c, 'd) nba"   and a1 :: "('c \<times> 'd) list"   and a2 :: "'d" shows "NBA.path A a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> (fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) \<and> NBA.path A r (snd a)))"
dc_crypto: shows "dc_crypto = Some ` {0..<n} \<times> {xs. length xs = n}"
exec_step_ind_Getstatic_NoField: fixes P :: "jvm_prog"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and h :: "heap"   and stk :: "val list"   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 "\<nexists>t b. P \<turnstile> C has F,b:t in D" shows "exec_step_ind (StepI (Getstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NoSuchFieldError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
tdghm_length: shows "vcard \<NN> = 5\<^sub>\<nat>"
assms: shows "R\<^sup>+\<^sup>+ x z"
clo4: shows "closedin (top_of_set (U - C)) H4"
distinct_insort: fixes f :: "'c \<Rightarrow> 't"   and x :: "'c"   and xs :: "'c list" shows "distinct (ord_term_lin.insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
H_def: shows "H \<equiv> \<lambda>z. exp (complex_of_real a * z) / (1 - exp z)"
assms: shows "\<exists>i<e_length e. e_nth e i \<noteq> 0"
ln_gt_zero_imp_gt_one: fixes x :: "real" assumes "0 < ln x"   and "0 < x" shows "1 < x"
fBex_triv_one_point1: fixes A :: "'b fset"   and a :: "'b" shows "(\<exists>x|\<in>|A. x = a) = (a |\<in>| A)"
i_shrink_eq_Msg_iEx_iAll_cut_greater_conv: fixes k :: "nat"   and m :: "'a message_af"   and s :: "nat \<Rightarrow> 'a message_af"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg" shows "((s \<div> k) t = m) = (\<diamond> t1 [t * k\<dots>,k - Suc 0]. s t1 = m \<and> (\<box> t2 [t * k\<dots>,k - Suc 0] \<down>> t1. s t2 = NoMsg))"
rphil_transitions_def: fixes N :: "nat"   and i :: "nat"   and s :: "nat" assumes "DiningPhilosophers N" shows "Tr\<^sub>r\<^sub>p i s = (if s = 0 then {picks i i} else if s = 1 then {picks i (i - 1)} else if s = 2 then {putsdown i (i - 1)} else if s = 3 then {putsdown i i} else {})"
set_of_add_inc_right: fixes B :: "'b interval"   and B' :: "'b interval"   and A :: "'b interval" assumes "set_of B \<subseteq> set_of B'" shows "set_of (A + B) \<subseteq> set_of (A + B')"
var2: fixes x :: "'a"   and y :: "'a"   and M :: "'a trm" assumes "x \<noteq> y" shows "substitutes (Var x) y M (Var x)"
finite_Ej: fixes j :: "nat"   and x :: "nat list" shows "finite (Ej j x)"