hcju/isabelleps huggingface dataset:
prob_space: shows "I.prob (space natural_extension_measure) = 1"
intro: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "'a"   and A :: "'a set"   and B :: "'a" assumes "Semilat.top r T"   and "T \<in> A"   and "bottom r B"   and "B \<in> A" shows "lbv_axioms A r T B"
ide_def: fixes a :: "'a" shows "Right_a.ide a \<equiv> Right_a.comp a a \<noteq> Right_a.null \<and> (\<forall>f. (Right_a.comp f a \<noteq> Right_a.null \<longrightarrow> Right_a.comp f a = f) \<and> (Right_a.comp a f \<noteq> Right_a.null \<longrightarrow> Right_a.comp a f = f))"
ln_ms_aux: fixes C :: "'a"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and L :: "'a" assumes "trimmed_pos C"   and "basis_wf (b # basis)"   and "is_expansion_aux (MSLCons (C, e) xs) f (b # basis)"   and "((\<lambda>x. ln (eval C x) + e * ln (b x)) expands_to L) basis" shows "is_expansion_aux (MSLCons (L, 0) (times_ms_aux (scale_shift_ms_aux (inverse C, - e) xs) (powser_ms_aux' (ln_series_stream_aux False 1) (scale_shift_ms_aux (inverse C, - e) xs)))) (\<lambda>x. ln (f x)) (b # basis)"
transition_def: shows "transition \<equiv> {x. \<exists>P P' \<alpha> P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P' \<and> P' \<longmapsto> \<alpha> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P''' \<and> x = (P, \<alpha> \<prec> P''')} \<union> {x. \<exists>P P' a y P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P' \<and> P' \<longmapsto> a<\<nu>y> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P''' \<and> x = (P, a<\<nu>y> \<prec> P''')}"
m_const_def: fixes x :: "const" shows "m_const x = (case x of Const x \<Rightarrow> 1::'b | Any \<Rightarrow> 0::'b)"
linear_conv_bounded_linear: fixes f :: "'a \<Rightarrow> 'b" shows "linear f = bounded_linear f"
set_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = x21 \<triangleright> set x22"
arr_char: fixes f :: "'b" shows "J.arr f = (f \<in> Collect J0.arr \<union> Collect J1.arr)"
ideal_generated_subset: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "ideal_generated S \<subseteq> ideal_generated T"
set_sorted_list_of_set: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" shows "list.set (linorder.sorted_list_of_set less_eq A) = A"
mult_float_interval: fixes A :: "float interval"   and B :: "float interval"   and prec :: "nat" shows "set_of (real_interval A) * set_of (real_interval B) \<subseteq> set_of (real_interval (mult_float_interval prec A B))"
virt_to_phys_unchanged2: fixes va :: "word32"   and s2 :: "('b, 'c) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" shows "virt_to_phys va (mmu s2) (mem s2) = virt_to_phys va (mmu s2) ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None)))"
simps_2: fixes h :: "'b"   and t :: "'b list" shows "uniq (h # t) = (h \<notin> set t \<and> uniq t)"
assms_3: shows "Y \<longlonglongrightarrow> y"
max_coeff_bound_right: fixes xs :: "nat list"   and ys :: "nat list" assumes "(xs, ys) \<in> local.Minimal_Solutions" shows "\<forall>x\<in>set xs. x \<le> maxne0 ys b"
IH: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and upds :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list" assumes "vc C Q"   and "finite (support Q)"   and "finite (varacom C)"   and "lesvars upds \<inter> varacom C = {}"   and "distinct (map fst upds)" shows "\<turnstile>\<^sub>1 {\<lambda>l s. pre C Q l s \<and> preList upds C l s} strip C { time C \<Down> \<lambda>l s. Q l s \<and> postList upds l s} \<and> (\<forall>l s. pre C Q l s \<longrightarrow> Q l (postQ C s))"
assms_2: fixes e :: "'a" assumes "e \<in> set_pmf q" shows "pmf q e = pmf p e"
prems_2: fixes i :: "'a" shows "dsafe (\<sigma> i)"
has_contour_integral_add: fixes f1 :: "complex \<Rightarrow> complex"   and i1 :: "complex"   and g :: "real \<Rightarrow> complex"   and f2 :: "complex \<Rightarrow> complex"   and i2 :: "complex" assumes "(f1 has_contour_integral i1) g"   and "(f2 has_contour_integral i2) g" shows "((\<lambda>x. f1 x + f2 x) has_contour_integral i1 + i2) g"
mod_ring_rel_def: fixes x :: "int"   and x' :: "'a mod_ring" shows "pm.mod_ring_rel x x' = (x = to_int_mod_ring x')"
hoare_refinement_post: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and y :: "'a" assumes "mono f" shows "\<Turnstile> x{| f |}y = ({. x .} \<circ> post_fun y \<sqsubseteq> f)"
subset_alts: fixes X :: "'alt set" assumes "X \<in> set_pmf ` A" shows "X \<subseteq> alts"
list_comb_cond_inj_1: fixes f :: "'c"   and xs :: "'c list"   and g :: "'c"   and ys :: "'c list" assumes "list_comb f xs = list_comb g ys"   and "left_nesting f = left_nesting g" shows "xs = ys"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) \<chi>.A.map local.map \<chi>"
cong_dvd_modulus: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and division_segment :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and m :: "'c"   and n :: "'c" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment"   and "unique_euclidean_semiring.cong modulo x y m"   and "dvd.dvd times n m" shows "unique_euclidean_semiring.cong modulo x y n"
smap0_conv_special_subgroups: shows "dual_order.poset_simplex_map \<P> 0 = (\<lambda>s. \<langle>S - {s}\<rangle>) ` S"
Fun_mapsto: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S f" shows "set_category_given_img.Fun S img f \<in> extensional (set_category_data.set S img (partial_magma.dom S f)) \<inter> (set_category_data.set S img (partial_magma.dom S f) \<rightarrow> set_category_data.set S img (partial_magma.cod S f))"
sub_addD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and s :: "'a" assumes "sub_add t"   and "sound P"   and "sound Q" shows "t P s + t Q s \<le> t (\<lambda>s'. P s' + Q s') s"
assms_4: shows "prv (eqv \<phi> \<chi>)"
fractrel_iff: 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 x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" assumes "class.idom times one plus zero minus uminus" shows "idom.fractrel times zero x y = (snd x \<noteq> zero \<and> snd y \<noteq> zero \<and> times (fst x) (snd y) = times (fst y) (snd x))"
nn_integral_return: fixes x :: "'b"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "x \<in> space M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (return M x) g = g x"
tendsto_within_open: fixes a :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "a \<in> S"   and "open S" shows "(f \<longlongrightarrow> l) (at a within S) = f \<midarrow>a\<rightarrow> l"
simps: fixes a :: "'c multiset" shows "local.well_base a = (\<exists>M. a = M \<and> (\<forall>n. count M n < base))"
Act\<^sub>\<alpha>_eq_iff: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "(Act\<^sub>\<alpha> f1 \<alpha>1 t1 = Act\<^sub>\<alpha> f2 \<alpha>2 t2) = (tAct f1 \<alpha>1 (rep_Tree\<^sub>\<alpha> t1) =\<^sub>\<alpha> tAct f2 \<alpha>2 (rep_Tree\<^sub>\<alpha> t2))"
lc_setI: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "f \<in> F"   and "f \<noteq> 0" shows "lc f \<in> lc_set F"
item_rule_of_inc_item: fixes x :: "item"   and k :: "nat" shows "item_rule (inc_item x k) = item_rule x"
assms: shows "S : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B"
integrable_on_cexp_minus_to_infinity: fixes a :: "complex"   and c :: "real" assumes "0 < Re a" shows "(\<lambda>x. exp (x *\<^sub>R - a)) integrable_on {c..}"
S1_14_def: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and uu :: "unit" shows "S1_14 msgs uu = (let (m00, m01, m10, m11) = msgs in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. S1_OT12 (S0, S1) () \<bind> (\<lambda>a. S1_OT12 (S2, S3) () \<bind> (\<lambda>b. S1_OT12 (S4, S5) () \<bind> (\<lambda>c. return_spmf (msgs, (S0, S1, S2, S3, S4, S5), a, b, c)))))))))))"
a91: shows "\<Union> (Acc level1 ` DAcc level1 sA91) = Acc level1 sA81"
preserves_comp: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and g :: "'a"   and f :: "'a" assumes "functor A B F"   and "partial_magma.arr A (A g f)" shows "F (A g f) = B (F g) (F f)"
that: fixes P'' :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> : R \<rhd> P \<Longrightarrow>\<alpha> \<prec> P''"   and "\<Psi> \<otimes> \<Psi>' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "(\<Psi> \<otimes> \<Psi>', P', Q') \<in> Rel" shows "thesis"
lbrules_2: shows "\<Union> (A ` {1..m}) = {1..Suc j}"
DSourceOfSource: fixes C :: "CSet"   and i :: "AbstrLevelsID"   and S :: "CSet"   and Z :: "CSet" assumes "C \<in> DSources i S"   and "S \<in> Sources i Z" shows "C \<in> Sources i Z"
assms: shows "\<not> is_zero_row_upt_k i k A"
assms: shows "inj_on f (Sup_llist (lmap g xs))"
3: shows "\<exists>x y z. f a * a + f c * c = x * a + y * c + z * c"
not_call_simps_10: fixes pn :: "'c" shows "\<not> not_call (call(pn))"
simps_1: fixes V :: "'c set" shows "wf\<^sub>s\<^sub>t V [] = True"
inducts: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> segment f a b"   and "f a \<noteq> b \<Longrightarrow> P (f a)"   and "\<And>x. \<lbrakk>x \<in> segment f a b; P x; f x \<noteq> b\<rbrakk> \<Longrightarrow> P (f x)" shows "P x"
less_mask_eq: fixes x :: "'b word"   and n :: "nat" assumes "x < 2 ^ n" shows "x AND mask n = x"
simps_7: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G (F \<phi>) w = \<G>\<^sub>F\<^sub>G \<phi> w"
fin: shows "emeasure lborel (A \<inter> {..a}) \<noteq> top"
in_carrier_3: fixes i :: "nat" shows "local.coeff p2 i \<in> carrier R"
select_convs_2: fixes nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
listgcd_greatest: fixes xs :: "'b list"   and y :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> y dvd x" shows "y dvd some_gcd.listgcd xs"
div_self: 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 a :: "'a" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide a a = one"
his_bc_empty: fixes n' :: "nat"   and nid' :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "(n', nid') \<in> his t n nid"   and "\<not> (\<exists>n''<n'. \<exists>nid''. (n'', nid'') \<in> his t n nid)" shows "bc (\<sigma>\<^bsub>nid'\<^esub>t n') = []"
step: fixes x :: "V" assumes "Ord x"   and "\<And>y. y \<in> elts x \<Longrightarrow> P y" shows "P x"
il_inf_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
p_3: shows "a_kernel (K [X]) R h = PIdl\<^bsub>K [X]\<^esub> p"
g_orbit_eq: fixes X :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'c set" shows "g_orbit X G U = {X t |t. t \<in> U \<and> (\<forall>\<tau>\<in>down U t. G (X \<tau>))}"
rank_of_le: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> carrier" shows "rank_of \<E> \<le> card \<E>"
add_smult_distrib_vec: fixes a :: "'b"   and b :: "'b"   and v :: "'b Matrix.vec" shows "(a + b) \<cdot>\<^sub>v v = a \<cdot>\<^sub>v v + b \<cdot>\<^sub>v v"
w1': shows "w1 = rconjby B (lconjby half2_st (pair_relator_freeword s t))"
splitAt_snd: fixes xs :: "'a list"   and a :: "'a list"   and v :: "'a"   and b :: "'a list" assumes "distinct xs"   and "xs = a @ v # b" shows "snd (splitAt v xs) = b"
simps_18: fixes uy :: "'b"   and v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Atom uy) (Times v va) = True"
measure_preserving_preserves_integral_2: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "T \<in> measure_preserving M N"   and "integrable N f" shows "integral\<^sup>L N f = LINT x|M. f (T x)"
inf_le2: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<le> y"
complete_closed_subset: fixes S :: "'b set"   and t :: "'b set" assumes "closed S"   and "S \<subseteq> t"   and "complete t" shows "complete S"
zip_option_None: fixes xs :: "'e list"   and ys :: "'f list" shows "(zip_option xs ys = None) = (length xs \<noteq> length ys)"
inorder_upto_split_3: fixes l :: "int tree"   and k :: "int"   and r :: "int tree"   and i :: "int"   and j :: "int" assumes "inorder \<langle>l, k, r\<rangle> = [i..j]" shows "i \<le> k"
ceiling_le_iff: fixes x :: "'a"   and z :: "int" shows "(\<lceil>x\<rceil> \<le> z) = (x \<le> of_int z)"
i: fixes vl :: "'value list"   and vl1 :: "'value list" assumes "B vl vl1" shows "hd \<Delta>s istate vl istate vl1"
nat_take_bit_eq: fixes k :: "int"   and n :: "nat" assumes "0 \<le> k" shows "nat (take_bit n k) = take_bit n (nat k)"
not_less_iff_gr_or_eq: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" shows "(\<not> x \<prec> y) = (y \<prec> x \<or> x = y)"
integrable_divide_zero: fixes c :: "'b"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x / c)"
h3: shows "prod_list (map ((!) sign) (a # xa)) = 1 \<longrightarrow> 0 < poly (qs ! a) x \<and> 0 < poly (prod_list (retrieve_polys qs xa)) x \<or> poly (qs ! a) x < 0 \<and> poly (prod_list (retrieve_polys qs xa)) x < 0"
eq_iff: fixes a :: "'e word"   and b :: "'e word" shows "(a = b) = (b \<le>s a \<and> a \<le>s b)"
prems_1: fixes A\<^sub>P' :: "name list"   and \<Psi>\<^sub>P' :: "'b"   and \<Psi>' :: "'b" assumes "extractFrame P'' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>"   and "\<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'"   and "A\<^sub>P' \<sharp>* P''"   and "A\<^sub>P' \<sharp>* C"   and "distinct A\<^sub>P'" shows "thesis"
infdist_eq_setdist: fixes x :: "'c"   and A :: "'c set" shows "infdist x A = setdist {x} A"
incseq_SucD: fixes A :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "incseq A" shows "A i \<le> A (Suc i)"
non_empty_spectrum: shows "Collect (eigen_value A) \<noteq> {}"
Pow_singleton_iff: fixes X :: "'c set"   and Y :: "'c set" shows "(Pow X = {Y}) = (X = {} \<and> Y = {})"
iso_Sup_supdistl: fixes f :: "'a \<Rightarrow> 'c" assumes "mono f" shows "Sup \<circ> (`) f \<le> f \<circ> Sup"
bounded_clinear_right: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'b" assumes "bounded_cbilinear prod" shows "bounded_clinear (prod a)"
coprime_pderiv_imp_rsquarefree: fixes p :: "'a poly" assumes "coprime p (pderiv p)" shows "rsquarefree p"
lookup_restrict_indets_pp: fixes t :: "'b option \<Rightarrow>\<^sub>0 nat" shows "lookup (restrict_indets_pp t) = (\<lambda>x. lookup t (Some x))"
simps: fixes a :: "nat \<Rightarrow> nat" shows "(a \<in> PrimRec1) = (a = (\<lambda>x. 0) \<or> a = Suc \<or> a = (\<lambda>x. x) \<or> (\<exists>f g. a = (\<lambda>x. f (g x)) \<and> f \<in> PrimRec1 \<and> g \<in> PrimRec1) \<or> (\<exists>f g h. a = (\<lambda>x. f (g x) (h x)) \<and> f \<in> PrimRec2 \<and> g \<in> PrimRec1 \<and> h \<in> PrimRec1) \<or> (\<exists>f g h k. a = (\<lambda>x. f (g x) (h x) (k x)) \<and> f \<in> PrimRec3 \<and> g \<in> PrimRec1 \<and> h \<in> PrimRec1 \<and> k \<in> PrimRec1))"
one_dim_iso_of_zero': fixes x :: "'c" assumes "one_dim_iso x = (0::'b)" shows "x = (0::'c)"
sim: shows "strong_reduction_simulation ((indRelLTPO TRel)\<inverse>) (STCal Source Target)"
mk_or_def: fixes x :: "'a ltln"   and y :: "'a ltln" shows "mk_or x y \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> y | _ \<Rightarrow> case y of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> x | _ \<Rightarrow> x or\<^sub>n y"
prime_imp_prime_elem: fixes p :: "'b poly" assumes "field_poly.prime p" shows "prime_elem p"
insertI: fixes x :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and y :: "'f"   and D :: "'f set"   and e :: "'f" assumes "x \<notin> A"   and "f x y \<in> D"   and "(A, y) \<in> foldSetD D f e" shows "(insert x A, f x y) \<in> foldSetD D f e"
Rep_circline_mat: fixes x :: "circline_mat" shows "Rep_circline_mat x \<in> hermitean_nonzero"
fps_nonzero_nth: fixes f :: "'a fps" shows "(f \<noteq> 0) = (\<exists>n. f $ n \<noteq> (0::'a))"
shrK_image_eq: fixes x :: "agent"   and AA :: "agent set" shows "(shrK x \<in> shrK ` AA) = (x \<in> AA)"
pre_finish_def: fixes u :: "'v"   and s0 :: "'s"   and s :: "'s" shows "pre_finish u s0 s \<equiv> post_get_pending u None s0 s"
evaluate_list_singleton_cases: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and res :: "'b state \<times> (v list, v) result" assumes "evaluate_list ck env s [e] res" obtains s' :: "'b state" and v :: "v" where "res = (s'__, Rval [v])" and "evaluate ck env s e (s'__, Rval v)"   | s' :: "'b state" and err :: "v error_result" where "res = (s', Rerr err)" and "evaluate ck env s e (s', Rerr err)"
mult_pos_neg2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times b a) zero"
map_poly_idom_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_idom_hom hom \<equiv> idom_hom hom"
matrix_inv_unique: fixes A :: "(('d, 'f) vec, 'e) vec"   and B :: "(('d, 'e) vec, 'f) vec" assumes "A ** B = mat (1::'d)"   and "B ** A = mat (1::'d)" shows "A\<^sup>-\<^sup>1 = B"
inverse_inverse_ennreal: fixes x :: "ennreal" shows "inverse (inverse x) = x"
prefix_save: fixes sbc :: "'a list"   and n' :: "nat"   and n'' :: "nat" assumes "prefix sbc (bc n')"   and "\<forall>n'''>n'. n''' \<le> n'' \<longrightarrow> length sbc \<le> length (bc n''')"   and "n' \<le> n''" shows "prefix sbc (bc n'')"
distinct_drop: fixes xs :: "'c list"   and i :: "nat" assumes "distinct xs" shows "distinct (drop i xs)"
create_graph_precond: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<le> SPEC (create_gba_from_nodes_precond \<phi>)"
unit_imp_val_Zp0: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "val_Zp x = 0"
assms_2: shows "set \<phi>1s \<subseteq> set \<phi>2s"
decrpoly: fixes t :: "'b poly"   and x :: "'b"   and bs :: "'b list" assumes "polybound0 t" shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
bintr_cat1: fixes k :: "nat"   and n :: "nat"   and b :: "int"   and a :: "int" shows "take_bit (k + n) (concat_bit n b a) = concat_bit n b (take_bit k a)"
cqt_5: fixes \<psi> :: "\<kappa> \<Rightarrow> \<o>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" assumes "SimpleExOrEnc \<psi>" shows "[[\<psi> (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x))]]"
upper_dim_implies_not_two_sides_one_side_aux: assumes "UpperDimAxiom" shows "\<forall>A B X Y PX. A \<noteq> B \<and> PX \<noteq> A \<and> A B Perp X PX \<and> Col A B PX \<and> \<not> Col X A B \<and> \<not> Col Y A B \<and> \<not> A B TS X Y \<longrightarrow> A B OS X Y"
translation_Int: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and s :: "'c set"   and t :: "'c set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (s \<inter> t) = plus a ` s \<inter> plus a ` t"
minus_carrier_vec: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 - v\<^sub>2 \<in> carrier_vec n"
poincare_circle_is_euclidean_circle: fixes z :: "complex_homo"   and r :: "real" assumes "z \<in> unit_disc"   and "0 < r" shows "let (Ze, Re) = poincare_circle_euclidean z r in of_complex ` circle Ze Re = poincare_circle z r"
iso_lunit: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.iso (Span.lunit f)"
verts_del_vert: fixes u :: "'a" shows "verts (del_vert u) = verts G - {u}"
1: shows "ARR f \<and> ARR g \<and> DOM f = COD g"
Set_inv_lemma': fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and e :: "'e" assumes "\<tau> \<Turnstile> \<delta> X"   and "e \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>" shows "\<tau> \<Turnstile> \<upsilon> (\<lambda>_. e)"
S_list_automorphism: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "ChamberComplexAutomorphism X ((\<rightarrow>) (sum_list ss))"
P'_def: fixes n' :: "V" shows "P' n' = (if n' \<subseteq>\<^sub>\<circ> n \<Rightarrow> P (f' n') | otherwise \<Rightarrow> True)"
assms_2: fixes n :: "nat" shows "(0::'a) \<le> f n"
him: shows "h ` sphere b s \<subseteq> sphere 0 1"
rep_eq: fixes x :: "'a key_order" shows "le_of_key_order x = ko.le x"
hyps_11: shows "length (euler_fas' DIM('a)) = 2 * DIM('a)"
norm_lt_square: fixes x :: "'c"   and a :: "real" shows "(norm x < a) = (0 < a \<and> x \<bullet> x < a\<^sup>2)"
mndet_FD_subset: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d \<Rightarrow> 'd process" assumes "A \<noteq> {}"   and "A \<subseteq> B" shows "mndet B P \<le> mndet A P"
lin_f': fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "linear (f' x)"
vdisjnt_vempty2: fixes A :: "V" shows "A \<inter>\<^sub>\<circ> []\<^sub>\<circ> = []\<^sub>\<circ>"
assms_3: fixes g :: "'a \<Rightarrow> 'd"   and h :: "'a \<Rightarrow> 'd" assumes "g \<in> \<Theta>[F](h)" shows "(\<lambda>x. t x (g x)) \<in> \<Theta>[F](\<lambda>x. t x (h x))"
code_rate_def: fixes e :: "'c list \<Rightarrow> bool list"   and Xo :: "'a \<Rightarrow> 'c" shows "code_rate e Xo = expectation (\<lambda>x. real (code_word_length e (Xo x)))"