hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'a list"   and xa :: "'a \<Rightarrow> 'b" shows "mapping.rep (Mapping.tabulate x xa) = map_of (map (\<lambda>k. (k, xa k)) x)"
merkle_nr_blindable: shows "merkle_interface (hash_blindable id) (blinding_of_blindable id (=)) (merge_blindable id merge_discrete)"
chine_composites_4: shows "Chn \<lparr>Chn = \<mu>_\<nu>_\<pi>\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>)\<rparr> = \<mu>_\<nu>_\<pi>\<rho>.chine_assoc"
INF_eqI: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and x :: "'b"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x (f i)"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y (f i)) \<Longrightarrow> less_eq y x" shows "Inf (f ` A) = x"
one_le_ceiling: fixes x :: "'a" shows "(1 \<le> \<lceil>x\<rceil>) = ((0::'a) < x)"
product_commutativity: fixes x :: "i"   and y :: "i" assumes "O x y" shows "x \<otimes> y = y \<otimes> x"
posk: shows "\<forall>n\<ge>k. positive (X n)"
norm_eff: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list" shows "norm_eff i P pc \<tau> = map (\<lambda>pc'. (pc', \<lfloor>eff\<^sub>i (i, P, \<tau>)\<rfloor>)) (succs i \<tau> pc)"
assms_5: shows "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"
subst_atm_mset_lists_comp_substs: fixes AAs :: "'a multiset list"   and \<tau>s :: "'s list"   and \<sigma>s :: "'s list" shows "AAs \<cdot>\<cdot>aml (\<tau>s \<odot>s \<sigma>s) = AAs \<cdot>\<cdot>aml \<tau>s \<cdot>\<cdot>aml \<sigma>s"
measurable_linear_image_interval: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" assumes "linear f" shows "f ` cbox a b \<in> lmeasurable"
all_zero_iff: fixes x :: "'a" shows "(\<forall>u. inner x u = 0) = (x = (0::'a))"
minimize_nonzero_solution: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b"   and "nonzero x" obtains u :: "nat list" and v :: "nat list" where "u @ v \<le>\<^sub>v x @ y" and "(u, v) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b"
explicationFWB1: shows "\<forall>c x xa. \<lfloor>NeedsForPurpose xa FWB x\<rfloor>\<^sub>c"
representation_diff: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d"   and u :: "'d" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis"   and "u \<in> module.span scale basis" shows "module.representation scale basis (u - v) = (\<lambda>b. module.representation scale basis u b - module.representation scale basis v b)"
decompose_rec: fixes ys :: "'b list"   and x :: "'b"   and y :: "'b" assumes "ys \<noteq> []" shows "decompose x (y # ys) = (case bezout_coefficients y (prod_list ys) of (a, b) \<Rightarrow> b * x # decompose (a * x) ys)"
partCompat_gen_aux: fixes theta :: "('a \<times> 'a) set"   and P :: "'b set set"   and f :: "'b \<Rightarrow> 'a"   and I :: "'b set"   and i :: "'b"   and j :: "'b" assumes "sym theta"   and "trans theta"   and "partCompat P theta f"   and "I \<in> P"   and "i \<in> I"   and "j \<in> gen P I"   and "i \<noteq> j" shows "(f i, f j) \<in> theta"
Pi_iff: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and X :: "'a \<Rightarrow> 'b set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
division_weak_lower_semilattice: shows "weak_lower_semilattice (division_rel G)"
defargs: fixes \<tau> :: "'\<AA> state \<times> '\<AA> state"   and x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" assumes "\<tau> \<Turnstile> f x y" shows "\<tau> \<Turnstile> \<upsilon> x \<and> \<tau> \<Turnstile> \<upsilon> y"
nonneg_Ax: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A"   and "x \<in> perron_frobenius.X" shows "\<forall>i. 0 $h i \<le> (A *v x) $h i"
on_curvep_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.on_curvep times one plus zero a b = (\<lambda>(x, y, z). z \<noteq> zero \<longrightarrow> times (power.power one times y 2) z = plus (plus (power.power one times x 3) (times (times a x) (power.power one times z 2))) (times b (power.power one times z 3)))"
uniqueD: fixes x :: "'d"   and y :: "'d" assumes "wrap_iso_arcs hom f x = wrap_iso_arcs hom f y" shows "x = y"
is_functor: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) A.map"
insertI: fixes A :: "'g set"   and a :: "'g" assumes "finite A" shows "finite (insert a A)"
\<tau>move1InstanceOfRed: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and U :: "ty" shows "\<tau>move1 P h (Val v instanceof U)"
sfilter_P: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" assumes "P (shd s)" shows "sfilter P s = shd s ## sfilter P (stl s)"
validVC: shows "isViewOn \<V> E\<^bsub>ES1 \<parallel> ES2\<^esub>"
path_independent_onI2: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f"   and "Chernoff_on A f"   and "consistency_on A f" shows "path_independent_on A f"
cvx: shows "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
inv_image_partition: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x"   and "\<And>y. y \<in> set ys \<Longrightarrow> \<not> P y" shows "partition P -` {(xs, ys)} = shuffles xs ys"
filtration_measurable: fixes f :: "bool stream \<Rightarrow> 'b"   and t :: "nat"   and N :: "'b measure" assumes "f \<in> G t \<rightarrow>\<^sub>M N" shows "random_variable N f"
set_tree_inter: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.set_tree (local.inter t1 t2) = Tree2.set_tree t1 \<inter> Tree2.set_tree t2"
WT_gpv_absorb: fixes \<I>' :: "('m, 'n) \<I>"   and gpv :: "('m, 'n) distinguisher"   and \<I> :: "('o, 'p) \<I>"   and conv :: "('m, 'n, 'o, 'p) converter" assumes "\<I>' \<turnstile>g gpv \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "\<I> \<turnstile>g absorb gpv conv \<surd>"
simps: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_ite i t e = (case lowest_tops [i, t, e] of None \<Rightarrow> case i of Trueif \<Rightarrow> t | Falseif \<Rightarrow> e | Some x \<Rightarrow> IFC x (ifex_ite (restrict_top i x True) (restrict_top t x True) (restrict_top e x True)) (ifex_ite (restrict_top i x False) (restrict_top t x False) (restrict_top e x False)))"
integral_measure_pmf: fixes A :: "'b set"   and M :: "'b pmf"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> (0::'c)\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. pmf M a *\<^sub>R f a)"
summable_divide: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "summable (\<lambda>n. f n / c)"
lub_least: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and z :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "M \<in> Chains leq"   and "M \<noteq> {}"   and "\<And>x. x \<in> M \<Longrightarrow> (x, z) \<in> leq" shows "(lub M, z) \<in> leq"
mersenne_mod2_nonneg_strong: fixes p :: "nat"   and a :: "int" assumes "- (2 ^ p) + 1 < a" shows "0 \<le> mersenne_mod2 a p"
Nmlize_Src_2: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Src t\<^bold>\<rfloor> = Src t"
lcm_div_le': fixes a :: "nat"   and b :: "nat" shows "lcm a b div a \<le> b"
dom_locals_lvar_mono: fixes s :: "state"   and vn :: "lname"   and s' :: "st"   and val :: "val" shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (snd (lvar vn s') val s)))"
Until: fixes \<phi> :: "formula"   and \<psi> :: "formula"   and I :: "\<I>" assumes "fv \<phi> \<subseteq> fv \<psi>"   and "safe_formula \<phi>"   and "safe_formula \<psi>"   and "P \<phi>"   and "P \<psi>" shows "P (Until \<phi> I \<psi>)"
rel_pred_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_pred R = rel_fun R (=)"
of_int_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (numeral k) = numeral.numeral one plus k"
fds_nth_const_not_Suc_0: fixes n :: "nat"   and c :: "'b" assumes "n \<noteq> 1" shows "fds_nth (fds_const c) n = (0::'b)"
path_split: fixes SES :: "('a, 'b) SES_rec"   and s1 :: "'a"   and t1 :: "'b list"   and t2 :: "'b list"   and sn :: "'a" assumes "s1 (t1 @ t2)\<Longrightarrow>\<^bsub>SES\<^esub> sn" shows "\<exists>s2. s1 t1\<Longrightarrow>\<^bsub>SES\<^esub> s2 \<and> s2 t2\<Longrightarrow>\<^bsub>SES\<^esub> sn"
hom_frag_diff: fixes h :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and S :: "'b set"   and T :: "'c set"   and a :: "'b \<Rightarrow>\<^sub>0 int"   and b :: "'b \<Rightarrow>\<^sub>0 int" assumes "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)"   and "Poly_Mapping.keys a \<subseteq> S"   and "Poly_Mapping.keys b \<subseteq> S" shows "h (a - b) = h a - h b"
hom_boundary_reduced_hom: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "hom_boundary p X S \<in> hom (relative_homology_group p X S) (reduced_homology_group (p - 1) (subtopology X S))"
weak_transition_stepI: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" assumes "P \<Rightarrow>{\<alpha>} P'" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle> P'"
prime_dvd_power_iff: fixes p :: "'a"   and n :: "nat"   and x :: "'a" assumes "prime p"   and "0 < n" shows "(p dvd x ^ n) = (p dvd x)"
th: shows "Var v \<lhd> \<theta> = t \<lhd> \<theta>"
deterministic: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and a :: "'f"   and a' :: "'f" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val"   and "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a \<noteq> targetnode a'" shows "\<exists>Q Q'. kind a = (Q)\<^sub>\<surd> \<and> kind a' = (Q')\<^sub>\<surd> \<and> (\<forall>s. (Q s \<longrightarrow> \<not> Q' s) \<and> (Q' s \<longrightarrow> \<not> Q s))"
gbinomial_Suc0: fixes a :: "'a" shows "a gchoose Suc 0 = a"
a_starI: fixes M :: "trm"   and M' :: "trm" assumes "M \<longrightarrow>\<^sub>a M'" shows "M \<longrightarrow>\<^sub>a* M'"
list_of_oalist_OAlist_tc: fixes xs :: "('a \<times> 'b) list" shows "list_of_oalist_tc (OAlist_tc xs) = sort_oalist_tc xs"
card_range_eval_sorter: fixes Rs :: "('a \<times> 'a) set set"   and e :: "'a sorter" assumes "finite Rs" shows "card ((\<lambda>R. eval_sorter R e) ` Rs) \<le> 2 ^ count_wc_queries Rs e"
simps_1: fixes \<mu> :: "'c" shows "Dom \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
path_count: fixes k :: "nat"   and v :: "'v"   and l :: "nat" assumes "\<And>v. v \<in> V \<Longrightarrow> card {n. adjacent v n} = k"   and "v \<in> V"   and "finite V"   and "0 < k" shows "card {ps. length ps = l \<and> adj_path v ps} = k ^ l"
S_Q_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "S_Q Sts = S_M S (Q_of_state (Liminf_state Sts))"
col_transpose: fixes i :: "nat"   and A :: "'a mat" assumes "i < dim_row A" shows "col A\<^sup>T i = row A i"
C7: shows "\<lfloor>\<lambda>w. \<forall>x. ((actualizedAt) x \<^bold>\<rightarrow> Necessary x \<^bold>\<and> Abstract x \<^bold>\<rightarrow> Explained x) w\<rfloor>"
ub1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
hext_def: fixes h :: "'heap"   and h' :: "'heap" shows "h \<unlhd> h' \<equiv> typeof_addr h \<subseteq>\<^sub>m typeof_addr h'"
fst_conv: fixes x1 :: "'c"   and x2 :: "'d" shows "fst (x1, x2) = x1"
on_curvep_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "cring R" shows "cring.on_curvep R a b = (\<lambda>(x, y, z). x \<in> carrier R \<and> y \<in> carrier R \<and> z \<in> carrier R \<and> (z \<noteq> \<zero>\<^bsub>R\<^esub> \<longrightarrow> y [^]\<^bsub>R\<^esub> 2 \<otimes>\<^bsub>R\<^esub> z = x [^]\<^bsub>R\<^esub> 3 \<oplus>\<^bsub>R\<^esub> a \<otimes>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> z [^]\<^bsub>R\<^esub> 2 \<oplus>\<^bsub>R\<^esub> b \<otimes>\<^bsub>R\<^esub> z [^]\<^bsub>R\<^esub> 3))"
first_countable_basis: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.first_countable_topology open" shows "\<exists>A. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
div_div_eq_right: 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 c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b"   and "dvd.dvd times b a" shows "divide a (divide b c) = times (divide a b) c"
nzero: shows "\<forall>x. d1 \<le> x \<and> x < c \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
pid_gstep_succ_def: fixes gc :: "('c, 'ls, 'gs) pid_global_config" shows "pid_gstep_succ gc \<equiv> let lcs = pid_global_config.processes gc in {0..<length lcs} \<bind> (\<lambda>pid. let lc = lcs ! pid; gs = pid_global_config.state gc; ls = local_config.state lc; cmd = command lc in {(a, cmd'). cstep cmd a cmd'} \<bind> (\<lambda>(a, cmd'). if en (ls, gs) a then {let (ls, gs) = ex (ls, gs) a; lc = \<lparr>command = cmd', state = ls\<rparr> in \<lparr>pid_global_config.processes = lcs[pid := lc], state = gs\<rparr>} else {}))"
nsub_RI_Vals: fixes vs :: "val list" shows "\<not> sub_RIs (map Val vs)"
universe_intro: fixes z :: "i" assumes "\<forall>w. P w z" shows "u = z"
simps_3: fixes a :: "atom" shows "clearQuantifiers (fm.Atom a) = simp_atom a"
sorted_isort_key: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "sorted (map f (isort_key f xs))"
ag_l_inv1: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> x \<plusminus>\<^bsub>A\<^esub> x = \<zero>\<^bsub>A\<^esub>"
reachable_Seq: fixes c1 :: "com"   and c2 :: "com" shows "reachable (c1;; c2) \<subseteq> (\<lambda>c1'. c1';; c2) ` reachable c1 \<union> reachable c2"
the_inv_into_into: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'e"   and B :: "'d set" assumes "inj_on f A"   and "x \<in> f ` A"   and "A \<subseteq> B" shows "the_inv_into A f x \<in> B"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c option" shows "map_option g (map_option f v) = map_option (g \<circ> f) v"
bigor: shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<or> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)) \<or> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
distr_C: fixes m :: "nat"   and m' :: "nat"   and x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "m \<le> m'"   and "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "C n m x = distr (C n m' x) (Pi\<^sub>M {0..<n + m} M) (\<lambda>x. restrict x {0..<n + m})"
transitive_acyclic_asymmetric: fixes x :: "'a" assumes "x \<sqinter> x \<le> x" shows "(x \<sqinter> top \<le> - top) = (x \<sqinter> id x = bot)"
elims: fixes x :: "'b heap"   and y :: "'b heap" assumes "del_min x = y" obtains "x = Empty" and "y = Empty"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "y = pass\<^sub>2 (pass\<^sub>1 hs)"
conc_assoc: fixes A :: "'b list set"   and B :: "'b list set"   and C :: "'b list set" shows "(A \<cdot> B) \<cdot> C = A \<cdot> B \<cdot> C"
iso_relative_homology_group_upper_hemisphere: fixes p :: "int"   and n :: "nat"   and k :: "nat" shows "hom_induced p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0} (nsphere n) {x. x k \<le> 0} id \<in> Group.iso (relative_homology_group p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0}) (relative_homology_group p (nsphere n) {x. x k \<le> 0})"
Longest_common_prefix_def: fixes L :: "'b list set" shows "Longest_common_prefix L = (ARG_MAX length ps. \<forall>xs\<in>L. prefix ps xs)"
summable_suminf_not_top: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ennreal (f i)) \<noteq> top" shows "summable f"
Body: fixes E :: "env"   and D :: "qtname"   and dt :: "loc \<Rightarrow> ty option"   and blk :: "stmt"   and T :: "ty" assumes "is_class (prg E) D"   and "E,dt\<Turnstile>blk\<Colon>\<surd>"   and "lcl E Result = Some T"   and "is_type (prg E) T" shows "E,dt\<Turnstile>Body D blk\<Colon>-T"
distinct_223: fixes x91 :: "Syntax.term"   and x92 :: "Syntax.term"   and x13 :: "Syntax.term" shows "App x91 x92 \<noteq> Roll x13"
io_reduction_from_atc_io_reduction: fixes M1 :: "('d, 'e, 'f) FSM"   and T :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set"   and M2 :: "('d, 'e, 'g) FSM" assumes "M1 \<preceq>\<lbrakk>T.\<Omega>\<rbrakk> M2"   and "finite T" shows "M1 \<preceq>\<lbrakk>T\<rbrakk> M2"
T: shows "T \<in> stream_space M \<rightarrow>\<^sub>M count_space UNIV"
vec_eq: fixes n :: "nat"   and v1 :: "'c list"   and v2 :: "'c list" assumes "vec n v1"   and "vec n v2" shows "(v1 = v2) = (\<forall>i<n. v1 ! i = v2 ! i)"
WellDefinedFn: fixes S :: "('a, 'b) Signature"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a"   and f :: "'b"   and C :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> e : B"   and "f \<in> Sig S : B \<rightarrow> C" shows "Sig S \<triangleright> Vx : A \<turnstile> f E@ e : C"
permute_fun_def: fixes p :: "perm"   and f :: "'b \<Rightarrow> 'c" shows "p \<bullet> f = (\<lambda>x. p \<bullet> f (- p \<bullet> x))"
\<delta>_def: shows "\<delta> \<equiv> var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set M)))"
join_cont1: fixes Y :: "nat \<Rightarrow> 'b"   and y :: "'b" assumes "chain Y"   and "\<And>i. compatible (Y i) y" shows "(\<Squnion>i. Y i) \<squnion> y = (\<Squnion>i. Y i \<squnion> y)"
zerozero_PooI: fixes x :: "nat \<times> nat" assumes "zerozero x" shows "pred.eval zerozero_Poo x"
distinct_93: fixes x7 :: "rat"   and x91 :: "rat"   and x92 :: "rat" shows "Pri_AgrK x7 \<noteq> Auth_Data x91 x92"
semilat_Def: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semilat (A, r, f) = (order r A \<and> closed A f \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
split_S: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "trace init t final" shows "trace (S t i) (drop i t) final"
\<psi>_img_set: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<psi> (b, a) ` Hom.set (b, a) = C.hom b a"
independent_basis_left_null_space: fixes A :: "(('a, 'c) vec, 'b) vec" shows "vec.independent (basis_left_null_space A)"
distinct_1: shows "Leq_Rel \<noteq> Lt_Rel"
lfp_def: fixes L :: "'b set"   and Glb :: "'b set \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "Complete_Lattice L Glb" shows "Complete_Lattice.lfp L Glb f = Glb {a \<in> L. f a \<le> a}"
spec2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "\<forall>x y. P x y" shows "P x y"
Cfg_def: shows "Cfg \<equiv> map_fun id (map_fun id (map_fun (map_fun id rep_cfg) abs_cfg)) (\<lambda>s D c. (s, Scheduler D (\<lambda>t. snd (c t))))"
delete_edge_is_path: fixes x :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and y :: "'v"   and a :: "'v"   and b :: "'w"   and c :: "'v" assumes "is_path_undir G x p y"   and "(a, b, c) \<notin> set p"   and "(c, b, a) \<notin> set p" shows "is_path_undir (delete_edge a b c G) x p y"
simps_1: fixes f :: "'b \<Rightarrow> 'c Heap" shows "for [] f = return ()"
eqn_inv1: fixes a :: "'a"   and x :: "'a" assumes "a \<in> carrier G"   and "x \<in> carrier G"   and "a = \<rho> x" shows "x = \<rho> a"
assms_8: shows "length slist_1 \<noteq> length slist_2"
convex_hull_exchange_Int: fixes S :: "'a set"   and a :: "'a"   and T :: "'a set"   and T' :: "'a set" assumes "\<not> affine_dependent S"   and "a \<in> convex hull S"   and "T \<subseteq> S"   and "T' \<subseteq> S" shows "convex hull insert a T \<inter> convex hull insert a T' = convex hull insert a (T \<inter> T')"
pop_Nil: fixes xs :: "int list"   and v :: "int" shows "pop xs [] v = []"
Ks_4: shows "\<not> set (take (length Us) Ks) \<subseteq> {\<zero>}"
openin_Int_open: fixes U :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "openin (top_of_set U) S"   and "open T" shows "openin (top_of_set U) (S \<inter> T)"
lookup_poly_eval_focus: fixes a :: "'d \<Rightarrow> 'c"   and X :: "'d set"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat" shows "lookup (poly_eval (\<lambda>x. monomial (a x) 0) (focus X p)) t = poly_eval a (lookup (focus (- X) p) t)"
has_derivative_subset: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "D f \<mapsto> f' at x within s"   and "t \<subseteq> s" shows "D f \<mapsto> f' at x within t"
of_nat_max_real: fixes a :: "nat"   and b :: "nat" shows "real (max a b) = max (real a) (real b)"