hcju/isabelleps huggingface dataset:
Rep_boffa_mon_cases: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> {f. True}" obtains x :: "('b, 'c) boffa_mon" where "y = Rep_boffa_mon x"
smult_one: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<one> \<odot>\<^bsub>M\<^esub> x = x"
rem_implicit_pres_ops_astG: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.astG (rem_implicit_pres_ops prob) = ast_problem.astG prob"
cop_F_closed_inv: fixes ds :: "'b set" shows "invariant ds cop_F_closed_inv"
assms: fixes a :: "'a"   and b\<^sub>1 :: "'b"   and b\<^sub>2 :: "'b" assumes "(a, b\<^sub>1) \<in> M"   and "(a, b\<^sub>2) \<in> M" shows "b\<^sub>1 = b\<^sub>2"
measure_measure_spmf_conv_measure_pmf: fixes p :: "'b spmf"   and A :: "'b set" shows "Sigma_Algebra.measure (measure_spmf p) A = measure_pmf.prob p (Some ` A)"
wf_weaken': fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T"
inter_restrict: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c set" assumes "finite A" shows "sum g (A \<inter> B) = (\<Sum>x\<in>A. if x \<in> B then g x else (0::'d))"
hd_replicate: fixes n :: "nat"   and x :: "'g" assumes "n \<noteq> 0" shows "hd (replicate n x) = x"
sum_leaves: fixes t :: "int tree"   and i :: "int"   and j :: "int"   and f :: "int \<Rightarrow> nat" assumes "inorder t = [i..j]"   and "i \<le> j + 1" shows "(\<Sum>x\<in>leaves i t. f (fst x)) = sum f {i..j + 1}"
nth_replicate: fixes i :: "nat"   and n :: "nat"   and x :: "'a" assumes "i \<sqsubset> n" shows "replicate n x ! i = x"
rcset_cases: fixes y :: "'c set" assumes "y \<in> {A. countable A}" obtains x :: "'c cset" where "y = rcset x"
elims: fixes x :: "role_t"   and y :: "role_t" assumes "role_comp x = y" obtains "x = Init" and "y = Resp"   | "x = Resp" and "y = Init"
sign_simps_17: fixes aa :: "'d"   and ba :: "'d"   and ca :: "'d" shows "(aa + ba) * ca = aa * ca + ba * ca"
antisymmetric_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "antisymmetric A r = antisymmetric A r'"
range: shows "h ` carrier g1 \<subseteq> carrier g2"
netmask_def: fixes I :: "nat set"   and s :: "(nat \<Rightarrow> 'd) \<times> 'e" shows "netmask I s \<equiv> (\<lambda>i. if i \<in> I then Some (fst s i) else None, snd s)"
addition_set_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" shows "addition_set f A = \<Union> {add_set f A n |n. 0 \<le> n}"
add_int_pow_monom: fixes a :: "'a"   and n :: "int"   and k :: "nat" assumes "a \<in> carrier R" shows "[n] \<cdot>\<^bsub>P\<^esub> monom P a k = monom P ([n] \<cdot> a) k"
leg0_simps_4: shows "D.dom (F f) = F f"
I_def: shows "I = \<integral>\<^sup>+ u. ennreal (indicat_real {0<..} u * u powr (b - 1) / (1 + u) powr (a + b)) \<partial>lborel"
simps_2: fixes x :: "'g"   and xs :: "'g list" shows "rev (x # xs) = rev xs @ [x]"
ord_spmf_bindI: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'a spmf"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "ord_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> ord_spmf P (f x) (g y)" shows "ord_spmf P (bind_spmf p f) (bind_spmf q g)"
spectral_radius_def: fixes A :: "complex mat" shows "spectral_radius A = Max (cmod ` spectrum A)"
get_lens_plus: fixes x :: "'e \<Longrightarrow> 'g"   and y :: "'f \<Longrightarrow> 'g"   and s :: "'g" shows "get\<^bsub>x +\<^sub>L y\<^esub> s = (get\<^bsub>x\<^esub> s, get\<^bsub>y\<^esub> s)"
B_def: shows "B \<equiv> \<lambda>p s. complex_of_real (ln (real p)) / of_nat p * A p s"
order_trans: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.preorder (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (y::'b::type) (z::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (z::'b::type)"
f: shows "(f \<longlongrightarrow> \<infinity>) F"
kgt: assumes "z < k" shows "0 < a * z\<^sup>2 + b * z + c"
Un_insert_left: fixes a :: "'e"   and B :: "'e set"   and C :: "'e set" shows "insert a B \<union> C = insert a (B \<union> C)"
arrI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "B.arr f"
dickson_grading_minus: fixes d :: "'c \<Rightarrow> nat"   and s :: "'c"   and t :: "'c" assumes "dickson_grading d"   and "s adds t" shows "d (t - s) \<le> d t"
valid: shows "length (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat) = length qs"
word_power_nonzero: fixes x :: "'b word"   and n :: "nat" assumes "x < 2 ^ (LENGTH('b) - n)"   and "n < LENGTH('b)"   and "x \<noteq> 0" shows "x * 2 ^ n \<noteq> 0"
Cond: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e0 :: "expr"   and P' :: "'b assn"   and e1 :: "expr"   and e2 :: "expr"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} e0-\<succ> {P'}"   and "\<forall>b. G,A\<turnstile>{P'\<leftarrow>=b} (if b then e1 else e2)-\<succ> {Q}" shows "G,A\<turnstile>{Normal P} e0 ? e1 : e2-\<succ> {Q}"
results_gpv_def: fixes \<Gamma> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv" shows "results_gpv \<Gamma> gpv \<equiv> {x. Generative_Probabilistic_Value.resultsp_gpv \<Gamma> x gpv}"
simps_2: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and s\<^sub>1 :: "'f net_state"   and s\<^sub>2 :: "'f net_state" shows "netgmap sr (SubnetS s\<^sub>1 s\<^sub>2) = (let (\<sigma>\<^sub>1, ss) = netgmap sr s\<^sub>1; (\<sigma>\<^sub>2, tt) = netgmap sr s\<^sub>2 in (\<sigma>\<^sub>1 ++ \<sigma>\<^sub>2, SubnetS ss tt))"
trimmed_pos_uminus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "multiseries.trimmed_neg trimmed dominant_term F" shows "multiseries.trimmed_pos trimmed dominant_term (uminus F)"
w: shows "\<forall>w. P w (\<midarrow> x) = (\<not> O w x)"
S4WriteInner: fixes p :: "PrIds"   and l :: "Locs"   and v :: "Vals"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> WriteInner rmCh mm ires p l v \<and> $S4 rmhist p \<and> unchanged (e p, c p, r p) \<and> HNext rmhist p \<longrightarrow> S4 rmhist p$ \<and> unchanged (rmhist!p)"
der_g: fixes x :: "(real, 'm) vec" assumes "x \<in> \<Union> (range F)" shows "(g has_derivative g' x) (at x within \<Union> (range F))"
eventually_at_right_less: fixes x :: "'a" shows "eventually ((<) x) (at_right x)"
wcode_fourtimes_case_le_def: shows "wcode_fourtimes_case_le \<equiv> inv_image Recursive.lex_pair wcode_fourtimes_case_measure"
SUP_mult_right_ennreal: fixes f :: "'c \<Rightarrow> ennreal"   and I :: "'c set"   and c :: "ennreal" shows "Sup (f ` I) * c = (SUP i\<in>I. f i * c)"
length_insort: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and f :: "'l \<Rightarrow> 'k"   and x :: "'l"   and xs :: "'l list" assumes "class.linorder less_eq less" shows "length (linorder.insort_key less_eq f x xs) = Suc (length xs)"
order_iff: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (a = max a b)"
correct_state_heap_change: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and frs :: "'addr frame list"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and xcp' :: "'addr option"   and h' :: "'heap"   and frs' :: "'addr frame list"   and t'' :: "'thread_id"   and xcp'' :: "'addr option"   and frs'' :: "'addr frame list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P,t \<turnstile> Normal (xcp, h, frs) -ta-jvmd\<rightarrow> Normal (xcp', h', frs')"   and "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>"   and "\<Phi> \<turnstile> t'':(xcp'', h, frs'') \<surd>" shows "\<Phi> \<turnstile> t'':(xcp'', h', frs'') \<surd>"
tendsto_at_top_withinI: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "s \<noteq> {}"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>x0\<in>s. \<forall>x\<in>{x0..} \<inter> s. dist (f x) l < e" shows "(f \<longlongrightarrow> l) (at_top_within s)"
wf_sub_rel_of_def: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "wf_sub_rel_of g subs \<equiv> sub_rel_of g subs \<and> wf_sub_rel subs"
RelS_Eg1Eg2_def: assumes "sifum_refinement_eg1_eg2" shows "RelS_Eg1Eg2 \<equiv> {(((xa, x), xb), (xd, xc), xe). RelS_Eg1Eg2p xa x xb xd xc xe}"
enat_minus_mono1: fixes x :: "enat"   and y :: "enat"   and m :: "enat" assumes "x \<le> y" shows "x - m \<le> y - m"
not_dbm_lt_lt_impl: fixes a :: "'b"   and b :: "'b" assumes "\<not> Lt a \<prec> Lt b" shows "b \<le> a"
simps: fixes h :: "'b \<Rightarrow> int"   and t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" shows "wrm h (T t1 x t2) = (if t2 = Tip then t1 else T t1 x (wrm h t2))"
supercls_lst_app: fixes P :: "'a prog"   and C :: "char list"   and Cs :: "char list list"   and C' :: "char list" assumes "supercls_lst P (C # Cs)"   and "P \<turnstile> C \<preceq>\<^sup>* C'" shows "supercls_lst P (C' # C # Cs)"
Bs_2: shows "B2 \<in> carrier_mat n1 n2"
restrict_PiE: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and S :: "'d \<Rightarrow> 'e set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
eWlsInp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'c option" shows "eWlsInp MOD delta (OKI inp) = igWlsInp MOD delta inp"
calculus_with_standard_redundancy_axioms: shows "calculus_with_standard_redundancy Inf {{#}} (\<TTurnstile>e)"
Rep_pdata: fixes x :: "'b pdata" shows "Rep_pdata x \<in> pdata"
integral_le_const: fixes M :: "'a pmf"   and f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable (measure_pmf M) f"   and "AE x in measure_pmf M. f x \<le> c" shows "measure_pmf.expectation M f \<le> c"
inject: fixes x1 :: "nat"   and y1 :: "nat" shows "(ONat x1 = ONat y1) = (x1 = y1)"
uniform_distributed_measure: fixes X :: "'a \<Rightarrow> real"   and a :: "real"   and b :: "real"   and t :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (indicat_real {a..b} x / content {a..b}))"   and "a \<le> t"   and "t \<le> b" shows "prob {x \<in> space M. X x \<le> t} = (t - a) / (b - a)"
VInter_vsubset: fixes A :: "V"   and B :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> x \<subseteq>\<^sub>\<circ> B"   and "A \<noteq> []\<^sub>\<circ>" shows "\<Inter>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> B"
simps_2: fixes x :: "('a, 'b) aformula" shows "nFOR [x] = x"
subset_of_def: fixes A :: "'c set"   and F :: "('c \<Rightarrow> 'b) set" shows "subset_of A F = univ (\<lambda>f. f ` A) F"
isomorphism_def: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set"   and f :: "nat \<Rightarrow> nat" shows "isomorphism G\<^sub>1 G\<^sub>2 f \<equiv> bij_betw f (uverts G\<^sub>1) (uverts G\<^sub>2) \<and> G\<^sub>2 = map_ugraph f G\<^sub>1"
dist_norm: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and x :: "'d"   and y :: "'d" assumes "class.dist_norm minus dist norm" shows "dist x y = norm (minus x y)"
modify_eval_def: fixes f :: "real \<Rightarrow> real"   and F :: "'a ms" shows "modify_eval f F = (case F of MS xs x \<Rightarrow> MS xs f)"
dvd_mod_iff: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c dvd b" shows "(c dvd a mod b) = (c dvd a)"
heap_is_wellformed_children_disc_nodes_different: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "set children \<inter> set disc_nodes = {}"
FCD_def: fixes \<Gamma> :: "'a Gamma"   and \<V> :: "'a V_rec"   and Tr :: "'a list set" shows "FCD \<Gamma> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<beta> @ [c, v] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>' \<delta>'. set \<delta>' \<subseteq> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub> \<and> \<beta> @ \<delta>' @ [v] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
fls_base_factor_to_fps_X_inv: shows "fls_base_factor_to_fps fls_X_inv = 1"
diagvector_def: fixes x :: "'b \<Rightarrow> 'a" shows "diagvector x = (\<chi>i j. if i = j then x i else (0::'a))"
eqvtRelE: fixes Rel :: "('a \<times> 'a) set"   and perm :: "(name \<times> name) list"   and P :: "'a"   and Q :: "'a" assumes "eqvt Rel"   and "(perm \<bullet> P, perm \<bullet> Q) \<in> Rel" shows "(P, Q) \<in> Rel"
dgn: fixes n :: "nat" shows "g n \<in> carrier_mat d d"
nth_default_Nil: fixes dflt :: "'c"   and n :: "nat" shows "nth_default dflt [] n = dflt"
assms_5: shows "\<not> is_Prim\<^sub>0 u"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H"
UNlift01_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n" shows "UNlift01 cl dl P F ` UNpart1 cl dl P = UNpart1 dl cl (\<lambda>n. F n ` P n)"
ode_deriv: fixes i :: "'c" assumes "i \<in> ODE_vars I ODE" shows "((\<lambda>t. sol t $ i) has_derivative (\<lambda>v'. v' *\<^sub>R ODE_sem I ODE (sol s) $ i)) (at s within {0..t})"
finite_fun_UNIVD2: assumes "finite UNIV" shows "finite UNIV"
intros_1: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and t :: "Term.term" shows "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<rightarrow> (undefined3 t, (Some xc, s))"
coalesce_less_eq: fixes a :: "real"   and c :: "real"   and b :: "real" assumes "a \<le> c"   and "c \<le> b" shows "lift_option (+) (if f integrable_on {a..c} then Some (integral {a..c} f) else None) (if f integrable_on {c..b} then Some (integral {c..b} f) else None) = (if f integrable_on {a..b} then Some (integral {a..b} f) else None)"
distinct_3: fixes x31 :: "com"   and x32 :: "com" shows "com.SKIP \<noteq> x31;; x32"
slice_intra_kind_in_slice: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "sourcenode a \<in> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)"   and "intra_kind (kind a)" shows "SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a = kind a"
assms_8: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
simps_9: fixes i :: "int"   and c :: "int"   and e :: "MIR.num" shows "iszlfm (Dvd i (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> 0 < i \<and> numbound0 e \<and> isint e bs)"
trace_def: fixes sig :: "'i signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
mplus_assoc: fixes x :: "'a match"   and y :: "'a match"   and z :: "'a match" shows "(x +++ y) +++ z = x +++ y +++ z"
run_one_step_basic_get_global_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x17 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Get_global x17) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
alpha: fixes a :: "name"   and x :: "'a"   and b :: "name"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
pw_eqI: fixes S :: "'c nres"   and S' :: "'c nres" assumes "nofail S = nofail S'"   and "\<And>x. inres S x = inres S' x" shows "S = S'"
lrev_simps_2: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "lrev (a ## r) = lrev r @@ a ## LNil"
is_ground_atm_def: fixes A :: "('c, 'd) Term.term" shows "is_ground_atm A = (\<forall>\<sigma>. A = subst_atm_abbrev A \<sigma>)"
fresh_finite_set_at_base: fixes S :: "'b set"   and a :: "'b" assumes "finite S" shows "atom a \<sharp> S = (a \<notin> S)"
sa: shows "Vx : A \<turnstile> sub d in d'\<equiv>sub e in d' : C \<in> Axioms.Theory T"
trans_elim: fixes s :: "('a, 'b, 'c) SLin_state"   and t :: "('a, 'b, 'c) SLin_state"   and a :: "('a, 'b, 'c, 'd) SLin_action"   and s' :: "('a, 'b, 'c) SLin_state"   and t' :: "('a, 'b, 'c) SLin_state" assumes "(s, t) \<midarrow>a\<midarrow>composition\<longrightarrow> (s', t')" obtains i :: "nat" and p :: "'b" and c :: "'c" where "Inv p__ c__ s s' \<and> t = t'" and "i__ < id1" and "a = Invoke i__ p__ c__"   | i :: "nat" and p :: "'b" and c :: "'c" where "Inv pa__ ca__ t t' \<and> s = s'" and "id1 \<le> ia__ \<and> ia__ < id2" and "a = Invoke ia__ pa__ ca__"   | p :: "'b" and c :: "'c" and av :: "'a" where "Abort pb__ cb__ av__ s s' \<and> Init pb__ cb__ av__ t t'" and "a = Switch id1 pb__ cb__ av__"   | p :: "'b" and c :: "'c" and av :: "'a" where "s = s' \<and> Abort pc__ c av t t'" and "a = Switch id2 pc__ c av"   | i :: "nat" and p :: "'b" and ou :: "'d" where "Resp pd__ ou__ s s' \<and> t = t'" and "ib__ < id1" and "a = Response ib__ pd__ ou__"   | i :: "nat" and p :: "'b" and ou :: "'d" where "Resp p ou t t' \<and> s = s'" and "id1 \<le> i \<and> i < id2" and "a = Response i p ou"   | "Lin s s' \<and> t = t'" and "a = Linearize 0"   | "Lin t t' \<and> s = s'" and "a = Linearize id1"   | "Reco t t' \<and> s = s'" and "a = Recover id1"
qiB: fixes b :: "'b" assumes "b \<in> B" shows "q (inv q b) = b"
CFG_SSA_Transformed_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and phis :: "'a \<Rightarrow> 'b \<times> 'e \<Rightarrow> 'e list option"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd" shows "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var \<equiv> CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses \<and> CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<and> CFG_SSA_Transformed_axioms \<alpha>n invar inEdges' oldDefs oldUses defs uses phis var"
Top_less_ivl: fixes x :: "ivl" assumes "\<top> \<sqsubseteq> x" shows "m_ivl x = 0"
blinfun_apply_cases: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> {f. bounded_linear f}" obtains x :: "'b \<Rightarrow>\<^sub>L 'c" where "y = blinfun_apply x"
even_iff_mod_2_eq_zero: fixes a :: "'a" shows "even a = (a mod (2::'a) = (0::'a))"
wset_thread_okI: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ws :: "'a \<Rightarrow> 'd wait_set_status option" assumes "\<And>t. ts t = None \<Longrightarrow> ws t = None" shows "wset_thread_ok ws ts"
times_empty_iff: fixes A :: "V"   and B :: "V \<Rightarrow> V" shows "(VSigma A B = 0) = (A = 0 \<or> (\<forall>x\<in>elts A. B x = 0))"
s_Let1: fixes x :: "Syntax.var"   and e\<^sub>1 :: "Syntax.term"   and e\<^sub>1' :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and \<pi>' :: "Syntax.term list"   and m :: "mode"   and e\<^sub>2 :: "Syntax.term" assumes "atom x \<sharp> (e\<^sub>1, e\<^sub>1', \<pi>, \<pi>')"   and "\<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater>" shows "\<lless>\<pi>, Syntax.Let e\<^sub>1 x e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', Syntax.Let e\<^sub>1' x e\<^sub>2\<ggreater>"
dec_string_of_word0_def: shows "dec_string_of_word0 \<equiv> string_of_word True 10 0"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "S.awalk u (e # es) w = (e \<in> parcs (mk_symmetric G) \<and> u = fst e \<and> S.awalk (snd e) es w)"
igbg_impl_rel_eext_internal_def: fixes Rm :: "('b \<times> 'd) set"   and Rv :: "('a \<times> 'c) set" shows "igbg_impl_rel_eext Rm Rv \<equiv> \<langle>Rm, Rv \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel\<rangle>gen_igbg_impl_rel_eext"