hcju/isabelleps huggingface dataset:
that: fixes as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'" shows "thesis"
synth_analz_idem_mono: fixes S :: "msg set"   and S' :: "msg set" assumes "S \<subseteq> synth (analz S')" shows "synth (analz S) \<subseteq> synth (analz S')"
fn_i: fixes i :: "nat" shows "flow_network \<Delta> (f_i i)"
monoid_axioms: shows "monoid (S \<rightarrow>\<^sub>E S) (compose S) (identity S)"
eventually_nonzero_ln_const_at_top: fixes b :: "real" assumes "0 < b" shows "eventually_nonzero at_top (\<lambda>x. ln (b * x))"
distinct_117: fixes x4 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Inverse x4 \<noteq> Power x151 x152"
dg_GRPH_components_1: fixes \<alpha> :: "V" shows "dg_GRPH \<alpha>\<lparr>Obj\<rparr> = ZFC_in_HOL.set {\<CC>. digraph \<alpha> \<CC>}"
block_Suc_1: fixes b :: "nat"   and i :: "nat"   and t :: "nat"   and k :: "nat" assumes "b = block i t"   and "k = gap i t"   and "change_on_0 b k" shows "block i (Suc t) = e_snoc b 0"
tangent_space_dim_eq: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and X :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c" assumes "c_manifold charts k"   and "X \<subseteq> c_manifold.tangent_space charts k p" shows "vector_space_on.dim (c_manifold.tangent_space charts k p) (*\<^sub>R) X = dim X"
prime_imp_power_coprime_nat: fixes p :: "nat"   and a :: "nat"   and m :: "nat" assumes "prime p"   and "\<not> p dvd a" shows "coprime a (p ^ m)"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.CC.arr f" shows "\<phi> f = D.null"
simps_5: fixes master :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and measuring :: "clock"   and slave :: "clock"   and i :: "nat" shows "\<lbrakk> master time-delayed by \<delta>\<tau> on measuring implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> i\<^esup> = {\<rho>. \<forall>n\<ge>i. hamlet (Rep_run \<rho> n master) \<longrightarrow> (let measured_time = time (Rep_run \<rho> n measuring) in \<forall>m\<ge>n. first_time \<rho> measuring m (measured_time + \<delta>\<tau>) \<longrightarrow> hamlet (Rep_run \<rho> m slave))}"
prefix_suffix: fixes x :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "x = prefix n x \<frown> suffix n x"
prems_1: shows "\<forall>a\<in>set L'. case a of (p, i) \<Rightarrow> p = 0"
rcos_imp_homeq: fixes a :: "'a"   and x :: "'a" assumes "a \<in> carrier R"   and "x \<in> a_kernel R S h +> a" shows "h x = h a"
set_def: fixes s' :: "'b" shows "local.set s' = State (\<lambda>_. ((), s'))"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "R.isomorphic f g" shows "R.isomorphic g f"
image_diff_subset: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'d set" shows "f ` A - f ` B \<subseteq> f ` (A - B)"
p2_ident: fixes p :: "int" assumes "mod_ring_locale TYPE('b) p" shows "int (CARD('b) - 2) = p - 2"
bind_distr: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and X :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and K :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M X"   and "N \<in> X \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}" shows "distr M X f \<bind> N = M \<bind> (\<lambda>x. N (f x))"
mset_neg_as_pos: fixes M :: "'a zmultiset" shows "zmset_of (mset_neg M) = zmset_of (mset_pos M) - M"
mkProc_cl_inv: fixes g :: "'a gState_scheme"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list"   and prog :: "'b program_scheme"   and sidx :: "nat"   and pidN :: "nat" assumes "cl_inv (g, p)" shows "cl_inv (mkProc g p name args (processes prog !! sidx) pidN)"
extend_in_C_Exists: fixes C :: "(nat, 'a) form set set"   and S :: "(nat, 'a) form set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" assumes "alt_consistency C"   and "infinite (- \<Union> (params ` S))"   and "extend S C f n \<union> {f n} \<in> C"   and "\<exists>p. f n = Exists p" shows "extend S C f (Suc n) \<in> C"
einterval_nonempty: fixes a :: "ereal"   and b :: "ereal" assumes "a < b" shows "\<exists>c. c \<in> einterval a b"
inf_add_K_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.join_semilattice plus less_eq less" shows "less_eq a (plus b a)"
vector_component_2: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec"   and i :: "'b" shows "(x + y) $ i = x $ i + y $ i"
bounded_subset: fixes T :: "'b set"   and S :: "'b set" assumes "bounded T"   and "S \<subseteq> T" shows "bounded S"
new_keys_not_analzd: fixes evs :: "event list"   and K :: "nat" assumes "evs \<in> kerbIV"   and "K \<in> symKeys"   and "Key K \<notin> used evs" shows "K \<notin> keysFor (analz (knows Spy evs))"
lookup_Mapping: fixes xs :: "('c \<times> 'd) list" shows "Mapping.lookup (AList_Mapping.Mapping xs) = map_of xs"
simps_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (ShuffleLR r s) = (\<oo> r \<and> \<oo> s)"
pair_sigma_finite_axioms: shows "pair_sigma_finite M2 M1"
card_size_filter_eq: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "card {a \<in> A. P a} = size (filter_mset P (mset_set A))"
zbeforey: shows "z < cs ! i in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))"
simple_function_If: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "simple_function M f"   and "simple_function M g"   and "{x \<in> space M. P x} \<in> sets M" shows "simple_function M (\<lambda>x. if P x then f x else g x)"
largest_contained_prefix_def: fixes a :: "'b word"   and r :: "'b wordinterval" shows "largest_contained_prefix a r = (let cs = map (PrefixMatch a) (pfxes TYPE('b)); cfs = find (\<lambda>s. valid_prefix s \<and> wordinterval_subset (prefix_to_wordinterval s) r) cs in cfs)"
nth_length_takeWhile: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "length (takeWhile P xs) < length xs" shows "\<not> P (xs ! length (takeWhile P xs))"
B2_def: shows "B2 = {x. jumpF f (at_left x) \<noteq> 0 \<and> b < x \<and> x \<le> c}"
fps_radical_def: fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a fps" shows "fps_radical r n a = Abs_fps (radical r n a)"
possible_steps_empty: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "(possible_steps e s r l i = {||}) = (\<forall>((origin, dest), t)\<in>fset e. origin \<noteq> s \<or> Label t \<noteq> l \<or> \<not> can_take_transition t i r)"
simps_6: fixes \<sigma> :: "nat \<Rightarrow> val"   and e :: "cexpr"   and t :: "pdf_type" shows "cexpr_sem \<sigma> (\<integral>\<^sub>c e \<partial>t) = RealVal (LINT x|stock_measure t. extract_real (cexpr_sem (case_nat x \<sigma>) e))"
fourtimes_lemma: fixes rs :: "nat" shows "rec_exec rec_fourtimes [rs] = 4 * rs"
iffs: fixes Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e"   and Pol' :: "('d \<times> 'd) set"   and Map' :: "'c \<Rightarrow> 'd"   and Dom' :: "'d"   and In' :: "'c list"   and Out' :: "'c list"   and more' :: "'e" shows "(\<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = \<lparr>Pol = Pol', Map = Map', Dom = Dom', In = In', Out = Out', \<dots> = more'\<rparr>) = (Pol = Pol' \<and> Map = Map' \<and> Dom = Dom' \<and> In = In' \<and> Out = Out' \<and> more = more')"
full_quicksort_ref_full_quicksort: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. R (h x) (h y) \<or> R (h y) (h x)" shows "(full_quicksort_ref R h, full_quicksort R h) \<in> \<langle>Id\<rangle>list_rel \<rightarrow>\<^sub>f \<langle>\<langle>Id\<rangle>list_rel\<rangle>nres_rel"
measurable_linorder_from_keys: shows "linorder_from_keys A \<in> Pi\<^sub>M A (\<lambda>_. borel) \<rightarrow>\<^sub>M count_space (Pow (B \<times> B))"
pr_dup_2: fixes a :: "'a" assumes "ide a" shows "\<pp>\<^sub>1[a, a] \<cdot> \<d>[a] = a"
hyps_3: shows "monic (\<Prod>c\<in>A. gcd f (h - [:c:]))"
case_prodD: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and b :: "'d" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
assms_1: shows "f piecewise_C1_differentiable_on i"
subcls1_subclsR_or_subclsS: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "P \<turnstile> C \<prec>\<^sub>R D \<or> P \<turnstile> C \<prec>\<^sub>S D"
simps: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat"   and e :: "nat" shows "f1 a b c d e = a + b + c + d + e"
tendsto_ine_in: fixes u :: "nat \<Rightarrow> 'c"   and N :: "'c quasinorm"   and x :: "'c" assumes "\<And>n. u n \<in> space\<^sub>N N"   and "x \<in> space\<^sub>N N" shows "tendsto_ine\<^sub>N N u x = tendsto_in\<^sub>N N u x"
sum_permutations_compose_right: fixes q :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and f :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c" assumes "q permutes S" shows "sum f {p. p permutes S} = (\<Sum>p | p permutes S. f (p \<circ> q))"
simps_3: fixes G :: "('a, 'b) cfg"   and N :: "'b"   and \<alpha> :: "('c + 'b) list" shows "\<oo>\<^sub>P G (Inr N # \<alpha>) = ([] |\<in>| cfg.prod G N \<and> \<oo>\<^sub>P G \<alpha>)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d \<times> ('e, 'f) stateful_strand_step) list" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
inverse_arrows_can_1: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "B.iso (B.can u t)"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "k (A i) = k (B i)"
retraction_cfcomp_strict: fixes f :: "'b \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'b" assumes "f oo g = ID" shows "f\<cdot>\<bottom> = \<bottom>"
le_iff_sup: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod x y = (sup_prod x y = y)"
d_delta_lnexp_cf5_nonneg: fixes x :: "real" assumes "0 < numer_cf5 x"   and "0 < numer_cf5 (- x)" shows "0 \<le> diff_delta_lnexp_cf5 x"
less_eq_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "xs \<le> ys \<equiv> prefix xs ys"
fdg: shows "(f12, f21, g') = splitFace g ram1 ram2 f []"
sum_mult_indicator: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "finite A" shows "(\<Sum>x\<in>A. f x * indicator B x) = sum f (A \<inter> B)"
size_add_mset: fixes a :: "'c"   and A :: "'c multiset" shows "size (add_mset a A) = Suc (size A)"
shift_Int: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and I1 :: "nat set"   and I2 :: "nat set" assumes "PL_Indis cval indis" shows "(PL_Indis.shift cl n I1 \<inter> PL_Indis.shift cl n I2 = {}) = (I1 \<inter> I2 = {})"
get_def: fixes h :: "heap"   and a :: "'b array" shows "Array.get h a = map from_nat (arrays h TYPEREP('b) (addr_of_array a))"
prems_3: shows "history_consistent \<theta> (last_prog p (x # xs)) ys"
p0p1: shows "a0xa1.has_as_binary_product p0 p1"
eq_FF_1: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and xs :: "'a"   and ys :: "'a" assumes "class.Eq_def eq" shows "(FF = eq\<cdot>xs\<cdot>ys) = (xs \<noteq> \<bottom> \<and> ys \<noteq> \<bottom> \<and> xs \<noteq> ys)"
pnop_axioms_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "pnop_axioms A n f \<equiv> n \<in>\<^sub>\<circ> \<omega> \<and> \<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> n \<and> \<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
applyLearnPreservedVariables: fixes state :: "State" shows "let state' = applyLearn state in getM state' = getM state \<and> getQ state' = getQ state \<and> getC state' = getC state \<and> getCl state' = getCl state \<and> getConflictFlag state' = getConflictFlag state \<and> getConflictClause state' = getConflictClause state \<and> getF state' = (if getC state = [opposite (getCl state)] then getF state else getF state @ [getC state])"
accepting_pair\<^sub>R_simp: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and F :: "('b \<times> 'c \<times> 'b) set"   and I :: "('b \<times> 'c \<times> 'b) set"   and w :: "nat \<Rightarrow> 'c" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 (F, I) w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> F = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {}"
cring0_is_zariski_open: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d" assumes "comm_ring R addition multiplication zero unit" shows "cring0.is_zariski_open = (\<lambda>U. U = {})"
Sup_union_distrib: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
divide_minus_right: 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.field divide inverse times one plus zero minus uminus" shows "divide a (uminus b) = uminus (divide a b)"
nn_transfer_operator_mono: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "AE x in M. f x \<le> g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_transfer_operator f x \<le> nn_transfer_operator g x"
length_set_of_appr: fixes xrs :: "real list"   and xs :: "(real \<times> real pdevs) list" assumes "xrs \<in> Joints xs" shows "length xrs = length xs"
cr_one: fixes t :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes " t \<longrightarrow>\<^sub>1\<^sup>* t1"   and " t \<longrightarrow>\<^sub>1 t2" shows "\<exists>t3.  t1 \<longrightarrow>\<^sub>1 t3 \<and>  t2 \<longrightarrow>\<^sub>1\<^sup>* t3"
length_append_singleton: fixes xs :: "'d list"   and x :: "'d" shows "length (xs @ [x]) = Suc (length xs)"
nat_le_numeral_power_cancel_iff: fixes a :: "int"   and x :: "num"   and n :: "nat" shows "(nat a \<le> numeral x ^ n) = (a \<le> numeral x ^ n)"
ltake_plus_conv_lappend: fixes n :: "enat"   and m :: "enat"   and xs :: "'c llist" shows "ltake (n + m) xs = lappend (ltake n xs) (ltake m (ldrop n xs))"
Minimal_Solutions_def: fixes a :: "nat list"   and b :: "nat list" shows "Linear_Diophantine_Equations.Minimal_Solutions a b = {(x, y). (x, y) \<in> Linear_Diophantine_Equations.Solutions a b \<and> nonzero x \<and> \<not> (\<exists>(u, v)\<in>Linear_Diophantine_Equations.Solutions a b. nonzero u \<and> u @ v <\<^sub>v x @ y)}"
uv_prop_def: fixes X :: "'b program set" shows "uv_prop X \<equiv> \<bottom> \<in> X \<and> (\<forall>F G. F ok G \<longrightarrow> F \<in> X \<and> G \<in> X \<longrightarrow> F \<squnion> G \<in> X)"
assms_2: shows "b \<longlonglongrightarrow> a"
Suc_5: shows "Span K (Us' @ Us) \<subseteq> Span K Vs"
empty_offending_contra: fixes m :: "'b SecurityInvariant_configured"   and F :: "('b \<times> 'b) set"   and G :: "'b graph" assumes "configured_SecurityInvariant m"   and "F \<in> c_offending_flows m G"   and "F = {}" shows "False"
assms_4: shows "has_greatest_postfixpoint (u \<circ> h)"
Skip: fixes Env :: "env"   and B :: "lname set" shows "Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
m1_inv3r_initI: fixes x :: "'a m1_state_scheme" assumes "\<And>A B Rb Kab. \<lbrakk>B \<notin> bad; A \<notin> bad; Kab \<notin> Domain (m1_state.leak x); runs x Rb = Some (Resp, [A, B], [aKey Kab, END])\<rbrakk> \<Longrightarrow> \<exists>Ra nla. runs x Ra = Some (Init, [A, B], aKey Kab # aNon (Rb $ nb) # nla)" shows "x \<in> m1_inv3r_init"
CEReq_def: shows "CEReq = (Geq (Differential (Var vid1)) (Differential (Const 0)) \<leftrightarrow> Geq ($' vid1) (Const 0))"
Cod_TensorDiag: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u" shows "Cod (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u"
map_simp: fixes f :: "'a" assumes "Adj.C.arr f" shows "Adj.C.map f = f"
transf_res_is_rev: fixes ns :: "nat list"   and n :: "nat" assumes "Sorted_Less.sorted (rev ns)"   and "hd ns < n" shows "Sorted_Less.sorted (rev (transf n ns))"
uVW: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "u = (#v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.v' q_ty b u v w)) * (#w + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.w' q_ty a b u v w))"
poly_shift_def: fixes n :: "nat"   and p :: "'b poly" shows "poly_shift n p = Abs_poly (\<lambda>i. coeff p (i + n))"
listE_set: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set" assumes "xs \<in> list n A" shows "set xs \<subseteq> A"
FG_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltln" shows "suffix i w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)"
process_rule_2_futures: fixes ys :: "'e list"   and ys' :: "'e list"   and Y :: "'e set"   and P :: "'e process"   and xs :: "'e list" assumes "(ys @ ys', Y) \<in> futures P xs" shows "(ys, {}) \<in> futures P xs"
sp_fundefs_def: fixes F :: "'f \<Rightarrow> ('g, 'h, 'f, 'op, 'opinl, 'opubx, 'i, 'j) Ubx.instr fundef option" shows "sp_fundefs F \<equiv> \<forall>f fd. F f = Some fd \<longrightarrow> sp_fundef F fd (body fd) = Ok [None]"
prec_first_IPC_action: fixes aseq :: "int_point_t list"   and s :: "rstate_t"   and d :: "thread_id_t" assumes "is_SK_IPC aseq" shows "rprecondition s d (hd aseq)"
polymap_pCons: fixes a :: "'f"   and p :: "'f poly" shows "polymap (pCons a p) = (\<lambda>v. (a \<cdot>\<cdot> If (v \<in> V) (id v)) (0::'v)) + (T \<circ> polymap p)"
opp_half_apartment_def: shows "Y \<equiv> {x \<in> X. \<exists>C\<in>\<C> - f \<turnstile> \<C>. order.greater_eq C x}"
thetaParZOT_sym: shows "sym thetaParZOT"
dec_def: fixes \<eta> :: "nat" shows "CIPHER.dec \<eta> \<equiv> CNV (stateless_callee (\<lambda>_. Pause (Inr ()) (\<lambda>c'. case c' of Inl a \<Rightarrow> Done None | Inr None \<Rightarrow> Done None | Inr (Some c) \<Rightarrow> Pause (Inl ()) Done \<bind> (\<lambda>k. Done (dec \<eta> (projl k) c))))) ()"
map_poly_comm_monoid_add_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "map_poly_comm_monoid_add_hom hom \<equiv> comm_monoid_add_hom hom"
asIMOD_igWlsDisj: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsNE SEM"   and "sWlsDisj SEM" shows "igWlsDisj (asIMOD SEM)"
length_upt: fixes i :: "nat"   and j :: "nat" shows "|[i..<j]| = j - i"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'c \<Rightarrow> 'c tree \<Rightarrow> 'b"   and tree :: "'c tree" shows "(case tree of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = (if tree = \<langle>\<rangle> then f1 else f2 (un_Node1 tree) (value tree) (un_Node3 tree))"
HK_3: shows "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso (subgroup_generated (homology_group p (subtopology X S)) K) (homology_group p X)"
red: fixes P :: "'a set \<Rightarrow> bool"   and T :: "'a set"   and F :: "'a set \<Rightarrow> 'a set" assumes "\<And>a. P a \<Longrightarrow> T \<subseteq> S \<inter> \<Inter> {F X |X. P X}" shows "T \<subseteq> \<Inter> {S \<inter> F X |X. P X}"
bex_Infinitesimal_iff2: fixes x :: "'b star"   and z :: "'b star" shows "(\<exists>y\<in>Infinitesimal. x = z + y) = (x \<approx> z)"
JN_Join_distrib: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program"   and G :: "'d \<Rightarrow> 'c program" shows "(\<Squnion>i\<in>I. F i \<squnion> G i) = JOIN I F \<squnion> JOIN I G"
sees_method_decl_above: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
OF_priority_match_def: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and flow_entries :: "('c, 'b) flow_entry_match list"   and packet :: "'d" shows "OF_priority_match \<gamma> flow_entries packet \<equiv> let m = filter (\<lambda>f. \<gamma> (ofe_fields f) packet) flow_entries; m' = filter (\<lambda>f. \<forall>fo\<in>set m. ofe_prio fo \<le> ofe_prio f) m in case m' of [] \<Rightarrow> NoAction | [s] \<Rightarrow> Action (ofe_action s) | s # aa # lista \<Rightarrow> Undefined"
comp_leq: fixes x :: "'a::refinement_lattice" shows "((F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)) (x::'a::refinement_lattice) \<sqsubseteq> ((H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> F) x"
measurable_on_spike_set: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a set" assumes "f measurable_on S"   and "negligible (sym_diff S T)" shows "f measurable_on T"
powerset_lift_def: fixes f :: "'b \<rightarrow> 'c set" shows "\<^ps> f = (\<Lambda> S. \<Union> (Rep_cfun f ` S))"
rel_msetI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and A :: "'c multiset"   and B :: "'d multiset" assumes "list_all2 R xs ys"   and "mset xs = A"   and "mset ys = B" shows "rel_mset R A B"
add_cancel_left_left: fixes b :: "'b"   and a :: "'b" shows "(b + a = a) = (b = (0::'b))"
size_multiset_overloaded_eq: fixes x :: "'k multiset" shows "size x = sum (count x) (set_mset x)"
times_right_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c"   and y :: "'c" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero x"   and "less_eq z y" shows "less_eq (times x z) (times x y)"
prod_vwb_lens: fixes X :: "'g \<Longrightarrow> 'h"   and Y :: "'i \<Longrightarrow> 'j" assumes "vwb_lens X"   and "vwb_lens Y" shows "vwb_lens (X \<times>\<^sub>L Y)"
connectedD_interval: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.linorder_topology less_eq less open"   and "topological_space.connected open U"   and "x \<in> U"   and "y \<in> U"   and "less_eq x z"   and "less_eq z y" shows "z \<in> U"
rtrancl_list_impl_def: shows "rtrancl_list_impl = rtrancl_impl (\<lambda>r as. remdups (map snd (filter (\<lambda>(a, b). a \<in> set as) r))) (\<lambda>xs ys. filter (\<lambda>x. x \<notin> set ys) xs @ ys) (\<lambda>x xs. x \<in> set xs) []"