hcju/isabelleps huggingface dataset:
rcp_specD3: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "rcp_spec rcp"   and "dickson_grading d" shows "dgrad_p_set_le d (fst ` set (fst (rcp gs bs ps sps data))) (args_to_set (gs, bs, sps))"
latestActNxt: fixes n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<langle>nid \<rightarrow> t\<rangle>\<^bsub>\<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> = \<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>"
add_pow_ldistr_int: fixes a :: "'a"   and b :: "'a"   and k :: "int" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "[k] \<cdot> a \<otimes> b = [k] \<cdot> (a \<otimes> b)"
d_approx_empty: fixes d :: "'a \<aproc> set" shows "{} \<lessapprox> d"
finite_FactGroup: fixes H :: "'a set" assumes "H \<lhd> G" shows "finite_group (G Mod H)"
simps_5: fixes v :: "poly"   and va :: "poly" shows "isnpolyh (Sub v va) = (\<lambda>k. False)"
Compl_eq_empty_iff: fixes A :: "'a set" shows "(- A = {}) = (A = UNIV)"
taylor_expansion_cf_closed: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a"   and i :: "nat" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "taylor_expansion R a p i \<in> carrier R"
simps_5: fixes V :: "char list" shows "fv (Var V) = {V}"
r_def: shows "r = (\<lambda>n. sum_upto (\<lambda>d. mangoldt d * (n / real d - real_of_int \<lfloor>n / real d\<rfloor>)) n)"
stake_smap: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and s :: "'e stream" shows "stake n (smap f s) = map f (stake n s)"
A: shows "acomp a\<^sub>1 @ acomp a\<^sub>2 @ [ADD] \<Turnstile> cfs\<box>"
beta_reduces_composition: fixes A' :: "'a trm"   and A'' :: "'a trm"   and A :: "'a trm" assumes "A' \<rightarrow>\<beta>\<^sup>* A''"   and "A \<rightarrow>\<beta>\<^sup>* A'" shows "A \<rightarrow>\<beta>\<^sup>* A''"
c_nc_comp1: shows "1\<^sub>\<pi> + nc = U"
IH: fixes gpv :: "('a, 'out, 'in) gpv"   and s :: "'b" assumes "\<I> \<turnstile>g gpv \<surd>" shows "exec_gpv' (local.restrict_gpv gpv) s = exec_gpv' gpv s"
rel_defs_1: fixes M :: "('b \<Rightarrow> 'b ref) set" shows "relS M = \<Union> (rel ` M)"
fin_ch_betw: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.fin_long_chain betw f a b c X" shows "betw a b c"
uminus_uminus_mat: fixes A :: "'c mat" shows "- (- A) = A"
ovalidNF_pre_imp: fixes P' :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>s. P' s \<Longrightarrow> P s"   and "ovalidNF P f Q" shows "ovalidNF P' f Q"
simps_1: fixes l :: "'b llist" shows "l !! 0 = (case l of LNil \<Rightarrow> None | x ## xs \<Rightarrow> Some x)"
bool_to_ternary_pullup2: fixes X1 :: "bool"   and X2 :: "bool" shows "eval_ternary_And (bool_to_ternary X1) (bool_to_ternary X2) = bool_to_ternary (X1 \<and> X2)"
affine_hull_halfspace_le: fixes a :: "'a"   and r :: "real" shows "affine hull {x. a \<bullet> x \<le> r} = (if a = (0::'a) \<and> r < 0 then {} else UNIV)"
f2: shows "Matrix.mat (2 ^ Suc n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ Suc n)"
vars_\<C>_Un: fixes X :: "'Var set"   and Y :: "'Var set" shows "vars_\<C> (X \<union> Y) = vars_\<C> X \<union> vars_\<C> Y"
abs_eq: fixes x :: "rat \<times> rat \<times> nat" assumes "eq_onp (\<lambda>uu_. \<exists>p q b. uu_ = (p, q, b) \<and> (q = 0 \<or> b \<in> sqrt_irrat)) x x" shows "real_of (Abs_mini_alg x) = (case x of (p, q, b) \<Rightarrow> real_of_rat p + real_of_rat q * sqrt (real b))"
not_coin: shows "\<nexists>i. e = (STR ''coin'', [i])"
index_transpose_mat_1: fixes i :: "nat"   and A :: "'a Matrix.mat"   and j :: "nat" assumes "i < dim_col A"   and "j < dim_row A" shows "A\<^sup>t $$ (i, j) = A $$ (j, i)"
leq_qe_inf_helper: fixes leq :: "(real \<times> real \<times> real) list"   and q :: "real" assumes "\<forall>(d, e, f)\<in>set leq. \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0" shows "\<exists>y'>q. \<forall>(d, e, f)\<in>set leq. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
eq_upair_def: shows "eq_upair = (\<lambda>(a, b) (c, d). a = c \<and> b = d \<or> a = d \<and> b = c)"
bounded_pos_less: fixes S :: "'b set" shows "bounded S = (\<exists>b>0. \<forall>x\<in>S. norm x < b)"
similar'_typed_2: fixes n :: "nat"   and f :: "Value \<rightarrow> Value"   and b :: "bool discr" shows "\<not> Fn\<cdot>f \<triangleleft>\<triangleright>\<^bsub>n\<^esub> CB\<cdot>b"
fbd_disj: fixes f :: "'c \<Rightarrow> 'd set" shows "sup_pres (bd\<^sub>\<F> f)"
simps_2: fixes n :: "nat"   and w :: "int"   and bl :: "bool list" shows "bin_to_bl_aux (Suc n) w bl = bin_to_bl_aux n (w div 2) (odd w # bl)"
A_0_j: shows "A $ (0::'rows) $ j = (0::'a)"
suffix_appendI: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "suffix xs ys" shows "suffix xs (zs @ ys)"
assms_2: shows "deps m \<subseteq> I ` insert_messages M"
hom_0_iff: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "zero_hom_0 hom" shows "(hom x = (0::'c)) = (x = (0::'b))"
memoized_correct: fixes \<O> :: "'c \<Rightarrow> 'd pmf"   and \<K> :: "'c \<Rightarrow> 'c pmf"   and state_list :: "'c list"   and os :: "'d iarray"   and a :: "'c"   and b :: "'c"   and c :: "nat" shows "HMM3_defs.forward_ix_rec \<O> \<K> state_list os a b c = fst (run_state (HMM3_defs.forward_ix\<^sub>m' \<O> \<K> state_list os a b c) Mapping.empty)"
the_equality: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
perp_per_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B A C" shows "Tarski_neutral_dimensionless.Per Bet Cong B A C"
trueArgs_def: fixes g :: "'g"   and \<phi> :: "'val" shows "trueArgs g \<phi> \<equiv> {\<phi>''. \<phi>'' \<notin> reachable g \<phi>} \<inter> {\<phi>''. \<exists>\<phi>'''\<in>reachable g \<phi>. phiArg g \<phi>''' \<phi>''}"
missing_cd_or_loop: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<nexists>k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'" obtains i :: "nat" and i' :: "nat" where "i < k" and "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'" and "\<pi> (Suc i) \<noteq> \<pi>' (Suc i')" and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<or> (\<forall>j'>i'. j' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i')"
rep_eq: shows "Rep_ell2 0 = (\<lambda>_. 0)"
filtration_def: fixes M :: "'d measure"   and F :: "'e \<Rightarrow> 'd measure" shows "Filtration.filtration M F = ((\<forall>t. subalgebra M (F t)) \<and> (\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)))"
degree_neville_aitken_main: fixes x :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "degree (neville_aitken_main x f i j) \<le> j - i"
distinct_19: fixes ccs1 :: "ccs"   and ccs2 :: "ccs"   and ccs1' :: "ccs"   and ccs2' :: "ccs" shows "ccs1 \<oplus> ccs2 \<noteq> ccs1' \<parallel> ccs2'"
assms_2: shows "\<not> i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
abs_eq: fixes xb :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and xa :: "'d \<Rightarrow> 'e"   and x :: "'d \<Rightarrow> 'f" shows "bop xb (mk\<^sub>e xa) (mk\<^sub>e x) = mk\<^sub>e (\<lambda>b. xb (xa b) (x b))"
lcliff_greek_less2: fixes r :: "('a \<times> 'a) set"   and cs :: "(accent \<times> 'a) list"   and a :: "'a"   and b :: "'a"   and a' :: "(accent \<times> 'a) list"   and bs :: "(accent \<times> 'a) list" assumes "trans r"   and "snd ` set cs \<subseteq> under r a \<union> under r b"   and "a' \<in> {[(Acute, a)], []}"   and "snd ` set bs \<subseteq> under r b" shows "(cs @ a' @ bs, [(Acute, a), (Macron, b)]) \<in> greek_less r"
ACI_nPlus_singleton_PLUS: fixes xs :: "'c rexp list"   and x :: "'c rexp" assumes "xs \<noteq> []"   and "sorted xs"   and "distinct xs"   and "\<forall>x\<in>{x} \<union> set xs. \<nexists>r s. x = Plus r s" shows "ACI_nPlus x (PLUS xs) = (if x \<in> set xs then PLUS xs else PLUS (insort x xs))"
Roots'_def: shows "Roots' = {X. X \<subseteq> {..<n} \<and> root' X \<noteq> 0}"
enat_of_option_simps_2: fixes x :: "nat option" shows "(Dijkstra_Abstract.enat_of_option x = \<infinity>) = (x = None)"
shrK_neq: fixes K :: "nat"   and evs :: "event list"   and B :: "agent" assumes "Key K \<notin> used evs" shows "shrK B \<noteq> K"
cod_MkArr: fixes A :: "'j \<Rightarrow> 'c"   and B :: "'j \<Rightarrow> 'c"   and F :: "'j \<Rightarrow> 'c" assumes "J_C.arr (J_C.MkArr A B F)" shows "J_C.cod (J_C.MkArr A B F) = J_C.MkIde B"
trap_no_progress: fixes es :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list" assumes "es = [Trap]" shows "\<not> \<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
lspasl_starl_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h0 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj A B h0 = (\<exists>h1 h2. heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> A h1 \<and> B h2)"
sinpoly_lower_nonneg: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "sinpoly (4 * Suc n) x \<le> sin x"
dif: fixes x :: "real" assumes "a < x"   and "x < b" shows "f differentiable at x"
BotL: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<bottom> \<in># \<Gamma>" shows "\<Gamma> \<Rightarrow> \<Delta>"
Lt_Le_dbm_lt_D: fixes a :: "'c"   and b :: "'c" assumes "Lt a \<prec> Le b" shows "a \<le> b"
fmdom_filter: fixes P :: "'e \<Rightarrow> bool"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmdom (fmfilter P m) = ffilter P (fmdom m)"
observable_move_return: fixes f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ms :: "'node list"   and S :: "'node SDG_node set"   and ms' :: "'node list" assumes "pred (f a) s"   and "transfer (f a) s = s'"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "\<forall>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "length ms = length s"   and "length s = Suc (length s')"   and "s' \<noteq> []"   and "hd ms = sourcenode a"   and "hd (tl ms) = targetnode a"   and "ms' = tl ms" shows "S,f \<turnstile> (ms,s) -a\<rightarrow> (ms',s')"
P2_e0_rewrite: fixes s :: "nat"   and r :: "nat"   and x :: "nat"   and xa :: "nat"   and \<alpha> :: "nat" assumes "s < order \<G>" shows "(\<^bold>g [^] (r * x + xa), \<^bold>g [^] (r * \<alpha> * x + xa * \<alpha>) \<otimes> \<^bold>g [^] x) = (\<^bold>g [^] (r * ((order \<G> - s + x) mod order \<G>) + (r * s + xa) mod order \<G>), \<^bold>g [^] (r * \<alpha> * ((order \<G> - s + x) mod order \<G>) + (r * s + xa) mod order \<G> * \<alpha>) \<otimes> \<^bold>g [^] ((order \<G> - s + x) mod order \<G> + s))"
simps_4: fixes c :: "'b"   and r1 :: "'b rexp"   and r2 :: "'b rexp" shows "deriv c (Plus r1 r2) = Plus (deriv c r1) (deriv c r2)"
callee: fixes x :: "'e"   and s :: "'b" assumes "x \<in> outs_\<I> \<I>" shows "plossless_gpv \<I>' (callee s x)"
exp_eq: fixes w :: "complex"   and z :: "complex" shows "(exp w = exp z) = (\<exists>n. w = z + complex_of_real (real_of_int (2 * n) * pi) * \<i>)"
hyps: fixes D :: "'a literal list" shows "wstate_of_dstate (N, P @ (D @ D', j) # P', Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P @ (D @ reduce [C] D D', j) # P', Q, n)"
Lcm_ratfps_def: shows "Lcm = normalization_euclidean_semiring_class.Lcm"
ideD_1: fixes a :: "'a" assumes "\<phi>.A.ide a" shows "\<phi>.A.arr a"
conf_decode1: fixes m :: "nat"   and n :: "nat"   and k :: "nat"   and l :: "nat"   and st :: "nat"   and r :: "nat" assumes "m \<noteq> n"   and "m \<noteq> k"   and "k \<noteq> n" shows "Max {u. Pi m ^ u dvd Pi m ^ l * Pi n ^ st * Pi k ^ r} = l"
power_minus_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.monoid_mult one times"   and "0 < n" shows "times (power.power one times a (n - 1)) a = power.power one times a n"
rlfm_I: fixes p :: "fm"   and x :: "real"   and bs :: "real list" assumes "qfree p"   and "0 \<le> x"   and "x < 1" shows "Ifm (x # bs) (rlfm p) = Ifm (x # bs) p \<and> isrlfm (rlfm p)"
diff_empty: fixes M :: "'c multiset" shows "M - {#} = M \<and> {#} - M = {#}"
simps_2: fixes y :: "('b, 'c) Combinators list" shows "removeShadowRules2 (DenyAll # y) = DenyAll # removeShadowRules2 y"
minimalize_offending_overapprox_boundnP: fixes m :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and nP :: "'b \<Rightarrow> 'c"   and fs :: "('b \<times> 'b) list"   and keeps :: "('b \<times> 'b) list"   and G :: "'b graph" shows "minimalize_offending_overapprox (\<lambda>G. m G nP) fs keeps G = SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox m fs keeps G nP"
cprank_max0: fixes ds :: "nat list" shows "cprank_max 0 (tensor0 ds)"
restrict_rel_def: fixes A :: "'b set"   and R :: "('b \<times> 'b) set" shows "restrict_rel A R = {(x, y). (x, y) \<in> R \<and> x \<in> A \<and> y \<in> A}"
map_sum_sel_2: fixes s :: "'e + 'f"   and f :: "'e \<Rightarrow> 'h"   and g :: "'f \<Rightarrow> 'g" assumes "\<not> isl s" shows "projr (map_sum f g s) = g (projr s)"
numeral_lt_\<omega>: fixes n :: "num" shows "numeral n < \<omega>"
mult_sm_distribution: fixes k1 :: "complex"   and k2 :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "k1 *\<^sub>s\<^sub>m (k2 *\<^sub>s\<^sub>m A) = (k1 * k2) *\<^sub>s\<^sub>m A"
order_iff: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (a = inf_class.inf a b)"
l: shows "l = (p, is, xs, sb, \<D>, \<O>, \<R>)"
freshSets_1: fixes C :: "'e" shows "[] \<sharp>* C"
assms_4: shows "YFtor C ## f = YFtor C ## g"
Quotient_stream: shows "Quotient (eq_onp (\<lambda>xs. \<not> lfinite xs)) stream_of_llist llist_of_stream stream_from_llist_setup.cr_stream"
mem: shows "(mset xs, mset ys) \<in> ns_mul_ext {(x, y). snd (P x y)} {(x, y). fst (P x y)}"
matrix_matrix_mult_def: fixes m :: "(('e, 'h) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec"   and m' :: "(('e, 'f) Finite_Cartesian_Product.vec, 'h) Finite_Cartesian_Product.vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $h i $h k * m' $h k $h j"
range_mapping_relax: fixes i :: "nat"   and j :: "nat"   and P :: "'a \<Rightarrow> nat option"   and i' :: "nat"   and j' :: "nat" assumes "range_mapping i j P"   and "i' \<le> i"   and "j \<le> j'" shows "range_mapping i' j' P"
iffs: fixes verts :: "'a set"   and arcs :: "('a \<times> 'a) set"   and more :: "'b"   and verts' :: "'a set"   and arcs' :: "('a \<times> 'a) set"   and more' :: "'b" shows "(\<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr> = \<lparr>verts = verts', arcs = arcs', \<dots> = more'\<rparr>) = (verts = verts' \<and> arcs = arcs' \<and> more = more')"
E: shows "map (inv q) (filter (\<lambda>x. x \<in> range q) xs') \<in> divergences Q"
minus_equality: fixes y :: "int"   and x :: "int" assumes "y + x = 0"   and "x \<in> UNIV"   and "y \<in> UNIV" shows "\<ominus>\<^bsub>\<Z>\<^esub> x = y"
viewRn_inv: fixes t :: "('c, 'd) FingerTreeStruc"   and n :: "nat"   and nd :: "('c, 'd) Node"   and s :: "('c, 'd) FingerTreeStruc" assumes "is_measured_ftree t"   and "is_leveln_ftree n t"   and "viewRn t = Some (nd, s)" shows "is_measured_ftree s \<and> is_measured_node nd \<and> is_leveln_ftree n s \<and> is_leveln_node n nd"
meld_queue_invar: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.queue_invar q"   and "SkewBinomialHeapStruc.queue_invar q'" shows "SkewBinomialHeapStruc.queue_invar (SkewBinomialHeapStruc.meld q q')"
oghoare_sound_Parallel_Normal_case: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and x :: "'d"   and c' :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and y :: "'d"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and P :: "('d, 'e, 'f) ann"   and Q :: "'d set"   and A :: "'d set" assumes "\<Gamma> \<turnstile> (Parallel cs, Normal x) \<rightarrow>\<^sup>* (c', t)"   and "t = Normal y"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Parallel cs) Q, A"   and "final (c', t)"   and "x \<in> pre P"   and "t \<notin> Fault ` F" shows "c' = Throw \<and> t \<in> Normal ` A \<or> c' = Skip \<and> t \<in> Normal ` Q"
well_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "well_ordered_set A less_eq \<equiv> antisymmetric A less_eq \<and> well_related_set A less_eq"
simps_5: fixes x :: "'sz"   and y :: "'sz"   and z :: "'sz"   and \<phi> :: "('sf, 'sc, 'sz) formula" shows "FUrename x y (Exists z \<phi>) = Exists (swap x y z) (FUrename x y \<phi>)"
recvmsg_simps_6: fixes P :: "'a \<Rightarrow> bool"   and d :: "nat" shows "recvmsg P (deliver d) = True"
corSim: shows "weak_reduction_correspondence_simulation (indRelTEQ TRel) (STCal Source Target)"
\<Phi>_inv: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "S.inverse_arrows (\<Phi>.map yx) (\<Psi>o yx)"
mono_Span_sublist: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> set Vs"   and "set Vs \<subseteq> carrier R" shows "ring.Span R K Us \<subseteq> ring.Span R K Vs"
compare_double_simps_1: fixes x :: "double"   and y :: "double" shows "(compare_double x y < 0) = (Code_Double.is_nan x \<and> \<not> Code_Double.is_nan y \<or> fcompare_double x y = ccode.Lt)"
f_Exec_Stream_Acc_Output_1: fixes output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output (Suc 0) output_fun trans_fun xs c = map output_fun (f_Exec_Comp_Stream trans_fun xs c)"
E\<^sub>2: shows "\<E> f\<^sub>2 = Edges (u # vs @ [v]) \<union> Edges (v # between (vertices f) v u @ [u])"
dvd_elements_mult_matrix_left_right: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and Q :: "'a mat"   and x :: "'a" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd A $$ (i, j)" shows "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd (P * A * Q) $$ (i, j)"
preserved_def: fixes f :: "('ap, 'aq, 'ar) prog"   and h :: "'ap"   and h' :: "'ap" shows "preserved f h h' = (\<forall>x. h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x)"
SR3nMT: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and p :: "'b list" assumes "\<not> not_MT C p" shows "rm_MT_rules C p = []"
simps_2: fixes cs :: "'edge list"   and as :: "'edge list"   and a :: "'edge" shows "valid_path_rev_aux cs (as @ [a]) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> valid_path_rev_aux [] as | c' # cs' \<Rightarrow> c' \<in> get_return_edges a \<and> valid_path_rev_aux cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> valid_path_rev_aux (a # cs) as | _ \<Rightarrow> valid_path_rev_aux cs as)"
p\<^sub>0_simps: shows "trg g\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)"
union_with_inside: fixes S :: "'b set" shows "S \<union> inside S = - outside S"
wp1ID: fixes p :: "('a, 'b) Combinators list" shows "wellformed_policy1_strong (insertDeny (removeShadowRules1 p))"
fact_prod_rev: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero n = semiring_1.of_nat one plus zero (prod ((-) n) {0..<n})"
is_inf_meet: fixes x :: "'a"   and y :: "'a" shows "is_inf x y (x \<sqinter> y)"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) env option) \<Rightarrow> 'f"   and env :: "('g, 'h, 'i) env" shows "P (case env of Val x \<Rightarrow> f1 x | Env x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. env = Val x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. env = Env x21 x22 \<longrightarrow> P (f2 x21 x22)))"
simps_2: fixes b :: "bexp"   and b' :: "bexp" shows "cnf (b and b') = cnf b \<union> cnf b'"
independent_mono: fixes A :: "'b set"   and B :: "'b set" assumes "independent A"   and "B \<subseteq> A" shows "independent B"
invex: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "\<exists>b\<in>carrier R. a \<otimes> b = \<one>"
output_consistent: shows "\<forall>a s t. vpeq (current s) s t \<and> current s = current t \<longrightarrow> output_f s a = output_f t a"
assoc: fixes a :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> bool" shows "((a \<and>* b) \<and>* c) = (a \<and>* b \<and>* c)"
nn_integral_substitution_aux: fixes f :: "real \<Rightarrow> ennreal"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "continuous_on {a..b} g'"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> g' x"   and "a < b" shows "set_nn_integral lborel {g a..g b} f = \<integral>\<^sup>+x\<in>{a..b}. (f (g x) * ennreal (g' x))\<partial>lborel"
big_step_fun: fixes \<rho> :: "(nat \<times> bval) list"   and e :: "exp"   and v :: "bval"   and v' :: "bval" assumes "\<rho> \<turnstile> e \<Down> v"   and "\<rho> \<turnstile> e \<Down> v'" shows "v = v'"
surjective: fixes r :: "('b, 'c) V_rec_scheme" shows "r = \<lparr>V = V r, N = N r, C = C r, \<dots> = V_rec.more r\<rparr>"
games_lfp_unfold: shows "games_lfp = fixgames games_lfp"
pow_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" shows "([^]\<^bsub>subgroup_generated G S\<^esub>) = ([^]\<^bsub>G\<^esub>)"
assms_3: fixes n :: "nat" shows "emeasure M (C n) < \<infinity>"