hcju/isabelleps huggingface dataset:
vcg_wlp_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and mods :: "char list set"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT_partial_mods \<pi> mods P c Q"   and "P s"   and "\<And>s'. \<lbrakk>modifies mods s' s; Q s s'\<rbrakk> \<Longrightarrow> Q' s'" shows "wlp \<pi> c Q' s"
neval_reachable_mode_states: fixes lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and n :: "nat"   and lc' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and cms :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and k :: "nat"   and mem :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "abs.neval lc n lc'"   and "lc = (cms ! k, mem)"   and "k < length cms" shows "map snd (cms[k := fst lc']) \<in> abs.reachable_mode_states (cms, mem)"
D: shows "Min (offs_set_prev ns (x # xs) index key mi ma i) \<in> offs_set_prev ns (x # xs) index key mi ma i"
is_left_adjoint_functor: shows "left_adjoint_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F"
oalist_inv_list_of_oalist_tc: fixes xs :: "('a, 'b) oalist_tc" shows "tc.oalist_inv_raw (list_of_oalist_tc xs)"
isAugmentingPath_def: fixes p :: "(nat \<times> nat) list" shows "f.isAugmentingPath p \<equiv> f.cf.isSimplePath s p t"
mes: shows "(\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. f N -` A i \<inter> space M) \<in> sets M"
bst_of_list_altdef: fixes xs :: "'a list" shows "bst_of_list xs = fold Tree_Set.insert xs \<langle>\<rangle>"
rel_U_neg_distr_imp: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "(('z, 'n, 'q, 't, 'w, 'aa) U \<times> (('z, 'n, 'q, 't, 'w, 'aa) U \<times> ('ab, 'p, 's, 'v, 'y, 'aa) U) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) U \<times> 'z \<times> ('z \<times> 'ab) \<times> 'ab \<times> 'aa) itself"   and tytok_T :: "('z \<times> 'ac \<times> 'ab \<times> 'aa) itself" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_U_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_T"
sheaf_spec_of_empty_is_singleton: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> 'b"   and t :: "'a set \<Rightarrow> 'b" assumes "U = {}"   and "s \<in> extensional U"   and "t \<in> extensional U" shows "s = t"
sum_list_map_remove1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_monoid_add plus zero"   and "x \<in> set xs" shows "monoid_add.sum_list plus zero (map f xs) = plus (f x) (monoid_add.sum_list plus zero (map f (remove1 x xs)))"
mem_carrier: fixes x :: "'a" assumes "x \<in> adjoin G H a" shows "x \<in> carrier G"
times_eint_simps_1: fixes m :: "int"   and n :: "int" shows "eint m * eint n = eint (m * n)"
effect_returnI: fixes h :: "heap"   and h' :: "heap"   and x :: "'a" assumes "h = h'" shows "effect (return x) h h' x"
linear_componentwise: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and j :: "'c" assumes "linear f" shows "f x \<bullet> j = (\<Sum>i\<in>Basis. x \<bullet> i * (f i \<bullet> j))"
bal_tree_tm_def: fixes n :: "nat"   and t :: "'b tree" shows "bal_tree_tm n t =1 inorder2_tm t [] \<bind> bal_list_tm n"
assms_3: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>"
IH: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "higher_differentiable_on T f n"   and "higher_differentiable_on S g n"   and "g ` S \<subseteq> T" shows "higher_differentiable_on S (f \<circ> g) n"
cases: fixes a :: "'a"   and r :: "('a \<times> 'b) set" assumes "a \<in> Domain r" obtains a :: "'a" and b :: "'b" where "ab__ = a" and "(a, b) \<in> r"
order_decomp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b list"   and a :: "'b" assumes "class.idom_char_0 one times plus zero minus uminus"   and "semiring_0.poly plus zero times p \<noteq> semiring_0.poly plus zero times []" shows "\<exists>q. semiring_0.poly plus zero times p = semiring_0.poly plus zero times (semiring_0.pmult plus zero times (semiring_1.pexp one times plus zero [uminus a, one] (ring_1.order one times plus zero uminus a p)) q) \<and> \<not> semiring_0.divides plus zero times [uminus a, one] q"
filter_mset_mset_set: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "finite A" shows "filter_mset P (mset_set A) = mset_set {x \<in> A. P x}"
set0_4: fixes x2 :: "('f \<times> 'e) set['g]" shows "set2_var0 (BPS x2) = \<Union> (Basic_BNFs.snds ` set_bset x2)"
infinite_union: fixes Y :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "finite Y"   and "infinite (\<Union> (f ` Y))" shows "\<exists>y. y \<in> Y \<and> infinite (f y)"
homogeneous_set_direct_decomp: fixes A :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ss :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> homogeneous_set s" shows "homogeneous_set A"
remove_max_def: fixes l :: "'b list" shows "remove_max l = foldl f (hd l, []) (tl l)"
T_endpow_list_apply_commute: fixes v :: "'v"   and n :: "nat" shows "map T (map (\<lambda>S. S v) (map endpow [0..<n])) = map (\<lambda>S. S v) (map ((\<circ>) T) (map endpow [0..<n]))"
g_def: shows "g \<equiv> \<lambda>v i. if i < n then representation B v (b i) else 0"
\<alpha>_def: shows "\<alpha> = Matrix.vec N (\<lambda>x. complex_of_real (if f x then 0 else 1 / sqrt (real (N - M))))"
integral_norm_bound: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> 'd" shows "norm (integral\<^sup>L M f) \<le> LINT x|M. norm (f x)"
coprime_dvd_mult': fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "coprime a b"   and "a dvd b * c" shows "a dvd c"
l_coset_eq_set_mult: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and H :: "'c set" shows "x <#\<^bsub>G\<^esub> H = {x} <#>\<^bsub>G\<^esub> H"
left_totalI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. \<exists>y. R x y" shows "left_total R"
g: fixes x :: "'a" assumes "x \<in> S" shows "norm ((\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i) - g x) < e / real (n + 2)"
segment_forall: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<in> segment D b"   and "b \<in> carrier D"   and "x \<in> segment D b"   and "x \<prec> a"   and "\<forall>y\<in>segment D b. x \<prec> y \<longrightarrow> \<not> y \<prec> a" shows "\<forall>y\<in>carrier D. x \<prec> y \<longrightarrow> \<not> y \<prec> a"
ct_2: fixes s :: "'s" assumes "s \<in> S"   and "s \<in> S\<^sub>r" shows "\<exists>t\<in>set_pmf (ct s). \<exists>n. t \<in> r n \<and> (\<forall>m. s \<in> r m \<longrightarrow> n < m)"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case final of init \<Rightarrow> f1 | loop \<Rightarrow> f2 | final \<Rightarrow> f3) = f3"
prems: fixes i :: "nat" assumes "i < length (v # vs)" shows "(0::'a) \<le> (u # us) ! i \<and> (u # us) ! i \<le> (v # vs) ! i"
pred_def: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" shows "Measurable.pred M P = ({x \<in> space M. P x} \<in> sets M)"
t2: shows "bdd_above {norm (f x) / norm x |x. True}"
equality_axioms_def: shows "equality_axioms = {refl_ax, eq_cong_ax, mem_cong_ax, eats_cong_ax}"
ARR_eqI: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "ARR f"   and "ARR g"   and "f \<inter> g \<noteq> {}" shows "f = g"
Fun: fixes f :: "'a"   and g :: "'a"   and ss :: "('a, 'b) Term.term list"   and ts :: "('a, 'c) Term.term list" assumes "f = g"   and "length ss = length ts"   and "\<forall>i<length ts. emv (ss ! i) (ts ! i)" shows "emv (Fun f ss) (Fun g ts)"
pickFT_pred_def: fixes cl :: "('g, 'h, 'i) cmd list"   and n :: "nat" shows "pickFT_pred cl n \<equiv> n < length cl \<and> finished (cl ! n)"
group_left_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "group f z inverse" shows "f z a = a"
hom_inverse: fixes x :: "rat" shows "of_rat (inverse x) = inverse (of_rat x)"
res_comp2: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "residual f \<circ> f \<circ> residual f = residual f"
subset_fimage_iff: fixes B :: "'a fset"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b fset" shows "(B |\<subseteq>| f |`| A) = (\<exists>AA. AA |\<subseteq>| A \<and> B = f |`| AA)"
map_sq_matrix_add: fixes f :: "'c \<Rightarrow> 'd"   and A :: "('c, 'e) sq_matrix"   and B :: "('c, 'e) sq_matrix" assumes "\<And>a b. f (a + b) = f a + f b" shows "map_sq_matrix f (A + B) = map_sq_matrix f A + map_sq_matrix f B"
cut_ge_empty: fixes t :: "'b" shows "{} \<down>\<ge> t = {}"
simps_19: fixes Vs :: "char list list"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "char list exp" shows "compE\<^sub>1 Vs (INIT C (Cs,b) \<leftarrow> e) = INIT C (Cs,b) \<leftarrow> compE\<^sub>1 Vs e"
insert_def: fixes g :: "'d" shows "\<iota> g = [(False, g)]"
simps: fixes M :: "('d, 'e, 'f) FSM" shows "completely_specified M = (\<forall>s1\<in>nodes M. \<forall>x\<in>inputs M. \<exists>y\<in>outputs M. \<exists>s2. s2 \<in> succ M (x, y) s1)"
len_sub_int: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "len v ts c \<le> ext v"
Gromov_completion_rel_to_const': fixes x :: "'a"   and u :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "Gromov_completion_rel (\<lambda>n. x) u" shows "u n = x"
eventually_at_topological: fixes "open" :: "'d set \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and a :: "'d"   and s :: "'d set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
ideals_set_is_comm_monoid: shows "Group.comm_monoid (ideals_set R)"
P''TransP': shows "P'' \<longmapsto> \<tau> \<prec> P'"
reachable_mode_states_def: fixes gc :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" shows "conc.reachable_mode_states gc \<equiv> {mdss. \<exists>cms' mem' sched. conc.meval_sched_abv gc sched (cms', mem') \<and> map snd cms' = mdss}"
Sup_fold_sup: fixes A :: "'a set" assumes "finite A" shows "\<Squnion> A = Finite_Set.fold (\<squnion>) \<bottom> A"
Max_atLeastAtMost: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "Max {m..n} = n"
qs_def: shows "qs = snd (split 0 X (lpp ` punit.reduced_GB F))"
invertible_left_inverse: fixes u :: "'b" assumes "A.additive.invertible u"   and "u \<in> A" shows "addA (A.additive.inverse u) u = zeroA"
insert_3: fixes p :: "'a \<Rightarrow> 'a" assumes "bij p"   and "\<forall>x. x \<notin> F \<longrightarrow> p x = x" shows "permutation p"
chain_mono_less: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "class.po below"   and "po.chain below Y"   and "i < j" shows "below (Y i) (Y j)"
hcomp_null_1: fixes \<mu> :: "'a" shows "null \<star> \<mu> = null"
left_neutral: fixes a :: "'d upred" shows "(true \<and> a) = a"
simps_1: fixes r :: "'b rexp" shows "derivs [] r = r"
butlast_list_update: fixes xs :: "'a list"   and k :: "nat"   and x :: "'a" shows "butlast (xs[k := x]) = (if k = length xs - 1 then butlast xs else (butlast xs)[k := x])"
iTL_defs_5: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iWeakUntil I P Q \<equiv> (\<box> t I. P t) \<or> (\<diamond> t I. Q t \<and> (\<box> t' I \<down>< t. P t'))"
ln_\<theta>_asymp_equiv: shows "(\<lambda>x. ln (\<theta> x)) \<sim>[at_top] ln"
uniformly_continuous_on_Cauchy: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'a" assumes "uniformly_continuous_on S f"   and "Cauchy X"   and "\<And>n. X n \<in> S" shows "Cauchy (\<lambda>n. f (X n))"
card_interleavings: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<inter> set ys = {}" shows "card {zs. interleavings xs ys zs} = length xs + length ys choose length xs"
to_nat_less_ncols: fixes a :: "'a"   and A :: "(('b, 'a) vec, 'c) vec" shows "to_nat a < ncols A"
ccpo_fun: shows "class.ccpo (fun_lub Sup) (fun_ord (\<le>)) (mk_less (fun_ord (\<le>)))"
verit_comp_simplify_1: fixes a :: "'a" shows "(a < a) = False"
edges_face_def: fixes f :: "face" shows "\<E> f \<equiv> {(a, f \<bullet> a) |a. a \<in> \<V> f}"
simps_1: fixes k :: "'a" shows "nnf (Atom k) = Atom k"
nvalids_SucD: fixes A :: "triple set"   and n :: "nat" assumes "Ball A (nvalid (Suc n))" shows "Ball A (nvalid n)"
ratio_def: shows "ratio = - (tval y - tval x) / (tval z - tval y)"
assms_2: shows "\<forall>i. local.gba.L (\<sigma> i) (\<xi> i)"
\<BB>_def: shows "\<BB> \<equiv> {(fold Deriv w (lang r), fold Deriv w (lang s)) |w. w \<in> lists (atoms r \<union> atoms s)}"
dcharacter_c2dc: fixes \<chi> :: "nat \<Rightarrow> complex" assumes "character G \<chi>" shows "dcharacter n (c2dc \<chi>)"
clock_simAction_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'e, 'd) GC list"   and envVal :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'f" shows "ClockView.clock_simAction jkbp envVal envObs \<equiv> \<lambda>a (Y, X). concat (map (\<lambda>gc. if eval envVal envObs (Y, X) (guard gc) then [action gc] else []) (jkbp a))"
simps_59: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x21 :: "char list"   and x22 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> Cast x21 x22"
simps_11: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a"   and x31 :: "'b pred"   and x32 :: "'b seq" shows "(case Join x31 x32 of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
simps_4: fixes b :: "expr"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd" shows "#:if (b) c\<^sub>1 else c\<^sub>2 = #:c\<^sub>1 + #:c\<^sub>2 + 1"
continuous_on_const: fixes s :: "'d set"   and c :: "'e" shows "continuous_on s (\<lambda>x. c)"
Cons_in_shuffles_rightI: fixes zs :: "'g list"   and xs :: "'g list"   and ys :: "'g list"   and z :: "'g" assumes "zs \<in> shuffles xs ys" shows "z # zs \<in> shuffles xs (z # ys)"
set_paths_def: shows "set_paths = {p. apath u p v}"
that: fixes A :: "nat \<Rightarrow> 'a set" assumes "\<And>i. open (A i)"   and "\<And>i. x \<in> A i"   and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F i in sequentially. A i \<subseteq> S" shows "thesis"
assms: fixes x :: "'a" shows "norm (apply_bcontfun f x) \<le> b"
arr_pr0_iff: fixes a :: "'a"   and b :: "'a" shows "arr \<pp>\<^sub>0[a, b] = (ide a \<and> ide b)"
COPY_def: shows "COPY \<equiv> \<mu> x. left`?`xa \<rightarrow> right`!`xa \<rightarrow> x"
iMODb_iTILL_subset_same: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<subseteq> [\<dots>r + m * c]"
nsv_hom_meet: fixes x :: "'a"   and y :: "'a" shows "(d x \<sqinter> d y) \<cdot> nc = d x \<cdot> nc \<sqinter> d y \<cdot> nc"
simps_298: fixes f1 :: "String.literal \<Rightarrow> trm list \<Rightarrow> 'a"   and f2 :: "String.literal \<Rightarrow> formula \<Rightarrow> formula \<Rightarrow> 'a"   and f3 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f4 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f5 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f6 :: "formula \<Rightarrow> 'a"   and f7 :: "formula \<Rightarrow> formula \<Rightarrow> 'a"   and f8 :: "formula \<Rightarrow> formula \<Rightarrow> 'a"   and f9 :: "formula list \<Rightarrow> 'a"   and f10 :: "formula \<Rightarrow> 'a"   and f11 :: "nat \<Rightarrow> agg_type \<times> event_data \<Rightarrow> nat \<Rightarrow> trm \<Rightarrow> formula \<Rightarrow> 'a"   and f12 :: "\<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f13 :: "\<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f14 :: "formula \<Rightarrow> \<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f15 :: "formula \<Rightarrow> \<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f16 :: "\<I> \<Rightarrow> formula Regex.regex \<Rightarrow> 'a"   and f17 :: "\<I> \<Rightarrow> formula Regex.regex \<Rightarrow> 'a"   and x9 :: "formula list" shows "(case Ands x9 of Pred x xa \<Rightarrow> f1 x xa | formula.Let x xa xb \<Rightarrow> f2 x xa xb | formula.Eq x xa \<Rightarrow> f3 x xa | Less x xa \<Rightarrow> f4 x xa | LessEq x xa \<Rightarrow> f5 x xa | Neg x \<Rightarrow> f6 x | Or x xa \<Rightarrow> f7 x xa | And x xa \<Rightarrow> f8 x xa | Ands x \<Rightarrow> f9 x | Exists x \<Rightarrow> f10 x | Agg x xa xb xc xd \<Rightarrow> f11 x xa xb xc xd | Prev x xa \<Rightarrow> f12 x xa | Next x xa \<Rightarrow> f13 x xa | Since x xa xb \<Rightarrow> f14 x xa xb | Until x xa xb \<Rightarrow> f15 x xa xb | MatchF x xa \<Rightarrow> f16 x xa | MatchP x xa \<Rightarrow> f17 x xa) = f9 x9"
stream_enc_enc_some: fixes \<Sigma> :: "'b list"   and I :: "(nat + nat set) list"   and w :: "'b list" assumes "formula \<Sigma>"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "formula.stream_enc \<Sigma> (w, I) = (SOME v. v \<in> formula.enc \<Sigma> (w, I)) @- sconst (formula.any \<Sigma>, replicate (length I) False)"
sel_1: fixes x1 :: "heap"   and x2 :: "nat set" shows "heapOf (pHeap x1 x2) = x1"
unit_simps_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src a" shows "partial_magma.dom V (\<i> a) = H a a"
hyps_2: shows "(\<S>1, \<A>1) \<Rightarrow>\<^sup>\<bullet> (\<S>2, \<A>2)"
toList_def: fixes t :: "('c, 'd) FingerTree" shows "toList t \<equiv> FingerTreeStruc.toList (Rep_FingerTree t)"
vertex_3_nat_def: shows "vertex_3 \<equiv> 3"
minus_insert_in_fset: fixes ys :: "'b fset"   and x :: "'b"   and xs :: "'b fset" assumes "x |\<in>| ys" shows "insert_fset x xs - ys = xs - ys"
streamsp_def: shows "streamsp \<equiv> \<lambda>A. gfp (\<lambda>p x. \<exists>a s. x = a ## s \<and> A a \<and> p s)"
ENR_sphere: fixes a :: "'b"   and r :: "real" shows "ENR (sphere a r)"
simulation_silent1I': fixes \<mu>1 :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and s1' :: "'h"   and s1 :: "'h"   and trsys2 :: "'g \<Rightarrow> 'i \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'g \<Rightarrow> 'i \<Rightarrow> 'g \<Rightarrow> bool"   and s2 :: "'g"   and bisim :: "'h \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<exists>s2'. (if \<mu>1 s1' s1 then (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* else (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>+\<^sup>+) s2 s2' \<and> bisim s1' s2'" shows "bisim s1' s2 \<and> \<mu>1\<^sup>+\<^sup>+ s1' s1 \<or> (\<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>+\<^sup>+ s2 s2' \<and> bisim s1' s2')"
construct_NofI_R_relation: fixes p :: "real poly"   and I1 :: "real poly list"   and I2 :: "real poly list" shows "construct_NofI_R p I1 I2 = construct_NofI (sum_list (map power2 (p # I1))) I2"
create_reservation_length_stable: fixes ts :: "traffic"   and d :: "cars"   and ts' :: "traffic"   and v :: "view"   and c :: "cars" shows "ts \<^bold>\<midarrow>r( d ) \<^bold>\<rightarrow> ts' \<longrightarrow> len v ts c = len v ts' c"
countable_web_axioms: shows "countable_web (quotient_web \<Gamma> f)"
wf_imp_nf_ex: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "wfP r\<inverse>\<inverse>" obtains b :: "'a" where "r\<^sup>*\<^sup>* a b" and "relation.is_final r b"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" assumes "a1 \<rightarrow>c a2" obtains c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1__ ;; c2__, s__)" and "a2 = (c1'__ ;; c2__, s'a__)" and "(c1__, s__) \<rightarrow>c (c1'__, s'a__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1a__ ;; c2a__, sa__)" and "a2 = (c2a__, s'b__)" and "(c1a__, sa__) \<rightarrow>t s'b__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tst__ then c1b__ else c2b__, sb__)" and "a2 = (c1b__, sb__)" and "tval tst__ sb__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tsta__ then c1c__ else c2c__, sc__)" and "a2 = (c2c__, sc__)" and "\<not> tval tsta__ sc__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, sd__)" and "a2 = (c ;; while tst do c, sd__)" and "tval tst sd__"   | c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1d__ | c2d__, se__)" and "a2 = (c1' | c2d__, s'c__)" and "(c1d__, se__) \<rightarrow>c (c1', s'c__)"   | c2 :: "('test, 'atom) com" and s :: "'state" and c2' :: "('test, 'atom) com" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1e__ | c2e__, sf__)" and "a2 = (c1e__ | c2', s'd__)" and "(c2e__, sf__) \<rightarrow>c (c2', s'd__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1f__ | c2f__, sg__)" and "a2 = (c2f__, s'e__)" and "(c1f__, sg__) \<rightarrow>t s'e__"   | c2 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1 | c2, s)" and "a2 = (c1, s')" and "(c2, s) \<rightarrow>t s'"
satC_tC: fixes c :: "('fsym, 'psym) lit list"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wtC c"   and "Ik.wtE \<xi>" shows "TE.satC (eenv \<xi>) (tC c) = Ik.satC \<xi> c"
scaleR_conv_of_real: fixes r :: "real"   and x :: "'a" shows "r *\<^sub>R x = of_real r * x"
conorm_componentwise: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "onorm f \<le> (\<Sum>i\<in>CBasis. norm (f i))"
up_def: fixes R :: "('c, 'd) ring_scheme" shows "up R = {f \<in> UNIV \<rightarrow> carrier R. \<exists>n. bound \<zero>\<^bsub>R\<^esub> n f}"
lossless_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> bool spmf" shows "lossless \<A> = (\<forall>\<alpha> \<beta> \<gamma>. lossless_spmf (\<A> \<alpha> \<beta> \<gamma>))"
lex_two: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and s2 :: "('c \<times> 'c) set" assumes "ns O s \<subseteq> s"   and "SN s"   and "SN s2" shows "SN (lex_two s ns s2)"
bit_revcast_iff: fixes w :: "'c word"   and n :: "nat" shows "bit (revcast w) n = (LENGTH('b) - LENGTH('c) \<le> n \<and> n < LENGTH('b) \<and> bit w (n + (LENGTH('c) - LENGTH('b)) - (LENGTH('b) - LENGTH('c))))"
upd_conv_take_nth_drop: fixes i :: "nat"   and xs :: "'a list"   and a :: "'a" assumes "i < length xs" shows "xs[i := a] = take i xs @ a # drop (Suc i) xs"
source_neq_sink: shows "source \<Delta> \<noteq> sink \<Delta>"
arith_simps_46: fixes m :: "num"   and n :: "num" shows "numeral m + - numeral n = neg_numeral_class.sub m n"