hcju/isabelleps huggingface dataset:
that: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < length fs"   and "v = lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j)) - fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i)))" shows "thesis"
diagvector_mult_right: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and x :: "'b \<Rightarrow> 'a" shows "A ** diagvector x = (\<chi>i j. A $h i $h j * x j)"
ibind_strict_2: fixes f :: "unit \<Rightarrow> 'maa" shows "abind top f = top"
is_SAT_PUSH'_def: fixes pr_operation' :: "pr_operation'" shows "is_SAT_PUSH' pr_operation' = (\<exists>x3. pr_operation' = SAT_PUSH' x3)"
rel_interior_convex_shrink: fixes S :: "'b set"   and c :: "'b"   and x :: "'b"   and e :: "real" assumes "convex S"   and "c \<in> rel_interior S"   and "x \<in> S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
assms_4: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
asym: fixes a :: "'b Word.word"   and b :: "'b Word.word" assumes "a <s b"   and "b <s a" shows "False"
circle_polar_def: fixes t :: "real" shows "circle_polar t = (d / 2 * cos (2 * pi * t), d / 2 * sin (2 * pi * t))"
assms: shows "\<forall>i\<in>net_ips s1 \<union> net_ips s2. the ((netmap s1 ++ netmap s2) i) = \<sigma> i"
f_add: fixes m1 :: "'c \<Rightarrow> 'a"   and m2 :: "'c \<Rightarrow> 'a" assumes "m1 \<in> carrier (func_space A)"   and "m2 \<in> carrier (func_space A)" shows "lincomb (m1 \<oplus>\<^bsub>func_space A\<^esub> m2) A = lincomb m1 A \<oplus>\<^bsub>M\<^esub> lincomb m2 A"
PRoject_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and x :: "'d" shows "\<pi>\<^bsub>I,A,x\<^esub> = (\<lambda>f\<in>carr_prodag I A. f x)"
basis_sub: fixes G :: "'b list set" shows "\<BB> G \<subseteq> G"
mono_on_greaterD: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "mono_on g A"   and "x \<in> A"   and "y \<in> A"   and "g y < g x" shows "y < x"
lowner_le_antisym: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A \<le>\<^sub>L B"   and "B \<le>\<^sub>L A" shows "A = B"
induct: fixes P :: "nterm \<Rightarrow> bool"   and a0 :: "nterm" assumes "\<And>x. P (Nconst x)"   and "\<And>t\<^sub>1 t\<^sub>2. \<lbrakk>P t\<^sub>1; P t\<^sub>2\<rbrakk> \<Longrightarrow> P (t\<^sub>1 $\<^sub>n t\<^sub>2)"   and "\<And>uu_ t. P t \<Longrightarrow> P (\<Lambda>\<^sub>n uu_. t)"   and "\<And>uv_. P (Nvar uv_)" shows "P a0"
simps_2: fixes atm :: "'atom"   and s :: "'state"   and i :: "nat" shows "wt_cont_eff (Atm atm) s i = (1, Done, aval atm s)"
count_image_mset_lt_imp_lt_raw: fixes A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "finite A"   and "A = set_mset M \<union> set_mset N"   and "count (image_mset f M) b < count (image_mset f N) b" shows "\<exists>x. f x = b \<and> count M x < count N x"
intro: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "safe_distance_normal_axioms \<delta>" shows "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"
strand_vars_unfold: fixes v :: "'d"   and S :: "('e, 'd) strand_step list" assumes "v \<in> vars\<^sub>s\<^sub>t S" shows "\<exists>S' x S''. S = S' @ x # S'' \<and> v \<in> vars\<^sub>s\<^sub>t\<^sub>p x"
DynPDG_path_Exit: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>d* n'"   and "n' = (_Exit_)" shows "n = (_Exit_)"
listsp_mono: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "A \<le> B" shows "listsp A \<le> listsp B"
less_imp_le: fixes x :: "'a list"   and y :: "'a list" assumes "strict_border x y" shows "border x y"
sep_extract_pure_2: fixes P :: "bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" shows "(\<up> P \<and>* Q) h = (P \<and> Q h)"
fresh_16: fixes a :: "coname"   and x2 :: "coname"   and x3 :: "trm"   and x1 :: "coname" shows "a \<sharp> OrR1 <x2>.x3 x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
has_derivative_transform_eventually: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') (at x within s)"   and "\<forall>\<^sub>F x' in at x within s. f x' = g x'"   and "f x = g x"   and "x \<in> s" shows "(g has_derivative f') (at x within s)"
resPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "P \<approx> Q" shows "<\<nu>x>P \<approx> <\<nu>x>Q"
Rel_def: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Rel r \<equiv> r"
min_max_xyz: shows "maximum i + 1 = minimum (j \<squnion> k)"
le_def: shows "le = liminf (\<lambda>n. eNorm (\<LL> p M) (f n))"
Mp_f_representative: fixes m :: "int"   and f :: "int poly" assumes "poly_mod_type TYPE('a) m" shows "poly_mod.Mp m f = to_int_poly (of_int_poly f)"
call_option_lgeom: fixes T :: "nat"   and K :: "real"   and w :: "bool stream" shows "call_option T K w = max (lgeom_proc u d init (stake T w) - K) 0"
lmeasurable_division: fixes \<D> :: "'c set set"   and S :: "'c set" assumes "\<D> division_of S" shows "S \<in> lmeasurable"
infdist_def: fixes x :: "'b"   and A :: "'b set" shows "infdist x A = (if A = {} then 0 else Inf (dist x ` A))"
ipresIGSubst_def: fixes h :: "'u \<Rightarrow> 'v"   and MOD :: "('w, 'x, 'varSort, 'sort, 'y, 'z, 'u, 'aa, 'ab) model_scheme"   and MOD' :: "('ac, 'ad, 'varSort, 'ae, 'af, 'z, 'v, 'ag, 'ah) model_scheme" shows "ipresIGSubst h MOD MOD' \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> h (igSubst MOD ys Y y X) = igSubst MOD' ys (h Y) y (h X)"
assms: shows "Q \<sqsubseteq> R \<sqinter> (Q ;; P)"
dep: fixes x :: "'a" assumes "x \<in> X" shows "rank_of (X - {x}) = rank_of X"
exists_bezout_ext_aux: fixes a :: "'a"   and b :: "'a" shows "\<exists>p q u v d. p * a + q * b = d \<and> d dvd a \<and> d dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd d) \<and> d * u = - b \<and> d * v = a"
sgn_of_real: fixes r :: "real" shows "sgn (of_real r) = of_real (sgn r)"
countably_compactI: fixes s :: "'b set" assumes "\<And>C. \<lbrakk>\<forall>t\<in>C. open t; s \<subseteq> \<Union> C; countable C\<rbrakk> \<Longrightarrow> \<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'" shows "countably_compact s"
set_replicate_conv_if: fixes n :: "nat"   and x :: "'c" shows "set (replicate n x) = (if n = 0 then {} else {x})"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "normalization_euclidean_semiring_class.gcd (normalization_euclidean_semiring_class.gcd a b) c = normalization_euclidean_semiring_class.gcd a (normalization_euclidean_semiring_class.gcd b c)"
join_pres_def: fixes X :: "('c, 'd) gorder_scheme"   and Y :: "('e, 'f) gorder_scheme"   and f :: "'c \<Rightarrow> 'e" shows "join_pres X Y f \<equiv> lattice X \<and> lattice Y \<and> (\<forall>x\<in>carrier X. \<forall>y\<in>carrier X. f (x \<squnion>\<^bsub>X\<^esub> y) = f x \<squnion>\<^bsub>Y\<^esub> f y)"
span_subspace: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> vs1.span A"   and "vs1.subspace B" shows "vs1.span A = B"
less_fps_def: fixes f :: "('a, 'b) fps"   and g :: "('a, 'b) fps" shows "(f < g) = (f \<le> g \<and> f \<noteq> g)"
pquvd: shows "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)"
simps_2: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and set_error_code :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s :: "'a"   and d :: "'d" assumes "Controllable_Interruptible_Separation_Kernel kstep output_f s0 current cswitch kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting set_error_code" shows "Controllable_Interruptible_Separation_Kernel.CISK_control aborting waiting set_error_code s d [[]] = (None, [], s)"
step_3: obtains w :: "'action list" where "path w p" and "target w p = q" and "set w \<inter> cuts = {}" and "w \<noteq> []"
L_zero_1: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "eNorm (\<LL> 0 M) f = (if f \<in> borel_measurable M then 0 else \<infinity>)"
assms_4: shows "h \<turnstile> get_elements_by_tag_name ptr name \<rightarrow>\<^sub>r results"
assms_3: fixes fdr :: "nat \<Rightarrow> nat \<Rightarrow> nat spmf" assumes "P fdr" shows "P (\<lambda>v c. if n \<le> v then if c < n then return_spmf c else fdr (v - n) (c - n) else coin_spmf \<bind> (\<lambda>b. fdr (2 * v) (2 * c + (if b then 1 else 0))))"
char_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "b \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "\<chi> (a \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> b) = \<chi> a * \<chi> b"
StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n: fixes x :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "x \<doteq> y \<equiv> StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y"
rep_eq: shows "Rep_\<I> \<I> = (\<lambda>x. if x \<in> A then A else {})"
assert_injective: fixes p :: "'c Assertion"   and q :: "'c Assertion" assumes "{\<cdot> p } = {\<cdot> q }" shows "p = q"
mem_fine2: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and u :: "real"   and z :: "real"   and v :: "real" assumes "fine \<delta> (a, b) D"   and "(u, z, v) \<in> set D" shows "a \<le> u \<and> v \<le> b"
iEx_empty: fixes P :: "nat \<Rightarrow> bool" shows "\<not> (\<diamond> t {}. P t)"
0: shows "\<exists>!document_ptr. document_ptr \<in> set |h \<turnstile> document_ptr_kinds_M|\<^sub>r \<and> node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
TNil_eq_tllist_of_llist: fixes b :: "'b"   and b' :: "'b"   and xs :: "'a llist" shows "(TNil b = tllist_of_llist b' xs) = (b = b' \<and> xs = LNil)"
H_x_def: shows "H_x \<equiv> remove_vertex new_last"
Abs_fset_inverse: fixes y :: "'b set" assumes "y \<in> {A. finite A}" shows "fset (Abs_fset y) = y"
epred_llcp: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs"   and "\<not> lnull ys"   and "lhd xs = lhd ys" shows "epred (llcp xs ys) = llcp (ltl xs) (ltl ys)"
simps_5: fixes v :: "'b"   and M :: "'b trm" shows "unify (Var v) M = (if Var v \<prec> M then None else Some [(v, M)])"
elims: fixes x :: "('i, 'g, 'h) generat"   and y :: "('g \<times> 'h) option" assumes "dest_IO x = y" obtains uu_ :: "'i" where "x = Pure uu_" and "y = None"   | out :: "'g" and c :: "'h" where "x = IO out c" and "y = Some (out, c)"
s_2: shows "learn_cons \<phi> U\<^bsub>LIM-CONS\<^esub> s"
simps_12: shows "cospan_shape.BB \<noteq> cospan_shape.AA"
assms_17: shows "InvariantClCurrentLevel (getCl state) (getM state)"
init: shows "path \<sigma> 0 = path \<sigma>' 0"
FinalOverriderMethodDef_def: fixes 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"   and R :: "char list \<times> char list list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list" shows "P \<turnstile> R has overrider M = mthd via Cs \<equiv> (Cs, mthd) \<in> OverriderMethodDefs P R M \<and> card (OverriderMethodDefs P R M) = 1"
simps_1: fixes I :: "nat set" shows "I \<rightarrow> 0 = iMin I"
assms: shows "\<exists>p r i. j = make_full_observation e None r [] i"
fmupd_alt_def: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) fmap" shows "fmupd k v m = m ++\<^sub>f fmap_of_list [(k, v)]"
finite_Ord_omega: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>" shows "finite (elts \<alpha>)"
repePos_simps: fixes r :: "real" shows "(POS_INF \<equiv>\<^sub>E r) = (2147483647 \<le> r)"
asscor2: shows "\<forall>n\<ge>6. real_of_int (a n) powr (1 / real_of_int (2 ^ n)) * (1 + 4 * (2 / 3) ^ n) \<le> A \<and> real_of_int (b n) \<le> 2 powr (4 / 3) ^ (n - 1)"
multiplicative_function'_axioms: shows "multiplicative_function' moebius_mu (\<lambda>p k. if k = 1 then - (1::'a) else (0::'a)) (\<lambda>_. - (1::'a))"
conv_invol: fixes x :: "'a" shows "x \<rhd> (1::'a) \<rhd> (1::'a) = x"
assms_4: shows "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"
rel_spmf_bind_reflI: fixes p :: "'b spmf"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c spmf"   and g :: "'b \<Rightarrow> 'd spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf P (f x) (g x)" shows "rel_spmf P (p \<bind> f) (p \<bind> g)"
A: shows "\<forall>\<^sub>F n in F. u n = inv_into S f (f (u n))"
simps_1: fixes uu :: "'a option" shows "less_eq_option None uu = True"
dvd_pderiv_iff: fixes p :: "'c poly" shows "(p dvd pderiv p) = (degree p = 0)"
outputTransition_def: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" shows "P \<Longrightarrow>a<\<nu>x> \<prec> P' \<equiv> \<exists>P''' P''. P \<Longrightarrow>\<^sub>\<tau> P''' \<and> P''' \<longmapsto> a<\<nu>x> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P'"
poly_number_rootat: fixes p :: "rat poly" shows "poly_inf (real_of_rat_poly p) = real_of_rat (poly_number_rootat p)"
ResF: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and x :: "name" assumes "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "<\<nu>x>P \<Longrightarrow>\<^sub>l \<alpha> \<prec> <\<nu>x>P'"
distincts_Cons: fixes xs :: "'b list"   and xss :: "'b list list" shows "distincts (xs # xss) = (xs \<noteq> [] \<and> distinct xs \<and> distincts xss \<and> set xs \<inter> \<Union> (sset xss) = {})"
cardB: shows "card ((\<lambda>x. (- 1 - x\<^sup>2) mod p) ` {0..n}) = card {0..n}"
mat_of_row_mult_append_rows: fixes y1 :: "'b vec"   and nr1 :: "nat"   and y2 :: "'b vec"   and nr2 :: "nat"   and A1 :: "'b mat"   and nc :: "nat"   and A2 :: "'b mat" assumes "y1 \<in> carrier_vec nr1"   and "y2 \<in> carrier_vec nr2"   and "A1 \<in> carrier_mat nr1 nc"   and "A2 \<in> carrier_mat nr2 nc" shows "mat_of_row (y1 @\<^sub>v y2) * (A1 @\<^sub>r A2) = mat_of_row y1 * A1 + mat_of_row y2 * A2"
in_homE: fixes f :: "'a1 \<times> 'a2"   and a :: "'a1 \<times> 'a2"   and b :: "'a1 \<times> 'a2" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 b\<guillemotright>" obtains "\<guillemotleft>fst f : fst a \<rightarrow>\<^sub>A\<^sub>1 fst b\<guillemotright>" and "\<guillemotleft>snd f : snd a \<rightarrow>\<^sub>A\<^sub>2 snd b\<guillemotright>"
trms_list\<^sub>s\<^sub>s\<^sub>t_is_trms\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('a, 'b) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S = set (trms_list\<^sub>s\<^sub>s\<^sub>t S)"
WellOrderedStrategies_def: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set" shows "WellOrderedStrategies G S p good r \<equiv> ParityGame G \<and> WellOrderedStrategies_axioms G S p good r"
compact_continuous_image: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "compact s" shows "compact (f ` s)"
\<sigma>_attracts: shows "strategy_attracts p \<sigma> (attractor p K) K"
collapse_1: fixes event :: "('d, 'e, 'f) event" assumes "isTrans event" shows "Trans (occurs_on event) (un_Trans2 event) (un_Trans3 event) = event"
rev_conv_fold: fixes xs :: "'c list" shows "rev xs = fold (#) xs []"
LIMSEQ_SEQ_conv1: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and a :: "'c" assumes "f \<midarrow>a\<rightarrow> l" shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
bounded_exec': fixes w :: "nat list"   and a :: "nat"   and b :: "nat list"   and ss :: "nat list list" assumes "w \<noteq> []"   and "(a, b) \<in> set (step (hd w) (ss ! hd w))"   and "\<forall>p\<in>set w. p < n" shows "a < n"
eval_powr: 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 abort :: "bool"   and F :: "'a"   and p :: "real" 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" shows "eval (powr_expansion abort F p) = (\<lambda>x. eval F x powr p)"
select_index_belongs: fixes X :: "'c set"   and U :: "'c set"   and index :: "real set"   and cover :: "real \<Rightarrow> 'c set"   and x :: "'c" assumes "cover_of_subset X U index cover"   and "x \<in> U" shows "cover_of_subset.select_index index cover x \<in> index"
rcos_ring_hom: shows "rupture_surj K p \<in> ring_hom (K [X]) (K [X] Quot PIdl\<^bsub>K [X]\<^esub> p)"
rerr_invalid_any_step_invariant: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), a, uu_). anycast (rerr_invalid (rt \<xi>)) a)"
iftrue: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and c1 :: "('exp, 'id) MWLfCom"   and c2 :: "('exp, 'id) MWLfCom" assumes "BMap (E b m) = True" shows "SRabbr (if b then c1 else c2 fi) m [c1] m"
subset_empty: fixes A :: "'d set" shows "(A \<subseteq> {}) = (A = {})"
dist_commute: fixes dist :: "'g \<Rightarrow> 'g \<Rightarrow> real"   and uniformity :: "('g \<times> 'g) filter"   and "open" :: "'g set \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.metric_space dist uniformity open" shows "dist x y = dist y x"
wlp_Loop: fixes body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and I :: "'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "well_def body"   and "unitary I"   and "wlp_inv G body I" shows "I \<tturnstile> wlp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) (\<lambda>s. \<guillemotleft> \<N> G \<guillemotright> s * I s)"
3: shows "ik\<^sub>s\<^sub>t (proj_unl n C) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>"
sup_bot_right: fixes x :: "'b multiset" shows "x \<union># \<Empt> = x"
Lt_le_LtI: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "Lt a \<le> Lt b"
fls_lr_inverse_eq_0_iff_2: fixes g :: "'a fls"   and y :: "'a" shows "(fls_right_inverse g y = 0) = (y = (0::'a))"
elims_3: fixes x :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "\<not> shift1_inv x" obtains qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" where "x = (qs, B)" and "\<not> (B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)} \<and> shift2_inv qs)"
ev_eq_suntil: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "ev P \<omega> = ((\<lambda>xs. \<not> P xs) suntil P) \<omega>"
A_ub: fixes t :: "'b tree"   and la :: "'b tree"   and a :: "'b"   and ra :: "'b tree" assumes "bst t"   and "\<langle>la, a, ra\<rangle> \<in> subtrees t" shows "A a t \<le> log \<alpha> (real (size1 t) / real (size1 la + size1 ra)) + 1"
a1: shows "fst (case load_word_mem s1 addra 8 of None \<Rightarrow> (None, s1) | Some w \<Rightarrow> (Some w, add_instr_cache s1 addra w 15)) = fst (case load_word_mem s2 addra 8 of None \<Rightarrow> (None, s2) | Some w \<Rightarrow> (Some w, add_instr_cache s2 addra w 15))"
prim_precondition_def: fixes g :: "'a"   and r :: "'a" shows "prim_precondition g r \<equiv> g = g\<^sup>T \<and> injective r \<and> vector r \<and> regular r"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vreal_plus"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vreal_plus" shows "b = c"
h1: fixes i :: "'a"   and x :: "'b" assumes "i \<in> I" shows "h i (1, x) = g i x"
\<I>_def: shows "\<I> \<equiv> \<I>_uniform (Set.Collect (valid_mac_query \<eta>)) (insert None (Some ` (nlists UNIV \<eta> \<times> nlists UNIV \<eta>))) \<oplus>\<^sub>\<I> (\<I>_uniform (vld \<eta>) UNIV \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` vld \<eta>)))"
inc_dot_end: fixes d :: "nat"   and x :: "item" shows "item_end (inc_dot d x) = item_end x"
eq1: shows "tensor_P (1\<^sub>m N) (mat_incr K) * Q2 * tensor_P (1\<^sub>m N) (adjoint (mat_incr K)) = matrix_sum d (\<lambda>l. tensor_P (proj_psi_l (l + 1)) (proj_k (l + 1))) R"
le_dag_trans: fixes x :: "dag"   and y :: "dag"   and z :: "dag" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
x: shows "\<forall>\<^sub>F x in F. f x \<le> l"
phi'_def: fixes m :: "nat" shows "phi' m = card {x. 1 \<le> x \<and> x \<le> m \<and> coprime x m}"
sigma_finite_disjoint: obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets M" and "\<Union> (range A) = space M" and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and "disjoint_family A"
assms: shows "r \<subseteq>\<^sub>\<circ> s"
aff_dim_basis_exists: fixes V :: "'a set" shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
simps_5: fixes et :: "('c, 'd, 'e, 'f) edge_kind" shows "local.CFG.pred et [] = False"
exI3: fixes Pred :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and z :: "'e" assumes "Pred x y z" shows "\<exists>x y z. Pred x y z"
set_permutation_bij_restrict1: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "bij_betw f A A" shows "bij (restrict1 f A)"
x_2: shows "Fun f T \<sqsubseteq> (\<theta>1 \<circ>\<^sub>s \<theta>2) x"