hcju/isabelleps huggingface dataset:
image_chamber_system_image: shows "ChamberComplex.chamber_system (f \<turnstile> X) = \<C> \<inter> (f \<turnstile> X)"
fps_distl: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps"   and h :: "('c, 'd) fps" shows "f \<cdot> (g + h) = f \<cdot> g + f \<cdot> h"
swap_swap: fixes p :: "'b \<times> 'c" shows "prod.swap (prod.swap p) = p"
approx_binE: fixes F :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and w :: "real"   and e :: "nat \<Rightarrow> real"   and W' :: "real \<times> real pdevs"   and errw :: "real"   and x :: "real"   and X' :: "real \<times> real pdevs"   and errx :: "real"   and p :: "nat"   and Ye :: "(real \<times> real pdevs) \<times> real" assumes "interval_extension2 F f"   and "w \<in> aform_err e (W', errw)"   and "x \<in> aform_err e (X', errx)"   and "approx_bin p F (W', errw) (X', errx) = Some Ye"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "f w x \<in> aform_err e Ye"
per_exp_pref: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r \<cdot> u" shows "u \<le>p r \<^sup>@ k \<cdot> u"
spmf_map_pmf_Some: fixes p :: "'b pmf"   and i :: "'b" shows "spmf (map_pmf Some p) i = pmf p i"
list_slice_nth_length: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "m < length xs div k" shows "length (list_slice xs k ! m) = k"
squarefree_part_le: fixes p :: "nat"   and n :: "nat" assumes "p \<in> squarefree_part n" shows "p \<le> n"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "\<And>t. P \<langle>\<rangle> t"   and "\<And>t. P t \<langle>\<rangle>"   and "\<And>l1 a1 r1 l2 a2 r2. \<lbrakk>a1 \<le> a2 \<Longrightarrow> P \<langle>l2, a2, r2\<rangle> r1; \<not> a1 \<le> a2 \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> r2\<rbrakk> \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> \<langle>l2, a2, r2\<rangle>" shows "P a0 a1"
length_ode_d_expr: fixes f :: "nat"   and n :: "nat" shows "length (ode_d_expr f n) = length ode_e"
smult_l_distr: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M" shows "(a \<oplus> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x"
simps_2: fixes U :: "complex mat"   and P :: "complex mat" shows "wlp (Utrans U) P = adjoint U * P * U"
fmap_error_simps_1: fixes f :: "'c \<rightarrow> 'a" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
closure_Int: fixes I :: "'a set set" shows "closure (\<Inter> I) \<subseteq> \<Inter> {closure S |S. S \<in> I}"
sum_mat_0: fixes nr :: "nat"   and nc :: "nat" shows "sum_mat (0\<^sub>m nr nc) = (0::'b)"
outstanding_volatile_writes_unowned_by_others: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "j < length ts\<^sub>s\<^sub>b"   and "i \<noteq> j"   and "ts\<^sub>s\<^sub>b ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts\<^sub>s\<^sub>b ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>i = {}"
msetext_huet_trans_from_irrefl: fixes A :: "'b set"   and zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "zs \<in> lists A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>x\<in>A. \<not> gt x x"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "msetext_huet gt zs ys"   and "msetext_huet gt ys xs" shows "msetext_huet gt zs xs"
cases: fixes a1 :: "state"   and a2 :: "com list"   and a3 :: "state" assumes "a1 =a2\<Rightarrow> a3" obtains s :: "state" where "a1 = s__" and "a2 = []" and "a3 = s__"   | s :: "state" and c :: "com" and t :: "state" and cs :: "com list" and u :: "state" where "a1 = s" and "a2 = c # cs" and "a3 = u" and "s -c\<rightarrow> t" and "t =cs\<Rightarrow> u"
mono: fixes y :: "real"   and x :: "real" assumes "exp (1 / 2) \<le> y"   and "y \<le> x" shows "ln x / x\<^sup>2 \<le> ln y / y\<^sup>2"
Set_by_Ordered_axioms: shows "Set_by_Ordered empty insert delete isin inorder inv"
xor_commute: fixes x :: "'l"   and y :: "'l" shows "x \<oplus> y = y \<oplus> x"
simps_67: fixes x101 :: "stmt"   and x102 :: "stmt"   and x31 :: "jump"   and x32 :: "stmt" shows "x101 Finally x102 \<noteq> x31\<bullet> x32"
plain_RETURN: fixes a :: "'c" shows "RETURN a \<le> RETURN a"
find_first_append: fixes x :: "'a"   and ys :: "'a list"   and zs :: "'a list" shows "find_first x (ys @ zs) = (case find_first x ys of None \<Rightarrow> map_option (\<lambda>i. i + length ys) (find_first x zs) | Some a \<Rightarrow> Some a)"
th0: shows "(fps_const c * fps_X) $ 0 = (0::'a)"
subdivision_genus_same_rev: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and HM :: "'d pre_map"   and m :: "int" assumes "subdivision (G, rev_G) (H, edge_rev HM)"   and "digraph_map H HM"   and "pre_digraph_map.euler_genus H HM = m" shows "\<exists>GM. digraph_map G GM \<and> pre_digraph_map.euler_genus G GM = m \<and> edge_rev GM = rev_G"
Inr_not_Inl: fixes b :: "'c"   and a :: "'b" shows "Inr b \<noteq> Inl a"
conv: shows "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) at_top"
reachable_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and b :: "'b"   and a :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "Q \<in> \<P>"   and "b \<in> \<E>"   and "a \<in> Q - MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b" shows "\<exists>R\<in>\<P>. a \<in> R \<and> b \<in> R"
group_sum_image_ker: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and g :: "'d \<Rightarrow> 'g"   and K :: "('g, 'h) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H K"   and "(g \<circ> f) ` carrier G = carrier K"   and "Group.group G"   and "Group.group H"   and "Group.group K" shows "f ` carrier G <#>\<^bsub>H\<^esub> kernel H K g = carrier H"
order_tendstoD_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and y :: "'b"   and F :: "'c filter"   and a :: "'b" assumes "class.order_topology less_eq less open"   and "filterlim f (topological_space.nhds open y) F"   and "less a y" shows "\<forall>\<^sub>F x in F. less a (f x)"
insert_reachable: fixes x :: "'a"   and y :: "'a"   and F :: "'edge set"   and e :: "'edge" assumes "x \<in> V"   and "y \<in> V"   and "F \<subseteq> E"   and "e \<in> E"   and "joins x y e" shows "connected (insert e F) = per_union (connected F) x y"
yx_1: shows "y = x \<or> y \<notin> Fvars \<phi>"
internal_is_path: fixes ds :: "bool list"   and T :: "tree" assumes "internal ds T" shows "path ds T"
fmdom'_map: fixes f :: "'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmdom' (fmmap f m) = fmdom' m"
mojmir_accept_to_Acc: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "mojmir_def.accept \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"   and "\<pi> (G \<psi>) = mojmir_def.smallest_accepting_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}" shows "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> (G \<psi>)) w"
distinct_2lcomprI: fixes as :: "'a list"   and bs :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "distinct as"   and "distinct bs"   and "\<And>a b e i. f a b = f e i \<Longrightarrow> a = e \<and> b = i" shows "distinct (concat (map (\<lambda>a. map (f a) bs) as))"
mktop_def: fixes L :: "(alt \<times> alt) set"   and b :: "alt" shows "mktop L b \<equiv> {(x, y). x <\<^bsub>L\<^esub> y \<and> x \<noteq> b \<and> y \<noteq> b} \<union> {(x, b) |x. x \<noteq> b}"
le_infI2: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and b :: "'b::type"   and x :: "'b::type"   and a :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (b::'b::type) (x::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type)) (x::'b::type)"
inv_in_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "D.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>D.inv f : b \<Rightarrow>\<^sub>D a\<guillemotright>"
H_iso_seq_var: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and y :: "'b"   and g :: "'d \<Rightarrow> 'b"   and z :: "'d" assumes "mono f"   and "x \<le> f y"   and "y \<le> g z" shows "x \<le> (f \<circ> g) z"
one_cinner_one: shows "(1::'a) \<bullet>\<^sub>C (1::'a) = 1"
is_generalisation_of_constrains_input: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" assumes "is_generalisation_of t' t i r" shows "\<exists>v. Eq (V (I i)) (L v) \<in> set (Guards t)"
thetaParZOL2_converse_ZOretr: shows "thetaParZOL2\<inverse> \<subseteq> ZOretr (thetaParZOL2\<inverse> \<union> ZObis)"
Assign: fixes e :: "exp"   and s :: "loc \<Rightarrow> nat"   and v :: "nat"   and s' :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set"   and x :: "loc" assumes "((e, s), v, s') \<in> eval" shows "(x := e, s) -[eval]-> s'(x := v)"
norm_bound_Basis_le: fixes b :: "'a"   and x :: "'a"   and e :: "real" assumes "b \<in> Basis"   and "norm x \<le> e" shows "\<bar>x \<bullet> b\<bar> \<le> e"
is_ground_atm_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'b" shows "substitution_ops.is_ground_atm subst_atm A = (\<forall>\<sigma>. A = subst_atm A \<sigma>)"
abs_residual_pair_eqvt: fixes p :: "perm"   and \<alpha> :: "'b"   and P :: "'c" shows "p \<bullet> \<langle>\<alpha>,P\<rangle> = \<langle>p \<bullet> \<alpha>,p \<bullet> P\<rangle>"
face_of_Times: fixes F :: "'b set"   and S :: "'b set"   and F' :: "'c set"   and S' :: "'c set" assumes "F face_of S"   and "F' face_of S'" shows "F \<times> F' face_of S \<times> S'"
fst_Sup: fixes A :: "('c \<times> 'd) set" shows "fst (Sup A) = Sup (fst ` A)"
bit_bshiftr1_iff: fixes b :: "bool"   and w :: "'b word"   and n :: "nat" shows "bit (bshiftr1 b w) n = (b \<and> n = LENGTH('b) - 1 \<or> bit w (Suc n))"
pm_pth_def: fixes m :: "('c, 'd) pointermap_scheme"   and p :: "nat" shows "pm_pth m p \<equiv> entries m ! p"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "B x \<in> sets M"
mat_carrier: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'c" shows "mat nr nc f \<in> carrier_mat nr nc"
noWhile_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "noWhile c"   and "(c, s) \<rightarrow>c (c', s')" shows "noWhile c'"
sub: shows "subresultant_prs dichotomous_Lazard G1 G2 = (Gk, hk)"
stable_on_Xd: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "PdXXX.stable_on ds X"   and "x \<in> X" shows "Xd x \<in> ds"
has_contour_integral_reverse_linepath: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex"   and b :: "complex" assumes "(f has_contour_integral i) (linepath a b)" shows "(f has_contour_integral - i) (linepath b a)"
irrefl_prec_F: fixes C :: "'f" shows "\<not> C \<prec>\<cdot> C"
bernoulli_def: fixes p :: "val" shows "bernoulli p = density (stock_measure BOOL) (bernoulli_density (extract_real p) \<circ> extract_bool)"
or_int_rec: shows "x BIT b OR y BIT c = of_bool (odd (x BIT b) \<or> odd (y BIT c)) + 2 * (x BIT b div 2 OR y BIT c div 2)"
simps_1: fixes r :: "'b rexp"   and s :: "'b rexp" shows "toplevel_summands (Plus r s) = toplevel_summands r \<union> toplevel_summands s"
Rep_name: fixes x :: "name" shows "Rep_name x \<in> {a. sort_of a = Sort ''name'' []}"
trans: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ordering_ow U le ls"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U"   and "le a b"   and "le b c" shows "le a c"
simps: fixes n :: "enat"   and xs :: "'c llist" shows "ldrop n xs = (case n of 0 \<Rightarrow> xs | eSuc n' \<Rightarrow> case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> ldrop n' xs')"
left_unique_rel_lcI: assumes "left_unique rel_c" shows "left_unique rel_lc"
convex_bound_le: fixes x :: "'a"   and a :: "'a"   and y :: "'a"   and u :: "'a"   and v :: "'a" assumes "x \<le> a"   and "y \<le> a"   and "(0::'a) \<le> u"   and "(0::'a) \<le> v"   and "u + v = (1::'a)" shows "u * x + v * y \<le> a"
diaR: fixes A :: "(C, BD) form" shows "g3mod2p [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] ( \<Empt> \<Rightarrow>* \<LM> \<diamond> A  \<RM>)"
Fvars_exi: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a"   and \<phi> :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "x \<in> var"   and "\<phi> \<in> fmla" shows "Fvars (exi x \<phi>) = Fvars \<phi> - {x}"
take_bit_eq_mod: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n a = modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
TESL_interp_stepwise_strictly_precedes_coind_unfold: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and n :: "nat" shows "\<lbrakk> K\<^sub>1 strictly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> K\<^sub>1 strictly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
R_def: shows "R = (\<lambda>s. stirling_integral m s / of_nat m)"
axioms_1: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "H \<lhd> G" shows "subgroup H G"
WHILE_le_WHILEI: fixes b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and s :: "'a"   and I :: "'a \<Rightarrow> bool" shows "WHILE b f s \<le> WHILE\<^bsup>I\<^esup> b f s"
list_all_simps_1: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "list_all P (x # xs) = (P x \<and> list_all P xs)"
linear_invertible_bounded_below_pos: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "linear f"   and "linear g"   and "g \<circ> f = id" obtains B :: "real" where "0 < B" and "\<And>x. B * norm x \<le> norm (f x)"
eq_diff_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(a = minus c b) = (plus a b = c)"
linmap_im_linspan1: fixes A :: "'b set"   and N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e"   and H :: "'a set"   and h :: "'a" assumes "ideal R A"   and "R module N"   and "f \<in> mHom R M N"   and "H \<subseteq> carrier M"   and "h \<in> linear_span R M A H" shows "f h \<in> linear_span R N A (f ` H)"
f_Exec_Stream_expand_aggregate_map_nth_eq_i_nth': fixes k :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun (input \<Down> Suc n \<odot> k) c)) k ag ! n = i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag n"
real_div_nat_eq_floor_of_divide: fixes a :: "nat"   and b :: "nat" shows "real (a div b) = real_of_int \<lfloor>real a / real b\<rfloor>"
cong_less_modulus_unique_poly: fixes x :: "'b poly"   and y :: "'b poly"   and m :: "'b poly" assumes "[x = y] (mod m)"   and "degree x < degree m"   and "degree y < degree m" shows "x = y"
maximalD1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "class.order less_eq less"   and "x \<in> order.maximal less_eq f S" shows "x \<in> S"
in_listsp_conv_set: fixes A :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "listsp A xs = (\<forall>x\<in>set xs. A x)"
threes_def: shows "threes = (SOME (x, y, z). x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z)"
assms_1: shows "\<guillemotleft>\<mu> : x \<Rightarrow>\<^sub>S y\<guillemotright>"
lunstream_unfold_llist_raw: fixes stop :: "'d \<Rightarrow> bool"   and head :: "'d \<Rightarrow> 'c"   and tail :: "'d \<Rightarrow> 'd"   and s :: "'d" shows "lunstream (unfold_llist_raw stop head tail) s = unfold_llist stop head tail s"
bind_spmf_cong: fixes p :: "'h spmf"   and q :: "'h spmf"   and f :: "'h \<Rightarrow> 'i spmf"   and g :: "'h \<Rightarrow> 'i spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
\<beta>_decomposed: shows "\<beta> = take j b @ drop i a"
dim_vec_of_list: fixes as :: "'c list" shows "dim_vec (vec_of_list as) = length as"
wfG: shows "wf_graph \<lparr>nodes = N, edges = E\<rparr>"
card_Un_disjoint: fixes A :: "'d set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "card (A \<union> B) = card A + card B"
zero_le_floor: fixes x :: "'a" shows "(0 \<le> \<lfloor>x\<rfloor>) = ((0::'a) \<le> x)"
A: shows "\<delta>\<^sub>p' = dist_code p\<^sub>0 p\<^sub>1'"
tsubstT_matcher_imp_raw_match'_unchanged: fixes T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ"   and U :: "typ" assumes "tsubstT T \<rho> = U" shows "raw_match' T U (\<lambda>(idx, S). Some (\<rho> idx S)) = Some (\<lambda>(idx, S). Some (\<rho> idx S))"
minus_int_code_1: fixes k :: "int" shows "k - 0 = k"
wf_mstate_msteps_stateless_UNIV: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and st :: "'a mstate"   and \<pi>' :: "(char list \<times> 'a list) prefix" assumes "wf_mstate \<phi> \<pi> UNIV st"   and "\<pi> \<le> \<pi>'" shows "msteps_stateless (pdrop (plen \<pi>) \<pi>') st = M \<pi>' - M \<pi>"
possible_both: fixes t :: "'a ttree"   and t' :: "'a ttree"   and x :: "'a" shows "possible (t \<otimes>\<otimes> t') x = (possible t x \<or> possible t' x)"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "lex_pp_strict x xa = lex_pm_strict (pp.mapping_of x) (pp.mapping_of xa)"
simps_3: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly"   and c' :: "'b" shows "CN c n p *\<^sub>p (c')\<^sub>p = (if c' = (0::'b) then 0\<^sub>p else CN (c *\<^sub>p (c')\<^sub>p) n (p *\<^sub>p (c')\<^sub>p))"
pred_spmf_calleeI: fixes s :: "'b"   and x :: "'e" assumes "I2 s"   and "x \<in> outs_\<I> \<I>_full" shows "pred_spmf (pred_prod (\<lambda>_. True) I2) (callee2 s x)"
is_ord_insrt: fixes t :: "'a tree\<^sub>0"   and x :: "'a" assumes "is_ord t" shows "is_ord (insrt\<^sub>0 x t)"
ground_vars_term_ms: fixes t :: "('b, 'c) Term.term" assumes "ground t" shows "vars_term_ms t = {#}"
lt_ex: fixes x :: "'b" shows "\<exists>y. y < x"
compl_eq_compl_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "(uminus x = uminus y) = (x = y)"
return_bind: fixes x :: "'a fset"   and f :: "'a fset \<Rightarrow> 'm" shows "bind (return x) f = f x"
delete_min_interval_inv: fixes t :: "interval_tree" assumes "tree_interval_inv t"   and "t \<noteq> Tip" shows "tree_interval_inv (snd (del_min t))"
locked_unique: fixes s :: "v_state"   and v :: "val"   and w :: "val" assumes "s \<in> Vinv2"   and "v \<in> locked s"   and "w \<in> locked s" shows "v = w"
fixgames_def: fixes A :: "ZF set" shows "fixgames A \<equiv> {Opair l r |l r. explode l \<subseteq> A \<and> explode r \<subseteq> A}"
comp_red_basis_aux_rec: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "comp_red_basis_aux (x # xs) ys = comp_red_basis_aux xs (trd (xs @ ys) x # ys)"
l2: fixes x :: "int" assumes "\<not> QuadRes (int p) a"   and "x \<in> local.S1" shows "[\<Prod>(local.f x) = a] (mod int p)"
smult_0_right: fixes a :: "'c" shows "smult a 0 = 0"
il_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
plus_right_mono: fixes z :: "'b"   and y :: "'b"   and x :: "'b" assumes "z \<le> y" shows "x + z \<le> x + y"
swap: fixes a :: "'item"   and b :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "ind a b" shows "u @ [a] @ [b] @ v =\<^sub>S u @ [b] @ [a] @ v"
any_unop_type_det: fixes op :: "any_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "any_unop_type op \<tau> \<sigma>\<^sub>1"   and "any_unop_type op \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
canceling_terminates: shows "wfP cancels_to_1\<inverse>\<inverse>"
vmc_path_lset_induction: fixes Q :: "'a llist \<Rightarrow> bool"   and S :: "'a set" assumes "Q P"   and "v0 \<in> S"   and "\<And>P v0. \<lbrakk>vmc_path_no_deadend G P v0 p \<sigma>; v0 \<in> S; Q P\<rbrakk> \<Longrightarrow> Q (ltl P) \<and> vmc_path_no_deadend.w0 P \<in> S" shows "lset P \<subseteq> S"