hcju/isabelleps huggingface dataset:
converse_rtrancl_induct_red: fixes e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and R :: "expr \<Rightarrow> (nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> expr \<Rightarrow> (nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"   and "\<And>e h l. R e h l e h l"   and "\<And>e\<^sub>0 h\<^sub>0 l\<^sub>0 e\<^sub>1 h\<^sub>1 l\<^sub>1 e' h' l'. \<lbrakk>P,E \<turnstile> \<langle>e\<^sub>0,(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1)\<rangle>; R e\<^sub>1 h\<^sub>1 l\<^sub>1 e' h' l'\<rbrakk> \<Longrightarrow> R e\<^sub>0 h\<^sub>0 l\<^sub>0 e' h' l'" shows "R e h l e' h' l'"
sort_oalist_ko_ntm'_def: fixes ko :: "'a nat_term_order" shows "sort_oalist_ko_ntm' ko \<equiv> comparator.sort_oalist (nat_term_compare_inv ko)"
set_sort: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" assumes "class.linorder less_eq less" shows "set (linorder.sort_key less_eq f xs) = set xs"
sigma_finite: fixes I :: "nat set" assumes "finite I" shows "sigma_finite_measure (Pi\<^sub>M I (\<lambda>x. stock_measure (\<Gamma> x)))"
decisive3: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x z" shows "decisive swf A Is {j} y z"
nn_integral_measure_pmf: fixes p :: "'h pmf"   and f :: "'h \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = (\<Sum>\<^sup>+ x. ennreal (pmf p x) * f x)"
not_in_arcs_G: shows "{uw, swap_in (parcs H) uw, vw, swap_in (parcs H) vw} \<inter> arcs (with_proj I') = {}"
inject_3: fixes x41 :: "PreSimplyTyped.type"   and x42 :: "PreSimplyTyped.type"   and y41 :: "PreSimplyTyped.type"   and y42 :: "PreSimplyTyped.type" shows "(TPair x41 x42 = TPair y41 y42) = (x41 = y41 \<and> x42 = y42)"
euclidean_ring_ops_axioms: shows "euclidean_ring_ops ops R"
tlookup_finds: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and y :: "'b" shows "sortedTree h t \<and> y \<in> setOf t \<longrightarrow> tlookup h (h y) t = Some y"
g_2: fixes v :: "'c" assumes "v \<in> C" shows "f (g v) = v"
coeff_map_poly_hom: fixes p :: "int poly"   and i :: "nat" shows "poly.coeff (of_int_poly p) i = of_int (poly.coeff p i)"
bit_c''1: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 75 / 10\<^sup>2 * real (length qs) - 5 / 10"
simps_2: fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "dual\<^sub>s\<^sub>t (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = send\<langle>t\<rangle>\<^sub>s\<^sub>t # dual\<^sub>s\<^sub>t S"
IH1: shows "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (T - t 0) + 1) - 1))"
inf_right_idem: 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 x :: "'b::type"   and y :: "'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)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (inf (x::'b::type) (y::'b::type)) y = inf x y"
induct: fixes x1 :: "cmd"   and x2 :: "nat"   and x3 :: "cmd"   and P :: "cmd \<Rightarrow> nat \<Rightarrow> cmd \<Rightarrow> bool" assumes "labels x1 x2 x3"   and "\<And>c. P c 0 c"   and "\<And>V e. P (V:=e) 1 Skip"   and "\<And>c\<^sub>1 l c c\<^sub>2. \<lbrakk>labels c\<^sub>1 l c; P c\<^sub>1 l c\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) l (c;; c\<^sub>2)"   and "\<And>c\<^sub>2 l c c\<^sub>1. \<lbrakk>labels c\<^sub>2 l c; P c\<^sub>2 l c\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) (l + #:c\<^sub>1) c"   and "\<And>c\<^sub>1 l c b c\<^sub>2. \<lbrakk>labels c\<^sub>1 l c; P c\<^sub>1 l c\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (l + 1) c"   and "\<And>c\<^sub>2 l c b c\<^sub>1. \<lbrakk>labels c\<^sub>2 l c; P c\<^sub>2 l c\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (l + #:c\<^sub>1 + 1) c"   and "\<And>c' l c b. \<lbrakk>labels c' l c; P c' l c\<rbrakk> \<Longrightarrow> P (while (b) c') (l + 2) (c;; while (b) c')"   and "\<And>b c'. P (while (b) c') 1 Skip"   and "\<And>p es rets. P (Call p es rets) 1 Skip" shows "P x1 x2 x3"
deg_const: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "deg Zp (monom (UP Zp) a 0) = 0"
ge: fixes x :: "'a" assumes "x \<in> set xs" shows "(0::'a) \<le> x"
rangeUinv_def: shows "rangeUinv = Uinv *\<^sub>S \<top>"
cut_same_def: fixes x :: "'c"   and s :: "'c stream" shows "cut_same x s = stake (LEAST n. sdrop n s = sconst x) s"
sin_plus_pi: fixes z :: "'a" shows "sin (z + of_real pi) = - sin z"
frac_1_eq: fixes x :: "'a" shows "frac (x + (1::'a)) = frac x"
transE: fixes r :: "('k \<times> 'k) set"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" obtains "(x, z) \<in> r"
eq: fixes x :: "nat"   and y :: "nat" assumes "x \<in> set xs" shows "(y = x) = (s y = x)"
finite_proj_image: fixes P :: "'h \<Rightarrow>\<^sub>F 'g"   and S :: "'h set" shows "finite ((P)\<^sub>F ` S)"
locally_closed: fixes S :: "'b set" shows "locally closed S = locally compact S"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Arr (t \<^bold>\<otimes> u) = (Arr t \<and> Arr u)"
RD_on_mono: fixes ds :: "'b set" shows "mono (Rf (CD_on ds))"
S_list_pgallery_decomp3: fixes ss :: "'a permutation list"   and Cs :: "'a set list" assumes "set ss = (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "Cs \<noteq> []"   and "pgallery (C0 # Cs)" shows "\<exists>rs s ts As B Bs. ss = rs @ s # ts \<and> Cs = As @ B # Bs \<and> (\<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> B \<in> g \<turnstile> \<C>) \<and> (\<forall>A\<in>set As. \<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> A \<in> f \<turnstile> \<C>) \<and> (\<forall>r\<in>set rs. \<forall>C\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> C \<in> f \<turnstile> \<C>)"
eventually_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>P = ((\<lambda>s. True) \<U> P)"
assms_2: shows "finite (reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0')"
strongly_accepts_imp_accepts: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set" assumes "strongly_accepts \<F> S M"   and "infinite M" shows "accepts \<F> S M"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (le a b \<and> le b a)"
leg1_commutes: shows "cod.leg1 \<cdot> chine = leg1"
hfref_def: fixes P :: "'g \<Rightarrow> bool"   and RS :: "('g \<Rightarrow> 'e \<Rightarrow> assn) \<times> ('g \<Rightarrow> 'e \<Rightarrow> assn)"   and T :: "'h \<Rightarrow> 'f \<Rightarrow> assn" shows "[P]\<^sub>a RS \<rightarrow> T \<equiv> {(f, g). \<forall>c a. P a \<longrightarrow> hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)}"
Amk_not_zero: shows "A $ m $ mod_type_class.from_nat k \<noteq> (0::'a)"
C1_differentiable_imp_piecewise: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "f C1_differentiable_on S" shows "f piecewise_C1_differentiable_on S"
qsort_permutes: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "set (qsort xs l) = set xs"
cod_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.cod f = B.VxVxV.cod f"
Rep_uint64'_def: shows "Rep_uint64' = Rep_uint64"
containID: fixes i :: "nat"   and \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable (ext R R' M N)"
sum_Un_nat: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> nat" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
zn1: shows "continuous_on (- {0}) (\<lambda>z. z ^ n)"
fls_divide_1': fixes f :: "'a fls" assumes "inverse (1::'a) = (1::'a)" shows "f / 1 = f"
triple_judgement_def: fixes pre :: "'c \<Rightarrow> bool"   and body :: "'d"   and property :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "triple_judgement pre body property = (\<forall>s. pre s \<longrightarrow> property s body)"
split_bottom_iff: fixes pat :: "[:'r:]"   and xs :: "[:'r:]" shows "(split\<cdot>pat\<cdot>xs = \<bottom>) = (pat = \<bottom> \<or> xs = \<bottom>)"
hsub: shows "h ` ({0..1} \<times> S) \<subseteq> S"
nn_integral_LIMSEQ: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure"   and u :: "'b \<Rightarrow> ennreal" assumes "incseq f"   and "\<And>i. f i \<in> borel_measurable M"   and "\<And>x. (\<lambda>i. f i x) \<longlonglongrightarrow> u x" shows "(\<lambda>n. integral\<^sup>N M (f n)) \<longlonglongrightarrow> integral\<^sup>N M u"
And_semantics: fixes w :: "nat \<Rightarrow> 'b set"   and xs :: "'b ltl list" shows "w \<Turnstile> And xs = (\<forall>x\<in>set xs. w \<Turnstile> x)"
disj_conj_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x OR y AND z = (x OR y) AND (x OR z)"
leq_seq_def: fixes xp :: "'a Predicate.seq"   and Q :: "'a Predicate.pred" shows "local.leq_seq xp Q = local.leq_pred (Predicate.pred_of_seq xp) Q"
size_mset_def: fixes M :: "('d \<times> 'e) multiset" shows "size_mset M = fold_mset ((+) \<circ> pair_size) 0 M"
le_sup_equiv1: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<squnion> b = b)"
compile_env_rel_compile_fundef: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'fun \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and F1 :: "'a"   and F2 :: "'f"   and f :: "'fun" assumes "compile_env \<A> \<O> F1 = Ok F2" shows "rel_option (\<lambda>fd1 fd2. compile_fundef \<A> (\<O> f) fd1 = Ok fd2) (Finca_get F1 f) (Fubx_get F2 f)"
disc_2: fixes x21 :: "variable"   and x22 :: "typ" shows "\<not> is_Ct (Fv x21 x22)"
s_trans: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and D :: "nat"   and E :: "nat" assumes "CT \<turnstile> C <: D"   and "CT \<turnstile> D <: E" shows "CT \<turnstile> C <: E"
countNonFinals_def: fixes g :: "graph" shows "countNonFinals g \<equiv> |nonFinals g|"
simps_10: fixes v :: "formula" shows "get_and_list (Exists v) = [Exists v]"
b_e_type_grow_memory: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Grow_memory" shows "\<exists>ts''. ts = ts'' @ [T_i32] \<and> ts = ts' \<and> (\<exists>n. memory \<C> = Some n)"
less_eq_idiff_eq_sum: fixes y :: "enat"   and x :: "enat"   and z :: "enat" assumes "y \<le> x" shows "(z \<le> x - y) = (z + y \<le> x)"
in_open_segment_iff_rot: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<noteq> b" shows "(x \<in> {a<--<b}) = ((x - a) \<bullet> rev.rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)})"
aligned\<^sub>d_0: fixes M :: "hmultiset" shows "local.aligned\<^sub>d 0 M"
x_1: shows "(\<lambda>xa. ereal (birkhoff_sum f xa x)) \<longlonglongrightarrow> \<infinity>"
effect__strips_def: fixes op :: "'a strips_operator" shows "effect__strips op = map (\<lambda>v. (v, True)) (add_effects_of op) @ map (\<lambda>v. (v, False)) (delete_effects_of op)"
simps: fixes v :: "(real, 'd) vec \<times> (real, 'd) vec"   and x :: "'d"   and r :: "real" shows "repv v x r = (\<chi>y. if x = y then r else fst v $ y, snd v)"
PseudoHA_Events: fixes SA :: "('e, 'd, 'f) seqauto"   and D :: "'f data" shows "HAEvents (PseudoHA SA D) = SAEvents SA"
A_def: fixes p :: "Player" shows "A p = attractor p (deadends p**)"
atS_def: fixes p :: "'f"   and ls :: "'g set"   and s :: "('h, 'g, 'f, 'i, 'j) system_state" shows "atS p ls s = (\<exists>l\<in>ls. at p l s)"
bind_def: fixes s :: "('d, 'f) state"   and k :: "'f \<Rightarrow> ('d, 'e) state" shows "s \<bind> k = State (\<lambda>M. let (a, M') = runState s M in runState (k a) M')"
PGWT: fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "public f"   and "arity f = length T"   and "\<And>t. t \<in> set T \<Longrightarrow> public_ground_wf_term t" shows "public_ground_wf_term (Fun f T)"
vector_component_simps_5: fixes g :: "'b \<Rightarrow> 'a"   and i :: "'b" shows "vec_lambda g $h i = g i"
size_multiset_overloaded_eq: fixes x :: "'c multiset" shows "size x = sum (count x) (set_mset x)"
Interleaves_equal_all_nil: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "xs \<cong> {ys, [], P}" shows "xs = ys"
order_refl_basic_type: fixes \<tau> :: "'a basic_type" shows "\<tau> \<le> \<tau>"
Int_space_eq1: fixes x :: "'a set" assumes "x \<in> A'" shows "\<Omega>' \<inter> x = x"
is_nan_uminus: fixes a :: "('a, 'b) IEEE.float" shows "IEEE.is_nan (- a) = IEEE.is_nan a"
commute: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "dbm_add a b = dbm_add b a"
simps_2: fixes y :: "'a option" shows "(None \<le> y) = True"
prod_less_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "(x < y) = (fst x < fst y \<or> fst x \<le> fst y \<and> snd x < snd y)"
ssubstoption_insert: fixes l :: "Label"   and f :: "Label \<Rightarrow> sterm option"   and x :: "char list"   and t' :: "sterm"   and t :: "sterm" assumes "l \<in> dom f" shows "(\<lambda>la. ssubst_option x t' (if la = l then Some t else f la)) = (\<lambda>la. ssubst_option x t' (f la))(l \<mapsto> [x \<rightarrow> t'] t)"
eqvtSubstClosed: fixes Rel :: "(pi \<times> pi) set" assumes "eqvt Rel" shows "eqvt (substClosed Rel)"
two_block_non_interfering_dims_2: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_col (two_block_non_interfering A B) = dim_col A + dim_col B"
B: fixes x :: "'a" assumes "x \<in> \<Union> \<F>" shows "norm x < B"
d_delta_lnexp_cf3: fixes x :: "real" assumes "0 < numer_cf3 x"   and "0 < numer_cf3 (- x)" shows "((\<lambda>x. ln (exp_cf3 x) - x) has_real_derivative diff_delta_lnexp_cf3 x) (at x)"
top_unique: fixes a :: "'a" shows "((1::'a) \<le> a) = (a = (1::'a))"
resultant_smult_left: fixes c :: "'a"   and f :: "'a poly"   and g :: "'a poly" assumes "c \<noteq> (0::'a)" shows "resultant (Polynomial.smult c f) g = c ^ degree g * resultant f g"
xs: shows "S = {x, y} \<union> S' \<and> x \<notin> S' \<and> y \<notin> S' \<and> x \<noteq> y"
sum_three_elements': fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "\<exists>f. (\<Sum>i\<in>{a, b, c}. f i * i) = x * a + y * b + z * c"
spr_repSim: fixes t :: "('a, 'es, 'as) BEState Trace" assumes "t \<in> SPR.jkbpC" shows "sim (spr_simMCt t) ((spr_repMC \<circ> sprCRel) (spr_sim t)) spr_repSim"
bisim1Sync12: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val"   and v' :: "'addr val" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v, v'], xs, 4 + length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
length_suffixes: fixes xs :: "'d list" shows "length (suffixes xs) = Suc (length xs)"
Sup_upper: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
label_5: shows "Lfilled k lholed es [$Call j]"
b: shows "(\<lambda>i. Sigma_Algebra.measure M (g i)) sums Sigma_Algebra.measure M (\<Union> (range g))"
index_map_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and y :: "'b"   and xs :: "'b list" assumes "inj_on f S"   and "y \<in> S"   and "set xs \<subseteq> S" shows "index (map f xs) (f y) = index xs y"
D: shows "(\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x\<^sub>0) = sign_changes ps x\<^sub>0"
put_closure: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'b" assumes "weak_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v \<in> \<S>\<^bsub>x\<^esub>"
simps_2: fixes at :: "('c, 'd) atm" shows "nvL (Neg at) = {}"
zero_closed: fixes R :: "'a set" assumes "Ring1 R" shows "(0::'a) \<in> R"
funcsetI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
init_atoms_equiv_bounds: fixes t :: "(nat \<times> linear_poly) list" shows "{} \<doteq> \<B> (init t)"
unl_after: fixes v :: "state \<Rightarrow> 'b"   and s :: "state"   and t :: "state" shows "(v$) (s, t) \<equiv> v t"
flatten_matrix_extract_matrix: fixes k :: "nat"   and m :: "nat"   and n :: "nat"   and a :: "nat \<Rightarrow> 'a" assumes "k < m * n" shows "flatten_matrix (extract_matrix a m n) k = a k"
log_less_cancel_iff: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "1 < a"   and "0 < x"   and "0 < y" shows "(log a x < log a y) = (x < y)"
intros_1: fixes F :: "'b formula"   and G :: "'b formula" shows "F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<in> AXH"
assms_3: shows "\<forall>j'<i. \<not> postrecording_event t j'"
rel_comm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<rho> x y z" shows "\<rho> x z y"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "A.ide (g \<cdot>\<^sub>A f)" obtains "A.seq g f" and "A.seq f g" and "g \<cdot>\<^sub>A f = A.dom f" and "g \<cdot>\<^sub>A f = A.cod g"
D_IFF_H: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and d :: "OBJ"   and b :: "BExpr" assumes "(G, c, HighSec) \<in> Deriv"   and "(G, d, HighSec) \<in> Deriv" shows "(G, Iff b c d, HighSec) \<in> Deriv"
entails_preI: fixes A :: "assn"   and B :: "assn" assumes "vassn_tag A \<Longrightarrow> A \<Longrightarrow>\<^sub>A B" shows "A \<Longrightarrow>\<^sub>A B"
equiv_rcong: assumes "Group.group G" shows "equiv (carrier G) rcong H"
assms_2: fixes a :: "'c" assumes "a \<notin> S" shows "f a = undefined"
lm042: fixes pair :: "'c \<times> 'd set" shows "omega pair = {fst pair} \<times> {{y} |y. y \<in> snd pair}"
FinalAllows_subseteq_in_doubt_allow: fixes \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and rs :: "'a rule list"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option" assumes "matcher_agree_on_exact_matches (\<gamma>::'a \<Rightarrow> 'b \<Rightarrow> bool) (\<beta>::'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue)"   and "good_ruleset (rs::'a rule list)" shows "_Collect (p::'b) ((\<Gamma>::char list \<Rightarrow> 'a rule list option),(\<gamma>::'a \<Rightarrow> 'b \<Rightarrow> bool),p\<turnstile> \<langle>(rs::'a rule list), Undecided\<rangle> \<Rightarrow> Decision FinalAllow) \<subseteq> _Collect (p::'b) ((\<beta>::'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow)"
pidgc_\<alpha>_def: fixes pgc :: "('a, 'b, 'c) Pid_Scheduler.pid_global_config" shows "pidgc_\<alpha> pgc \<equiv> \<lparr>global_config.processes = mset (pid_global_config.processes pgc), state = pid_global_config.state pgc\<rparr>"
nxt_wrong_hyp: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and b' :: "nat"   and i :: "nat" assumes "total_cons s"   and "eval (total_cons.r_next s) [b] \<down>= b'"   and "s b \<down>= i" shows "\<exists>y<e_length b'. \<phi> i y \<down>\<noteq> e_nth b' y"
lm044: fixes a :: "('a \<times> 'b set) set" shows "pseudoAllocation a = \<Union> {{(fst pair, {y}) |y. y \<in> snd pair} |pair. pair \<in> a}"
par_beta_preserves_FV: fixes t :: "sterm"   and t' :: "sterm"   and x :: "char list" assumes "t => t'"   and "x \<notin> FV t" shows "x \<notin> FV t'"
SNa_preserved_renaming1: fixes M :: "trm"   and a :: "coname"   and b :: "coname" assumes "SNa M" shows "SNa (M[a\<turnstile>c>b])"
right_neutral: fixes a :: "'b" shows "a \<sqinter> top = a"
map_2: fixes f :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "map_option f (Some x2) = Some (f x2)"
Osum_minus_Id1: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "r \<subseteq> Id" shows "(r \<union>o r') - Id \<subseteq> r' - Id \<union> Field r \<times> Field r'"
atLeastAtMost_borel: fixes a :: "'a"   and b :: "'a" shows "{a..b} \<in> sets borel"
altc_env_def: fixes altc :: "'i cset \<Rightarrow> ('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and C :: "'i cset"   and f :: "'i \<Rightarrow> ('g, 'h) envT" shows "altc_env altc C f = EnvT (\<lambda>r. altc C (\<lambda>c. run_env (f c) r))"
congruent_triangle_asa_7: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "angle a1 b1 c1 = angle a2 b2 c2"   and "dist a1 b1 = dist a2 b2"   and "angle b1 a1 c1 = angle b2 a2 c2"   and "\<not> collinear {a1, b1, c1}" shows "dist a1 b1 = dist b2 a2"
ldropn_trans_fusion: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and n :: "nat"   and s :: "'d" shows "lunstream (drop_raw g) (n, s) = ldropn n (lunstream g s)"
roofed_def: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set" shows "roofed_gen G B S = {x. \<forall>p. \<forall>y\<in>B. path G x p y \<longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S}"
simple_function_indicator: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M" shows "simple_function M (indicator A)"
length_remdups_card_conv: fixes xs :: "'g list" shows "length (remdups xs) = card (set xs)"
sum_clauses_2: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite S" shows "sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
reduced_word_for_arg_min: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "class.monoid_add plus zero"   and "as \<in> lists A"   and "monoid_add.sum_list plus zero as = a" shows "monoid_add.reduced_word_for plus zero A a (ARG_MIN length as. as \<in> lists A \<and> monoid_add.sum_list plus zero as = a)"