hcju/isabelleps huggingface dataset:
inf_prefix_take_correct: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list" assumes "n \<le> length (xs \<sqinter> ys)" shows "xs \<down> n = ys \<down> n"
get_min_def: fixes t :: "'a trm"   and E :: "('a trm \<times> 'a trm) set" shows "get_min t E = (if min_trms E = {} then t else SOME x. x \<in> min_trms E)"
MkIde_Cod: fixes a :: "('a, 'a set) arr" assumes "arr a" shows "MkIde (local.Cod a) = cod a"
assms_1: fixes a :: "'a" assumes "a \<in> S" shows "y a \<sqsubseteq> z a"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "embedding_functor A B F"   and "full_functor A B F" shows "full_embedding_functor A B F"
ln_divisor_count_primorial'_asymp_equiv: shows "(\<lambda>k. ln (real (divisor_count (primorial' k)))) \<sim>[sequentially] (\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))"
non_empty_query_def: fixes Q :: "(nat set \<times> 'b option list set) set" shows "non_empty_query Q = (\<forall>X\<in>Q. 1 \<le> card (fst X))"
surj: fixes f :: "'c \<Rightarrow> 'c" assumes "bijection f" shows "surj f"
abs_eq: shows "bot = Abs_stone_phi_pair stone_phi.pairs_bot"
NoMessageLoss: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg2 :: "('p, 'v, 's) configuration"   and p :: "'p"   and m' :: "('p, 'v) message" assumes "qReachable cfg1 (Proc - Q) cfg2"   and "p \<in> Q"   and "isReceiverOf p m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
refl_onD1: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and x :: "'d"   and y :: "'d" assumes "refl_on A r"   and "(x, y) \<in> r" shows "x \<in> A"
tab_in_vhom': shows "\<guillemotleft>\<rho> : g \<Rightarrow>\<^sub>C r \<star>\<^sub>C f\<guillemotright>"
Min_gr_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Min less_eq A) = (\<forall>a\<in>A. less x a)"
multpw_implies_mult2_alt_ns: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> multpw ns" shows "(X, Y) \<in> mult2_alt_ns ns s"
conjugation_prop1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a" assumes "is_conjugation f g" shows "f y \<cdot> z \<le> f (y \<cdot> g z)"
roofed_circI: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and T :: "'a set" assumes "x \<in> RF\<^bsub>\<Gamma>\<^esub> T"   and "x \<in> T \<Longrightarrow> \<not> essential \<Gamma> (B \<Gamma>) T x" shows "x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> T"
parts_insert: fixes X :: "msg"   and H :: "msg set" shows "parts (insert X H) = parts {X} \<union> parts H"
ereal_eq_1_2: fixes r :: "real" shows "(1 = ereal r) = (r = 1)"
punit_adds_term: shows "MPoly_Type_Class.punit.adds_term = (adds)"
comp_null_2: fixes f :: "'a" shows "f \<cdot> B.null = B.null"
cut_less_Max_eq_Diff: fixes I :: "'a set" assumes "finite I" shows "I \<down>< Max I = I - {Max I}"
possible_steps_alt: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and d :: "nat"   and t :: "transition" shows "(possible_steps e s r l i = {|(d, t)|}) = (ffilter (\<lambda>((origin, dest), t). origin = s \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)) e = {|((s, d), t)|})"
abs_eq: shows "chart_eucl = Abs_chart (UNIV, UNIV, \<lambda>x. x, \<lambda>x. x)"
C4_a: fixes x :: "'a" shows "-r (-l x) = x"
Im_divide_real: fixes b :: "complex"   and a :: "complex" assumes "is_real b"   and "b \<noteq> 0" shows "Im (a / b) = Im a / Re b"
finite_total_preorder_on_axioms: shows "finite_total_preorder_on carrier le"
linear_fun_on_scaleR: fixes S :: "'c set"   and x :: "'c \<Rightarrow> real"   and c :: "real" assumes "linear_fun_on S x"   and "subspace S" shows "linear_fun_on S (c *\<^sub>R x)"
elts_eq_empty_iff: fixes x :: "V" shows "(elts x = {}) = (x = 0)"
COND: fixes l1i :: "'a list"   and l2i :: "'a list"   and l1 :: "'b list"   and l2 :: "'b list"   and si :: "'c"   and s :: "'d" assumes "li = l1i @ l2i"   and "l = l1 @ l2"   and "(l1i, l1) \<in> \<langle>S\<rangle>list_rel"   and "(l2i, l2) \<in> \<langle>S\<rangle>list_rel"   and "I l1i l2i si"   and "(si, s) \<in> R" shows "(ci si, c s) \<in> bool_rel"
assms_2: fixes r :: "complex"   and x :: "'a" shows "f (r *\<^sub>C x) = cnj r *\<^sub>C f x"
monad_fail_contT: fixes fail :: "'b" shows "monad_fail return_cont bind_cont (fail_cont fail)"
small_vreal_le: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> real_of_vreal a \<le> real_of_vreal b}"
quasi_inverses_isomorphic_right: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.quasi_inverses f g" shows "Span.quasi_inverses f g' = Span.isomorphic g g'"
try_gpv_cong: fixes gpv :: "('a, 'b, 'c) gpv"   and gpv'' :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv"   and gpv''' :: "('a, 'b, 'c) gpv" assumes "gpv = gpv''"   and "\<not> colossless_gpv \<I>_full gpv'' \<Longrightarrow> gpv' = gpv'''" shows "TRY gpv ELSE gpv' = TRY gpv'' ELSE gpv'''"
deep_model_correct_params_y_axioms_def: fixes rs :: "nat list"   and y :: "nat" shows "deep_model_correct_params_y_axioms rs y \<equiv> y < rs ! 0"
ind2: fixes xs :: "nat list" assumes "length xs = Suc (Suc n)" shows "terminate g xs"
rev_to_bl_eq: fixes w :: "'a word" shows "rev (to_bl w) = map (bit w) [0..<LENGTH('a)]"
pkt_def: shows "pkt \<equiv> \<lambda>(d, dip, sip). Pkt d dip sip"
mono_ccInf: fixes A :: "'a set" assumes "countable A" shows "f (Inf A) \<le> Inf (f ` A)"
cqt_further_1: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) in v]"
not_outside_2: fixes x :: "'b"   and y :: "'b" assumes "Pareto R x y" shows "y \<in> alts"
Arccos_range_lemma: fixes z :: "complex" assumes "\<bar>Re z\<bar> < 1" shows "0 < Im (z + \<i> * csqrt (1 - z\<^sup>2))"
proj_finmap_of: fixes inds :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite inds" shows "(finmap_of inds f)\<^sub>F = restrict f inds"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G F \<tau>'.map"
simps_2: fixes n :: "nat"   and s :: "variable"   and S :: "variable set" shows "variant_variables (Suc n) s S = (let (s', S') = variant_variable s S; (ss, y) = variant_variables n s' S' in (s' # ss, y))"
q_min: fixes t :: "real" assumes "0 \<le> t" shows "s\<^sub>e \<le> ego.q t"
mult_right_cancel: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times a c = times b c) = (a = b)"
hcomp_in_vhomE: fixes \<nu> :: "'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<guillemotleft>\<nu> \<star> \<mu> : f \<Rightarrow> g\<guillemotright>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = src f" and "src \<mu> = src g" and "trg \<nu> = trg f" and "trg \<nu> = trg g"
le: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x < y" shows "cmod (\<Sum>n\<in>real -` {x<..y}. \<chi> n * of_nat n powr - s) \<le> real (totient n) * (2 + cmod s / \<sigma>) / x powr \<sigma>"
trivial_group_subset: shows "trivial_group G = (carrier G \<subseteq> {\<one>})"
arr_iff_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "partial_magma.arr C f = partial_magma.in_hom C f (partial_magma.dom C f) (partial_magma.cod C f)"
simps: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "m\<tau>move1 a1 a2 a3 = (\<exists>s t x s' x' ln' ta. a1 = s \<and> a2 = (t, ta) \<and> a3 = s' \<and> thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> thr s' t = \<lfloor>(x', ln')\<rfloor> \<and> \<tau>move1 (x, shr s) ta (x', shr s'))"
n_le_v: fixes s :: "'s"   and cfg :: "'s cfg" assumes "s \<in> S"   and "cfg \<in> cfg_on s" shows "n s \<le> v cfg"
pseudominimal_inI: fixes x :: "'a"   and P :: "'a set" assumes "x \<in> P"   and "x \<noteq> domain.bottom P"   and "\<And>z. \<lbrakk>z \<in> P; x \<succ> z\<rbrakk> \<Longrightarrow> z = domain.bottom P" shows "domain.pseudominimal_in P x"
subspace_row_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec.subspace (row_space A)"
ccw'_scale_origin: fixes e :: "real \<times> real \<Rightarrow> real"   and x :: "(real \<times> real) \<times> real \<times> real"   and Pc :: "real \<times> real"   and P :: "real \<times> real"   and QRRs :: "(real \<times> real) list" assumes "e \<in> UNIV \<rightarrow> {0<..<1}"   and "x \<in> set (polychain_of Pc (P # QRRs))"   and "ccw'.sortedP 0 (P # QRRs)"   and "ccw' (fst x) (snd x) (P + (Pc + (\<Sum>P\<in>set QRRs. e P *\<^sub>R P)))" shows "ccw' (fst x) (snd x) (e P *\<^sub>R P + (Pc + (\<Sum>P\<in>set QRRs. e P *\<^sub>R P)))"
Un_Diff_cancel: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> (B - A) = A \<union> B"
le_infI2: fixes x :: "'c"   and b :: "'c"   and a :: "'c" assumes "x \<le> b" shows "x \<le> a \<squnion> b"
elims_3: fixes x :: "'b trm" assumes "\<not> is_a_variable x" obtains xa :: "'b" where "x = Const xa__"   | xa :: "'b trm" and y :: "'b trm" where "x = xa \<cdot> y"
row_to_poly_smult: fixes ts :: "'c list"   and r :: "'d vec"   and c :: "'d" assumes "distinct ts"   and "dim_vec r = length ts" shows "row_to_poly ts (c \<cdot>\<^sub>v r) = c \<cdot> row_to_poly ts r"
A: shows "gauss_jordan_single A \<in> carrier_mat nr nc"
assms_1: shows "bal c\<^sub>1 (c\<^sub>2 # T) c\<^sub>5"
vfsequence_vappend_vempty_vfsequence: shows "[]\<^sub>\<circ> @\<^sub>\<circ> xs = xs"
weak_least_unique: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and A :: "'c set"   and y :: "'c" assumes "weak_partial_order L"   and "least L x A"   and "least L y A" shows "x .=\<^bsub>L\<^esub> y"
letters_def: shows "letters = ''abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ_0123456789&;:-''"
pmf_map_outside: fixes x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf" assumes "x \<notin> f ` set_pmf M" shows "pmf (map_pmf f M) x = 0"
rep_eq: fixes x :: "'node"   and xa :: "'inPort" shows "fset (hyps_for x xa) = hyps_for' x xa"
map_eq_Cons_conv: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and y :: "'b"   and ys :: "'b list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
reflection: shows "rel_weakly_reflects_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)"
assoc_add_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a"   and r'' :: "'a"   and s'' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel"   and "(r'', s'') \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = (r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> ((r' |\<^bsub>rel\<^esub> s') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s''))"
Always_includes_reachable: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> Always A" shows "reachable F \<subseteq> A"
Reflect_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Reflect Bet Cong P' P A B \<equiv> A \<noteq> B \<and> Tarski_neutral_dimensionless.ReflectL Bet Cong P' P A B \<or> A = B \<and> Tarski_neutral_dimensionless.Midpoint Bet Cong A P P'"
init_fin_simulation_silent2: fixes t :: "'t"   and sxm1 :: "(status \<times> 'x1) \<times> 'm1"   and sxm2 :: "(status \<times> 'x2) \<times> 'm2"   and sxm2' :: "(status \<times> 'x2) \<times> 'm2" assumes "t \<turnstile> sxm1 \<approx>i sxm2"   and "\<tau>trsys.silent_move (r2.init_fin t) r2.init_fin_\<tau>move sxm2 sxm2'" shows "\<exists>sxm1'. (\<tau>trsys.silent_move (r1.init_fin t) r1.init_fin_\<tau>move)\<^sup>*\<^sup>* sxm1 sxm1' \<and> t \<turnstile> sxm1' \<approx>i sxm2'"
bij_betw_from_nat_into: fixes S :: "'b set" assumes "countable S"   and "infinite S" shows "bij_betw (from_nat_into S) UNIV S"
AE_iff_null_sets: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" shows "almost_everywhere (completion M) P = ({x \<in> space (completion M). \<not> P x} \<in> null_sets (completion M))"
supp_eqvt: fixes pi :: "(atom \<times> atom) list"   and x :: "'a" shows "pi \<bullet> supp x = supp (pi \<bullet> x)"
filterlim_uminus_at_bot_at_top: shows "filterlim uminus at_bot at_top"
divide_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "x div y = x * inverse y"
leadsETo_cancel2: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A leadsTo[CC] (A' \<union> B)"   and "F \<in> B leadsTo[CC] B'" shows "F \<in> A leadsTo[CC] (A' \<union> B')"
set_apply_ocrit_iff: fixes p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps' :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "((p, q) \<in> set (apply_ocrit crit data hs ps' ps)) = ((p, q) \<in> set ps \<and> \<not> crit data hs ps' p q)"
fine_covers_all: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and c :: "real"   and D :: "(real \<times> real \<times> real) list"   and x :: "real" assumes "fine \<delta> (a, c) D"   and "a < x"   and "x \<le> c" shows "\<exists>N<length D. \<forall>d t e. D ! N = (d, t, e) \<longrightarrow> d < x \<and> x \<le> e"
span_Un: fixes S :: "'a set"   and T :: "'a set" shows "More_Modules.ideal (S \<union> T) = {x + y |x y. x \<in> More_Modules.ideal S \<and> y \<in> More_Modules.ideal T}"
npowf_inverse: fixes x :: "'a"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x\<^bsub>K\<^esub>\<^bsup>j\<^esup> = x\<^bsup>\<hyphen> K\<^esup>\<^bsub>K\<^esub>\<^bsup>- j\<^esup>"
p1: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All i' (SyntaxN.All si (SyntaxN.All li (SyntaxN.All j (SyntaxN.All j' (SyntaxN.All sj (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP (Var li) (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))))))))))(i::=Var i)"
ennreal_inverse_positive: fixes x :: "ennreal" shows "(0 < inverse x) = (x \<noteq> top)"
ext: fixes g :: "'a"   and f :: "'a" assumes "Fop.Bop.comp g f \<noteq> Fop.Bop.null" shows "Fop.Bop.seq g f"
nsqn_quality_increases_dhops: fixes i :: "nat"   and \<xi> :: "state"   and \<xi>' :: "state" assumes "i \<in> kD (rt \<xi>)"   and "quality_increases \<xi> \<xi>'"   and "nsqn (rt \<xi>) i = nsqn (rt \<xi>') i" shows "the (dhops (rt \<xi>') i) \<le> the (dhops (rt \<xi>) i)"
space_restrict_space2: fixes \<Omega> :: "'b set"   and M :: "'b measure" assumes "\<Omega> \<in> sets M" shows "space (restrict_space M \<Omega>) = \<Omega>"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "'b"   and a2 :: "('b \<times> color) tree" assumes "\<And>t1 a t2 b t3. P t1 a (R t2 b t3)"   and "\<And>t1 a t2 b. P (B t1 a t2) b \<langle>\<rangle>"   and "\<And>t1 a t2 b v vc vb. P (B t1 a t2) b (B v vc vb)"   and "\<And>t1 a t2 b t3 c. P (R t1 a (B t2 b t3)) c \<langle>\<rangle>"   and "\<And>t1 a t2 b t3 c v vc vb. P (R t1 a (B t2 b t3)) c (B v vc vb)"   and "\<And>a. P \<langle>\<rangle> a \<langle>\<rangle>"   and "\<And>v vc a. P (R v vc \<langle>\<rangle>) a \<langle>\<rangle>"   and "\<And>v vc va vf ve a. P (R v vc (R va vf ve)) a \<langle>\<rangle>"   and "\<And>a v vc vb. P \<langle>\<rangle> a (B v vc vb)"   and "\<And>va vf a v vc vb. P (R va vf \<langle>\<rangle>) a (B v vc vb)"   and "\<And>va vf vd vi vh a v vc vb. P (R va vf (R vd vi vh)) a (B v vc vb)" shows "P a0 a1 a2"
perfect_def: fixes e :: "cars"   and ts :: "traffic"   and c :: "cars" shows "perfect e ts c \<equiv> physical_size ts c + braking_distance ts c"
map_Sigma_def: fixes M1 :: "'d \<Rightarrow> 'f option"   and F2 :: "'f \<Rightarrow> 'e set" shows "map_Sigma M1 F2 \<equiv> {(x, y). \<exists>v. M1 x = Some v \<and> y \<in> F2 v}"
finite_nList: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "finite (nList A n)"
assms: shows "\<forall>b\<in>set B. \<exists>a\<in>set A. \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
Lxy_append: fixes xs :: "'b list"   and ys :: "'b list"   and S :: "'b set" shows "Lxy (xs @ ys) S = Lxy xs S @ Lxy ys S"
T_inorder2_def: fixes t :: "'b tree"   and xs :: "'b list" shows "T_inorder2 t xs = time (inorder2_tm t xs)"
adh_closure_iff: fixes A :: "'a llist set"   and x :: "'a llist" assumes "A \<subseteq> carrier"   and "x \<in> carrier" shows "(x adh A) = (x \<in> closure A)"
strong_intersecttwo: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'" shows "\<exists>f. ChamberComplexIsomorphism A A' f \<and> fixespointwise f (\<Union> (A \<inter> A'))"
Zp_residue_eq2: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a k = b k"   and "a \<noteq> b" shows "eint (int k) \<le> val_Zp (a \<ominus> b)"
safe_RMW_common: fixes \<O>s :: "nat set list"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and i :: "nat"   and a :: "nat"   and t :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" assumes "\<O>s,\<R>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>" shows "(a \<in> \<O> \<or> a \<in> dom \<S>) \<and> (\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> (\<R>s ! j) a \<noteq> Some False)"
E_TApp: fixes t :: "trm"   and t' :: "trm"   and T :: "POPLmark.type" assumes "t \<longmapsto> t'" shows "t \<bullet>\<^sub>\<tau> T \<longmapsto> t' \<bullet>\<^sub>\<tau> T"
Abs1_eq_iff'_3: fixes a :: "'b"   and x :: "'a"   and b :: "'b"   and y :: "'a" shows "([[atom a]]lst. x = [[atom b]]lst. y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> (b \<leftrightarrow> a) \<bullet> x = y \<and> atom b \<sharp> x)"
open_left: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.linorder_topology less_eq less open"   and "open S"   and "x \<in> S"   and "less y x" shows "\<exists>b. less b x \<and> ord.greaterThanAtMost less_eq less b x \<subseteq> S"
count_roots_interval_def: fixes p :: "real poly" shows "count_roots_interval p = (let ps = sturm p in (\<lambda>a b. sign_changes ps a - sign_changes ps b + (if poly p a = 0 then 1 else 0), \<lambda>a. sign_changes_neg_inf ps - sign_changes ps a))"
bdd_below_Icc: fixes a :: "'a"   and b :: "'a" shows "bdd_below {a..b}"
\<psi>': fixes x :: "real" assumes "x \<in> {a..b}" shows "(\<psi> has_real_derivative \<psi>' x) (at x)"
simps_2: fixes p :: "'a \<times> 'b"   and ps :: "('a \<times> 'b) list" shows "clearjunk (p # ps) = p # clearjunk (delete (fst p) ps)"
br_run_conv: fixes \<alpha> :: "('c, 'ls, 'gs) pid_global_config \<Rightarrow> ('c, 'ls, 'gs) pid_global_config"   and I :: "('c, 'ls, 'gs) pid_global_config \<Rightarrow> bool"   and rb :: "nat \<Rightarrow> ('c, 'ls, 'gs) pid_global_config" assumes "Id = br \<alpha> I" shows "pid.is_run rb = (\<exists>ra. rb = \<alpha> \<circ> ra \<and> ga.is_run ra)"
u1_bound: fixes x :: "'a" shows "\<bar>u 1 x\<bar> \<le> K + F x"
comm_monoid_axioms: shows "Group.comm_monoid (add_monoid R)"
exhaust_sel: fixes tree :: "'a tree" obtains "tree = Node (tree.root tree) (cont tree)"
wf_mbuf2'_def: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'c list set"   and \<phi> :: "'c MFOTL.formula"   and \<psi> :: "'c MFOTL.formula"   and buf :: "'c option list set list \<times> 'c option list set list" shows "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf = wf_mbuf2 (min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)) (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j) (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) buf"
inv_implications_nonneg_def: fixes c :: "('d, 'e, 'f) configuration_scheme" shows "inv_implications_nonneg c = (\<forall>loc t. 0 \<le> zcount (c_imp c loc) t)"
Res_def: fixes x1 :: "name"   and x2 :: "pi" shows "<\<nu>x1>x2 \<equiv> Abs_pi (Res_Rep ([x1].Rep_pi x2))"
cSup_singleton: fixes x :: "'b multiset" shows "Sup {x} = x"
r_code_const1_def: shows "r_code_const1 \<equiv> r_shrink (Pr 1 Z r_code_const1_aux)"
Dom_assume: fixes b :: "'b upred" shows "Dom ?[b] = b"
less_atm_ground: fixes A :: "'a"   and B :: "'a" assumes "is_ground_atm A"   and "is_ground_atm B"   and "less_atm A B" shows "A < B"
downset_onD: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "downset_on A r"   and "(x, y) \<in> r"   and "y \<in> A" shows "x \<in> A"
frac_in_carrier_local: fixes \<pp> :: "'a set"   and r :: "'a"   and s :: "'a" assumes "\<pp> \<in> Spec"   and "r \<in> R"   and "s \<in> R"   and "s \<notin> \<pp>" shows "quotient_ring.frac (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> r s \<in> R \<^bsub>\<pp> (+) (\<cdot>) \<zero>\<^esub>"
dist_eq_0_iff: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "(dist x y = 0) = (x = y)"