hcju/isabelleps huggingface dataset:
Hom_DopxG_simp: fixes gf :: "'d \<times> 'c" assumes "DopxC.arr gf" shows "Hom_DopxG.map gf = S.mkArr (HomD.set (D.cod (fst gf), G (C.dom (snd gf)))) (HomD.set (D.dom (fst gf), G (C.cod (snd gf)))) (inD \<circ> (\<lambda>h. G (snd gf) \<cdot>\<^sub>D h \<cdot>\<^sub>D fst gf) \<circ> HomD.\<psi> (D.cod (fst gf), G (C.dom (snd gf))))"
rcm_D: fixes a :: "'b" shows "a l\<rightarrow> a = (1::'b)"
F_def: shows "F \<equiv> \<lambda>n. \<Prod>{p. prime p \<and> p \<le> n}"
img_in_bij: fixes g :: "'a" assumes "g \<in> carrier G" shows "\<phi> g \<in> Bij M"
hbex_succ: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" shows "HBex (succ k) P = (P k \<or> HBex k P)"
mod_ring_inverse32: shows "(mod_ring_rel32 ===> mod_ring_rel32) (inverse_p32 pp) inverse"
iterate_add_to_map_def: fixes m :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" shows "iterate_add_to_map m update it = it (\<lambda>_. True) (\<lambda>(k, v). update k v) m"
weak_ring_morphism_range: fixes f :: "'a \<Rightarrow> 'd"   and I :: "'a set"   and a :: "'a" assumes "weak_ring_morphism f I R"   and "a \<in> carrier R" shows "f ` (I +> a) = {f a}"
UN_iff: fixes b :: "'g"   and B :: "'h \<Rightarrow> 'g set"   and A :: "'h set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
1: fixes x :: "brick list"   and y :: "brick list" shows "basic x \<otimes> basic y = basic (x \<otimes> y)"
list_add_hi: fixes t :: "'b tree"   and x :: "'b" assumes "braun t" shows "list (update1 (Suc (size t)) x t) = list t @ [x]"
alphaAbs_qAbs_imp_alphaAbs_all_qAFresh: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_qAFresh xs x X xs' x' X'"
bind_mono_1: fixes M :: "'a nres"   and M' :: "'a nres"   and f :: "'a \<Rightarrow> 'b nres"   and f' :: "'a \<Rightarrow> 'b nres" assumes "M \<le> M'"   and "\<And>x. RETURN x \<le> M \<Longrightarrow> f x \<le> f' x" shows "M \<bind> f \<le> M' \<bind> f'"
eventually_not_equal_at_infinity: fixes a :: "'a" shows "\<forall>\<^sub>F x in at_infinity. x \<noteq> a"
mat_extension_id: fixes dims :: "nat list"   and vars :: "nat set" shows "mat_extension dims vars (1\<^sub>m (partial_state.d1 dims vars)) = 1\<^sub>m (state_sig.d dims)"
new_compl: fixes n :: "'c"   and p :: "('c, 'd) form" assumes "new n p" shows "new n (compl p)"
exec_strip: shows "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
simps_9: fixes f1 :: "ct list \<Rightarrow> 'a"   and f2 :: "t list \<Rightarrow> 'a"   and f3 :: "'a"   and x1 :: "ct list" shows "(case TopType x1 of TopType x \<Rightarrow> f1 x | Type x \<Rightarrow> f2 x | Bot \<Rightarrow> f3) = f1 x1"
RP_derivation_imp_GC_derivation: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>RP) Sts" shows "chain (\<leadsto>GC) (lmap lclss_of_state Sts)"
stable_on_blocking_onD: fixes ds :: "'b set"   and X :: "'a set"   and X'' :: "'a set"   and h :: "'c" assumes "Bossy'.stable_on ds X"   and "X'' = BCh h (X \<union> X'')"   and "X'' \<subseteq> Bossy'.CD_on ds (X \<union> X'')" shows "X'' = BCh h X"
aux1: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> E"   and "(v, u) \<notin> E" shows "f' (u, v) = (0::'capacity)"
span_diff: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> ideal S"   and "y \<in> ideal S" shows "x - y \<in> ideal S"
keys_pp_iff: fixes x :: "'b"   and t :: "('b, 'c) pp" shows "(x \<in> keys_pp t) = (lookup_pp t x \<noteq> (0::'c))"
digraph_iso_mk_iso_nomulti: fixes H :: "('e, 'f) pre_digraph"   and fv :: "'a \<Rightarrow> 'e" assumes "nomulti_digraph H"   and "inj_on fv (verts G)"   and "verts H = fv ` verts G"   and "arcs_ends H = pairself fv ` arcs_ends G" shows "digraph_iso G H"
summands_in: fixes x :: "i"   and y :: "i" shows "P x (x \<oplus> y) \<and> P y (x \<oplus> y)"
p1: shows "\<guillemotleft>p1 : local.dom p0 \<rightarrow> a1\<guillemotright>"
simps_2: shows "bool_to_ternary False = TernaryFalse"
abs_conv_abscissa_prod_le: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b fds"   and d :: "ereal" assumes "\<And>x. x \<in> A \<Longrightarrow> abs_conv_abscissa (f x) \<le> d" shows "abs_conv_abscissa (prod f A) \<le> d"
substitute_cong': fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and A :: "'b set"   and f' :: "'b \<Rightarrow> 'b ttree" assumes "xs \<in> paths (substitute f T t)"   and "\<And>x n. x \<in> A \<Longrightarrow> carrier (f x) \<subseteq> A"   and "carrier t \<subseteq> A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "xs \<in> paths (substitute f' T t)"
iMODb_add_neg': fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and k :: "nat" shows "[ r, mod m, c ] \<oplus>- k = (if k \<le> r then [ r - k, mod m, c ] else if k \<le> r + m * c then if k mod m \<le> r mod m then [ r mod m - k mod m, mod m, c + r div m - k div m ] else [ m + r mod m - k mod m, mod m, c + r div m - Suc (k div m) ] else {})"
type_less_x_OclAny_intro: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<noteq> OclAny" shows "\<tau> < OclAny"
semiring_norm_5: fixes m :: "num" shows "num.Bit0 m + num.One = num.Bit1 m"
lift_Suc_mono_less_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "class.order less_eq less"   and "\<And>n. less (f n) (f (Suc n))" shows "less (f n) (f m) = (n < m)"
WTNil: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option" shows "P,E \<turnstile> [] [::] []"
call_1\<^sub>C_def: shows "call_1\<^sub>C = call\<^sub>C"
prems_1: shows "\<delta>n = \<delta> \<union> \<delta>'"
hyps_2: assumes "\<not> lnull xs" shows "llast (lmap f xs) = f (llast xs)"
col_vec_mat_Tensor: fixes j :: "nat"   and M :: "'a list list"   and v :: "'a list" assumes "j < length M" shows "col (vec_mat_Tensor v M) j = vec_vec_Tensor v (col M j)"
case_eq_if: fixes f :: "real interval \<Rightarrow> real interval \<Rightarrow> 'a"   and complex_interval :: "complex_interval" shows "(case complex_interval of Complex_Interval x xa \<Rightarrow> f x xa) = f (Re_interval complex_interval) (Im_interval complex_interval)"
12: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e * (v * v\<^sup>T * e \<sqinter> g) = bot"
map_simp: fixes Fg :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr Fg" shows "E.map Fg = A_B.Map (fst Fg) (snd Fg)"
subdegree_le_imp_dvd_right_divring: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "subdegree f \<le> subdegree g" shows "\<exists>k. g = k * f"
assms_5: shows "\<forall>x\<in>set les. case x of (d, e, f) \<Rightarrow> d * ((- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a))\<^sup>2 + e * (- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a) + f < 0"
behaviourally_equiv_def: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'd) Protocol"   and jp' :: "'a \<Rightarrow> ('obs, 'aAct, 'e) Protocol" shows "behaviourally_equiv jp jp' \<equiv> \<forall>t\<in>jkbpC. set \<circ> actJP jp t = set \<circ> actJP jp' t"
simps_2: fixes xs :: "'g list"   and y :: "'h"   and ys :: "'h list" shows "zip xs (y # ys) = (case xs of [] \<Rightarrow> [] | z # zs \<Rightarrow> (z, y) # zip zs ys)"
Lambert_W_eq_0_iff: fixes x :: "real" shows "(Lambert_W x = 0) = (x = 0)"
h_f: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n + 1" shows "f' n x = sum (h n x) {..n + 1}"
conv_invol: fixes x :: "'a" shows "(x\<^sup>\<smile>)\<^sup>\<smile> = x"
of_nat_eq_0_iff_char_dvd: fixes n :: "nat" shows "(of_nat n = (0::'a)) = (CHAR dvd n)"
newton_poly_impl_int_def: fixes xs :: "int list"   and fs :: "int list" shows "newton_poly_impl_int xs fs \<equiv> case newton_coefficients_int xs fs of None \<Rightarrow> None | Some nc \<Rightarrow> Some (horner_composition (rev nc) xs)"
poly_of_vec_0_iff: fixes v :: "'b vec" shows "(poly_of_vec v = 0) = (v = 0\<^sub>v (dim_vec v))"
ACI_nPlus_flatten_PLUS: fixes X1 :: "'c rexp set"   and X2 :: "'c rexp set" assumes "X1 \<noteq> {}"   and "X2 \<noteq> {}"   and "finite X1"   and "finite X2"   and "\<forall>x\<in>X1 \<union> X2. \<nexists>r s. x = Plus r s" shows "ACI_nPlus (PLUS (sorted_list_of_set X1)) (PLUS (sorted_list_of_set X2)) = PLUS (sorted_list_of_set (X1 \<union> X2))"
elims: fixes x :: "'b"   and xa :: "'b tree"   and y :: "nat" assumes "T_insert x xa = y" obtains a :: "'b" and h :: "'b tree" where "x = a" and "xa = h" and "y = 1"
integral_combine_division_bottomup: fixes \<D> :: "'a set set"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<D> division_of S"   and "\<And>k. k \<in> \<D> \<Longrightarrow> f integrable_on k" shows "integral S f = (\<Sum>i\<in>\<D>. integral i f)"
snoc: shows "\<Union> {propos y |y. y = i \<or> y \<in> set xs} = propos (foldl f i xs)"
bool_split: assumes "\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> \<noteq> null \<tau>"   and "\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> \<noteq> \<bottom> \<tau>"   and "\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> \<noteq> false \<tau>" shows "\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = true \<tau>"
no_offending: assumes "c_sinvar m \<lparr>nodes = V, edges = E\<rparr>" shows "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> = {}"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (call_instr instr s1)) \<and> t2 = snd (fst (call_instr instr s2))"
mutual_information_def: fixes b :: "real"   and S :: "'d measure"   and T :: "'e measure"   and X :: "'b \<Rightarrow> 'd"   and Y :: "'b \<Rightarrow> 'e" shows "X.mutual_information b S T X Y = KL_divergence b (distr (distr M S X) S X \<Otimes>\<^sub>M distr (distr M S X) T Y) (distr (distr M S X) (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
partition_without_def: fixes elem :: "'b"   and P :: "'b set set" shows "partition_without elem P = (\<lambda>X. X - {elem}) ` P - {{}}"
inv_greek_append: fixes s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list" shows "inv_greek (s @ t) = inv_greek t @ inv_greek s"
lcompare_double_0_folds_1: fixes x :: "double"   and y :: "double" shows "(lcompare_double x y \<le> 0) = lle_double x y"
lossless_init: fixes h :: "'a"   and w :: "nat \<times> nat" shows "lossless_spmf (init h w)"
submonoid_inverse_closed: fixes u :: "'a" assumes "additive.sub.invertible u"   and "u \<in> I" shows "- u \<in> I"
rep_interface_def: shows "rep_interface \<equiv> quot_type.rep Rep_interface"
coeff_pcompose_linear: fixes p :: "'b poly"   and a :: "'b"   and i :: "nat" shows "coeff (p \<circ>\<^sub>p [:0::'b, a:]) i = a ^ i * coeff p i"
subseq_append_iff: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "subseq xs (ys @ zs) = (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> subseq xs1 ys \<and> subseq xs2 zs)"
reaching_path_length: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and q2 :: "'f \<times> 'g"   and q1 :: "'f \<times> 'g" assumes "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "q2 \<in> reachable AB q1"   and "q2 \<noteq> FAIL"   and "q1 \<in> nodes AB" shows "\<exists>p. path AB p q1 \<and> target p q1 = q2 \<and> length p < |A| * |B|"
ide_cod: fixes f :: "'a" assumes "C.arr f" shows "C.ide (C.cod f)"
and_extract_pure_left_iff: fixes b :: "bool"   and Q :: "assn" shows "\<up> b \<and>\<^sub>A Q = (emp \<and>\<^sub>A Q) * \<up> b"
force_enabled: fixes x :: "'b"   and \<omega> :: "'b stream" assumes "K2.enabled x \<omega>" shows "K2.force_enabled x \<omega> = \<omega>"
power_down_nonneg: fixes x :: "real"   and p :: "nat"   and n :: "nat" assumes "0 \<le> x" shows "0 \<le> power_down p x n"
vimage_Int: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'d set" shows "f -` (A \<inter> B) = f \<^sup>\<inverse> f -` B A"
assms_1: fixes x :: "complex" assumes "x \<in> S" shows "(f has_field_derivative f' x) (at x within S)"
assms_2: fixes x :: "'a"   and xs :: "'a list" shows "P (x # xs) []"
fresh_fresh_rename_ns: fixes G :: "variable set"   and n :: "nat"   and B :: "Core.term"   and insts :: "((variable \<times> typ) \<times> Core.term) list" assumes "finite G" shows "\<forall>nm\<in>set (fresh_rename_idns n B insts G). nm \<notin> fst ` (fv B \<union> \<Union> (fv ` snd ` set insts) \<union> fst ` set insts) \<union> G"
set_mode_locs_def: shows "set_mode_locs = (\<lambda>shadow_root_ptr. all_args (put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode_update))"
ncols_one_matrix: fixes n :: "nat" shows "ncols (one_matrix n) = n"
ev_cball: shows "\<forall>\<^sub>F y in at x. y \<in> cball x e"
prems: shows "\<Gamma>\<turnstile> \<langle>sequence Seq (flatten (Seq c1 c2)),s\<rangle> =n\<Rightarrow> t"
preserves_comp: fixes g :: "'j"   and f :: "'j" assumes "J.seq g f" shows "D (g \<cdot>\<^sub>J f) = D g \<cdot> D f"
advantage_obsf_distinguisher: fixes \<A> :: "('m, 'n) distinguisher"   and ideal_resource :: "('m, 'n) resource"   and real_resource :: "('m, 'n) resource" shows "advantage (obsf_distinguisher \<A>) (obsf_resource ideal_resource) (obsf_resource real_resource) = advantage \<A> ideal_resource real_resource"
1: fixes y :: "'a" assumes "y \<in> C" shows "ord x y \<or> ord y x"
inf_concat_surj_zero: fixes n :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "0 < n i" shows "\<exists>k. inf_concat n k = (i, 0)"
ns_mul_extI: fixes A :: "'a multiset"   and A1 :: "'a multiset"   and A2 :: "'a multiset"   and B :: "'a multiset"   and B1 :: "'a multiset"   and B2 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "A = A1 + A2"   and "B = B1 + B2"   and "(A1, B1) \<in> multpw ns"   and "\<And>b. b \<in># B2 \<Longrightarrow> \<exists>a. a \<in># A2 \<and> (a, b) \<in> s" shows "(A, B) \<in> ns_mul_ext ns s"
Image_mono: fixes r' :: "('a \<times> 'b) set"   and r :: "('a \<times> 'b) set"   and A' :: "'a set"   and A :: "'a set" assumes "r' \<subseteq> r"   and "A' \<subseteq> A" shows "r' `` A' \<subseteq> r `` A"
that: fixes C :: "'a set" assumes "countable C"   and "C \<subseteq> K"   and "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"   and "S \<subseteq> (\<Union>i\<in>C. cball (a i) (5 * r i))" shows "thesis"
splus_mono: fixes v :: "'a"   and w :: "'a"   and t :: "'a" assumes "v \<preceq> w" shows "local.punit.splus t v \<preceq> local.punit.splus t w"
simple_functionD_2: fixes M :: "'a measure"   and g :: "'a \<Rightarrow> 'b"   and X :: "'b set" assumes "simple_function M g" shows "g -` X \<inter> space M \<in> sets M"
fp_cop_F_range_inv': fixes x :: "'a"   and ds :: "'b set" assumes "x \<in> Singleton_for_d.fp_cop_F ds" shows "x \<in> Field (Pd_singletons_for_ds X {Xd x} (Xd x)) \<and> Xd x \<in> ds"
Ax: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> synth H"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "\<phi>.A.ide (g \<cdot>\<^sub>A f)" obtains "\<phi>.A.seq g f" and "\<phi>.A.seq f g" and "g \<cdot>\<^sub>A f = \<phi>.A.dom f" and "g \<cdot>\<^sub>A f = \<phi>.A.cod g"
homotopic_paths_def: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_paths s p q \<equiv> homotopic_with_canon (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
ide_char: fixes a :: "'a \<times> 'a" shows "H.VV.ide a = (H.VV.arr a \<and> H.VxV.ide a)"
0: shows "I.int \<xi> T = Ik.int (transE \<xi>) T"
spr_simObsC: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState"   and aec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a"   and X :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) set" assumes "t \<leadsto> s \<in> SPR.jkbpC"   and "toSet aec = Restr (rel_ext (envObs_rel (envObs a))) X `` {(tFirst t, s)}"   and "X = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = (tFirst t', s) \<and> t' \<leadsto> s \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t)" shows "spr_simObsC envObsC aec = envObsC (es s)"
dom_1: shows "dom \<tau> = vars_term t \<union> dom \<sigma>"
apex_def: shows "\<nu>.cod.apex \<equiv> C.dom \<nu>.cod.leg0"
simps_4: fixes p :: "('b, 'c) fm" shows "FOL_Axiomatic.size_fm (\<^bold>\<forall> p) = (1::'a) + FOL_Axiomatic.size_fm p"
sint_minus1: fixes x :: "'b word" shows "(sint x = - 1) = (x = - 1)"
reflpD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "reflp r" shows "r x x"
convex_imp_unicoherent: fixes U :: "'a set" assumes "convex U" shows "unicoherent U"
gcsort_stab_intro: fixes f :: "'c \<Rightarrow> 'd list"   and key :: "'d \<Rightarrow> 'c"   and t :: "nat \<times> nat list \<times> 'd list"   and k :: "'c" assumes "stab_inv f key t" shows "filter (\<lambda>x. key x = k) (gcsort_out t) = f k"
drinks_1_rejects: fixes a :: "String.literal \<times> value list"   and r :: "nat \<Rightarrow>f value option" assumes "fst a = STR ''coin'' \<longrightarrow> length (snd a) \<noteq> 1"   and "a \<noteq> (STR ''vend'', [])" shows "possible_steps drinks 1 r (fst a) (snd a) = {||}"
has_least_prefixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "has_least_prefixpoint f \<equiv> \<exists>x. is_least_prefixpoint f x"
rows_to_keep_hyp: shows "\<forall>y. y \<in> set (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) \<longrightarrow> y < length subsets"
map_ident: fixes t :: "'a option" shows "map_option (\<lambda>x. x) t = t"
assms_2: shows "\<exists>y. (g \<longlongrightarrow> y) F"
inorder_splay: fixes x :: "'c"   and t :: "('c \<times> 'd) tree" shows "inorder (Splay_Map.splay x t) = inorder t"
assms_3: shows "AE x in M. 0 \<le> D x"
\<delta>_statesI_2: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set" assumes "q \<rightarrow> l qs \<in> \<delta>" shows "set qs \<subseteq> \<delta>_states \<delta>"
fimage_def: fixes r :: "V"   and A :: "V" shows "r `\<^sub>\<bullet> A = \<R>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A)"
rs_2: shows "s' = foldl (\<lambda>s x. f x s) s rs"
bet_obtuse__acute: fixes A :: "'p"   and B :: "'p"   and A' :: "'p"   and C :: "'p" assumes "Bet A B A'"   and "A' \<noteq> B"   and "Obtuse A B C" shows "Acute A' B C"
wca_on_warp_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "wca_on A f = warp_on A f"
unif: shows "Unifier \<sigma> (s \<cdot> \<theta>) (t \<cdot> \<theta>)"
map_fst_zip: fixes xs :: "'e list"   and ys :: "'f list" assumes "length xs = length ys" shows "map fst (xs || ys) = xs"
fresh_2: fixes a :: "name"   and x2 :: "name"   and x3 :: "trm"   and x1 :: "stack" shows "a \<sharp> [x2]x3\<ggreater>x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
not_None_eq: fixes x :: "'b option" shows "(x \<noteq> None) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
nnegI2: fixes P :: "'b \<Rightarrow> 'c" assumes "(\<lambda>s. 0::'c) \<le> P" shows "nneg P"
rword_def: fixes x :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and y :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "rword x y = (\<not> rword_strict y x)"
proj_arcs_H_def: fixes x :: "'b" shows "proj_arcs_H x \<equiv> if x = uw \<or> x = rev_H vw then uv else if x = vw \<or> x = rev_H uw then rev_G uv else x"
assms_2: shows "H \<turnstile> SyntaxN.All i (OrdP (Var i) IMP All2 j (Var i) (A(i::=Var j)) IMP A)"
distinct_length_2_or_more: fixes a :: "'c"   and b :: "'c"   and xs :: "'c list" shows "cycle (a # b # xs) = (a \<noteq> b \<and> cycle (a # xs) \<and> cycle (b # xs))"
nvars_SNOC: fixes x :: "'x"   and \<AA> :: "'i" shows "#\<^sub>V (SNOC x \<AA>) = #\<^sub>V \<AA>"
gf_F_def: fixes x :: "'s"   and y :: "'s"   and z :: "real" shows "gf_F x y z = (\<Sum>n. f x y n * z ^ n)"
lexmax_decompose: fixes \<alpha> :: "'a"   and r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list" assumes "\<alpha> \<in># r|\<sigma>|" shows "\<exists>\<sigma>1 \<sigma>3. \<sigma> = \<sigma>1 @ [\<alpha>] @ \<sigma>3 \<and> \<alpha> \<notin> r \<down>l \<sigma>1"
WtFT_le_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'e) cmd list" assumes "PL cval" shows "WtFT cl \<le> 1"
unsigned_word_eqI: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and v :: "'c word"   and w :: "'c word" assumes "class.semiring_char_0 one times plus zero"   and "semiring_1.unsigned one plus zero v = semiring_1.unsigned one plus zero w" shows "v = w"
total_ordering_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "total_ordering A less_eq less \<equiv> compatible_ordering A less_eq less \<and> total_ordered_set A less_eq"
filterout_untame_succs: fixes P :: "graph \<Rightarrow> bool"   and f :: "graph \<Rightarrow> graph list"   and U :: "graph \<Rightarrow> bool"   and f' :: "graph \<Rightarrow> graph list"   and g :: "graph"   and g' :: "graph" assumes "invariant P f"   and "invariant (\<lambda>g. P g \<and> U g) f"   and "untame (\<lambda>g. P g \<and> U g)"   and "\<And>g g'. \<lbrakk>P g; g' \<in> set (f g); g' \<notin> set (f' g)\<rbrakk> \<Longrightarrow> U g'"   and "g [f]\<rightarrow>* g'"   and "P g"   and "final g'"   and "tame g'" shows "g [f']\<rightarrow>* g'"
2: shows "w\<^sup>\<star> * e\<^sup>T * top \<le> w\<^sup>T\<^sup>\<star> * v"
limsup_ereal_mult_left: fixes c :: "real"   and f :: "nat \<Rightarrow> ereal" assumes "0 \<le> c" shows "limsup (\<lambda>n. ereal c * f n) = ereal c * limsup f"
coprime_P: shows "\<forall>p1 p2. p1 \<in> P \<and> p2 \<in> P \<and> p1 \<noteq> p2 \<longrightarrow> comm_monoid_mult_class.coprime p1 p2"
float_round_down_le: fixes x :: "float"   and z :: "real"   and prec :: "nat" assumes "real_of_float x \<le> z" shows "real_of_float (float_round_down prec x) \<le> z"
monomial_inj: fixes s :: "'d"   and c :: "'e"   and t :: "'d"   and d :: "'e" assumes "monomial c s = monomial d t" shows "c = (0::'e) \<and> d = (0::'e) \<or> c = d \<and> s = t"
\<phi>L_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   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 prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Loeb_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P" shows "Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi> \<equiv> Repr_SelfSubst.diag var fmla Fvars subst cnj imp bprv enc (imp P \<phi>)"
density_density_eq: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "density (density M f) g = density M (\<lambda>x. f x * g x)"