hcju/isabelleps huggingface dataset:
\<oo>\<^sub>P_in_cfl: fixes G :: "('a, 'b) cfg"   and \<alpha> :: "('a + 'b) list" assumes "\<oo>\<^sub>P G \<alpha>" shows "in_cfl G \<alpha> []"
S_reduced_imp_distinct_lconjseq: fixes ss :: "'w list" assumes "S_reduced ss" shows "distinct (lconjseq ss)"
rts1: shows "\<forall>p t i. correct p t \<and> te p (i + 1) \<le> t \<longrightarrow> rmin \<le> t - te p i"
FunctionalConjunctive_def: fixes H :: "'b upred \<Rightarrow> 'b upred" shows "FunctionalConjunctive H = (\<exists>F. \<forall>P. H P = (P \<and> F P) \<and> Monotonic F)"
rel_distinct_1: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and y21 :: "'h"   and y22 :: "'h list" shows "\<not> list_all2 R [] (y21 # y22)"
index_row_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "row A i $ j = A $$ (i, j)"
card_gr0_imp_not_empty: fixes A :: "'b set" assumes "0 < card A" shows "A \<noteq> {}"
params_lift_1: fixes t :: "'a FOL_Fitting.term" shows "paramst (liftt t) = paramst t"
uniformly_convergent_cong: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and B :: "'a set" assumes "\<forall>\<^sub>F x in sequentially. \<forall>y\<in>A. f x y = g x y"   and "A = B" shows "uniformly_convergent_on A f = uniformly_convergent_on B g"
simps: fixes g1 :: "name"   and g2 :: "name"   and f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm" assumes "atom g1 \<sharp> (g2, f1, k1, f2, k2, g)"   and "atom g2 \<sharp> (f1, k1, f2, k2, g)" shows "SeqAppendP f1 k1 f2 k2 g = SyntaxN.Ex g1 (SyntaxN.Ex g2 (RestrictedP f1 k1 (Var g1) AND ShiftP f2 k2 k1 (Var g2) AND UnionP (Var g1) (Var g2) g))"
list_decode_singleton: fixes x :: "nat" shows "Partial_Recursive.list_decode (singleton_encode x) = [x]"
inres_simps_1: shows "inres FAIL = (\<lambda>_. True)"
assms_3: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (Gs i)"
and_def: fixes P :: "bool"   and Q :: "bool" shows "P \<and> Q \<equiv> \<forall>R. (P \<longrightarrow> Q \<longrightarrow> R) \<longrightarrow> R"
is_lub_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below" shows "po.is_lub below S x = (po.is_ub below S x \<and> (\<forall>u. po.is_ub below S u \<longrightarrow> below x u))"
t_stop_nonneg: shows "0 \<le> delayed_safe_distance.other.t_stop"
simps_1: fixes uu :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and uv :: "'d" shows "no_matching_Goto uu uv [] = True"
rev_append_invert: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs @ ys = rev zs) = (zs = rev ys @ rev xs)"
clean_no_outstanding_volatile_Write\<^sub>s\<^sub>b: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<not> \<D>" shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"
Sigp_def: shows "Sigp \<equiv> \<lambda>f i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / real (p + 2)"
parent_edge_cases: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a"   and v :: "'a"   and w :: "'a" assumes "RootedTree G root"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "w = RootedTree.parent G root v \<or> v = RootedTree.parent G root w"
of_int_le_numeral_iff: fixes z :: "int"   and n :: "num" shows "(of_int z \<le> numeral n) = (z \<le> numeral n)"
reachable_succs: fixes x :: "'a" shows "reachable (set (succs x)) \<subseteq> reachable {x}"
sup_idem: fixes sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type" assumes "class.semilattice_sup (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) x = x"
axiom10_F: fixes x :: "('l, 'm, 'k, 'n) F" shows "|F'set1 x| \<le>o F'bd"
ccSUP_inf: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c" assumes "class.countable_complete_distrib_lattice Inf Sup inf less_eq less sup bot top"   and "countable B" shows "inf (Sup (f ` B)) a = Sup ((\<lambda>b. inf (f b) a) ` B)"
cong_let: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "Let y f = f y"
init_def: fixes y :: "'grp"   and w :: "nat \<times> nat" shows "init y w = (let (x1, x2) = w in sample_uniform (order \<G>) \<bind> (\<lambda>r1. sample_uniform (order \<G>) \<bind> (\<lambda>r2. return_spmf ((r1, r2), \<^bold>g [^] r1 \<otimes> g' [^] r2))))"
compute_bound_poly_correct: fixes p :: "float poly"   and I :: "float interval list"   and e :: "nat \<Rightarrow> real"   and a :: "float interval list"   and prec :: "nat" assumes "num_params p \<le> length I"   and "develops_at_within e a I" shows "insertion e (map_poly real_of_float p) \<in>\<^sub>r compute_bound_poly prec (map_poly interval_of p) I a"
scaleR_nonneg_nonneg: fixes a :: "real"   and x :: "'a" assumes "0 \<le> a"   and "(0::'a) \<le> x" shows "(0::'a) \<le> a *\<^sub>R x"
hyps_5: shows "fps\<langle>l\<rangle>\<^sub>? = \<bottom>"
check_monom_strict_mono_def: fixes pm :: "bool"   and m :: "'c monom"   and v :: "'c" shows "check_monom_strict_mono pm m v \<equiv> case univariate_power v m of None \<Rightarrow> False | Some p \<Rightarrow> pm \<or> p = 1"
sim_fI: fixes M :: "('a, 'c, 'b) KripkeStructure"   and f :: "'b \<Rightarrow> 'd"   and M' :: "('a, 'c, 'd) KripkeStructure" assumes "\<And>a u v. (u, v) \<in> relations M a \<Longrightarrow> (f u, f v) \<in> relations M' a" shows "sim_f M M' f"
C_arr_expansion_2: fixes f :: "'c" assumes "C.arr f" shows "C.inv (\<epsilon> (C.cod f)) \<cdot>\<^sub>C f \<cdot>\<^sub>C \<epsilon> (C.dom f) = F (G f)"
3: assumes "ll \<noteq> \<langle>\<rangle>" shows "mset_tree (del_min ll) = mset_tree ll - {#get_min ll#}"
steps_is_node: fixes q :: "'a"   and w :: "'b list" assumes "is_node q"   and "list_all is_alpha w" shows "is_node (foldl trans q w)"
prems_2: shows "P,ba___ \<turnstile>\<^sub>b ({V:T; Throw a},b_) \<surd>"
nlistToTree_list: fixes nl :: "('c, 'd) Node list" shows "toList (nlistToTree nl) = nlistToList nl"
map_upd_eqD1: fixes m :: "'b \<Rightarrow> 'c option"   and a :: "'b"   and x :: "'c"   and n :: "'b \<Rightarrow> 'c option"   and y :: "'c" assumes "m(a \<mapsto> x) = n(a \<mapsto> y)" shows "x = y"
Sup_subset_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
preimage_identity_self: fixes A :: "'a set"   and B :: "'a set" shows "identity A \<^sup>\<inverse> A B = B \<inter> A"
2: shows "top * e * w\<^sup>T\<^sup>+ \<le> top * v\<^sup>T"
h_def: shows "h \<equiv> \<lambda>p. g p / f p + 1"
plus_monom_zero: fixes p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "p + MPoly_Type.monom m (0::'a) = p"
real2_of_real_def: fixes p :: "nat"   and x :: "real" shows "real2_of_real p x = (truncate_down (Suc p) x, truncate_up (Suc p) x)"
b: shows "prime (3 * 2 ^ (4 - 1) - 1)"
longest_Lyndon_suf_Lyndon: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "Lyndon (LynSuf w)"
res_i2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<le> y" shows "f (residual f x) \<le> f (residual f y)"
trancl_impl_def: fixes gen_succ :: "('c \<times> 'c) list \<Rightarrow> 'c list \<Rightarrow> 'c list"   and un :: "'c list \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and emp :: "'d"   and rel :: "('c \<times> 'c) list" shows "trancl_impl gen_succ un memb emp rel = (let succ = gen_succ rel; n = length rel in (\<lambda>as. relpow_impl succ un memb (succ as) emp n))"
8: shows "e * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
BB: shows "dim_row B = dim_row B' \<and> dim_col B = dim_col B' \<and> (\<forall>i j. i < dim_row B' \<longrightarrow> j < dim_col B' \<longrightarrow> R (B $$ (i, j)) (B' $$ (i, j)))"
f_1: shows "f \<in> U \<rightarrow> carrier_vec n"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'k list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
le_None: fixes x :: "'a option" shows "(x \<le> None) = (x = None)"
contractible_space_alt: fixes X :: "'b topology" shows "contractible_space X = (\<forall>a\<in>topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a))"
assms_7: shows "\<not> is_sig_red (\<prec>\<^sub>t) (=) G p"
step_hd_b: shows "\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). \<tau> = hd b # a ! hd b"
stream_pred__sconst: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "pred_stream P (sconst x) = P x"
insert_inter_insert: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" shows "insert a A \<inter> insert a B = insert a (A \<inter> B)"
simps_9: fixes p :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Exists p[s/k] = Exists (p[liftt s/Suc k])"
unique_dimension: fixes K :: "'c set"   and E :: "'c set" assumes "subfield K S"   and "S.finite_dimension K E" shows "\<exists>!n. S.dimension n K E"
fmdom'_restrict_set: fixes A :: "'a set"   and m :: "('a, 'b) fmap" shows "fmdom' (fmrestrict_set A m) \<subseteq> A"
impl_refines_spec: shows "COPY \<sqsubseteq> SYSTEM"
orient2: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> set (filter (\<lambda>(x, y). weak_gt x y) (concat (map (\<lambda>AB. concat (map (\<lambda>i. map (\<lambda>j. (fst AB $$ (i, j), snd AB $$ (i, j))) [0..<n]) [0..<n])) ABs)))" shows "gt x y"
leq_path_last: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   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 Cs :: "char list list"   and Cs' :: "char list list"   and C :: "char list" assumes "wf_prog wf_md P"   and "P,C \<turnstile> Cs \<sqsubseteq> Cs'"   and "last Cs = last Cs'" shows "Cs = Cs'"
AW: shows "A \<inter> W - U \<subseteq> carrier V"
TS_step_d_def: fixes s :: "'d list \<times> 'd list"   and q :: "'d" shows "TS_step_d s q = ((let li = index (snd s) q in if li = length (snd s) then 0 else let sincelast = take li (snd s); S = {x. x < q in fst s \<and> count_list sincelast x \<le> 1} in if S = {} then 0 else index (fst s) q - Min (index (fst s) ` S), []), q # snd s)"
ipv4addr_upto_def: fixes i :: "32 word"   and j :: "32 word" shows "ipv4addr_upto i j \<equiv> map (word_of_nat \<circ> nat) [int (unat i)..int (unat j)]"
measurable_bounded_differentiable_image: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and B :: "real" assumes "S \<in> lmeasurable"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> \<bar>det (matrix (f' x))\<bar> \<le> B" shows "f ` S \<in> lmeasurable"
no_sing: fixes t :: "real" shows "exp (complex_of_real r * cis t) \<noteq> 1"
lower_hemicontinuous: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and T :: "'b set" assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T" shows "(\<forall>U. closedin (top_of_set T) U \<longrightarrow> closedin (top_of_set S) {x \<in> S. f x \<subseteq> U}) = (\<forall>U. openin (top_of_set T) U \<longrightarrow> openin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}})"
int_eq_sint: fixes x :: "nat" assumes "x < 2 ^ (LENGTH('a) - 1)" shows "sint (word_of_nat x) = int x"
valuate_def: fixes x :: "(nat \<Rightarrow> real) \<Rightarrow> 'a" shows "valuate x = x ` (UNIV \<rightarrow> {- 1..1})"
poly_deg_extend_indets: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_deg (extend_indets p) = poly_deg p"
eint_add_left_cancel_less: fixes a :: "eint"   and b :: "eint"   and c :: "eint" shows "(a + b < a + c) = (a \<noteq> \<infinity> \<and> b < c)"
a_AuthT_value_inv: fixes v :: "Syntax.term"   and vP :: "Syntax.term"   and vV :: "Syntax.term"   and \<tau> :: "ty" assumes "{$$} \<turnstile> v, vP, vV : AuthT \<tau>"   and "value v"   and "value vP"   and "value vV" obtains vP' :: "Syntax.term" where "vP = Hashed (hash \<lparr>vP'\<rparr>) vP'" and "vV = Hash (hash \<lparr>vP'\<rparr>)" and "value vP'"
well_founded_ordered_set_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "well_founded_ordered_set A less \<equiv> well_founded A less \<and> transitive A less"
phils_transitions_def: fixes n :: "nat"   and ps :: "nat list" shows "Tr\<^sub>P n ps = Tr\<^sub>l\<^sub>p (ps ! 0) \<union> (\<Union>i\<in>{1..<n}. Tr\<^sub>r\<^sub>p i (ps ! i))"
PO_m1_inv1r_cache_init: shows "init m1 \<subseteq> m1_inv1r_cache"
OrdP_HPairEH_1: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (OrdP (HPair x y)) H \<turnstile> A"
has_densityD_2: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "M = density N f"
has_derivative_def: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (netlimit F) - f' (y - netlimit F)) /\<^sub>R norm (y - netlimit F)) \<longlongrightarrow> (0::'b)) F)"
simps_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a" shows "ins3_tm n d x \<langle>\<rangle> =1 (if bal_i (n + 1) (d + 1) then return (Bal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle>) else return (Unbal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> 1 1))"
real_root_eq_iff: fixes n :: "nat"   and x :: "real"   and y :: "real" assumes "0 < n" shows "(root n x = root n y) = (x = y)"
dom_simp: fixes f :: "'d \<times> 'd" assumes "D.VV.arr f" shows "D.VV.dom f = D.VxV.dom f"
filtermap_tan_at_right_inf: fixes a :: "real" assumes "cos a = 0" shows "filtermap tan (at_right a) = at_bot"
initQ_def: shows "initQ \<equiv> [s \<mapsto> 0::'c]"
finiteListDecompose: fixes l :: "'b list" shows "finite {(a, b). l = a @ b}"
IH1: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "wf_net_tree p1"   and "s \<in> reachable (pnet np p1) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np p1)" shows "(\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np (i:deliver(d)) i p1 s s'"
simps_2: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and c :: "color"   and r :: "('b \<times> color) tree" shows "bheight \<langle>l, (x, c), r\<rangle> = (if c = Black then bheight l + 1 else bheight l)"
Areflassump1: shows "Arefl R1 \<inter> Arefl R2 \<subseteq> {[]}"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (\<not> (nat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. nat = Suc x2 \<and> \<not> P (f2 x2))))"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> 'f" assumes "functor A B F"   and "functor B C G" shows "composite_functor A B C F G"
right_unique_def: fixes R :: "'j \<Rightarrow> 'k \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
assms: shows "lfinite (\<pi>\<^bsub>c\<^esub>inf_llist t)"
equiintegrable_diff: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set"   and G :: "('a \<Rightarrow> 'b) set" assumes "F equiintegrable_on I"   and "G equiintegrable_on I" shows "(\<Union>f\<in>F. \<Union>g\<in>G. {\<lambda>x. f x - g x}) equiintegrable_on I"
simps_1: fixes M :: "'d"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "M\<lparr>N\<rparr> \<prec> P = RIn M N P"
d2: shows "fresh_fun (pi \<bullet> h) = (pi \<bullet> h) (pi \<bullet> a')"
typ_of1_subst_bv_gen': fixes uty :: "typ"   and Ts :: "typ list"   and t :: "Core.term"   and tty :: "typ"   and u :: "Core.term" assumes "typ_of1 (uty # Ts) t = Some tty"   and "typ_of1 Ts u = Some uty" shows "typ_of1 Ts (subst_bv1 t 0 u) = Some tty"
xmax: fixes z :: "complex" assumes "z \<in> closure S" shows "cmod (f z) \<le> cmod (f x)"
rep_eq: fixes x :: "('a \<times> 'b) set" shows "pfun_of (graph_ffun x) = (if finite (Domain x) then graph_pfun x else {}\<^sub>p)"
ordinal_plus_minus2: fixes x :: "ordinal"   and y :: "ordinal" assumes "x \<le> y" shows "x + (y - x) = y"
set_bit_unfold: fixes w :: "'b word"   and n :: "nat"   and b :: "bool" shows "set_bit_class.set_bit w n b = (if b then ring_bit_operations_class.set_bit n w else unset_bit n w)"
alw_S: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "almost_everywhere (MDP.T cfg) (pred_stream (\<lambda>s. s \<in> S))"
token_run_mergepoint: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x < y"   and "token_run x (y + n) = token_run y (y + n)" obtains m :: "nat" where "x \<le> Suc m" and "y \<le> Suc m" and "y = Suc m \<or> token_run x m \<noteq> token_run y m" and "token_run x (Suc m) = token_run y (Suc m)"
h0_b: fixes n :: "nat" shows "d_IN (h 0) b \<le> d_IN (h n) b"
wprepare_loop2: fixes b :: "cell list"   and mr :: "nat"   and a :: "nat"   and lista :: "nat list"   and m :: "nat"   and lm :: "nat list"   and list :: "cell list"   and rn :: "nat" assumes "rev b @ Oc \<up> mr @ Bk # <a # lista> = Oc \<up> Suc m @ Bk # Bk # <lm>"   and "b \<noteq> []"   and "Oc # list = Oc \<up> mr @ Bk # <a # lista> @ Bk \<up> rn" shows "wprepare_loop_start_in_middle m lm (Oc # b, list)"
Inf_closed_clop: fixes X :: "'b set" assumes "Inf_closed_set X" shows "clop (\<lambda>y. \<Sqinter> {x \<in> X. y \<le> x})"
subspace_0: fixes S :: "'c set" assumes "m2.subspace S" shows "(0::'c) \<in> S"
period_def: fixes C :: "'s set" shows "period C = (SOME d. \<forall>x\<in>C. d = Gcd (period_set x))"
apartments_are_coxeter: fixes A :: "'a set set" assumes "A \<in> \<A>" shows "\<exists>S. CoxeterComplex S \<and> (\<exists>\<psi>. ChamberComplexIsomorphism A (CoxeterComplex.TheComplex S) \<psi>)"
lm147: fixes A :: "'b set" assumes "card (Pow A) = 2" shows "card A = 1"
selection_completion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A" shows "selection K \<phi> \<subseteq> Tarskian_logic.completion Cn selection K \<phi>"
\<alpha>_def: shows "f \<bullet> \<alpha> = \<zero>"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and a :: "('fun, 'b) stateful_strand_step" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>))"
scalar_vector_matrix_assoc: fixes k :: "'b"   and x :: "('b, 'd) vec"   and A :: "(('b, 'c) vec, 'd) vec" shows "k *s x v* A = k *s (x v* A)"
sorted_on_Cons: fixes x :: "'a"   and xs :: "'a list" shows "sorted_on r (x # xs) = (x \<in> Field r \<and> sorted_on r xs \<and> (\<forall>y\<in>set xs. (x, y) \<in> r))"
field_inverse_zero: shows "inverse (0::'a) = (0::'a)"
is_tree_def: fixes g :: "('c, 'd) rgraph_scheme" shows "is_tree g \<equiv> \<forall>l\<in>Graph.vertices g. \<exists>!p. path g p l"
lenx: shows "a * \<^bold>|x\<^bold>| < \<^bold>|z\<^bold>| + \<^bold>|x\<^bold>|"
add_mset_not_empty: fixes a :: "'c"   and A :: "'c multiset" shows "add_mset a A \<noteq> {#}"
monomial_pmdl_field: fixes B :: "('a \<Rightarrow>\<^sub>0 'b) set"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and v :: "'a"   and c :: "'b" assumes "is_monomial_set B"   and "p \<in> MPoly_Type_Class.punit.pmdl B"   and "v \<in> keys p" shows "monomial c v \<in> MPoly_Type_Class.punit.pmdl B"
ex_n_lists_def: fixes P :: "'b list \<Rightarrow> bool"   and n :: "nat" shows "ex_n_lists P n = (\<exists>xs\<in>set (List.n_lists n enum). P xs)"
add: fixes f :: "'q \<Rightarrow> 'r"   and x :: "'q"   and y :: "'q" assumes "ring_homomorphism f" shows "f (x + y) = f x + f y"
Seq_def: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a ;; b \<equiv> \<lambda>ab. a ab \<circ> b ab"
sep_conjI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and Q :: "'a \<Rightarrow> bool"   and y :: "'a"   and h :: "'a" assumes "P x"   and "Q y"   and "x ## y"   and "h = x + y" shows "(P \<and>* Q) h"
alpha_Tree_relI: fixes x :: "('b, 'c, 'd, 'e) Tree"   and y :: "('b, 'c, 'd, 'e) Tree" assumes "x =\<^sub>\<alpha> y" shows "(x, y) \<in> FL_Validity.alpha_Tree_rel"
fresh_Cons: fixes a :: "atom"   and x :: "'b"   and xs :: "'b list" shows "a \<sharp> x # xs = (a \<sharp> x \<and> a \<sharp> xs)"
entail_unions: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and I :: "'b set"   and N :: "'a set"   and Ni :: "'b \<Rightarrow> 'a set" assumes "consequence_relation Bot entails" shows "(\<forall>i\<in>I. entails N (Ni i)) = entails N (\<Union> (Ni ` I))"
sorted_list_of_set_remove: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A" shows "linorder.sorted_list_of_set less_eq (A - {x}) = remove1 x (linorder.sorted_list_of_set less_eq A)"
tendsto_Pair: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
assms_2: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
vsv_vdomain_vsingleton_vrange_vsingleton: fixes a :: "V" assumes "\<D>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = ZFC_in_HOL.set {a}" obtains b :: "V" where "\<R>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = ZFC_in_HOL.set {b}"
privateKey_notin_image_publicKey: fixes b :: "bool"   and A :: "agent"   and c :: "bool"   and AS :: "agent set" shows "invKey (publicKey b A) \<notin> publicKey c ` AS"
assms_4: fixes m :: "nat" assumes "m < k" shows "f m = (\<Sum>n<k. g n * unity_root k (int (m * n)))"
get_ancestors_si_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_ancestors_si ptr) h"
A: shows "Y \<in> ipurge_tr_rev_t_set X"
D_summable: fixes x :: "real" assumes "0 < x" shows "summable (\<lambda>n. D (real n + x))"
PPs_singleton: fixes x :: "'a" shows ".[{x}] = range (Poly_Mapping.single x)"