hcju/isabelleps huggingface dataset:
lookup_delete': fixes t :: "('c, 'd) Trie.trie"   and ks :: "'c list" assumes "invar_trie t" shows "lookup_trie (delete_trie ks t) = (lookup_trie t)(ks := None)"
assms_2: shows "\<guillemotleft>g : b \<cong> c\<guillemotright>"
small_Hom: fixes \<CC> :: "V"   and a :: "V"   and b :: "V" shows "small {f. f : a \<mapsto>\<^bsub>\<CC>\<^esub> b}"
the_inv_f_o_f_id: fixes f :: "'b \<Rightarrow> 'c"   and z :: "'b" assumes "inj f" shows "(the_inv f \<circ> f) z = id z"
eq_iff_15: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Auth term = Auth terma) = (term = terma)"
mult_rel_well_defined: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c set \<times> 'd"   and y :: "'c set \<times> 'd"   and z :: "'c set"   and z' :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "x \<in> Sigma I \<FF>"   and "y \<in> Sigma I \<FF>"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y"   and "z' \<in> I"   and "z' \<subseteq> fst x"   and "z' \<subseteq> fst y" shows "direct_lim.class_of \<FF> \<rho> I z (mult_str z (\<rho> (fst x) z (snd x)) (\<rho> (fst y) z (snd y))) = direct_lim.class_of \<FF> \<rho> I z' (mult_str z' (\<rho> (fst x) z' (snd x)) (\<rho> (fst y) z' (snd y)))"
in_hom: shows "\<guillemotleft>\<tau> \<cdot> \<mu> : local.dom \<mu> \<Rightarrow> cod \<tau>\<guillemotright>"
fcard_finsert_le: fixes A :: "'b fset"   and x :: "'b" shows "fcard A \<le> fcard (finsert x A)"
support_on_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.monoid_add plus zero" shows "monoid_add.support_on zero S f = {x \<in> S. f x \<noteq> zero}"
z_notin_ed: shows "z \<notin> closed_segment (a + complex_of_real e) (a - complex_of_real d)"
set_prevs_def: fixes base :: "'b"   and tail :: "'a list"   and n :: "nat" shows "set_prevs base tail n \<equiv> {(i, foldr f (map ((!) tail) is) base) |i is. subseq_of_length n [0..<length tail] is \<and> i = (if n = 0 then length tail else hd is)}"
is_final_def: fixes a :: "'a" shows "is_final a \<equiv> \<nexists>b. a \<rightarrow> b"
continuous_on_norm_id: fixes S :: "'d set" shows "continuous_on S norm"
digit_to_char_simps_6: shows "digit_to_char 5 = CHR ''5''"
mult_scaleR_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "real"   and x :: "'d"   and y :: "'d" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times (scaleR a x) y = scaleR a (times x y)"
Cons_5: shows "set lss = valid_states (prob_proj PROB vs)"
n\<^sub>3_def: shows "n\<^sub>3 = Max {LEAST m. stable_rank_at x m |x. x < n\<^sub>1 \<and> token_squats x}"
spr_sim_def: shows "spr_sim \<equiv> \<lambda>t. (spr_abs t, tLast t)"
geodesic_segment_between_x_x_2: fixes x :: "'b" shows "geodesic_segment {x}"
pgen_lossless_gpv_stop: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and fail :: "ennreal" assumes "\<I> \<turnstile>g gpv \<surd>"   and "fail \<le> 1" shows "pgen_lossless_gpv fail (stop_\<I> \<I>) (gpv_stop gpv) = pgen_lossless_gpv fail \<I> gpv"
elims_2: fixes x :: "'b literal" assumes "negative_literal x" obtains xa :: "'b equation" where "x = Neg xa"
takefill_Suc_Nil: fixes fill :: "'b"   and n :: "nat" shows "takefill fill (Suc n) [] = fill # takefill fill n []"
eval_dioph_mult: fixes ks :: "int list"   and xs :: "nat list"   and n :: "nat" shows "eval_dioph ks xs * int n = eval_dioph ks (map (\<lambda>x. x * n) xs)"
log_base_10_eq1: fixes x :: "real" assumes "0 < x" shows "log 10 x = ln (exp 1) / ln 10 * ln x"
eq_iff: fixes a :: "'c list"   and b :: "'c list" shows "(a = b) = (subseq b a \<and> subseq a b)"
divide_inverse: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a b = times a (inverse b)"
BRBangCases: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and F :: "ccs \<Rightarrow> bool" assumes "(P, !Q) \<in> bangRel Rel"   and "\<And>P. (P, Q) \<in> Rel \<Longrightarrow> F !P" shows "F P"
pure_tree_unique: fixes f :: "'b tree"   and x :: "'b" assumes "f = Node x f f" shows "f = pure x"
meval_intro: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and n :: "nat"   and mem :: "'Var \<Rightarrow> 'Val"   and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms ! n, mem) \<leadsto> (cm', mem')"   and "n < length cms" shows "(cms, mem) \<leadsto>\<^bsub>n\<^esub> (cms[n := cm'], mem')"
exhaust: fixes y :: "'c llist" obtains "y = LNil"   | x21 :: "'c" and x22 :: "'c llist" where "y = LCons x21 x22"
keys_single: fixes k :: "'a"   and v :: "'b" shows "keys (monomial v k) = (if v = (0::'b) then {} else {k})"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'c" assumes "SimplicialComplex X"   and "SimplicialComplexIsomorphism_axioms X f" shows "SimplicialComplexIsomorphism X f"
Limit_omega_oexp: fixes \<delta> :: "V" assumes "Ord \<delta>"   and "\<delta> \<noteq> 0" shows "Limit (\<omega> \<up> \<delta>)"
simps_1: fixes x1 :: "'c match_expr"   and x2 :: "action"   and y1 :: "'c match_expr"   and y2 :: "action" shows "(Rule x1 x2 = Rule y1 y2) = (x1 = y1 \<and> x2 = y2)"
times_empty_iff: fixes A :: "hf"   and B :: "hf" shows "(A * B = 0) = (A = 0 \<or> B = 0)"
fresh_trans_tm_iff: fixes i :: "atom"   and e :: "name list"   and t :: "tm" shows "i \<sharp> trans_tm e t = (i \<sharp> t \<or> i \<in> atom ` set e)"
matrix_vector_mul_bounded_linear: fixes A :: "(('a, 'b) vec, 'c) vec" shows "bounded_linear ((*v) A)"
suffix_commute: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" shows "s |\<^sub>s n |\<^sub>s m = s |\<^sub>s m |\<^sub>s n"
sat_sub_by_sat: fixes c\<^sub>2 :: "('c, 'd) conf"   and c\<^sub>1 :: "('c, 'd) conf" assumes "Conf.sat c\<^sub>2"   and "c\<^sub>2 \<sqsubseteq> c\<^sub>1" shows "Conf.sat c\<^sub>1"
INF_eq: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (g j) (f i)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (f i) (g j)" shows "Inf (f ` A) = Inf (g ` B)"
segment_def: shows "segment \<equiv> \<lambda>f a b. {x. segmentp f a b x}"
reflect_poly_def: fixes p :: "'a poly" shows "reflect_poly p = Poly (rev (coeffs p))"
vSwS: fixes n :: "nat" assumes "n < length cl" shows "1 / real (length cl) = 1 / real (length dl)"
statement60_one_way: fixes J :: "cltn2"   and p :: "proj2" assumes "is_K2_isometry J"   and "p \<in> hyp2" shows "apply_cltn2 p J \<in> hyp2"
if_NewHeapElem_typed: fixes t :: "'thread_id"   and xh :: "(status \<times> 'x) \<times> 'heap"   and ta :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action"   and x'h' :: "(status \<times> 'x) \<times> 'heap"   and ad :: "'addr"   and CTn :: "htype" assumes "t \<turnstile> xh -ta\<rightarrow>i x'h'"   and "NormalAction (NewHeapElem ad CTn) \<in> set \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"   and "typeof_addr ad \<noteq> None" shows "typeof_addr ad = \<lfloor>CTn\<rfloor>"
ind: fixes s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (s, l, r) A stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps (s + off, l, r) (shift A off, off) stp = (s' + off, l', r')"
simps_1: fixes k :: "'a"   and v :: "'c" shows "update_by_pair (k, v) [] = (if v = (0::'c) then [] else [(k, v)])"
initial_segment_ne: fixes xs :: "'a list"   and ys :: "'a list" assumes "initial_segment xs ys"   and "xs \<noteq> []" shows "ys \<noteq> [] \<and> hd ys = hd xs"
reach_Suc: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and n :: "nat" shows "reach A w (Suc n) = (if w !! n \<in> alphabet A then \<Union> (transition A (w !! n) ` reach A w n) else {})"
is_arg_minI: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and m :: "'b \<Rightarrow> 'c" assumes "P x"   and "\<And>y. P y \<Longrightarrow> \<not> m y < m x" shows "is_arg_min m P x"
rbt_lookup_rbt_less: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and t :: "('a, 'b) rbt" assumes "ord.rbt_less less k t" shows "ord.rbt_lookup less t k = None"
zmult_pos_pos: fixes i :: "int"   and j :: "int" assumes "0 < i"   and "0 < j" shows "0 < i * j"
d\<^sub>1_def: shows "d\<^sub>1 \<equiv> map_fun eval\<Pi>\<^sub>1 id Some"
norm_le_componentwise: fixes x :: "'b"   and y :: "'b" assumes "\<And>b. b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> \<bar>y \<bullet> b\<bar>" shows "norm x \<le> norm y"
iTILL_sub: fixes k :: "nat"   and n :: "nat" shows "k \<ominus> [\<dots>n] = (if n \<le> k then [k - n\<dots>,n] else [\<dots>k])"
prod_emb_PiE_same_index: fixes I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)" shows "prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E"
axioms_1: fixes M :: "'b measure" assumes "finite_measure M" shows "sigma_finite_measure M"
assms: fixes xs :: "'a list" assumes "xs \<in> set xss" shows "length xs = l"
assms: fixes \<sigma> :: "(name list \<times> 'a list) list" assumes "wellFormedSubst \<sigma>" shows "(\<Psi>, P[<\<sigma>>], Q[<\<sigma>>]) \<in> Rel"
capacity_A_weight: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_OUT cap \<langle>x\<rangle> \<le> weight \<Gamma> x"
contradict_def: fixes as :: "atom list"   and cs :: "real list" shows "contradict as cs = (length cs = length as \<and> (\<forall>c\<in>set cs. 0 \<le> c) \<and> (case cs \<odot>\<^sub>a as of Less r cs \<Rightarrow> 0 \<le> r \<and> (\<forall>c\<in>set cs. c = 0) | Eq r cs \<Rightarrow> r \<noteq> 0 \<and> (\<forall>c\<in>set cs. c = 0)))"
valid_hist: shows "valid_history program_step ts\<^sub>s\<^sub>b"
card_isolated_verts_da: shows "card H.isolated_verts = Suc (card G.isolated_verts)"
simps_1: fixes \<theta> :: "trm"   and \<eta> :: "trm" shows "Geqo (Aterm \<theta>) (Aterm \<eta>) = Afml (Geq \<theta> \<eta>)"
has_integral_def: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'c"   and s :: "'b set" shows "(f has_integral I) s = (if \<exists>a b. s = cbox a b then (sum (\<lambda>(x, k). Henstock_Kurzweil_Integration.content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter s) else \<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'b) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. (sum (\<lambda>(x, k). Henstock_Kurzweil_Integration.content k *\<^sub>R (if x \<in> s then f x else (0::'c))) \<longlongrightarrow> z) (division_filter (cbox a b)) \<and> norm (z - I) < e))"
g'_integrable: shows "\<exists>a. \<forall>b\<ge>a. integrable (\<lambda>u. g' u / u powr (p + 1)) a b"
integral_map_pmf: fixes g :: "'g \<Rightarrow> 'f"   and p :: "'g pmf"   and f :: "'f \<Rightarrow> 'e" shows "measure_pmf.expectation (map_pmf g p) f = measure_pmf.expectation p (\<lambda>x. f (g x))"
simps_2: fixes v :: "'b" shows "Infty + Num v = Infty"
Cons_1: fixes A :: "real mpoly list"   and L' :: "atom list" assumes "find_eq var L = (A, L')" shows "set (map Eq A @ L') = set L"
spr_sim: shows "sim SPR.MC spr_simMC spr_sim"
OKI_def: fixes inp :: "'c \<Rightarrow> 'd option" shows "OKI inp = lift OK inp"
add_leaves_mono: fixes xs :: "'a list"   and ys :: "'a list"   and tr :: "'a bdd" assumes "set xs \<subseteq> set ys" shows "set (add_leaves tr xs) \<subseteq> set (add_leaves tr ys)"
try_spmf_bind_spmf_lossless: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'd spmf"   and q :: "'d spmf" assumes "lossless_spmf p" shows "TRY p \<bind> f ELSE q = p \<bind> (\<lambda>x. TRY f x ELSE q)"
vimage2p_Grp: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\<inverse>\<inverse>"
simps_2: fixes u :: "'b"   and d :: "'b"   and v :: "'b"   and x :: "bool"   and xs :: "bool list" shows "lrev_geom u d v (x # xs) = (if x then u else d) * lrev_geom u d v xs"
matcher_def: fixes r :: "'b rexp"   and s :: "'b list" shows "matcher r s = nullable (derivs s r)"
assms: fixes x :: "'a" assumes "x \<in> S" shows "B *\<^sub>V x = \<phi> x"
bounded_byD2: fixes b :: "real"   and P :: "'b \<Rightarrow> real" assumes "bounded_by b P" shows "P \<tturnstile> \<lambda>s. b"
image_mset_empty: fixes f :: "'l \<Rightarrow> 'k" shows "image_mset f {#} = {#}"
Outside_def: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "Outside' X R = R - X \<times> Range R"
simple_function_iff_borel_measurable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "simple_function M f = (finite (f ` space M) \<and> f \<in> borel_measurable M)"
P: shows "P \<in> Units (ring_mat TYPE('a) n undefined)"
integrable_from_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "subalgebra M F"   and "integrable (restr_to_subalg M F) f" shows "integrable M f"
simps_1: fixes \<A> :: "'a ltln set" shows "\<A> \<Turnstile>\<^sub>P true\<^sub>n = True"
assms: shows "run_one_step d i (s, vs, ves, $Loop x61 x62) = (s', vs', res)"
upd_cond_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'a option"   and u :: "'v"   and v' :: "'v" shows "upd_cond Q \<pi> u v' \<equiv> (v', u) \<in> edges g \<and> v' \<noteq> r \<and> (Q v' = \<infinity> \<longrightarrow> \<pi> v' = None) \<and> enat (w {v', u}) < Q v'"
prim_invar1_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set" shows "Prim.prim_invar1 w g r A \<equiv> is_subset_MST w (Prim.rg g r) A \<and> (\<forall>(u, v)\<in>A. (v, r) \<in> A\<^sup>*)"
det_interchange_same_rows: fixes i :: "'b"   and j :: "'b"   and A :: "(('c, 'b) vec, 'b) vec" assumes "i = j" shows "det (interchange_rows A i j) = det A"
to_mini_is_mini: fixes F :: "'b formula" shows "is_mini_formula (to_mini_formula F)"
G_def: shows "G \<equiv> \<lambda>z. \<Sum>n. g n * z ^ n"
DBM_reset_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and k :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" shows "DBM_reset M n k d M' \<equiv> (\<forall>j\<le>n. 0 < j \<and> k \<noteq> j \<longrightarrow> M' k j = \<infinity> \<and> M' j k = \<infinity>) \<and> M' k 0 = Le d \<and> M' 0 k = Le (- d) \<and> M' k k = M k k \<and> (\<forall>i\<le>n. \<forall>j\<le>n. i \<noteq> k \<and> j \<noteq> k \<longrightarrow> M' i j = min (dbm_add (M i k) (M k j)) (M i j))"
equivalence_in_bicategory_axioms: shows "internal_equivalence g' f \<phi> \<psi>'"
simps_3: fixes a :: "poscheckvariant"   and t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = \<langle>a: (t \<cdot> \<theta>) \<doteq> t' \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t"
f_nn_integral_harm_2: shows "\<integral>\<^sup>+ x. ennreal (f (x + 1)) \<partial>count_space UNIV \<le> ennreal (- harm n / ln q)"
c: shows "rawpsubst (imp \<phi>1 \<phi>2) (zip (map Var us) (map snd txs)) = imp (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (rawpsubst \<phi>2 (zip (map Var us) (map snd txs)))"
rep_eq: fixes x :: "'b Filters.filter"   and xa :: "'b Filters.filter" shows "Filters.filter.Rep_filter (x \<squnion> xa) = filter_sup (Filters.filter.Rep_filter x) (Filters.filter.Rep_filter xa)"
elims: fixes x :: "bool \<Rightarrow> String.literal \<Rightarrow> integer option \<Rightarrow> 'b \<Rightarrow> 'c gState_scheme \<Rightarrow> pState \<Rightarrow> 'a"   and xa :: "varRef"   and xb :: "'b"   and xc :: "'c gState_scheme"   and xd :: "pState"   and y :: "'a" assumes "liftVar x xa xb xc xd = y" obtains f :: "bool \<Rightarrow> String.literal \<Rightarrow> integer option \<Rightarrow> 'b \<Rightarrow> 'c gState_scheme \<Rightarrow> pState \<Rightarrow> 'a" and gl :: "bool" and v :: "String.literal" and idx :: "expr option" and argm :: "'b" and g :: "'c gState_scheme" and p :: "pState" where "x = f" and "xa = VarRef gl v idx" and "xb = argm" and "xc = g" and "xd = p" and "y = f gl v (map_option (exprArith g p) idx) argm g p"
gcd_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "somegcd G a b \<in> carrier G"
Cb_def: shows "Cb = (6 / 1) ^ DIM('b)"
intros: fixes P1 :: "'a \<Rightarrow> bool"   and a :: "'a"   and P2 :: "'b \<Rightarrow> bool"   and b :: "'b" assumes "P1 a"   and "P2 b" shows "pred_prod P1 P2 (a, b)"
q_def: fixes p :: "real" shows "random_skip_list.q p = 1 - p"
a3: shows "(\<Union>x\<in>set Xi. case x of (x, s) \<Rightarrow> \<alpha>' x) \<subseteq> a (\<alpha> sctni)"
topology_eq: fixes T1 :: "'a topology"   and T2 :: "'a topology" shows "(T1 = T2) = (\<forall>S. openin T1 S = openin T2 S)"
seq_comp_failuresp_seq_comp_failures_eq: fixes P :: "'c option process"   and Q :: "'c option process" shows "seq_comp_failuresp P Q = (\<lambda>x xa. (x, xa) \<in> seq_comp_failures P Q)"
add_le_same_cancel2: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus a b) b = less_eq a zero"
floor_real_alg: fixes x :: "real_alg" shows "\<lfloor>real_of x\<rfloor> = \<lfloor>x\<rfloor>"
Un_left_absorb: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> (A \<union> B) = A \<union> B"
prems_4: fixes x :: "real \<times> real" assumes "x \<in> set (x__ # xs)" shows "0 < snd x"
actDn_def: fixes k :: "cnf" shows "actDn k \<equiv> {nid. \<parallel>nid\<parallel>\<^bsub>k\<^esub> \<and> \<not> honest nid}"
inv_solve_left: fixes a :: "'a set"   and b :: "'a set"   and c :: "'a set" assumes "a \<in> carrier (R Quot I)"   and "b \<in> carrier (R Quot I)"   and "c \<in> carrier (R Quot I)" shows "(a = \<ominus>\<^bsub>R Quot I\<^esub> b \<oplus>\<^bsub>R Quot I\<^esub> c) = (c = b \<oplus>\<^bsub>R Quot I\<^esub> a)"
toplevel_summands_flatten_ACI_norm_image: fixes r :: "'b rexp" shows "toplevel_summands (flatten PLUS (ACI_norm ` toplevel_summands r)) = ACI_norm ` toplevel_summands r"
induct_rulify_fallback_4: fixes A :: "bool"   and B :: "bool" shows "HOL.induct_conj A B \<equiv> A \<and> B"
hta_reindex_def: fixes H :: "('b, 'a) hashedTa" shows "hta_reindex H \<equiv> hta_remap (reindex_map H) H"
tangent_space_dim_eq: fixes X :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "X \<subseteq> src.tangent_space p" shows "src.tangent_space.dim p X = dim X"
succ_smallest: fixes r :: "('a \<times> 'a) set"   and i :: "'a"   and j :: "'a" assumes "wo_rel r"   and "(i, j) \<in> r"   and "i \<noteq> j" shows "(succ r i, j) \<in> r"
A3_DAcc_level0: shows "DAcc level0 sA3 = {sA4}"
span_scale: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'a \<Rightarrow>\<^sub>0 'c" assumes "x \<in> pmdl S" shows "c \<odot> x \<in> pmdl S"
prems: shows "Suc k < min (nrows A) (ncols A)"
psiFreshVec_2: fixes xvec :: "name list"   and M :: "'d"   and I :: "('d, 'e, 'f) input" shows "xvec \<sharp>* M\<lparr>I = (xvec \<sharp>* M \<and> xvec \<sharp>* I)"
sum_SDG_slice1_def: shows "sum_SDG_slice1 \<equiv> \<lambda>n. {x. sum_SDG_slice1p n x}"
x0_hb_bound3: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "x - (bs ! i * x + (hs ! i) x) \<le> x"
geod_distance_successor: fixes n :: "nat" shows "\<forall>a<2 ^ n. dist (geod n a) (geod n (Suc a)) = dist x0 y0 / 2 ^ n"
count_def: fixes t :: "('a, 'b) FingerTree" shows "FingerTree.count t \<equiv> FingerTreeStruc.count (Rep_FingerTree t)"
in_cosets_aux: fixes x :: "nat" assumes "euler_witness (int x) p"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<in> rcosets\<^bsub>G\<^esub> H"
simps_4: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and t :: "('b, 'pred, 'act, 'effect) Tree" shows "FL_valid_Tree P (tAct f \<alpha> t) = (\<exists>\<alpha>'' t' P'. tAct f \<alpha> t =\<^sub>\<alpha> tAct f \<alpha>'' t' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> FL_valid_Tree P' t')"
inv_f_f: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "inj f" shows "inv f (f x) = x"
Lyndon_unbordered: fixes w :: "'a list" assumes "Lyndon w" shows "\<not> bordered w"
assoc: fixes a :: "'g list"   and b :: "'g list"   and c :: "'g list" shows "(a @ b) @ c = a @ b @ c"
sumIdempLeft: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<leadsto>[Rel] P \<oplus> P"
ap_tree_strong_extensional: fixes f :: "('b \<Rightarrow> 'c) tree"   and g :: "('b \<Rightarrow> 'c) tree" assumes "\<And>x. f \<diamondop> pure x = g \<diamondop> pure x" shows "f = g"
Busemann_function_liminf: fixes u :: "nat \<Rightarrow> 'b"   and xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi" shows "ereal (Busemann_function_at xi x y) \<le> liminf (\<lambda>x. ereal (dist x (u x) - dist y (u x))) + ereal (2 * deltaG TYPE('b))"
image_Int: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj f" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
\<sigma>_def: fixes P :: "(int \<times> int) list" shows "R.\<sigma> P = map R.indexed_const P"
Irel_induct: fixes P1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and P2 :: "'d IF1 \<Rightarrow> 'e IF1 \<Rightarrow> bool"   and P3 :: "'d IF2 \<Rightarrow> 'e IF2 \<Rightarrow> bool" assumes "\<forall>x y. F1rel P1 P2 P3 x y \<longrightarrow> P2 (ctor1 x) (ctor1 y)"   and "\<forall>x y. F2rel P1 P2 P3 x y \<longrightarrow> P3 (ctor2 x) (ctor2 y)" shows "IF1rel P1 \<le> P2 \<and> IF2rel P1 \<le> P3"
alphaAbs_imp_asAbs_equal: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and qB :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs qA"   and "qA $= qB" shows "asAbs qA = asAbs qB"
select_convs_2: fixes V :: "'a set"   and N :: "'a set"   and C :: "'a set"   and more :: "'b" shows "N \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = N"
alpha_refl: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood X" shows "X #= X"
frameEqSupp: fixes x :: "name"   and F :: "'d frame"   and y :: "name"   and G :: "'d frame" assumes "\<lparr>\<nu>x\<rparr>F = \<lparr>\<nu>y\<rparr>G"   and "x \<in> supp F" shows "y \<in> supp G"
welldefined_subsets2: shows "all_list_constr subsets2 (length qs2)"
sets_Collect_countable_Ex1': fixes I :: "'b set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> {x \<in> \<Omega>. P i x} \<in> M"   and "countable I" shows "{x \<in> \<Omega>. \<exists>!i. i \<in> I \<and> P i x} \<in> M"
Uint_def: shows "Uint \<equiv> map_fun int_of_integer Abs_uint word_of_int"