hcju/isabelleps huggingface dataset:
assms_5: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"
set_tree_inter: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.set_tree (Set2_Join.inter join t1 t2) = Tree2.set_tree t1 \<inter> Tree2.set_tree t2"
unfold: shows "enumFromToL = (\<Lambda> va v. Fixrec.run\<cdot>(match_up\<cdot>va\<cdot>(\<Lambda> x. match_up\<cdot>v\<cdot>(\<Lambda> y. Fixrec.succeed\<cdot>(if x \<le> y then LCons\<cdot>(up\<cdot>x)\<cdot>(enumFromToL\<cdot>(up\<cdot>(x + 1))\<cdot>(up\<cdot>y)) else LNil)))))"
simps_1: fixes x :: "'b equation" shows "get_sign (Pos x) = pos"
simps_1: fixes uu :: "'a word"   and uv :: "'a wordinterval" shows "matching_srcs uu [] uv = Empty_WordInterval"
lex_ext_cong: fixes xs1 :: "'b list"   and ys1 :: "'c list"   and m1 :: "nat"   and m2 :: "nat"   and xs2 :: "'b list"   and ys2 :: "'c list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and g :: "'c \<Rightarrow> 'c \<Rightarrow> bool \<times> bool" assumes "length xs1 = length ys1"   and "m1 = m2"   and "length xs2 = length ys2"   and "\<And>i. \<lbrakk>i < length ys1; i < length ys2\<rbrakk> \<Longrightarrow> f (xs1 ! i) (xs2 ! i) = g (ys1 ! i) (ys2 ! i)" shows "lex_ext f m1 xs1 xs2 = lex_ext g m2 ys1 ys2"
dWk: fixes k :: "nat" shows "denote_while_n_iter M0 M1 DS k \<rho> \<in> carrier_mat d d"
leftright_associativity: fixes x :: "brick list"   and y :: "brick list"   and z :: "brick list" shows "(x \<otimes> y) \<otimes> z = x \<otimes> y \<otimes> z"
fconverse_fconverse: shows "(r\<^sub>1\<inverse>\<^sub>\<bullet>)\<inverse>\<^sub>\<bullet> = r\<^sub>1"
d\<mu>: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs"   and "ii < length fs" shows "rat_of_int (d\<mu> ii j) = rat_of_int (d fs (Suc j)) * gs.\<mu> ii j"
entries_lookup: fixes t1 :: "('a, 'b) RBT.rbt"   and t2 :: "('a, 'b) RBT.rbt" shows "(RBT.entries t1 = RBT.entries t2) = (RBT.lookup t1 = RBT.lookup t2)"
less_not_sym: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.preorder less_eq less"   and "less x y" shows "\<not> less y x"
hyps: assumes "distinct ms"   and "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]" shows "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>star \<circ> f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]"
intro: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and N :: "'c set" assumes "Group_Theory.monoid M composition unit"   and "submonoid_axioms N M composition unit" shows "submonoid N M composition unit"
lookup_Trie_Mapping: shows "Mapping.lookup (Trie_Mapping t) = lookup t"
Iod_same: fixes A :: "'a set"   and B :: "'a set" assumes "A = B" shows "Iod D A = Iod D B"
borel_adapt_stoch_proc_borel_measurable: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a" assumes "borel_adapt_stoch_proc F X" shows "\<forall>n. random_variable borel (X n)"
subgroup_self: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "subgroup (carrier G) G"
to_bl_unfold: fixes w :: "'a word" shows "to_bl w = rev (map (bit w) [0..<LENGTH('a)])"
proj_eq_def: shows "proj_eq = (\<lambda>(x\<^sub>1, y\<^sub>1, z\<^sub>1) (x\<^sub>2, y\<^sub>2, z\<^sub>2). (z\<^sub>1 = \<zero>) = (z\<^sub>2 = \<zero>) \<and> x\<^sub>1 \<otimes> z\<^sub>2 = x\<^sub>2 \<otimes> z\<^sub>1 \<and> y\<^sub>1 \<otimes> z\<^sub>2 = y\<^sub>2 \<otimes> z\<^sub>1)"
set_intros_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "x21 \<in>. x21 # x22"
presVar_def: fixes h :: "('e, 'f, 'varSort, 'var, 'g) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) Recursion.model_scheme" shows "presVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = gVar MOD xs x"
fsyms_satA_cong: fixes intT :: "'tp \<Rightarrow> 'b \<Rightarrow> bool"   and intF1 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and intP :: "'psym \<Rightarrow> 'b list \<Rightarrow> bool"   and intF2 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and at :: "('fsym, 'psym) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'b" assumes "Struct wtFsym wtPsym arOf resOf intT intF1 intP"   and "Struct wtFsym wtPsym arOf resOf intT intF2 intP"   and "\<And>f. f \<in> fsymsA at \<Longrightarrow> intF1 f = intF2 f" shows "Struct.satA intF1 intP \<xi> at = Struct.satA intF2 intP \<xi> at"
pr_def: fixes m :: "nat \<times> 'b \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'a"   and i :: "'b" shows "local.pr m x i \<equiv> \<Sum>k = 0..<n. x k * m (k, i)"
head: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "sum g {m..n} = g m + sum g {m<..n}"
unsigned_drop_bit_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and w :: "'c word" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "semiring_1.unsigned one plus zero (drop_bit n w) = drop_bit n (take_bit LENGTH('c) (semiring_1.unsigned one plus zero w))"
that: fixes i :: "nat" assumes "xs ! i = y"   and "i < length xs - 1" shows "thesis"
isometry_inverse_2: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "bij f"
path2_last: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "m = last ns"
partial_equiv_iff: fixes A :: "'b set"   and R :: "('b \<times> 'b) set" shows "partial_equiv A R = (\<exists>A'\<subseteq>A. equiv A' R)"
is_unit_normalize: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a one" shows "normalize a = one"
inj_eq: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "inj f" shows "(f x = f y) = (x = y)"
that: fixes p' :: "real \<Rightarrow> 'a" assumes "polynomial_function p'"   and "\<And>x. (p has_vector_derivative p' x) (at x)" shows "thesis"
nyinitcls_emptyD: fixes G :: "prog"   and s :: "state"   and C :: "qtname" assumes "nyinitcls G s = {}"   and "is_class G C" shows "initd C s"
assms_7: shows "bijective (x * top) \<and> bijective (x[[top]])"
ent_true_drop_2: fixes P :: "assn"   and Q :: "assn" assumes "P \<Longrightarrow>\<^sub>A Q" shows "P \<Longrightarrow>\<^sub>A Q * true"
finite_Collect_disjI: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" shows "finite {x. P x \<or> Q x} = (finite {x. P x} \<and> finite {x. Q x})"
div_gcd_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<noteq> zero \<or> b \<noteq> zero" shows "algebraic_semidom.coprime times one (divide a (gcd a b)) (divide b (gcd a b))"
simps_2: fixes uv :: "'e option" shows "None <*> uv = None"
SeqFormP_imp_LstSeqP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqFormP s k t} \<turnstile> LstSeqP s k t"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sup>s\<^sub>l <\<nu>x>P \<parallel> Q"
simps_17: fixes sh :: "sheap"   and e :: "'a exp" shows "iconf sh (throw e) = iconf sh e"
S4: shows "{x. p x = 0} = {x. \<exists>y. q x y = 0}"
inject_9: fixes x9 :: "formula list"   and y9 :: "formula list" shows "(Ands x9 = Ands y9) = (x9 = y9)"
inj_on_one_iff': fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)"
exists_unique_def: fixes forall :: "('a \<Rightarrow> \<o>) \<Rightarrow> \<o>"   and identity :: "'a \<Rightarrow> 'a \<Rightarrow> \<o>" assumes "class.quantifiable_and_identifiable forall identity" shows "quantifiable_and_identifiable.exists_unique forall identity \<equiv> \<lambda>\<phi>. quantifiable.exists forall (\<lambda>\<alpha>. \<phi> \<alpha> \<^bold>& forall (\<lambda>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> identity \<beta> \<alpha>))"
intros_4: fixes p :: "'a pfm"   and i :: "'a" assumes "\<turnstile>\<^sub>! p" shows "\<turnstile>\<^sub>! K\<^sub>! i p"
hyps_1: fixes p :: "real poly" assumes "x = degree p" shows "changes_poly_neg_inf (pders p) = int (degree p)"
mult_mat_vec_hom: fixes hom :: "'a \<Rightarrow> 'b"   and A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a vec" assumes "semiring_hom hom"   and "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "map_vec hom (A *\<^sub>v v) = map_mat hom A *\<^sub>v map_vec hom v"
inline_aux_Inr: fixes rpv :: "('b, 'call, 'ret) rpv"   and oracl :: "('ret \<times> 's, 'call', 'ret') gpv" shows "local.inline_aux (Inr (rpv, oracl)) = oracl \<bind> (\<lambda>(x, s). local.inline (rpv x) s)"
select_convs_2: fixes orderA :: "('c, 'd) gorder_scheme"   and orderB :: "('a, 'b) gorder_scheme"   and lower :: "'c \<Rightarrow> 'a"   and upper :: "'a \<Rightarrow> 'c"   and more :: "'e" shows "\<Y>\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = orderB"
Fishb_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set"   and A :: "'a set" shows "Fishb R B A = ((\<forall>a\<in>A. \<forall>b\<in>B - A. R b a) \<and> (\<forall>a\<in>A - B. \<forall>b\<in>B. R b a))"
opp_Point: fixes x :: "'a"   and y :: "'a" shows "opp (Point x y) = Point x (\<ominus> y)"
coreflexive_idempotent: fixes x :: "'a" assumes "coreflexive x" shows "idempotent x"
ideal_mult_in_subgroup_generated: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<aa> :: "'c set"   and \<bb> :: "'c set"   and a :: "'c"   and b :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "ideal \<aa> R addition multiplication zero unit"   and "ideal \<bb> R addition multiplication zero unit"   and "a \<in> \<aa>"   and "b \<in> \<bb>" shows "multiplication a b \<in> comm_ring.ideal_gen_by_prod R addition multiplication zero \<aa> \<bb>"
pw_THROW_3: fixes e :: "'c"   and x :: "'d" shows "\<not> inres (THROW e) (Inr x)"
eq_Apps_conv_1: fixes m :: "char list \<times> Unary_PCF.type"   and f :: "expr"   and ts :: "expr list" shows "(\<langle>m\<rangle> = f \<bullet> ts) = (\<langle>m\<rangle> = f \<and> ts = [])"
prems_1: shows "length xs \<le> l (Node i j a t1 t2) - 1"
p_t_stop: shows "ego2.p ego2.t_stop = ego2.p_max"
assms_1: shows "\<parallel>nid\<parallel>\<^bsub>t n'\<^esub>"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x - xa) = int_of_integer x - int_of_integer xa"
safe_reach_def: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and safe :: "'b \<Rightarrow> bool"   and cfg :: "'b" shows "safe_reach step safe cfg \<equiv> \<forall>cfg'. step\<^sup>*\<^sup>* cfg cfg' \<longrightarrow> safe cfg'"
simps: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c dres"   and s :: "'c" shows "dfoldli l c f s = (case l of [] \<Rightarrow> dRETURN s | x # ls \<Rightarrow> if c s then f x s \<bind> dfoldli ls c f else dRETURN s)"
hoare: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
trans_fm_abs: fixes e :: "name list"   and name :: "name"   and A :: "fm" shows "trans_fm (e @ [name]) A = abst_dbfm name (length e) (trans_fm e A)"
mset_ran_combine2: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'a" assumes "finite r"   and "i \<notin> r" shows "add_mset (a i) (mset_ran a r) = mset_ran a (Set.insert i r)"
num_reachable_norefl_def: fixes G :: "'a graph"   and v :: "'a" shows "num_reachable_norefl G v = card (succ_tran G v - {v})"
elims_2: fixes x :: "'d ifex"   and xa :: "'d \<Rightarrow> bool" assumes "val_ifex x xa" obtains s :: "'d \<Rightarrow> bool" where "x = Trueif" and "xa = s__"   | n :: "'d" and t1 :: "'d ifex" and t2 :: "'d ifex" and s :: "'d \<Rightarrow> bool" where "x = IF n t1 t2" and "xa = s" and "if s n then val_ifex t1 s else val_ifex t2 s"
cong_hcomp1: fixes P :: "conc_interface"   and Q :: "conc_interface"   and P' :: "conc_interface" assumes "P \<simeq> Q" shows "P' \<otimes>\<^sub>c P \<simeq> P' \<otimes>\<^sub>c Q"
le_0: fixes x :: "V" shows "[]\<^sub>\<circ> \<subseteq>\<^sub>\<circ> x"
is_map_from_is_homomorphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "Set_Theory.map (id.im_sheaf_morphisms U V) (id.im_sheaf U) (id.im_sheaf V)"
word_le_not_less: fixes b :: "'a word"   and a :: "'a word" shows "(b \<le> a) = (\<not> a < b)"
ext_inject: fixes root :: "'a"   and edges :: "'a edge set"   and more :: "'b"   and root' :: "'a"   and edges' :: "'a edge set"   and more' :: "'b" shows "(\<lparr>root = root, edges = edges, \<dots> = more\<rparr> = \<lparr>root = root', edges = edges', \<dots> = more'\<rparr>) = (root = root' \<and> edges = edges' \<and> more = more')"
quotient_map_def: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "quotient_map X X' f = (f ` topspace X = topspace X' \<and> (\<forall>U\<subseteq>topspace X'. openin X {x \<in> topspace X. f x \<in> U} = openin X' U))"
infinite_sphere: fixes r :: "real"   and a :: "complex" assumes "0 < r" shows "infinite (sphere a r)"
dim_update_mat_2: fixes A :: "'b mat"   and ij :: "nat \<times> nat"   and a :: "'b" shows "dim_col (A |\<^sub>m ij \<mapsto> a) = dim_col A"
e_2: shows "S \<subseteq> cball (0::'a) e"
lowner_is_lub_def: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and M :: "complex mat" assumes "matrix_seq dim f" shows "matrix_seq.lowner_is_lub f M = ((\<forall>n. f n \<le>\<^sub>L M) \<and> (\<forall>M'. (\<forall>n. f n \<le>\<^sub>L M') \<longrightarrow> M \<le>\<^sub>L M'))"
charslength_appendix_is_empty: fixes p :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "charslength (p @ ts) = charslength p"   and "t \<in> set ts" shows "chars_of_token t = []"
prems: shows "inj_on f (set_mset (add_mset x M))"
simps_2: fixes p :: "fm"   and q :: "fm" shows "iszlfm (Or p q) = (\<lambda>bs. iszlfm p bs \<and> iszlfm q bs)"
AB: shows "A \<or> B \<or> C \<or> D"
retraction_is_epi: fixes g :: "'s" assumes "retraction g" shows "epi g"
Aexp_pre_simps_2: fixes x :: "Vars.var"   and e :: "exp"   and a :: "Arity" shows "Aexp_syn' a (Lam [x]. e) = Aexp_syn' (pred\<cdot>a) e f|` fv (Lam [x]. e)"
e2i_def: fixes e :: "nat" shows "e2i e \<equiv> pdec1 (e2frame e)"
cases: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'f \<Rightarrow> bool"   and a :: "'e \<times> 'f" assumes "pred_prod P1 P2 a" obtains aa :: "'e" and b :: "'f" where "a = (aa, b)" and "P1 aa" and "P2 b"
inject_3: fixes x3 :: "'f"   and y3 :: "'f" shows "(tPred x3 = tPred y3) = (x3 = y3)"
cases: fixes x :: "'a list" obtains "x = []"   | n :: "'a" where "x = [n__]"   | n :: "'a" and v :: "'a" and va :: "'a list" where "x = n # v # va"
disc_eq_case_1: fixes sum :: "'a + 'b" shows "isl sum = (case sum of Inl uu_ \<Rightarrow> True | Inr uu_ \<Rightarrow> False)"
le_emeasure_sup_measure'2: fixes B :: "'b measure"   and A :: "'b measure"   and X :: "'b set" assumes "sets B = sets A"   and "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"
1: shows "\<tau> \<sigma> i + b < \<tau> \<sigma> i'"
inv\<^sub>1E_2: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2) = U - V"
eval_Assoc': fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Arr v" shows "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<rbrace> = D.assoc' (D.cod \<lbrace>t\<rbrace>) (D.cod \<lbrace>u\<rbrace>) (D.cod \<lbrace>v\<rbrace>) \<cdot>\<^sub>D (\<lbrace>t\<rbrace> \<otimes>\<^sub>D \<lbrace>u\<rbrace> \<otimes>\<^sub>D \<lbrace>v\<rbrace>)"
set_plus_closed: fixes A :: "'r set"   and B :: "'r set" assumes "A \<subseteq> R"   and "B \<subseteq> R" shows "A + B \<subseteq> R"
floor_add: fixes x :: "'d"   and y :: "'d" shows "\<lfloor>x + y\<rfloor> = (if frac x + frac y < (1::'d) then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> + 1)"
f2: fixes a :: "'a"   and aa :: "'a" shows "ad (a \<cdot> aa) \<cdot> (a \<cdot> ad aa) = ad (a \<cdot> aa) \<cdot> a"
k2: shows "Suc k < ncols (Gram_Schmidt_upt_k A k)"
trm_subst_ident'': fixes t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<And>x. x \<in> fv t \<Longrightarrow> \<theta> x = Var x" shows "t \<cdot> \<theta> = t"
attach_shadow_root_def: fixes element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode" shows "attach_shadow_root element_ptr shadow_root_mode = Heap_Error_Monad.bind (get_tag_name element_ptr) (\<lambda>tag. Heap_Error_Monad.bind (if tag \<notin> safe_shadow_root_element_types then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>prev_shadow_root. Heap_Error_Monad.bind (if prev_shadow_root \<noteq> None then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M (\<lambda>new_shadow_root_ptr. Heap_Error_Monad.bind (set_mode new_shadow_root_ptr shadow_root_mode) (\<lambda>_. Heap_Error_Monad.bind (set_shadow_root element_ptr (Some new_shadow_root_ptr)) (\<lambda>_. return new_shadow_root_ptr)))))))"
salt: shows "(sign ! a = - 1) = (poly (qs ! a) x < 0)"
msg_def: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'a list" shows "msg n s \<equiv> \<forall>t. length (s t) \<le> n"
ab: shows "nat (\<bar>a\<bar> * b) = sq * sq * fact"
next_performop_vacant: fixes s :: "('d, 'c) configuration stream"   and t :: "'c" assumes "NrecVacantUpto (shd s) t"   and "next_performop s" shows "NrecVacantUpto (shd (stl s)) t"
map_upds_append1: fixes xs :: "'a list"   and ys :: "'b list"   and m :: "'a \<Rightarrow> 'b option"   and x :: "'a" assumes "length xs < length ys" shows "m(xs @ [x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys, x \<mapsto> ys ! length xs)"
qtable_def: fixes n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Q :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set" shows "qtable n A P Q X = (table n A X \<and> (\<forall>x. (x \<in> X \<and> P x \<longrightarrow> Q x) \<and> (wf_tuple n A x \<and> P x \<and> Q x \<longrightarrow> x \<in> X)))"
natUnion_superset: fixes g :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "g i \<subseteq> natUnion g"
pax5: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "|~ \<circle>\<box>F \<longrightarrow> \<box>[\<circle>F]_v"
measure_map_pmf: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf"   and X :: "'b set" shows "measure_pmf.prob (map_pmf f M) X = measure_pmf.prob M (f -` X)"
d_sub_id_ax: fixes x :: "'a" shows "d x \<le> 1\<^sub>\<sigma>"
sums_mult: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b" assumes "f sums a" shows "(\<lambda>n. c * f n) sums (c * a)"
smooth_on_scaleR_fun: fixes k :: "enat"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and c :: "real" assumes "k-smooth_on S g"   and "open S" shows "k-smooth_on S (c *\<^sub>R g)"
Un_singleton_iff: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" shows "(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
inverse_divide: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "inverse (divide a b) = divide b a"
plain_nres_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>plain_nres_rel \<equiv> {(c, a). RETURN c \<le> \<Down> R a}"
extreme_point_of_convex_hull_insert: fixes S :: "'b set"   and a :: "'b" assumes "finite S"   and "a \<notin> convex hull S" shows "a extreme_point_of convex hull insert a S"
bigo_powr: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and p :: "real" assumes "f \<in> O[F](g)"   and "0 \<le> p" shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> O[F](\<lambda>x. \<bar>g x\<bar> powr p)"
that: fixes v0 :: "'c" assumes "v = iso_verts hom v0"   and "v0 \<in> verts G" shows "thesis"
add_order_equality: fixes n :: "nat"   and a :: "'a" assumes "order.greater n 0"   and "a +^ n = (0::'a)"   and "\<And>m. \<lbrakk>order.greater m 0; a +^ m = (0::'a)\<rbrakk> \<Longrightarrow> order.greater_eq m n" shows "add_order a = n"
gauss_jordan_integer_inverse_def: fixes n :: "'a"   and A :: "rat mat"   and B :: "rat mat"   and I :: "rat mat" shows "gauss_jordan_integer_inverse n A B I = (case gauss_jordan A B of (C, D) \<Rightarrow> C = I \<and> list_all is_int_rat (concat (mat_to_list D)))"
sl1: shows "(abscissa B - abscissa C)\<^sup>2 \<in> radical_sqrt"
h: shows "\<guillemotleft>h : F y \<rightarrow>\<^sub>C x\<guillemotright>"
weak_bisimilarity_implies_weak_equivalence: fixes P :: "'state"   and Q :: "'state" assumes "P \<approx>\<cdot> Q" shows "P \<equiv>\<cdot> Q"
assms_3: shows "set (Abs_State.dom S2) \<subseteq> X"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<tau> \<prec> P'"   and "guarded P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) P'" shows "Prop C \<Psi> (!P) P'"
L4_8': fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and \<xi> :: "nat \<Rightarrow> 'a set"   and \<psi> :: "'a ltlr" assumes "ipath gba.E \<sigma>"   and "gba.is_acc \<sigma>"   and "\<forall>i. local.gba.L (\<sigma> i) (\<xi> i)"   and "\<psi> \<in> old (\<sigma> 0)" shows "\<xi> \<Turnstile>\<^sub>r \<psi>"