hcju/isabelleps huggingface dataset:
inj: shows "inj_on (\<lambda>k. cis (2 * pi * real k / real n)) {..<n}"
Conseq: fixes P :: "'e set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set"   and c :: "('e, 'f, 'g) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
A1n_commute: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.A1n A ** A = A ** perron_frobenius.A1n A"
perm_10: fixes pi :: "(name \<times> name) list"   and x1 :: "pi" shows "pi \<bullet> !x1 = !(pi \<bullet> x1)"
nxt_sdrop: fixes n :: "nat"   and \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" shows "(nxt ^^ n) \<phi> xs = \<phi> (sdrop n xs)"
U_l: shows "\<forall>(c, s)\<in>set (remdups (uset (simpfm p))). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "DoI.map f = \<F>\<^sub>SC.null"
assms_5: shows "(\<^bold>\<diamond> Nom j) at i in (ps, i) # branch"
ModPonP_def: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "ModPonP x y z = y EQ Q_Imp x z"
alt_consistency: fixes C :: "('a, 'b) form set set" assumes "consistency C" shows "alt_consistency (mk_alt_consistency C)"
degree_zero_imp_splitted: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "degree p = 0" shows "splitted p"
mst_extends_old_tree: fixes t :: "'a"   and w :: "'a"   and v :: "'a"   and e :: "'a" assumes "t \<le> w"   and "t \<le> v * v\<^sup>T"   and "vector v" shows "t \<le> prim_W w v e"
prod_zero_iff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e" assumes "class.semidom plus minus zero times one"   and "finite A" shows "(comm_monoid_mult.prod times one f A = zero) = (\<exists>a\<in>A. f a = zero)"
Ide_Diagonalize_Can: fixes t :: "'a local.term" assumes "Can t" shows "Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
a_consistent_if\<^sub>1: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, scrut ? e1 : e2, S)" shows "ArityAnalysisLetSafe.a_consistent Aexp (ae, 0, a # as) (\<Gamma>, scrut, Alts e1 e2 # S)"
ps_prod_times_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "ps_prod_times x y = (fst x \<oplus> fst y, snd x \<oplus> snd y)"
produces_imp_Max_atom: fixes C :: "'a literal multiset"   and A :: "'a" assumes "production C = {A}" shows "A = Max (atms_of C)"
card_of_ordLeqI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A"   and "\<And>a. a \<in> A \<Longrightarrow> f a \<in> B" shows "ordLeq3 (card_of A) (card_of B)"
agree_fresh_env_fresh_term: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and a :: "Syntax.var" assumes "\<Gamma> \<turnstile> e, eP, eV : \<tau>"   and "atom a \<sharp> \<Gamma>" shows "atom a \<sharp> (e, eP, eV)"
if_doesnt_read': fixes x :: "'Var"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" assumes "x \<notin> bexp_vars e" shows "doesnt_read (Stmt.If e c\<^sub>1 c\<^sub>2) x"
lpd_match: fixes i :: "nat"   and j :: "nat"   and test :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and r :: "'a Regex.regex" assumes "i < j" shows "Regex.match test r i j = (\<Squnion> (Regex.match test ` Regex.lpd test i r)) (i + 1) j"
intro: fixes state_list :: "'c list" assumes "distinct state_list" shows "HMM4_axioms state_list"
square_rtrancl: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "square R S S T" shows "square R\<^sup>*\<^sup>* S S T\<^sup>*\<^sup>*"
simps: fixes s :: "('f, 'a) Term.term"   and t :: "('f, 'a) Term.term" shows "kbo s t = (if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s then if weight t < weight s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict (f, length ss) (g, length ts) then (True, True) else if pr_weak (f, length ss) (g, length ts) then lex_ext_unbounded kbo ss ts else (False, False) else (False, False))"
bezout_iterate_zero_column_k: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and i :: "'f"   and n :: "nat"   and a :: "'f" assumes "echelon_form_upt_k A k"   and "is_bezout_ext bezout"   and "A $ i $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "n < nrows A"   and "i < a"   and "k < ncols A"   and "mod_type_class.to_nat a \<le> n"   and "is_zero_row_upt_k i k A" shows "bezout_iterate A n i (mod_type_class.from_nat k) bezout $ a $ mod_type_class.from_nat k = (0::'d)"
assms_1: shows "(\<^bold>\<not> (\<^bold>@ a p)) at b in (ps, a) # branch"
C_def: shows "C = max 0 (Sup (cmod ` (\<lambda>x. \<Prod>k = 1..n. x - of_nat k) ` closed_segment 0 (of_nat n)))"
norm_sum: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (comm_monoid_add.sum plus zero f A) \<le> (\<Sum>i\<in>A. norm (f i))"
assms_1: shows "\<forall>t. length (a t) \<le> Suc 0"
word_size_bl: fixes w :: "'b word" shows "size w = length (to_bl w)"
pfun_minus_plus: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "pdom f \<inter> pdom g = {}" shows "f + g - g = f"
simps_119: fixes x101 :: "stmt"   and x102 :: "stmt"   and x7 :: "jump" shows "x101 Finally x102 \<noteq> Jmp x7"
add_divide_distrib: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> '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"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
cases: fixes a1 :: "'d set"   and a2 :: "'d set"   and a3 :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set" assumes "(a1, a2, a3) \<in> br'_initial \<delta>" obtains rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" where "a1 = br_iq \<delta>" and "a2 = br_iq \<delta>" and "a3 = rcm" and "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm r = Some (card (set (rhsq r)))"
IH: fixes A :: "nat set" shows "nths is A \<lhd> nths ds A"
remove_redundant_edge: fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "(u, v) \<in> (edges g - {(u, v), (v, u)})\<^sup>*" shows "(edges g - {(u, v), (v, u)})\<^sup>* = (edges g)\<^sup>*"
neg_pf_exists: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "stock_portfolio Mkt pf"   and "trading_strategy pf"   and "1 + r < u"   and "d < 1 + r"   and "val_process Mkt pf n w \<le> 0"   and "pf stk (Suc n) w \<noteq> 0 \<or> pf risk_free_asset (Suc n) w \<noteq> 0" shows "\<exists>y. cls_val_process Mkt pf (Suc n) y < 0"
ReadInner_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs" shows "ReadInner ch mm rs p l \<equiv> $RdRequest ch p l \<and> (GoodRead mm rs p l \<or> BadRead mm rs p l) \<and> unchanged (rtrner ch!p)"
eq_refl: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x = y" shows "x \<preceq> y"
double_compl: fixes x :: "'a upred" shows "(\<not> \<not> x) = x"
comp_same: fixes x :: "'b" shows "c1 x x = Eq"
the_equality: fixes P :: "'f \<Rightarrow> bool"   and a :: "'f" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
dghm_cn_comp_components_2: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ArrMap\<rparr> = \<GG>\<lparr>ArrMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>"
lift_RealVal_eq: fixes f :: "real \<Rightarrow> real"   and x :: "real" shows "lift_RealVal f (RealVal x) = RealVal (f x)"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong x" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "xa = DenyAll \<and> DenyAll \<notin> set xs"
analytic_higher_deriv: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and n :: "nat" assumes "f analytic_on S" shows "(deriv ^^ n) f analytic_on S"
less_imp_Max_less_iMin: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "\<forall>a\<in>A. \<forall>b\<in>B. a < b" shows "Max A < iMin B"
simps: fixes f :: "int \<Rightarrow> (bool \<times> term list) option"   and d :: "int"   and i :: "int" shows "Quickcheck_Exhaustive.exhaustive_int' f d i = (if d < i then None else Quickcheck_Exhaustive.orelse (f i) (Quickcheck_Exhaustive.exhaustive_int' f d (i + 1)))"
absolutely_integrable_measurable: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue" shows "(f absolutely_integrable_on S) = (f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (norm \<circ> f))"
set_listset_el_subset: fixes xs :: "'a list"   and Xs :: "'a set list"   and A :: "'a set" assumes "xs \<in> listset Xs"   and "\<forall>X\<in>set Xs. X \<subseteq> A" shows "set xs \<subseteq> A"
ring_n_eSum_K_eSum: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "distinct_pds K n P"   and "\<forall>j\<le>m. f j \<in> carrier (O\<^bsub>K P n\<^esub>)" shows "\<Sigma>\<^sub>e O\<^bsub>K P n\<^esub> f m = \<Sigma>\<^sub>e K f m"
7: shows "Ind {a} (set ([b] @ v\<^sub>1))"
iT_Mult_singleton: fixes a :: "nat"   and k :: "nat" shows "{a} \<otimes> k = {a * k}"
binary_chinese_remainder_nat: fixes m1 :: "nat"   and m2 :: "nat"   and u1 :: "nat"   and u2 :: "nat" assumes "coprime m1 m2" shows "\<exists>x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
bisim1LAss1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and V :: "nat" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,V:=e,h \<turnstile> (V:=e', xs) \<leftrightarrow> (stk, loc, pc, xcp)"
disc_tree_orders': shows "h' \<turnstile> map_M (to_tree_order \<circ> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r) disc_nodes_h' \<rightarrow>\<^sub>r disc_tree_orders'"
thetaZOSeqD_def: shows "thetaZOSeqD \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. proper c1 \<and> proper d1 \<and> proper c2 \<and> proper d2 \<and> discr c2 \<and> discr d2 \<and> c1 \<approx>01 d1}"
sublist_take: fixes n :: "nat"   and xs :: "'a list" shows "sublist (take n xs) xs"
fact_less_mono: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m < n" shows "fact m < fact n"
prems_4: shows "closed_except (name $$ ts) (fmdom \<Gamma>)"
simps: fixes a1 :: "'a"   and a2 :: "'a \<Rightarrow> real"   and a3 :: "Regions.intv" shows "Regions.intv_elem a1 a2 a3 = ((\<exists>u x d. a1 = x \<and> a2 = u \<and> a3 = Regions.intv.Const d \<and> u x = real d) \<or> (\<exists>d u x. a1 = x \<and> a2 = u \<and> a3 = Regions.intv.Intv d \<and> real d < u x \<and> u x < real (d + 1)) \<or> (\<exists>c u x. a1 = x \<and> a2 = u \<and> a3 = Regions.intv.Greater c \<and> real c < u x))"
undirected: fixes v :: "'c"   and w :: "'c" shows "v \<rightarrow>\<^bsub>T\<^esub> w = w \<rightarrow>\<^bsub>T\<^esub> v"
span_mono: fixes A :: "('t \<Rightarrow>\<^sub>0 'e) set"   and B :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "A \<subseteq> B" shows "pmdl A \<subseteq> pmdl B"
finite_Un: fixes F :: "'c set"   and G :: "'c set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
sync_def: fixes P :: "'c process"   and A :: "'c set"   and Q :: "'c process" shows "P \<lbrakk>A\<rbrakk> Q \<equiv> Abs_process ({(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}, {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)})"
outs_gpv_def: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" shows "outs_gpv \<I> gpv \<equiv> {x. outsp_gpv \<I> x gpv}"
assms: shows "x \<in> cipher_queries (ose(n \<mapsto> (ek, dk, ciphers)))"
f: shows "f \<in> U \<rightarrow> carrier_vec n"
bij_if_span_eq_span_bases: fixes B :: "'b set"   and C :: "'b set" assumes "independent B"   and "independent C"   and "span B = span C" shows "\<exists>f. bij_betw f B C"
SubstForm_subst_dbfm_eq: fixes v :: "hf"   and i :: "name"   and ux :: "hf"   and e :: "(name, hf) finfun"   and u :: "dbtm"   and A' :: "dbfm"   and A :: "dbfm" assumes "v = q_Var i"   and "Term ux"   and "ux = \<lbrakk>quot_dbtm u\<rbrakk>e"   and "A' = subst_dbfm u i A" shows "SubstForm v ux \<lbrakk>quot_dbfm A\<rbrakk>e \<lbrakk>quot_dbfm A'\<rbrakk>e"
Wbis_MtransT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>w d"   and "s \<approx> t"   and "(c, s) \<rightarrow>*t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'a__" and "s' \<approx> t'a__"   | d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "discr d'"
p: shows "\<turnstile> p \<rhd> t \<Rightarrow> ts\<^sub>1"
c_eq: shows "column k (Gram_Schmidt_column_k A (mod_type_class.to_nat k)) = column i (Gram_Schmidt_column_k A (mod_type_class.to_nat k))"
assms_9: fixes s1 :: "'s1" shows "\<I> \<turnstile>c oracle1 s1 \<surd>"
val_def: fixes tm :: "'a tm" shows "tm.val tm = (case tm of ??.Time_Monad.tm.TM x1 x2 \<Rightarrow> x1)"
step_k: shows "\<Gamma>\<turnstile> p k \<rightarrow> p (Suc k)"
h3: shows "lose t = [False] \<longrightarrow> ack t = [y] \<and> st_out t = y"
mem_adjoin: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c"   and a :: "'c" assumes "finite_comm_group G"   and "subgroup H G"   and "x \<in> H"   and "a \<in> carrier G" shows "x \<in> adjoin G H a"
intpow_mult_combine: fixes x :: "'c"   and m :: "int"   and n :: "int"   and y :: "'c" assumes "(0::'c) < x" shows "x ^\<^sub>Z m \<cdot> (x ^\<^sub>Z n \<cdot> y) = x ^\<^sub>Z (m + n) \<cdot> y"
satisfies_atom_set_def: fixes v :: "nat \<Rightarrow> 'b"   and as :: "'b atom set" shows "v \<Turnstile>\<^sub>a\<^sub>s as \<equiv> \<forall>a\<in>as. v \<Turnstile>\<^sub>a a"
eq: fixes n :: "nat" shows "- 2 * \<i> * sin (s * complex_of_real pi) * I n + smallc + bigc n = - 2 * \<i> * complex_of_real pi * complex_of_real (2 * pi) powr - s * (F1 n + F2 n)"
cspan_to_conjugate_space: fixes X :: "'c set" shows "cspan (to_conjugate_space ` X) = to_conjugate_space ` cspan X"
vimageI2: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e"   and A :: "'d set" assumes "f a \<in> A" shows "a \<in> f -` A"
coherent_Lunit_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "coherent \<^bold>\<l>\<^bold>[a\<^bold>]"
fresh_star_delete: fixes S :: "atom set"   and \<Gamma> :: "('a \<times> 'b) list"   and v :: "'a" assumes "S \<sharp>* \<Gamma>" shows "S \<sharp>* delete v \<Gamma>"
rawpsubstT_compose_freshVar: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list"   and us :: "'var list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "distinct (map snd txs)"   and "\<And>i j. \<lbrakk>i < j; j < length txs\<rbrakk> \<Longrightarrow> snd (txs ! j) \<notin> FvarsT (fst (txs ! i))"   and "set us \<subseteq> var"   and "set us \<inter> FvarsT r = {}"   and "set us \<inter> \<Union> (FvarsT ` fst ` set txs) = {}"   and "set us \<inter> snd ` set txs = {}"   and "length us = length txs"   and "distinct us" shows "rawpsubstT (rawpsubstT r (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubstT r txs"
abs_o_rep: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c" assumes "Quotient3 R Abs Rep" shows "Abs \<circ> Rep = id"
the1_equality: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<exists>!x. P x"   and "P a" shows "(THE x. P x) = a"
assms: shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>"
l11_22: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and P :: "'a"   and A :: "'a"   and C :: "'a"   and B' :: "'a"   and P' :: "'a"   and A' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet B P A C \<and> Tarski_neutral_dimensionless.TS Bet B' P' A' C' \<or> Tarski_neutral_dimensionless.OS Bet B P A C \<and> Tarski_neutral_dimensionless.OS Bet B' P' A' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B P A' B' P'"   and "Tarski_neutral_dimensionless.CongA Bet Cong P B C P' B' C'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"
order_finite_set_exists_foundation: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set"   and t :: "'b" assumes "class.order less_eq less"   and "finite M"   and "t \<in> M" shows "\<exists>s\<in>M. less_eq s t \<and> (\<forall>u\<in>M. \<not> less u s)"
preserves_dom: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "local.dom (R f) = R (local.dom f)"
assms_4: shows "h \<turnstile> get_element_by_id ptr idd \<rightarrow>\<^sub>r Some result"
frameImpEAux: fixes F :: "'b frame"   and \<Phi> :: "'c"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "F \<turnstile>\<^sub>F \<Phi>"   and "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"   and "A\<^sub>F \<sharp>* \<Phi>" shows "\<Psi>\<^sub>F \<turnstile> \<Phi>"
suminf_exist_split2: fixes f :: "nat \<Rightarrow> 'c" assumes "summable f" shows "(\<lambda>n. \<Sum>k. f (k + n)) \<longlonglongrightarrow> (0::'c)"
linorder_chain: fixes Y :: "'a set" shows "Complete_Partial_Order.chain (\<le>) Y"
callee': fixes x :: "'a"   and s :: "'d" assumes "x \<in> outs_\<I> \<I>" shows "set_spmf (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"
joinable_sym: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'b) set" assumes "(x, y) \<in> joinable R" shows "(y, x) \<in> joinable R"
r_badblock_last: fixes s :: "nat \<Rightarrow> nat option"   and n :: "nat"   and v :: "nat" assumes "total_cons s" shows "e_nth (the (eval total_cons.r_badblock [n, v])) n = 1 - v"
closedin_Hausdorff_finite: fixes X :: "'c topology"   and S :: "'c set" assumes "Hausdorff_space X"   and "S \<subseteq> topspace X"   and "finite S" shows "closedin X S"
II2: shows "II2 \<in> UNpart1 cl dl P"
initd_def2: fixes C :: "qtname"   and s :: "state" shows "initd C s = inited C (globs (snd s))"
tdghm_dghm_comp_NTMap_vdomain: fixes \<NN> :: "V" shows "\<D>\<^sub>\<circ> ((\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<KK>)\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
slo_defs_2: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
exe_wf_sig_imp_wf_sig': fixes \<Sigma> :: "exesignature" assumes "exe_sig_conds \<Sigma>"   and "exe_wf_sig \<Sigma>" shows "wf_sig (translate_signature \<Sigma>)"
match_pres_submap: fixes pres :: "('a \<times> 'b) list"   and s :: "'a \<Rightarrow> 'b option" shows "Solve_SASP.match_pres (Tree2.inorder (M.tree_map_of' RBT_Set.empty pres)) s = (Map.map_of pres \<subseteq>\<^sub>m s)"
zero_divisor_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.mult_zero times zero" shows "mult_zero.zero_divisor times zero a \<equiv> \<exists>b. b \<noteq> zero \<and> times a b = zero"
inv_baliR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "invc l"   and "invc2 r"   and "bheight l = bheight r" shows "invc (baliR l a r) \<and> invh (baliR l a r) \<and> bheight (baliR l a r) = Suc (bheight l)"
dg_is_arrD_5: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
eq_resource_onD: fixes A :: "'a set"   and res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource"   and a :: "'a" assumes "A \<turnstile>\<^sub>R res \<sim> res'"   and "a \<in> A" shows "rel_spmf (rel_prod (=) (eq_resource_on A)) (run_resource res a) (run_resource res' a)"
mem_ball: fixes y :: "'b"   and x :: "'b"   and e :: "real" shows "(y \<in> ball x e) = (dist x y < e)"
admissible_kbo': shows "admissible_kbo w w0 pr_strict' pr_weak' least scf"
doubleRestriction: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" shows "(P || X) || Y = P || (X \<inter> Y)"
IO: fixes x :: "'b"   and c :: "('a, 'b, 'c) rpv"   and gpv :: "('a, 'b, 'c) gpv"   and \<I> :: "('b, 'c) \<I>" assumes "IO x c \<in> set_spmf (the_gpv gpv)" shows "x \<in> outs_gpv \<I> gpv"
Var_injD: fixes x :: "'var"   and y :: "'var" assumes "Var x = Var y"   and "x \<in> var"   and "y \<in> var" shows "x = y"
lnth_lappend1: fixes n :: "nat"   and xs :: "'a llist"   and ys :: "'a llist" assumes "enat n < llength xs" shows "lnth (lappend xs ys) n = lnth xs n"
dickson_gradingI: fixes d :: "'a \<Rightarrow> nat" assumes "\<And>s t. d (s + t) = max (d s) (d t)"   and "\<And>n. almost_full_on (adds) {x. d x \<le> n}" shows "dickson_grading d"
EmptyMap_def: fixes S :: "'d set" shows "EmptyMap S = (\<lambda>a. if a \<in> S then Some {} else None)"