hcju/isabelleps huggingface dataset:
extra_axiom_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e extra_axiom"
equivP_def: fixes A :: "'d \<Rightarrow> bool"   and \<phi> :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "equivP A \<phi> \<equiv> reflP A \<phi> \<and> symP \<phi> \<and> transP \<phi>"
ID2: fixes f :: "'c \<rightarrow> 'd" shows "f oo ID = f"
ctr_simps_mem_Collect_eq: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
c_pair_sum_mono: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "x1 + y1 < x2 + y2" shows "c_pair x1 y1 < c_pair x2 y2"
pair_measure_count_space: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B" shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)"
base_case_info_def: shows "base_case_info = (mat_of_rows_list 2 [[1, 1], [1, - 1]], [[], [0]], [[1], [- 1]])"
pmaxsimpchainI_maxsimpchain: fixes xs :: "'a set list" assumes "gallery xs"   and "distinct xs" shows "pgallery xs"
closure_aff_dim: fixes S :: "'c set" shows "aff_dim (closure S) = aff_dim S"
vmc2_path_def: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" shows "vmc2_path G P v0 p \<sigma> \<sigma>' \<equiv> vmc_path G P v0 p** \<sigma>' \<and> vmc_path G P v0 p \<sigma>"
tranclp_into_tranclp2: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
BuildSeq2_exI: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> bool"   and t :: "hf"   and t' :: "hf"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "B t t'" shows "\<exists>s k. BuildSeq2 B C s k t t'"
frameEq_2: fixes \<Psi>' :: "'d"   and A\<^sub>F :: "name list"   and \<Psi> :: "'d" shows "(\<langle>\<epsilon>, \<Psi>'\<rangle> = \<langle>A\<^sub>F, \<Psi>\<rangle>) = (A\<^sub>F = [] \<and> \<Psi> = \<Psi>')"
permutes_natset_ge: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S"   and "\<forall>i\<in>S. i \<le> p i" shows "p = id"
Times_Diff_distrib1: fixes A :: "'c set"   and B :: "'c set"   and C :: "'d set" shows "(A - B) \<times> C = A \<times> C - B \<times> C"
oalist_inv_update_by_fun_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv xs" shows "oalist_inv (update_by_fun_raw k f xs)"
simps_1: fixes r :: "real"   and x :: "octo" shows "Ree (r *\<^sub>R x) = r * Ree x"
hyps_3: fixes u :: "dB" assumes "s' => u" shows "\<exists>t'. s => t' \<and> t' \<rightarrow>\<^sub>\<eta>\<^sup>* u"
i2: shows "i < dim_row A + dim_row (0\<^sub>m (dim_row B) 0)"
row_def: fixes i :: "'d"   and A :: "(('b, 'c) vec, 'd) vec" shows "row i A = (\<chi>j. A $ i $ j)"
nf_ACI_D_4: fixes \<phi> :: "('a, 'b) aformula" assumes "nf_ACI \<phi>" shows "distinct (conjuncts_list \<phi>)"
strand_sem_Receive_map_1: fixes M :: "('fun, 'var) Term.term set"   and T :: "('fun, 'var) Term.term list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<lbrakk>M; map Receive T\<rbrakk>\<^sub>c \<I>"
PO_a0i_agreement: shows "reach a0i \<subseteq> a0i_agreement"
faces_subset_def: fixes g :: "graph" shows "faces_subset g \<equiv> \<forall>f\<in>\<F> g. \<V> f \<subseteq> \<V> g"
embed_ge_0: fixes G :: "'b \<Rightarrow> bool"   and s :: "'b" shows "0 \<le> \<guillemotleft> G \<guillemotright> s"
WTrtCall: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and es :: "char list exp list"   and Ts' :: "ty list" assumes "P,E,h \<turnstile> e : Class C"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"   and "P,E,h \<turnstile> es [:] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts" shows "P,E,h \<turnstile> e\<bullet>M(es) : T"
ivl: shows "ll.existence_ivl 0 (\<chi>i. x) = UNIV"
nxtActLe: fixes n :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
is_first_drop_length: fixes k :: "nat"   and a :: "'b list"   and u :: "'b list"   and v :: "'b list"   and X :: "'b"   and w :: "'b list" assumes "k \<le> length a"   and "length u < k"   and "v = X # w"   and "take k a = take k (u @ v)" shows "is_first X (drop (length u) a)"
deg_not_zero_imp_not_unit: fixes f :: "'b poly" assumes "0 < degree f" shows "\<not> is_unit f"
dec_induct: fixes i :: "nat"   and j :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i \<le> j"   and "P i"   and "\<And>n. \<lbrakk>i \<le> n; n < j; P n\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P j"
Delay_lengthOut1: fixes x :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and y :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and t :: "nat" assumes "\<forall>t. length (x t) \<le> Suc 0"   and "Delay x i1 d y i2" shows "length (y t) \<le> Suc 0"
redT_updW_simps_6: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option" shows "redT_updW t ws Notified ws' = (ws' = ws(t := None))"
lepoll_trans1: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" assumes "A \<approx> B"   and "B \<lesssim> C" shows "A \<lesssim> C"
verit_eq_simplify_9: fixes x3 :: "num"   and y3 :: "num" shows "(num.Bit1 x3 = num.Bit1 y3) = (x3 = y3)"
measure: shows "\<forall>\<sigma>. cond \<sigma> \<and> P \<sigma> \<longrightarrow> M \<sigma> \<noteq> None \<and> f (snd (the (M \<sigma>))) < f \<sigma>"
DiffI: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
Lq_def: shows "Lq \<equiv> exi zz (eql (Var yy) (pls (Var zz) (Var xx)))"
trivial_subset_non_empty: fixes t :: "'c set"   and X :: "'c set" assumes "trivial t"   and "t \<inter> X \<noteq> {}" shows "t \<subseteq> X"
exp_times_self_inj': shows "inj_on (\<lambda>x. x * exp x) {..- 1}"
intros_4: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and w :: "'c" shows "redT_updW t ws (Suspend w) (ws(t \<mapsto> InWS w))"
nxt_substitute: fixes t :: "'a ttree"   and x :: "'a"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set" assumes "possible t x" shows "nxt (substitute f T t) x = substitute (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x)"
simps_3: fixes m1 :: "'a convnet"   and m2 :: "'a convnet" shows "input_sizes (Pool m1 m2) = input_sizes m1 @ input_sizes m2"
completeD: fixes CC :: "'b set set"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "CC-complete A r"   and "X \<subseteq> A"   and "X \<in> CC" shows "Ex (extreme_bound A r X)"
equiv_chain_1a: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[[.. a .. b .. c ..]X]" shows "\<exists>f. ch_by_ord f X \<and> a \<in> X \<and> b \<in> X \<and> c \<in> X \<and> a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"
leq_trans: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
critical_empty_intro: fixes Ib :: "'a \<Rightarrow> intv"   and vb :: "'a \<Rightarrow> real"   and rb :: "('a \<times> 'a) set"   and tb :: "real" assumes "\<forall>x\<in>X. \<not> isConst (Ib x)"   and "vb \<in> region X Ib rb"   and "0 \<le> tb"   and "{x \<in> X. \<exists>c. Ib x = Intv c \<and> real (c + 1) \<le> vb x + tb} = {}" shows "vb \<oplus> tb \<in> region X Ib rb"
px: shows "p \<cdot> p\<inverse>\<^sup>>x = x"
sel_rews_1: shows "unbox\<cdot>\<bottom> = \<bottom>"
terminates_strip_guards_to_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>strip_guards F c \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
assms_1: shows "G1 \<in> {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"
dvd_eq_mod_eq_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times a b = (modulo b a = zero)"
sat_models_pow_pow: fixes P :: "'b ltln set" shows "{sat_models (abs_ltln\<^sub>P \<phi>) \<inter> Pow P |\<phi>. prop_atoms \<phi> \<subseteq> P} \<subseteq> Pow (Pow P)"
distinct_5: shows "T_i32 \<noteq> T_f64"
ostep_invariant_def: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool" shows "(A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. \<forall>a s'. (s, a, s') \<in> automaton.trans A \<and> S (fst s) (fst s') a \<longrightarrow> P (s, a, s'))"
poly_gt_def: fixes p :: "('c monom \<times> 'a) list"   and q :: "('c monom \<times> 'a) list" shows "p >p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> eval_poly \<alpha> p \<succ> eval_poly \<alpha> q)"
assms_3: shows "\<forall>k\<ge>N. f $$ k = g $$ k"
SUP_ereal_mult_left: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ereal"   and c :: "ereal" assumes "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"   and "0 \<le> c" shows "(SUP i\<in>I. c * f i) = c * Sup (f ` I)"
hyp3: shows "fst (snd (foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k])) \<le> nrows A"
Sigma_Diff_distrib1: fixes I :: "'d set"   and J :: "'d set"   and C :: "'d \<Rightarrow> 'e set" shows "Sigma (I - J) C = Sigma I C - Sigma J C"
length_enumerate: fixes n :: "nat"   and xs :: "'b list" shows "length (enumerate n xs) = length xs"
Funcs_mkinterp: fixes C :: "char \<Rightarrow> real"   and F :: "char \<Rightarrow> real \<Rightarrow> real"   and P :: "char \<Rightarrow> real \<Rightarrow> bool"   and G :: "char \<Rightarrow> (variable \<Rightarrow> real) set \<Rightarrow> (variable \<Rightarrow> real) set" shows "Funcs (mkinterp (C, F, P, G)) = F"
mult_L_circ_mult: fixes x :: "'a"   and y :: "'a" shows "(x * L)\<^sup>\<circ> * y = y \<squnion> x * L"
rep_eq: fixes x :: "int" shows "generator (upto_prod x) = upto_raw x"
vsubset_vsingleton_leftD: fixes a :: "V"   and A :: "V" assumes "ZFC_in_HOL.set {a} \<subseteq>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> A"
discI_4: fixes stateful_strand_step :: "('c, 'd) stateful_strand_step"   and x41 :: "('c, 'd) Term.term"   and x42 :: "('c, 'd) Term.term" assumes "stateful_strand_step = insert\<langle>x41,x42\<rangle>" shows "is_Insert stateful_strand_step"
path_Entry_Cons: fixes as :: "'edge list"   and n' :: "'node" assumes "(_Entry_) -as\<rightarrow>* n'"   and "n' \<noteq> (_Entry_)" obtains n :: "'node" and a :: "'edge" where "sourcenode a = (_Entry_)" and "targetnode a = n" and "n -tl as\<rightarrow>* n'" and "valid_edge a" and "a = hd as"
object_ptr_kinds_eq_h2: shows "object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h2"
elims_2: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "pred x xa" obtains f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = \<Up>f__" and "xa = cf__ # cfs__"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Qa__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "Qa__ (fst cfa__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qb__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "Qb__ (fst cfb__, r)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and p :: "'pname" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "Q cf \<and> cfs \<noteq> []"
card_UNIV_fun_def: shows "card_UNIV_class.card_UNIV = Phantom('a \<Rightarrow> 'b) (let ca = of_phantom card_UNIV_class.card_UNIV; cb = of_phantom card_UNIV_class.card_UNIV in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
ge_convert: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a = b"   and "c \<le> a" shows "c \<le> b"
exhaust: fixes y :: "'a negation_type" obtains x1 :: "'a" where "y = Pos x1"   | x2 :: "'a" where "y = Neg x2"
asymptotics1: fixes c :: "real" shows "\<forall>\<^sub>F x in at_top. 1 + ln x powr (- e / 2) \<le> (1 + c * inverse b * ln x powr - (1 + e)) powr p * (1 + ln (b * x + hb * x / ln x powr (1 + e)) powr (- e / 2))"
cd_in_cs: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat" assumes "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m" shows "\<exists>ns. cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> m @ ns @ [\<pi> n]"
vars_genericJoin: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and Q_I_pos :: "(nat set \<times> 'a option list set) set"   and Q_I_neg :: "(nat set \<times> 'a option list set) set"   and R_I :: "'a option list set"   and Q_J_neg :: "(nat set \<times> 'a option list set) set"   and Q_J_pos :: "(nat set \<times> 'a option list set) set"   and X :: "('a option list \<times> 'a option list set) set"   and R :: "'a option list set"   and n :: "nat" assumes "2 \<le> card V"   and "(I, J) = getIJ Q Qn V"   and "Q_I_pos = projectQuery I (filterQuery I Q)"   and "Q_I_neg = filterQueryNeg I Qn"   and "R_I = genericJoin I Q_I_pos Q_I_neg"   and "Q_J_neg = filterQuery J (Qn - Q_I_neg)"   and "Q_J_pos = filterQuery J Q"   and "X = {(t, genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) |t. t \<in> R_I}"   and "R = (\<Union>(t, x)\<in>X. {merge xx t |xx. xx \<in> x})"   and "rwf_query n V Q Qn" shows "R = genericJoin V Q Qn"
meas: fixes P :: "'b \<Rightarrow> bool" assumes "Measurable.pred N P" shows "Measurable.pred N (F P)"
N_dec: shows "\<forall>n\<ge>N. c (n + 1) \<le> c n"
list_all2_shortcircuit_weaken: fixes P :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "('b, 'c) result list"   and Q :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool" assumes "list_all2_shortcircuit P xs ys"   and "\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys" shows "list_all2_shortcircuit Q xs ys"
simps_3: shows "local.map J.TT = C.cod f0"
is_bintree_list_triv_1: fixes ts :: "('c, 'd) bintree list" shows "is_bintree_list 0 ts = (ts = [])"
eucl_ordinal_utility_axioms: shows "eucl_ordinal_utility carrier relation u"
associated_refl: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.monoid G"   and "a \<in> carrier G" shows "a \<sim>\<^bsub>G\<^esub> a"
DOWN_UP: fixes x :: "'e" shows "SC.DOWN (SC.UP x) = x"
par_sep_swap: fixes P :: "('d, 'e) urel"   and Q :: "('d, 'e) urel" shows "(P ;; U0 \<and> Q ;; U1 \<and> $\<Sigma>\<^sub><\<acute> =\<^sub>u $\<^bold>v) ;; swap\<^sub>m = (Q ;; U0 \<and> P ;; U1 \<and> $\<Sigma>\<^sub><\<acute> =\<^sub>u $\<^bold>v)"
mismatchPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<sim> Q" shows "[a\<noteq>b]P \<sim> [a\<noteq>b]Q"
one_le_ub_sin_cos_aux: fixes n :: "nat"   and prec :: "nat"   and i :: "nat" assumes "odd n" shows "1 \<le> ub_sin_cos_aux prec n i (Suc 0) 0"
length_dropWhile_takeWhile: fixes x :: "nat"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "x < length (dropWhile P xs)" shows "x + length (takeWhile P xs) < length xs"
prems: shows "k < length (trace_parallel_plan_strips I (a # \<pi>)) - 1"
integral_by_parts_near_bounded_convex_set: fixes f :: "'b \<Rightarrow> real"   and I :: "real"   and a :: "'b"   and b :: "'b"   and g :: "'b \<Rightarrow> 'c"   and P :: "'c"   and G :: "'c set" assumes "(f has_integral I) (cbox a b)"   and "((\<lambda>x. f x *\<^sub>R g x) has_integral P) (cbox a b)"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> g x \<in> G"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> 0 \<le> f x"   and "convex G"   and "bounded G" shows "infdist P ((*\<^sub>R) I ` G) = 0"
var_pow_finsum_decomp: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p = (\<Oplus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub>t\<in>set (UP.dense_repr p). [fst t] \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> X\<^bsub>K [X]\<^esub> [^]\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> snd t)"
r_le: shows "real_of_int r \<le> 2 powr real_of_int k - 1"
sup_Sup: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = \<Squnion>{x, y}"
proj_subseq: fixes Di :: "('a strand_label \<times> 'b) list"   and j :: "'a"   and D :: "('a strand_label \<times> 'b) list"   and i :: "'a" assumes "Di \<in> set (subseqs (dbproj ln j D))"   and "j \<noteq> i" shows "filter (\<lambda>d. d \<notin> set Di) (proj i D) = proj i D"
entail_set_all_formulas: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile>\<inter>\<G> N2) = (\<forall>C\<in>N2. N1 \<Turnstile>\<inter>\<G> {C})"
is_none_code_2: fixes x :: "'a" shows "Option.is_none (pure x) = False"
fls_inverse_X_inv_power': fixes n :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "inverse (fls_X_inv ^ n) = fls_X ^ n"
add_cancel_left_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
atMostAtLeast_subset_convex: fixes C :: "real set"   and x :: "real"   and y :: "real" assumes "convex C"   and "x \<in> C"   and "y \<in> C"   and "x < y" shows "{x..y} \<subseteq> C"
replace_comp_stocks: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('e, 'c) discrete_market"   and x :: "'c"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" assumes "support_set pf1 \<subseteq> stocks Mkt \<union> {x}"   and "support_set pf2 \<subseteq> stocks Mkt" shows "support_set (qty_replace_comp pf1 x pf2) \<subseteq> stocks Mkt"
elims_2: fixes x :: "'a"   and xa :: "'a list" assumes "exhaustive_above x xa" obtains xb :: "'a" where "x = xb__" and "xa = []" and "\<not> proper_interval (Some xb__) None"   | xb :: "'a" and y :: "'a" and ys :: "'a list" where "x = xb" and "xa = y # ys" and "\<not> proper_interval (Some xb) (Some y) \<and> exhaustive_above y ys"
l_p: shows "D \<in> N_of_state (lnth Sts l) \<and> D \<notin> N_of_state (lnth Sts (Suc l)) \<and> i \<le> l \<and> enat (Suc l) < llength Sts"
Domainp_rel: fixes R1 :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'h \<Rightarrow> bool" shows "Domainp (rel_gpv R1 R2) = pred_gpv (Domainp R1) (Domainp R2)"
finite_less_Inf_iff: fixes X :: "'a set"   and a :: "'a" assumes "finite X"   and "X \<noteq> {}" shows "(a < Inf X) = (\<forall>x\<in>X. a < x)"
ide_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "replete_set_category S"   and "A \<subseteq> Collect (category.terminal S)" shows "partial_magma.ide S (set_category_given_img.mkIde S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) A)"
insort_insert_insort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "x \<notin> set xs" shows "linorder.insort_insert_key less_eq (\<lambda>x. x) x xs = linorder.insort_key less_eq (\<lambda>x. x) x xs"
assms_2: fixes s :: "'t" assumes "s \<in> set ss'" shows "is_syz_sig dgrad s"
change_respecting_doesnt_modify_dma: fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set"   and x :: "'Var" assumes "change_respecting (cms, mem) (cms', mem') X"   and "(cms, mem) \<leadsto> (cms', mem')"   and "x \<in> X" shows "dma mem x = dma mem' x"
supp_heapToAssn: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "supp (heapToAssn \<Gamma>) = supp \<Gamma>"
valid_path_drop: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist"   and n :: "nat" assumes "Digraph G"   and "Digraph.valid_path G P" shows "Digraph.valid_path G (ldropn n P)"
terminates_strip_to_terminates: fixes F :: "'f set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "strip F \<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
nucleus_alt_def_cor2: fixes f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "nucleus f" shows "f (y \<leftarrow> x) \<le> f y \<leftarrow> x"
sorted_wrt_rev_linord: fixes l :: "'a list" shows "sorted_wrt (\<lambda>x y. y \<le> x) l = sorted (rev l)"
numeral_neq_neg_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "numeral.numeral one plus m \<noteq> uminus (numeral.numeral one plus n)"
2: fixes y :: "nat" assumes "y < x" shows "lookup_pp (rep_nat_pp t) y = lookup_pp (rep_nat_pp s) y"
indeps_7: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a"   and v :: "char list \<Longrightarrow> 'a" assumes "myss3 x y a z v" shows "v \<bowtie> z"
relAPP_def: fixes f :: "('d \<times> 'e) set \<Rightarrow> 'c"   and x :: "('d \<times> 'e) set" shows "\<langle>x\<rangle>f \<equiv> f x"
hyps: shows "\<tau>.Q \<longmapsto> \<alpha> \<prec> Q'"
perpendicular_moebius: fixes H1 :: "circline"   and H2 :: "circline"   and M :: "moebius" assumes "perpendicular H1 H2" shows "perpendicular (moebius_circline M H1) (moebius_circline M H2)"
power2_norm_eq_cinner': fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(norm x)\<^sup>2 = Re (cinner x x)"