hcju/isabelleps huggingface dataset:
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "\<And>c e. P (Gt (CN 0 c e))"   and "\<And>c e. P (Ge (CN 0 c e))"   and "\<And>i c e. P (Dvd i (CN 0 c e))"   and "\<And>i c e. P (NDvd i (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Gt (C va))"   and "\<And>va. P (Gt (Bound va))"   and "\<And>vd vb vc. P (Gt (CN (Suc vd) vb vc))"   and "\<And>va. P (Gt (Neg va))"   and "\<And>va vb. P (Gt (Add va vb))"   and "\<And>va vb. P (Gt (Sub va vb))"   and "\<And>va vb. P (Gt (Mul va vb))"   and "\<And>va. P (Ge (C va))"   and "\<And>va. P (Ge (Bound va))"   and "\<And>vd vb vc. P (Ge (CN (Suc vd) vb vc))"   and "\<And>va. P (Ge (Neg va))"   and "\<And>va vb. P (Ge (Add va vb))"   and "\<And>va vb. P (Ge (Sub va vb))"   and "\<And>va vb. P (Ge (Mul va vb))"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>v vb. P (Dvd v (C vb))"   and "\<And>v vb. P (Dvd v (Bound vb))"   and "\<And>v ve vc vd. P (Dvd v (CN (Suc ve) vc vd))"   and "\<And>v vb. P (Dvd v (Neg vb))"   and "\<And>v vb vc. P (Dvd v (Add vb vc))"   and "\<And>v vb vc. P (Dvd v (Sub vb vc))"   and "\<And>v vb vc. P (Dvd v (Mul vb vc))"   and "\<And>v vb. P (NDvd v (C vb))"   and "\<And>v vb. P (NDvd v (Bound vb))"   and "\<And>v ve vc vd. P (NDvd v (CN (Suc ve) vc vd))"   and "\<And>v vb. P (NDvd v (Neg vb))"   and "\<And>v vb vc. P (NDvd v (Add vb vc))"   and "\<And>v vb vc. P (NDvd v (Sub vb vc))"   and "\<And>v vb vc. P (NDvd v (Mul vb vc))"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
assms: shows "\<exists>n'\<ge>n\<^sub>s. latestAct_cond nid t n n'"
hyps_1: shows "n i-ns\<rightarrow>\<^sub>d* nx"
order_gcd: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "order x (gcd p q) = min (order x p) (order x q)"
rep_eq: fixes x :: "'a"   and xa :: "'b" shows "lookup (Poly_Mapping.single x xa) = (\<lambda>k'. xa when x = k')"
rep_eq: fixes x :: "'a" shows "list_of_dnelist (single x) = [x]"
height_bound: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Height_Bounded_Labeling c s t f l" shows "\<forall>u\<in>Graph.V c. l u \<le> 2 * card (Graph.V c) - 1"
and_or_or_or_or_s: fixes w1 :: "word32"   and w2 :: "word32"   and w3 :: "word32"   and w4 :: "word32"   and w5 :: "word32" assumes "get_S w1 = 0 \<and> w2 AND 128 = 0 \<and> w3 AND 128 = 0 \<and> w4 AND 128 = 0 \<and> w5 AND 128 = 0" shows "get_S ((((w1 AND 4279238655 OR w2) OR w3) OR w4) OR w5) = 0"
lhopital_right: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and F :: "real filter" assumes "(f \<longlongrightarrow> 0) (at_right x)"   and "(g \<longlongrightarrow> 0) (at_right x)"   and "\<forall>\<^sub>F x in at_right x. g x \<noteq> 0"   and "\<forall>\<^sub>F x in at_right x. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at_right x. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_right x. (g has_real_derivative g' x) (at x)"   and "LIM x at_right x. f' x / g' x :> F" shows "LIM x at_right x. f x / g x :> F"
filterlim_sequentially_Suc: fixes f :: "nat \<Rightarrow> 'b"   and F :: "'b filter" shows "(LIM x sequentially. f (Suc x) :> F) = filterlim f F sequentially"
dbl_inc_simps_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one zero = one"
uint_word_of_int: fixes k :: "int" shows "uint (word_of_int k) = k mod 2 ^ LENGTH('a)"
filterlim_INF: fixes f :: "'a \<Rightarrow> 'b"   and G :: "'c \<Rightarrow> 'b filter"   and B :: "'c set"   and F :: "'a filter" shows "filterlim f (Inf (G ` B)) F = (\<forall>b\<in>B. filterlim f (G b) F)"
emeasure_scale_measure: fixes r :: "ennreal"   and M :: "'b measure"   and A :: "'b set" shows "emeasure (scale_measure r M) A = r * emeasure M A"
div_mult_mult2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
set_iterator_abs_genord_def: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" shows "set_iterator_abs_genord \<alpha> invar iti S0 R \<equiv> \<exists>lc. (\<forall>xc\<in>set lc. invar xc) \<and> distinct (map \<alpha> lc) \<and> S0 = set (map \<alpha> lc) \<and> sorted_wrt R (map \<alpha> lc) \<and> iti = foldli lc"
simps: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat"   and \<rho> :: "complex mat" shows "denote_while_n M0 M1 DS n \<rho> = M0 * denote_while_n_iter M0 M1 DS n \<rho> * adjoint M0"
basis_in_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>' \<subseteq> \<E>" shows "indep_system.basis_in (\<E>.indep_in \<E>) \<E>' B = \<E>.basis_in \<E>' B"
tuple_is_extensional: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "\<not> commutative_square f g h k" shows "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = null"
assms: shows "all_trms_irreducible (subst_set (trms_ecl C) \<sigma>) (\<lambda>t. trm_rep t S)"
index_mat_multcol_4: fixes k :: "nat"   and a :: "'d"   and A :: "'d mat" shows "dim_row (multcol k a A) = dim_row A"
annil: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and x :: "'c" assumes "class.ab_near_semiring_one_zerol plus times one zero" shows "times zero x = zero"
assms_4: shows "\<pi> n pd\<rightarrow> \<pi> 0"
assms: shows "\<downharpoonright>\<alpha>\<^sub>1\<downharpoonleft> = \<downharpoonright>\<alpha>\<^sub>2\<downharpoonleft>"
vimageD: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "a \<in> f -` A" shows "f a \<in> A"
bezout_matrix_preserves_previous_columns: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and i :: "'e"   and j :: "'e"   and A :: "(('d, 'f) vec, 'e) vec"   and k :: "'f"   and b :: "'f"   and a :: "'e" assumes "is_bezout_ext bezout"   and "i \<noteq> j"   and "A $ i $ k \<noteq> (0::'d)"   and "b < k"   and "is_zero_row_upt_k i (mod_type_class.to_nat k) A"   and "is_zero_row_upt_k j (mod_type_class.to_nat k) A" shows "(bezout_matrix A i j k bezout ** A) $ a $ b = A $ a $ b"
ct: shows "r1.cond_action_oks (ls, (ts1, m1), ws, is) t cts"
poly_shift_eq: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f = monom P (f 0) 0 \<oplus>\<^bsub>P\<^esub> X_poly R \<otimes>\<^bsub>P\<^esub> poly_shift f"
path_connected_path_component: fixes s :: "'a set"   and x :: "'a" shows "path_connected (path_component_set s x)"
sel_2: fixes x31 :: "'b local.term"   and x32 :: "'b local.term" shows "un_Tensor1 (x31 \<^bold>\<otimes> x32) = x31"
assms_2: assumes "a \<in># B"   and "A = B - {#a#} + C" shows "P"
LIM_equal: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "\<forall>x. x \<noteq> a \<longrightarrow> f x = g x" shows "f \<midarrow>a\<rightarrow> l = g \<midarrow>a\<rightarrow> l"
etp_def: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "etp I domain range F F\<^sub>i\<^sub>n\<^sub>v \<equiv> ((\<forall>y. y \<in> set_spmf I \<longrightarrow> domain (fst y) = range (fst y)) \<and> (\<forall>y. y \<in> set_spmf I \<longrightarrow> finite (range (fst y))) \<and> (\<forall>y. y \<in> set_spmf I \<longrightarrow> range (fst y) \<noteq> {})) \<and> (\<forall>y. y \<in> set_spmf I \<longrightarrow> bij_betw (F (fst y)) (domain (fst y)) (range (fst y))) \<and> lossless_spmf I \<and> (\<forall>y x. y \<in> set_spmf I \<longrightarrow> x \<in> range (fst y) \<longrightarrow> F\<^sub>i\<^sub>n\<^sub>v (fst y) (snd y) (F (fst y) x) = x)"
Suc_2: shows "\<rho> \<in> carrier_mat d d"
assms_3: shows "\<not> (a \<or> c \<and> \<not> c) \<or> b"
stable_Join_Always1: fixes F :: "'c program"   and A :: "'c set"   and G :: "'c program" assumes "F \<in> stable A"   and "G \<in> invariant A" shows "F \<squnion> G \<in> Always A"
assms_3: shows "A \<inter> {a. en a s} \<inter> sticky = {}"
minus_vec_def: fixes v\<^sub>1 :: "'a vec"   and v\<^sub>2 :: "'a vec" shows "v\<^sub>1 - v\<^sub>2 \<equiv> vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $ i - v\<^sub>2 $ i)"
Liminf_set_filter_commute: fixes p :: "'a \<Rightarrow> bool"   and Xs :: "'a set llist" shows "Liminf_llist (lmap (\<lambda>X. {x \<in> X. p x}) Xs) = {x \<in> Liminf_llist Xs. p x}"
hyps_2: fixes t1 :: "lam" assumes " t \<longrightarrow>\<^sub>1\<^sup>* t1" shows "\<exists>t3.  t1 \<longrightarrow>\<^sub>1\<^sup>* t3 \<and>  s1 \<longrightarrow>\<^sub>1\<^sup>* t3"
disjointed_def: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "disjointed A n = A n - \<Union> (A ` {0..<n})"
elims: fixes x :: "term"   and y :: "(term \<times> term) option" assumes "unapp x = y" obtains t :: "term" and u :: "term" where "x = t $ u" and "y = Some (t, u)"   | v :: "name" where "x = Const v__" and "y = None"   | v :: "name" where "x = Free va__" and "y = None"   | v :: "term" where "x = \<Lambda> vb__" and "y = None"   | v :: "nat" where "x = Bound v" and "y = None"
one_in_Basis: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and Basis :: "'a set" assumes "class.dirichlet_series one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power Basis" shows "one \<in> Basis"
Alexander_subbase_alt: fixes U :: "'c set"   and \<B> :: "'c set set"   and X :: "'c topology" assumes "U \<subseteq> \<Union> \<B>"   and "\<And>C. \<lbrakk>C \<subseteq> \<B>; U \<subseteq> \<Union> C\<rbrakk> \<Longrightarrow> \<exists>C'. finite C' \<and> C' \<subseteq> C \<and> U \<subseteq> \<Union> C'"   and "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to U)) = X" shows "compact_space X"
Fvars_instInp: fixes \<phi> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "Fvars \<phi> = {inp}" shows "Fvars (instInp \<phi> t) = FvarsT t"
is_partition_def: fixes cs :: "'b set list" shows "is_partition cs = (\<forall>j<length cs. \<forall>i<j. cs ! i \<inter> cs ! j = {})"
axioms_3: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "functor A B F"
left_idem: fixes a :: "'b"   and b :: "'b" shows "inf a (inf a b) = inf a b"
induct: fixes x :: "'e"   and G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and P :: "'e \<Rightarrow> bool" assumes "x \<in> generate G H"   and "P \<one>\<^bsub>G\<^esub>"   and "\<And>h. h \<in> H \<Longrightarrow> P h"   and "\<And>h. h \<in> H \<Longrightarrow> P (inv\<^bsub>G\<^esub> h)"   and "\<And>h1 h2. \<lbrakk>h1 \<in> generate G H; P h1; h2 \<in> generate G H; P h2\<rbrakk> \<Longrightarrow> P (h1 \<otimes>\<^bsub>G\<^esub> h2)" shows "P x"
R_lin_independent_imp_hd_n0: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b"   and ms :: "'b list" assumes "RModule R smult M"   and "R_scalar_mult.R_lin_independent R smult (m # ms)" shows "m \<noteq> (0::'b)"
epred_llength: fixes xs :: "'g llist" shows "epred (llength xs) = llength (ltl xs)"
intros_4: fixes T :: "typ"   and Ts :: "typ list"   and t :: "Core.term"   and T' :: "typ" assumes "T # Ts \<turnstile>\<^sub>\<tau> t : T'" shows "Ts \<turnstile>\<^sub>\<tau> Abs T t : T \<rightarrow> T'"
in_CS: shows "X \<in> UNIV // \<approx>A"
rpr_refl: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a" assumes "rpr A r"   and "x \<in> A" shows "x \<^bsub>r\<^esub>\<preceq> x"
norm_bound_component_le_cart: fixes x :: "(real, 'a) vec"   and e :: "real"   and i :: "'a" assumes "norm x \<le> e" shows "\<bar>x $ i\<bar> \<le> e"
index_to_residue_def: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat"   and m :: "nat" assumes "padic_integers p" shows "index_to_residue s k m = s m k"
lfp_lowerbound: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'b::type \<Rightarrow> 'b::type"   and A :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((f::'b::type \<Rightarrow> 'b::type) (A::'b::type)) A" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (complete_lattice.lfp (Inf::'b::type set \<Rightarrow> 'b::type) less_eq (f::'b::type \<Rightarrow> 'b::type)) (A::'b::type)"
Pd_shuffle_to_top_range: fixes ds' :: "'b set"   and Y :: "'a set"   and d :: "'b" shows "Field (TruePref.Pd_shuffle_to_top ds' Y d) \<subseteq> {x. Xd x = d}"
VInterE2: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A" obtains x :: "V" where "a \<in>\<^sub>\<circ> x" and "x \<in>\<^sub>\<circ> A"
bounded: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "g \<in> Y" shows "g e \<le> f e"
F: shows "msgs cJ -# <pJ, inM False> \<union># {# <pJ, inM True> } = msgs cJ1 -# <pJ, inM True> \<union># {# <pJ, inM True> }"
card_eq_dim_span_indep: fixes A :: "('b, 'c) vec set" assumes "vec.dim (vec.span A) = card A"   and "finite A" shows "vec.independent A"
witness'_valid: fixes Y :: "nat"   and rs :: "nat list" shows "valid_net' (witness' Y rs)"
m: shows "m = \<lparr>iiface = iif, oiface = oif, src = sip, dst = dip, proto = protocol, sports = (sps1, sps2), dports = (dps1, dps2)\<rparr>"
Inf_spec_def: fixes X :: "'b set" shows "Inf_spec X = SPEC (\<lambda>r. \<forall>x\<in>X. r \<le> x)"
dres_ne_bot_iterateo: fixes f :: "'x \<Rightarrow> 'b \<Rightarrow> 'b dres"   and r :: "'s"   and c :: "'b dres \<Rightarrow> bool"   and s :: "'b" assumes "\<And>x s. f x s \<noteq> dSUCCEED" shows "iterateoi r c (\<lambda>x s. s \<bind> f x) (dRETURN s) \<noteq> dSUCCEED"
coalesce_eq_Err_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [\<squnion>\<^bsub>f\<^esub>] ys) = Err" shows "\<not> (\<exists>zs\<in>list n A. xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs \<and> ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs)"
lead_coeff_g: shows "cmod (lead_coeff (\<Prod>a\<leftarrow>complex_roots_complex f. if 1 < cmod a then [:- 1, cnj a:] else [:- a, 1:])) = (\<Prod>a\<leftarrow>complex_roots_complex f. max 1 (cmod a))"
Inst: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned"   and T :: "ref_ty" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>e InstOf T\<rangle>\<guillemotright> A"
sbelow_sctn_nres: fixes X :: "'c set"   and sctn :: "'c sctn" shows "Sup_inner X (normal sctn) \<bind> (\<lambda>si. RETURN (si < pstn sctn)) \<le> sbelow_sctn X sctn"
TRACES_def: fixes P :: "('b event list \<times> 'b event set) set \<times> 'b event list set" shows "TRACES P = {tr. \<exists>a. a \<in> FAILURES P \<and> tr = fst a}"
fv_simps_2: fixes x21 :: "'b"   and x22 :: "('b, 'a) Term.term list" shows "fv (Fun x21 x22) = \<Union> (fv ` set x22)"
assms_2: assumes "\<Phi>" shows "m' \<le>\<^sub>n m"
eint_ord_simps_2: fixes m :: "int"   and n :: "int" shows "(eint m < eint n) = (m < n)"
RUV_split_el: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and q :: "'b"   and Ml :: "'d set"   and Me :: "'d set"   and h :: "'d \<Rightarrow> 'd set"   and V :: "'a set"   and Ml' :: "'d set"   and Me' :: "'d set"   and h' :: "'d \<Rightarrow> 'd set"   and U :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 1"   and "q \<in># P"   and "(v, Ml, Me, h) \<in> RU_cs fg V"   and "(entry fg q, Ml', Me', h') \<in> RU_cs fg U"   and "(mon_n fg u \<union> mon fg p) \<inter> (Me \<union> Ml' \<union> Me') = {}"   and "h [*] h'" shows "(u, mon fg p \<union> M \<union> Ml, Me \<union> Ml' \<union> Me') \<in> RUV_cs fg U V"
ipresCons_imp_ipresSubst: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.ipresCons Delta h hA MOD"   and "FixSyn.igSubstCls Delta MOD"   and "FixSyn.igConsIPresIGWls Delta MOD"   and "FixSyn.igFreshCls Delta MOD" shows "FixSyn.ipresSubst Delta h MOD"
proj_unl_cons_2: fixes l :: "'a"   and l' :: "'a"   and a :: "'b"   and A :: "('a strand_label \<times> 'b) list" assumes "l \<noteq> l'" shows "proj_unl l' ((ln l, a) # A) = proj_unl l' A"
greedy_conforming_path_properties_3: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v0 \<in> V\<^bsub>G\<^esub>"   and "ParityGame.strategy G p \<sigma>"   and "ParityGame.strategy G p** \<sigma>'" shows "Digraph.valid_path G (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0)"
run_axioms_def: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" shows "run_axioms s\<^sub>0 lstep s \<equiv> df_system.is_run s\<^sub>0 lstep s"
abs_eq: shows "Tree_wf\<^sub>p = map_prod abs_Tree\<^sub>p abs_Tree\<^sub>p ` Tree_wf"
le_ref2: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<le> Q" shows "F Q \<subseteq> F P"
vimage_UN: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "f -` \<Union> (B ` A) = (\<Union>x\<in>A. f -` B x)"
field_frac2: fixes x :: "'a" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x = 1\<^sub>r \<cdot>\<^sub>r x\<^bsup>\<hyphen> K\<^esup>\<^bsup>\<hyphen> K\<^esup>"
diffeomorphism_image_eq_2: fixes k :: "enat"   and S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "diffeomorphism k S T f g" shows "g ` T = S"
pair_scale_spmf1: fixes r :: "real"   and p :: "'c spmf"   and q :: "'d spmf" assumes "r \<le> 1" shows "pair_spmf (scale_spmf r p) q = scale_spmf r (pair_spmf p q)"
ide_exp: fixes b :: "'a"   and c :: "'a" assumes "ide b"   and "ide c" shows "ide (exp b c)"
conj_bot_left: fixes c :: "'b::refinement_lattice" shows "\<bottom> \<iinter> (c::'b::refinement_lattice) = \<bottom>"
pre_between_dist: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "distinct vs"
closure_mono: fixes a :: "'b set"   and b :: "'b set"   and T :: "'b set set" assumes "a \<subseteq> b" shows "closure\<^bsub>T\<^esub> a \<subseteq> closure\<^bsub>T\<^esub> b"
diff_Pair: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "(a, b) - (c, d) = (a - c, b - d)"
v: shows "- v = - fst (snd (snd (snd (bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)))))"
indep_subset_carrier: fixes X :: "'a set" assumes "indep_in carrier X" shows "X \<subseteq> carrier"
\<E>_E: fixes x :: "'a"   and S :: "'a set"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "x \<in> \<E>\<^bsub>\<Gamma>\<^esub> S" obtains p :: "'a list" and y :: "'a" where "path \<Gamma> x p y" and "y \<in> B \<Gamma>" and "\<And>z. \<lbrakk>x \<noteq> y; z \<in> set p\<rbrakk> \<Longrightarrow> z = x \<or> z \<notin> S"
arg_min_natI: fixes P :: "'b \<Rightarrow> bool"   and k :: "'b"   and m :: "'b \<Rightarrow> nat" assumes "P k" shows "P (arg_min m P)"
nontriv_solution_pell_nat_mul2: fixes D :: "nat"   and z1 :: "nat \<times> nat"   and z2 :: "nat \<times> nat" assumes "pell D"   and "pell.nontriv_solution D z1"   and "pell.solution D z2" shows "pell.nontriv_solution D (pell.pell_mul D z1 z2)"
vint_of_int_transfer: fixes a :: "int" shows "cr_vint (a\<^sub>\<int>) a"