hcju/isabelleps huggingface dataset:
inputInduct: fixes P :: "pi"   and a :: "name"   and u :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto> a<u> \<prec> P'"   and "\<And>a x P u C. \<lbrakk>x \<sharp> C; x \<noteq> u; x \<noteq> a\<rbrakk> \<Longrightarrow> F C (a<x>.P) a u (P[x::=u])"   and "\<And>P a u P' b C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a u P'"   and "\<And>P a u P' b c C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a u P'"   and "\<And>P a u P' Q C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a u P'"   and "\<And>Q a u Q' P C. \<lbrakk>Q \<longmapsto> a<u> \<prec> Q'; \<And>C. F C Q a u Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a u Q'"   and "\<And>P a u P' Q C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a u (P' \<parallel> Q)"   and "\<And>Q a u Q' P C. \<lbrakk>Q \<longmapsto> a<u> \<prec> Q'; \<And>C. F C Q a u Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a u (P \<parallel> Q')"   and "\<And>P a u P' x C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; x \<noteq> a; x \<noteq> u; x \<sharp> C; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a u (<\<nu>x>P')"   and "\<And>P a u P' C. \<lbrakk>P \<parallel> !P \<longmapsto> a<u> \<prec> P'; \<And>C. F C (P \<parallel> !P) a u P'\<rbrakk> \<Longrightarrow> F C (!P) a u P'" shows "F C P a u P'"
twice_conv_BIT: fixes x :: "int" shows "2 * x = x BIT False"
inj: shows "inj_on (\<lambda>k. a ^ k mod n) {..<d}"
scale_cancel_left: fixes a :: "'b"   and x :: "'b"   and y :: "'b" shows "(a * x = a * y) = (x = y \<or> a = (0::'b))"
ceq_prod_def: shows "CEQ('b \<times> 'c) \<equiv> case ID CEQ('b) of None \<Rightarrow> None | Some eq_'a \<Rightarrow> case ID CEQ('c) of None \<Rightarrow> None | Some eq_'b \<Rightarrow> Some (equality_prod eq_'a eq_'b)"
setdist_subset_left: fixes S :: "'b set"   and T :: "'b set"   and u :: "'b set" assumes "S \<noteq> {}"   and "S \<subseteq> T" shows "setdist T u \<le> setdist S u"
merge_conv: fixes xs :: "('d \<times> 'c) list"   and ys :: "('d \<times> 'c) list"   and k :: "'d" shows "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
map_of_eq_None_iff: fixes xys :: "('b \<times> 'a) list"   and x :: "'b" shows "(map_of xys x = None) = (x \<notin> fst ` list.set xys)"
convergent_def: fixes X :: "nat \<Rightarrow> 'c" shows "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
x_U_prop2: fixes x :: "'a" shows "x \<cdot> nc = d (x \<sqinter> nc) \<cdot> nc + x \<cdot> (0::'a)"
assms_1: shows "frees t\<^sub>2 |\<subseteq>| frees t\<^sub>1"
comp_of_ords_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "comp_of_ords le lt x y = (if lt x y then Lt else if le x y then Eq else Gt)"
funpow_simps_right_2: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a" shows "f ^^ Suc n = f ^^ n \<circ> f"
vinsert_4: shows "vinsert x F = \<D>\<^sub>\<circ> r"
intros_2: fixes x :: "name"   and M :: "trm"   and a :: "coname" assumes "x \<sharp> M" shows "fin (NotL <a>.M x) x"
surjective_sum: fixes f :: "'b + 'c \<Rightarrow> 'd" shows "case_sum (\<lambda>x. f (Inl x)) (\<lambda>y. f (Inr y)) = f"
distinct_8: fixes x5 :: "char list" shows "Class x5 \<noteq> Void"
wf_mul: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "wf r" shows "wf (mul r)"
horiz_vector_helper_pos_ind_R1: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x"   and "list_constr I (length qs)" shows "(prod_list (map ((!) (map_sgas sign)) I) = 1) = (\<forall>p\<in>set (retrieve_polys qs I). poly p x = 0)"
R_def: fixes x :: "'a"   and p :: "'a"   and q :: "'a" shows "(x \<le> R p q) = (d p \<le> wp x q)"
well_analyzed_singleton_3: fixes a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term" shows "well_analyzed [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t]"
smc_is_arrD_1: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "semicategory \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
x_c_U_split_d: fixes x :: "'a" shows "d x \<cdot> U = x \<cdot> U + x \<cdot> (0::'a) \<parallel> U"
that: fixes l :: "'a"   and r :: "nat \<Rightarrow> nat" assumes "l \<in> S"   and "strict_mono r"   and "(f \<circ> r) \<longlonglongrightarrow> l" shows "thesis"
Re_winding_number_half_right: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "\<forall>p\<in>path_image \<gamma>. Re z \<le> Re p"   and "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "Re (winding_number \<gamma> z) = (Im (Ln (pathfinish \<gamma> - z)) - Im (Ln (pathstart \<gamma> - z))) / (2 * pi)"
subset: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" assumes "subgroup H G" shows "H \<subseteq> carrier G"
distinct_set_eq_Singl: fixes al :: "'a list"   and a :: "'a" assumes "distinct al" shows "(set al = {a}) = (al = [a])"
simps_1: fixes lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat"   and n :: "nat" shows "divide_poly_main lc q r d dr (Suc n) = (let cr = coeff r dr; a = cr div lc; mon = monom a n in if False \<or> a * lc = cr then divide_poly_main lc (q + mon) (r - mon * d) d (dr - 1) n else 0)"
God_starNecExists_v1: shows "universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v = xa x)))"
is_IntgE: fixes v :: "'a val" assumes "is_Intg v" obtains i :: "32 word" where "v = Intg i"
oarrivemsg_simps_5: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and i1 :: "nat"   and i2 :: "nat" shows "oarrivemsg P \<sigma> (disconnect(i1, i2)) = True"
foo2: fixes a :: "'a"   and ps :: "'a list"   and ss :: "'a list"   and p :: "'a list"   and s :: "'a list" assumes "a \<notin> set ps"   and "a \<notin> set ss"   and "set p = set s"   and "p = a # ps"   and "s = a # ss" shows "set ps = set ss"
relto_absorb_2: fixes E :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" shows "E\<^sup>* O E\<^sup>* O R O E\<^sup>* = E\<^sup>* O R O E\<^sup>*"
rbt_max_def: fixes t :: "('b, unit) RBT_Impl.rbt" shows "rbt_max t = rbt_fold1_keys max t"
sappend_bottom_iff: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" shows "(xs :@ ys = \<bottom>) = (xs = \<bottom> \<or> ys = \<bottom>)"
fn_norm_works: fixes f :: "'a \<Rightarrow> real" assumes "Function_Norm.continuous V f norm" shows "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
bernoulli_stream_def: fixes p :: "real" shows "bernoulli_stream p = stream_space (measure_pmf (bernoulli_pmf p))"
ag_l_inv1: fixes x :: "'a" assumes "x \<in> carrier M" shows "-\<^sub>a x \<plusminus> x = \<zero>"
assms_2: shows "ps1 = apply_ncrit ncrit data gs bs hs (apply_icrit icrit data gs bs hs (np gs bs hs data))"
list_combE_2: fixes f :: "'c"   and xs :: "'c list"   and S :: "name fset"   and x :: "'c" assumes "closed_except (list_comb f xs) S"   and "x \<in> set xs" shows "closed_except x S"
codomains_char: fixes f :: "'a" shows "codomains f = (if arr f then {cod f} else {})"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "Span.arr (SPN f)"
slim_verts_def: shows "S.slim_verts \<equiv> verts3 (with_proj slim) \<union> (\<Union>(u, p, uu_)\<in>S.slim_paths. set (pawalk_verts u p))"
epath_decompose': fixes G :: "'b uprod set"   and v :: "'b"   and p :: "'b uprod list"   and v' :: "'b"   and a :: "'b"   and b :: "'b" assumes "epath G v p v'"   and "Upair a b \<in> set p" shows "\<exists>u u' p' p''. Upair a b = Upair u u' \<and> epath G v p' u \<and> epath G u' p'' v' \<and> length p' < length p \<and> length p'' < length p"
sp_rewrite_eq_Ok: fixes n :: "nat"   and xs :: "('a, 'b, 'c, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr list"   and F :: "'c \<Rightarrow> 'f fundef option"   and x :: "('a, 'b, 'c, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr"   and ys :: "Unboxed.type option list"   and zs :: "Unboxed.type option list" assumes "n < length xs"   and "\<forall>\<Sigma>. Subx.sp_instr F (xs ! n) \<Sigma> = Subx.sp_instr F x \<Sigma>"   and "Subx.sp F xs ys = Ok zs" shows "Subx.sp F (rewrite xs n x) ys = Ok zs"
Inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" shows "z \<le> \<Sqinter>A"
sym_sum_mset_inter2_sets_count: fixes blv :: "'a set"   and p :: "'a set" assumes "blv \<in># \<B>"   and "p \<subseteq> blv"   and "card p = 2" shows "int (count (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2}) p) = \<Lambda> - 1"
coreflexive_comp_top_inf_one: fixes x :: "'a" assumes "x \<le> top" shows "x \<sqinter> top \<sqinter> top = x"
D2: shows "\<forall>mo. Deliver (i, Store e2 mo i) \<notin> set pre"
extend_infpath: fixes f :: "nat \<Rightarrow> bool list"   and C :: "fterm literal set" assumes "wf_infpath f"   and "\<forall>n. \<not> falsifies\<^sub>c (f n) C"   and "finite C" shows "eval\<^sub>c HFun (extend f) C"
assms: shows "f\<^sub>1 = \<tau> i(gap i t \<mapsto> 1)"
le_word_or2: fixes x :: "'b word"   and y :: "'b word" shows "x \<le> x || y"
size_gen_1: fixes x :: "'a \<Rightarrow> nat" shows "size_list x [] = 0"
interior_convex_hull_eq_empty: fixes S :: "'b set" assumes "card S = Suc DIM('b)" shows "(interior (convex hull S) = {}) = affine_dependent S"
distrib_comp_sum_left: fixes S :: "'a \<Rightarrow> 'v"   and S' :: "'a \<Rightarrow> 'v" assumes "range S \<subseteq> V"   and "range S' \<subseteq> V" shows "T \<circ> S + S' = (T \<circ> S) + (T \<circ> S')"
test_double_comp_var: fixes p :: "'a" assumes "test p" shows "p = t p"
cond_distr: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and x :: "'a"   and y :: "'a" assumes "test p"   and "test q"   and "test r" shows "(p \<cdot> q + !p \<cdot> r) \<cdot> (p \<cdot> x + !p \<cdot> y) = p \<cdot> q \<cdot> x + !p \<cdot> r \<cdot> y"
simps_4: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "sym.implSecure A B M = Enc \<langle>SecureTag, M\<rangle> (shrK A B)"
absorb_iff1: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a \<le>s b) = (signed.min a b = a)"
simps_10: fixes a :: "('f, 'g, 'h) exp"   and i :: "('f, 'g, 'h) exp"   and e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (a\<lfloor>i\<rceil> := e) A = (\<D> a A \<and> \<D> i (A \<squnion> \<A> a) \<and> \<D> e (A \<squnion> \<A> a \<squnion> \<A> i))"
scale_right_imp_eq: fixes x :: "('b, 'c) vec"   and a :: "'b"   and b :: "'b" assumes "x \<noteq> 0"   and "a *s x = b *s x" shows "a = b"
divide_cancel_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide c a = divide c b) = (c = zero \<or> a = b)"
conjugate_exponent_real_2: fixes p :: "real" assumes "1 < p" shows "1 < conjugate_exponent p"
prems_2: fixes u :: "'b poly" assumes "u \<in> set (x # P)" shows "0 < degree u \<and> monic u"
closedin_topspace: fixes U :: "'c topology" shows "closedin U (topspace U)"
add_diff_cancel_enat: fixes x :: "enat"   and y :: "enat" assumes "x \<noteq> \<infinity>" shows "x + y - x = y"
Collect_split_Suc_1: fixes P :: "nat \<Rightarrow> bool" assumes "\<not> P 0" shows "{i. P i} = Suc ` {i. P (Suc i)}"
call_zero_iff: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'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 sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(\<forall>u. cinner x u = 0) = (x = zero)"
comp_reduce: fixes f :: "'a2"   and g :: "'a2"   and k :: "'a2"   and h :: "'a2" assumes "f \<cdot>\<^sub>2 g = k"   and "C2.seq f g"   and "C2.seq g h" shows "f \<cdot>\<^sub>2 g \<cdot>\<^sub>2 h = k \<cdot>\<^sub>2 h"
is_aligned_neg_mask_eq: fixes p :: "'b word"   and n :: "nat" assumes "is_aligned p n" shows "p AND NOT (mask n) = p"
assms: shows "f \<in> V\<^sub>0\<^sub>1"
p1p3: fixes s :: "'c" assumes "G s" shows "I s"
inj_on_image_Int: fixes f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
rel_uprod_simps: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'c"   and d :: "'c" shows "rel_uprod R (Upair a b) (Upair c d) = (R a c \<and> R b d \<or> R a d \<and> R b c)"
irreducible_def: fixes p :: "'c" shows "comm_semiring_1_class.irreducible p = (p \<noteq> (0::'c) \<and> \<not> p dvd (1::'c) \<and> (\<forall>a b. p = a * b \<longrightarrow> a dvd (1::'c) \<or> b dvd (1::'c)))"
sound_def: fixes R :: "fml list \<times> fml" shows "sound R \<equiv> (\<forall>k\<ge>0. k < length (fst R) \<longrightarrow> valid (fst R ! k)) \<longrightarrow> valid (snd R)"
pick_alpha: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X" shows "pick X #= pick X"
pp_np_rv_iff: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "(f \<in> rv M) = (pp f \<in> rv M \<and> np f \<in> rv M)"
mv_box_ok_suc_simp: fixes xs :: "nat list"   and n :: "nat"   and f :: "recf"   and fft :: "nat"   and gft :: "nat"   and x :: "nat"   and anything :: "nat list" assumes "length xs = n" shows "{\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything} mv_box n (Suc n) {\<lambda>nl. nl = xs @ 0 # rec_exec f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ x # anything}"
jw_Roll_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty"   and \<alpha> :: "Syntax.tvar" assumes "\<Gamma> \<turnstile>\<^sub>W Roll e : \<tau>"   and "atom \<alpha> \<sharp> (\<Gamma>, \<tau>)" obtains \<tau>'' :: "ty" where "\<tau> = Mu \<alpha> \<tau>''" and "\<Gamma> \<turnstile>\<^sub>W e : subst_type \<tau>'' (Mu \<alpha> \<tau>'') \<alpha>"
simps_2: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and n :: "nat"   and c'' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "P \<turnstile> c \<rightarrow>^(Suc n) c'' = (\<exists>c'. P \<turnstile> c \<rightarrow> c' \<and> P \<turnstile> c' \<rightarrow>^n c'')"
inj_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'a" assumes "inj f"   and "inj g" shows "inj (f \<circ> g)"
Lim_within_LIMSEQ: fixes a :: "'b"   and T :: "'b set"   and X :: "'b \<Rightarrow> 'c"   and L :: "'c" assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" shows "(X \<longlongrightarrow> L) (at a within T)"
Rep_lifted_pair_inject: fixes x :: "('c, 'd) lifted_pair"   and y :: "('c, 'd) lifted_pair" shows "(Rep_lifted_pair x = Rep_lifted_pair y) = (x = y)"
eqvt_bound: fixes p :: "perm"   and x :: "'c" shows "p \<bullet> unpermute p x \<equiv> x"
assms: fixes x :: "'a \<times> 'a" assumes "x \<in>\<^sub>p A" shows "x \<in>\<^sub>p B"
word_rcat_def: shows "word_rcat = word_of_int \<circ> horner_sum uint (2 ^ LENGTH('c)) \<circ> rev"
F_ide: fixes y :: "'d" assumes "D.ide y" shows "F y = Fo y"
standard_lifting_axioms: shows "standard_lifting Inf_F Bot_G Inf_G (\<Turnstile>G) Red_I_G Red_F_G Bot_F \<G>_F \<G>_I"
UN_insert: fixes B :: "'d \<Rightarrow> 'c set"   and a :: "'d"   and A :: "'d set" shows "\<Union> (B ` insert a A) = B a \<union> \<Union> (B ` A)"
vertices_constr: fixes a :: "'a"   and b :: "'a"   and e :: "'edge" assumes "joins a b e" shows "{a, b} \<subseteq> vertices e"
IH: fixes w' :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "(xs \<frown> w') \<Turnstile>\<^sub>n \<phi> = w' \<Turnstile>\<^sub>n af \<phi> xs"
scalar_product_times: fixes n :: "nat"   and x :: "'a"   and y :: "'a" shows "\<forall>w1 w2. length w1 = length w2 \<and> length w1 = n \<longrightarrow> x * y * scalar_product w1 w2 = scalar_product (local.times x w1) (local.times y w2)"
ext: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "C g f \<noteq> partial_magma.null C" shows "partial_magma.arr C (C g f)"
obt1: shows "e \<in> incoming s \<and> f e \<noteq> (0::'capacity)"
absn_dist_eta_free: fixes t :: "dB"   and n :: "nat" assumes "\<forall>i\<in>frees t. n \<le> i" shows "(Abs ^^ n) (var_dist (rev [0..<n]) t) \<leftrightarrow> strip_context n t 0"
simps_29: fixes vg :: "'a rexp"   and vh :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Plus vg vh) (Times v va) = True"
hypreal_sqrt_mult_distrib: fixes x :: "real star"   and y :: "real star" assumes "0 < x"   and "0 < y" shows "(*f* sqrt) (x * y) = (*f* sqrt) x * (*f* sqrt) y"
dvd_times_left_cancel_iff: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero" shows "dvd.dvd times (times a b) (times a c) = dvd.dvd times b c"
Leaf_inject: fixes x :: "'b"   and y :: "'b" assumes "Leaf x = Leaf y" shows "x = y"
msg_defs_7: shows "Enc \<equiv> map_fun Re (map_fun Re Ab) cEnc"
infsetsum_geq0_complex: fixes f :: "'c \<Rightarrow> complex"   and M :: "'c set" assumes "f abs_summable_on M"   and "\<And>x. x \<in> M \<Longrightarrow> 0 \<le> f x" shows "0 \<le> infsetsum f M"
m2r_succ_def: fixes s :: "'b \<Rightarrow> 'b option set" shows "m2r_succ s \<equiv> \<lambda>x. case x of None \<Rightarrow> {} | Some a \<Rightarrow> s a"
Rep_iso_inverse: fixes x :: "'b iso" shows "Abs_iso (Rep_iso x) = x"
geodesic_subsegment_subset: fixes G :: "'b set"   and x :: "'b"   and s :: "real"   and t :: "real" shows "geodesic_subsegment G x s t \<subseteq> G"
inv_comp: fixes f :: "'c"   and g :: "'c" assumes "C.iso f"   and "C.iso g"   and "C.seq g f" shows "C.inv (g \<cdot>\<^sub>C f) = C.inv f \<cdot>\<^sub>C C.inv g"
ex_br_conj_iff: fixes y :: "'e"   and a :: "'e \<Rightarrow> 'd"   and I :: "'e \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" shows "(\<exists>x. (y, x) \<in> br a I \<and> P x) = (I y \<and> P (a y))"
ord: fixes a :: "nat"   and n :: "nat" shows "[a ^ Pocklington.ord n a = 1] (mod n)"
Id_def: fixes n :: "nat" shows "Quantum.Id n \<equiv> 1\<^sub>m (2 ^ n)"
corres: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and u' :: "'a" assumes "valid_unMultigraph G" shows "((v, w, u') \<in> edges G) = ((u', w, v) \<in> edges G)"
ordinal_oLog_less: fixes x :: "ordinal"   and b :: "ordinal"   and y :: "ordinal" assumes "0 < x"   and "oSuc 0 < b"   and "x < b ** y" shows "oLog b x < y"
diff2_3: shows "local.rev_curr g m \<in> rs.FINFUNC"
convp_conv_eq: fixes ars :: "('a \<times> 'b \<times> 'a) set" shows "convp (\<lambda>x xa xb. (x, xa, xb) \<in> ars) = (\<lambda>x xa. (x, xa) \<in> conv ars)"
elem_set: fixes x :: "'b"   and xo :: "'b option" shows "(x \<in> set_option xo) = (xo = \<lfloor>x\<rfloor>)"
dbproj_subseq_subset: fixes Di :: "('a \<times> 'b) list"   and i :: "'a"   and D :: "('a \<times> 'b) list" assumes "Di \<in> set (subseqs (dbproj i D))" shows "set Di \<subseteq> set D"
int_decode_inverse: fixes n :: "nat" shows "int_encode (ni_bij n) = n"
HFinite_HInfinite_iff: fixes x :: "'b star" shows "(x \<in> HFinite) = (x \<notin> HInfinite)"
exec_gpv_invariant: fixes x :: "'b"   and s' :: "bool list \<Rightarrow> bool list option"   and gpv :: "('b, bool list, bool list \<times> bool list) gpv"   and s :: "bool list \<Rightarrow> bool list option" assumes "(x, s') \<in> set_spmf (exec_gpv rf_encrypt gpv s)"   and "finite (dom s)"   and "\<I>_full \<turnstile>g gpv \<surd>" shows "finite (dom s')"
arr_char: fixes f :: "'a" shows "hom'.arr f = \<guillemotleft>f : src e \<rightarrow> src (trg f)\<guillemotright>"
set_of_uminus: fixes X :: "'b interval" shows "set_of (- X) = {- x |x. x \<in>\<^sub>i X}"
erase_mkt: fixes n :: "'a"   and l :: "'a tree"   and r :: "'a tree" shows "erase (mkt n l r) = MKT\<^sub>0 n (erase l) (erase r)"
tailrec_transform1: fixes m :: "'d nres"   and a :: "'d \<Rightarrow> 'd nres"   and b :: "'d \<Rightarrow> 'c nres" shows "m \<bind> REC (\<lambda>D s. sup (a s \<bind> D) (b s)) \<le> lfp (\<lambda>x. sup m (x \<bind> a)) \<bind> b"
osc_conclusion: shows "s2 \<in> S\<^bsub>SES\<^esub> \<and> \<delta> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> \<delta> \<upharpoonleft> V\<^bsub>\<V>\<^esub> = [e1] \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> s1 \<delta>\<Longrightarrow>\<^bsub>SES\<^esub> s2 \<and> (s2', s2) \<in> ur"
cinfinite_def: fixes r :: "('c \<times> 'c) set" shows "cinfinite r = infinite (Field r)"
d_comm: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y = d y \<cdot> d x"
matroid_axioms: shows "matroid \<E> (indep_in \<E>)"
times_int_code_2: fixes l :: "int" shows "0 * l = 0"
max_list_mem: fixes xs :: "nat list" assumes "xs \<noteq> []" shows "Utility.max_list xs \<in> set xs"
d2_def: shows "st1.d2 = prod_list st1.dims2"
a': shows "aa_1 \<otimes> aa_2 \<sim> a'"
ordering_axioms: shows "ordering (\<preceq>\<^sub>p) (\<prec>\<^sub>p)"
fresh_at_snoc: fixes t :: "('form, 'rule, 'a, 'b) itnode rose_tree"   and "is" :: "nat list"   and i :: "nat" shows "fresh_at t (is @ [i]) = freshenLC (iAnnot (tree_at t is)) ` a_fresh (inPorts' (iNodeOf (tree_at t is)) ! i)"
Step1_4_4b1: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and l :: "Locs"   and v :: "Vals" shows "\<turnstile> $S4 rmhist p \<and> S4 rmhist p$ \<and> WriteInner rmCh mm ires p l v \<and> unchanged (e p, c p, r p, rmhist!p) \<longrightarrow> WriteInner memCh mm (resbar rmhist) p l v"
maxsimpD_simplex: fixes x :: "'a set" assumes "folding.chamber x" shows "x \<in> X"
oppositeLiteralListNonempty: fixes literalList :: "Literal list" shows "(literalList \<noteq> []) = (oppositeLiteralList literalList \<noteq> [])"
random_serial_dictatorship_singleton: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and x :: "'b" assumes "pref_profile_wf agents alts R"   and "finite agents"   and "finite alts"   and "agents' \<subseteq> agents"   and "x \<in> alts" shows "random_serial_dictatorship agents' {x} R = return_pmf x"
fin_succs': fixes v :: "nat" assumes "v < length (g_V G)" shows "finite (succs v)"
lang_derivs: fixes s :: "'b list"   and r :: "'b rexp" shows "lang (derivs s r) = Derivs s (lang r)"
singleton_iff: fixes b :: "'f"   and a :: "'f" shows "(b \<in> {a}) = (b = a)"
inflstsI: fixes x :: "'b llist"   and A :: "'b set" assumes "x \<in> A\<^sup>\<infinity>"   and "x \<in> UNIV\<^sup>\<star> \<Longrightarrow> False" shows "x \<in> A\<^sup>\<omega>"
hc: fixes x :: "real" shows "c = insertion (nth_default 0 (xs' @ x # xs)) C"