hcju/isabelleps huggingface dataset:
sup_assoc_lgroup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup" shows "sup (sup a b) c = sup a (sup b c)"
r_child_lower_bound: assumes "l \<le> p" shows "p < r_child l p"
card_distinct: fixes xs :: "'b list" assumes "card (set xs) = length xs" shows "distinct xs"
src_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "src (src \<mu>) = src \<mu>"
sorted_Cons_le: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "ASSUMPTION (Sorted_Less.sorted (x # xs))"   and "y \<le> x" shows "Sorted_Less.sorted (y # xs)"
elementary_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and s :: "'b set" assumes "elementary slices W" shows "elementary.elementary slices W s \<equiv> stellar.quorum slices W s \<and> (\<forall>s'\<subset>s. \<not> stellar.quorum slices W s')"
elims_2: fixes x :: "'a list" assumes "local.strict_sorted x" obtains "xa__ = []"   | x :: "'a" and ys :: "'a list" where "xa__ = x # ys" and "Ball (set ys) (lt x) \<and> local.strict_sorted ys"
gbinomial_ge_n_over_k_pow_k: fixes k :: "nat"   and a :: "'a" assumes "of_nat k \<le> a" shows "(a / of_nat k) ^ k \<le> a gchoose k"
diff_zero: fixes plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and a :: "'q" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
stream_all2_SCons2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b stream"   and y :: "'c"   and ys :: "'c stream" shows "stream_all2 P xs (y ## ys) = (\<exists>z zs. xs = z ## zs \<and> P z y \<and> stream_all2 P zs ys)"
assms_3: shows "prv (neg (exi x \<phi>))"
not_eq_order_implies_strict: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" assumes "a \<noteq> b"   and "a \<unrhd> b" shows "a \<rhd> b"
farkas_coefficients_ns_unsat: fixes ns :: "'a ns_constraint set"   and C :: "(rat \<times> 'a ns_constraint) list" assumes "farkas_coefficients_ns ns C" shows "\<nexists>v. v \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  ns"
is_parser_def: fixes p :: "char list \<Rightarrow> char list + 'b \<times> char list" shows "is_parser p = (\<forall>s r x. p s = Inr (x, r) \<longrightarrow> length r \<le> length s)"
M_def: fixes k :: "nat" shows "M k = mat k k (\<lambda>(x, y). \<mu> x y)"
UP_car_memI': fixes g :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "\<And>x. g x \<in> carrier R"   and "\<And>x. k < x \<Longrightarrow> g x = \<zero>" shows "g \<in> carrier (UP R)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. \<lbrakk>x = y \<Longrightarrow> P (x # xs); x \<noteq> y \<Longrightarrow> P (y # xs)\<rbrakk> \<Longrightarrow> P (x # y # xs)" shows "P a0"
cases: fixes a1 :: "'state"   and a2 :: "(('act, 'pred) S_action, 'state) residual" assumes "a1 \<rightarrow>\<^sub>S a2" obtains P :: "'state" and \<alpha> :: "'act" and P' :: "'state" where "a1 = Pa__" and "a2 = \<langle>S_action.Act \<alpha>,P'\<rangle>" and "Pa__ \<rightarrow> \<langle>\<alpha>,P'\<rangle>"   | P :: "'state" and \<phi> :: "'pred" where "a1 = P" and "a2 = \<langle>S_action.Pred \<phi>,P\<rangle>" and "P \<turnstile> \<phi>"
filter_min_aux_relE: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "transp rel"   and "x \<in> set xs"   and "x \<notin> set (local.filter_min_aux xs ys)" obtains y :: "'a" where "y \<in> set (local.filter_min_aux xs ys)" and "rel y x"
assms: shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
map_SSup_def: fixes F :: "'a downset \<Rightarrow> 'b downset" shows "map_SSup F = SSup \<circ> F \<circ> ds"
fps_power_zeroth_eq_one: fixes a :: "'a fps"   and n :: "nat" assumes "fps_nth a 0 = (1::'a)" shows "fps_nth (a ^ n) 0 = (1::'a)"
iter_induct_isolate: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "iteration_finite_conjunctive (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_iteration.fiter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (d::'b::refinement_lattice) \<sqinter> infinite_iteration.infiter seq c = lfp (\<lambda>x::'b::refinement_lattice. d \<sqinter> seq c x)"
Z_i: fixes R :: "('c \<Rightarrow> real) set" assumes "R \<in> U" shows "Closure\<^sub>\<alpha> (Z \<inter> R)\<^bsub>r \<rightarrow> 0\<^esub> \<subseteq> Closure\<^sub>\<alpha> (Z \<inter> R)\<^bsub>r \<rightarrow> 0\<^esub>"
bit_set_bit_word_iff: fixes w :: "'b word"   and m :: "nat"   and b :: "bool"   and n :: "nat" shows "bit (set_bit_class.set_bit w m b) n = (if m = n then n < LENGTH('b) \<and> b else bit w n)"
lem: fixes z :: "'b" assumes "z \<in> cball (f x) (e / 2)" shows "\<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
simps_2: shows "\<bar>- \<infinity>\<bar> = \<infinity>"
P2: shows "to_fun f (to_fun g a) = b \<otimes> to_fun g a [^] n"
process_prop_1_def: fixes P :: "('e list \<times> 'e set) set \<times> 'e list set" shows "process_prop_1 P \<equiv> ([], {}) \<in> fst P"
inject_3: fixes x31 :: "'b trm"   and x32 :: "'b trm"   and y31 :: "'b trm"   and y32 :: "'b trm" shows "(x31 \<cdot> x32 = y31 \<cdot> y32) = (x31 = y31 \<and> x32 = y32)"
list_rw: shows "[0..<degree f + degree g] = [0..<degree g] @ [degree g..<degree f + degree g]"
span_neg: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> span S" shows "- x \<in> span S"
act1: fixes x :: "'a"   and y :: "'a"   and p :: "'b" shows "\<alpha> (x \<cdot> y) p = \<alpha> x (\<alpha> y p)"
refl_joinable_iff: fixes R :: "('b \<times> 'c) set" shows "(\<forall>x. \<exists>y. (x, y) \<in> R) = refl (joinable R)"
matrix_to_iarray_transpose: fixes A :: "(('d, 'f) vec, 'e) vec" shows "matrix_to_iarray (Finite_Cartesian_Product.transpose A) = transpose_iarray (matrix_to_iarray A)"
in_refl: fixes n :: "nat" shows "n \<^bold>\<in> Abs_nat_int {n}"
product_functor_axioms: shows "product_functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B\<^sub>1) (\<cdot>\<^sub>B\<^sub>2) F1 F2"
trms\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('a, 'b) stateful_strand_step list"   and B :: "('a, 'b) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t (A @ B) = trms\<^sub>s\<^sub>s\<^sub>t A \<union> trms\<^sub>s\<^sub>s\<^sub>t B"
one_plus_x_powr_Taylor2: fixes p :: "real" obtains k :: "real" where "\<And>x. \<bar>x\<bar> \<le> 1 / 2 \<Longrightarrow> \<bar>(1 + x) powr p - 1 - p * x\<bar> \<le> k * x\<^sup>2"
param: shows "(rel_spmf R ===> rel_spmf R) F G"
simps: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'a + 'c"   and a2 :: "'b + 'd" shows "rel_sum R1 R2 a1 a2 = ((\<exists>a c. a1 = Inl a \<and> a2 = Inl c \<and> R1 a c) \<or> (\<exists>b d. a1 = Inr b \<and> a2 = Inr d \<and> R2 b d))"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'b list" assumes "i < \<^bold>|xs\<^bold>|" shows "take (Suc i) xs = take i xs \<cdot> [xs ! i]"
less_le_trans: fixes x :: "complex"   and y :: "complex"   and z :: "complex" assumes "x <\<^sub>\<complex> y"   and "y \<le>\<^sub>\<complex> z" shows "x <\<^sub>\<complex> z"
split_scaleR_pos_le: fixes a :: "real"   and b :: "'b" assumes "0 \<le> a \<and> (0::'b) \<le> b \<or> a \<le> 0 \<and> b \<le> (0::'b)" shows "(0::'b) \<le> a *\<^sub>R b"
gen_length_code_2: fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "List.gen_length n (x # xs) = List.gen_length (Suc n) xs"
assms_7: shows "path_image g \<subseteq> s - pts"
injectivity: fixes x :: "'a"   and d :: "nat"   and y :: "'a" assumes "Polynomial.monom x d = Polynomial.monom y d" shows "x = y"
dghm_ArrMap_app_in_HomCod_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<phi> i\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>"
dividesI: fixes c :: "'c"   and G :: "('c, 'd) monoid_scheme"   and b :: "'c"   and a :: "'c" assumes "c \<in> carrier G"   and "b = a \<otimes>\<^bsub>G\<^esub> c" shows "a divides\<^bsub>G\<^esub> b"
measure_eq_0I: fixes A :: "'a set"   and M :: "'a measure" assumes "A = {}" shows "Sigma_Algebra.measure M A = 0"
thm_cont_prop_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[NonContingent F \<^bold>\<equiv> NonContingent (F\<^sup>-) in v]"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "I.int \<xi> (Var x) = \<xi> x"
powsum_split_var1: fixes x :: "'a"   and n :: "nat" shows "x\<^bsub>0\<^esub>\<^bsup>n + 1\<^esup> = (1::'a) + x\<^bsub>1\<^esub>\<^bsup>n\<^esup>"
z_min: fixes z' :: "'a" assumes "z' \<in> set R_pre" shows "\<not> Q.hitting_paths z'"
UP_m_assoc: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
obind_return: fixes m :: "'c \<Rightarrow> 'd option" shows "m |>> oreturn = m"
PBD_by_adjoining_point: fixes x :: "'a" assumes "x \<notin> \<V>"   and "\<And>k. k \<in> {\<k>} \<Longrightarrow> 2 \<le> k" shows "PBD (add_point x) (\<B> + mset_set {insert x g |g. g \<in> \<G>}) ({\<k>} \<union> {int (card g + 1) |g. g \<in> \<G>})"
OrR_sim: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "\<^bold>\<not> F \<triangleright> \<^bold>\<not> G \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<^bold>\<not> (G \<^bold>\<or> F) \<triangleright> \<Gamma> \<turnstile> \<bottom>"
lunit'_in_vhom: fixes f :: "'d"   and b :: "'d" assumes "D.ide f"   and "trg\<^sub>D f = b" shows "\<guillemotleft>\<l>\<^sub>D\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>D b \<star>\<^sub>D f\<guillemotright>"
mat_of_cblinfun_vector_to_cblinfun: fixes \<psi> :: "'b" shows "mat_of_cblinfun (vector_to_cblinfun \<psi>) = mat_of_cols (length canonical_basis) [vec_of_basis_enum \<psi>]"
map_runs_def: fixes h :: "role_t \<Rightarrow> atom list \<Rightarrow> atom list"   and runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and rid :: "fid_t" shows "map_runs h runz rid \<equiv> case runz rid of None \<Rightarrow> None | Some (rol, agts, al) \<Rightarrow> Some (rol, agts, h rol al)"
rt_fresher_def: shows "rt_fresher \<equiv> \<lambda>dip rt rt'. the (rt dip) \<sqsubseteq> the (rt' dip)"
WhileFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> (While b c # cs, css, Normal s) \<rightarrow> (cs, css, Normal s)"
part2: shows "\<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = 2 \<longrightarrow> f X < s"
non_empty_rbt_keys: fixes t :: "('a, 'b) RBT_Impl.rbt" assumes "t \<noteq> rbt.Empty" shows "RBT_Impl.keys t \<noteq> []"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>tyvrs. P (Tvar tyvrs)"   and "P Top"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)"   and "\<And>tyvrs ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (\<forall>tyvrs<:ty2. ty1)" shows "P ty"
rep_eq: fixes x :: "'a dual set" shows "undual (Sup x) = Inf (undual ` x)"
same_prefix_nil: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (xs @ ys) xs = (ys = [])"
prodM_carr: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'a, 'h) Module_scheme" assumes "\<forall>i\<in>I. R module M i" shows "carrier (m\<Pi>\<^bsub>R I\<^esub> M) = carr_prodag I M"
supp_fun_app_eqvt: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "eqvt f" shows "supp (f x) \<subseteq> supp x"
fst_swap: fixes x :: "'c \<times> 'b" shows "hamlet (prod.swap x) = time x"
ground_Fvars: fixes t :: "tm" shows "ground t = (Fvars t = {})"
simps_10: fixes f :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "map_literal f (Neg x2) = Neg (f x2)"
cinner_diff_left: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "cinner (x - y) z = cinner x z - cinner y z"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a0 :: "'b"   and a1 :: "'b" assumes "partial_magma.ide C a0 \<and> partial_magma.ide C a1" shows "binary_product_diagram_axioms C a0 a1"
map_cong0: fixes x :: "'a fset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>z. z \<in> fset x \<Longrightarrow> f z = g z" shows "f |`| x = g |`| x"
elims: fixes x :: "'b rexp"   and xa :: "'b trm"   and y :: "'b trm" assumes "Append_rexp x xa = y" obtains r :: "'b rexp" and rexp :: "'b rexp" where "x = r__" and "xa = Lam rexp__" and "y = Lam (Times rexp__ r__)"   | r :: "'b rexp" and X :: "'b list set" and rexp :: "'b rexp" where "x = r" and "xa = Trn X rexp" and "y = Trn X (Times rexp r)"
is_shadow_root_ptr_kind_obtains: fixes document_ptr :: "(_) document_ptr" assumes "is_shadow_root_ptr_kind\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr" obtains shadow_root_ptr :: "(_) shadow_root_ptr" where "document_ptr = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr"
l13: fixes p :: "'node"   and Q :: "'node set"   and p' :: "'node" assumes "quorum_of p Q"   and "p \<in> W"   and "reachable_through p Q p'" shows "quorum_of p' Q"
prefix_refl: fixes xs :: "'a list" shows "xs \<le> xs"
simps_1: fixes \<xi> :: "nat \<Rightarrow> 'a set" shows "\<xi> \<Turnstile>\<^sub>n true\<^sub>n = True"
measurable_iff_sets: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" shows "(f \<in> M \<rightarrow>\<^sub>M N) = (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"
cycle_decomp: fixes u :: "'a list" assumes "u \<noteq> []" shows "cycle u = u @- cycle u"
abs_eq: fixes xa :: "'b"   and x :: "'b multiset \<times> 'b multiset" shows "add_zmset xa (abs_zmultiset x) = abs_zmultiset (case x of (Mp, Mn) \<Rightarrow> (add_mset xa Mp, Mn))"
sat_def: fixes e :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "Bexp.sat e = (\<exists>\<sigma>. e \<sigma>)"
assms_5: shows "range_vars \<theta>\<^sub>1 \<inter> bvars\<^sub>s\<^sub>t S\<^sub>1 = {}"
\<N>covered: fixes C :: "'a set" assumes "C \<in> \<N>" shows "\<exists>D. D \<in> \<M> \<and> C \<subseteq> D"
alphaAbs_qSwapAbs_preserves_qGoodAbs2: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "A $[[y \<and> x]]_zs $= A' $[[y' \<and> x']]_zs'"   and "qGoodAbs A'" shows "qGoodAbs A"
all_trie_eq_ran: fixes t :: "('a, 'b) trie"   and P :: "'b \<Rightarrow> bool" assumes "invar_trie t" shows "all_trie P t = (\<forall>x\<in>ran (lookup_trie t). P x)"
exhaust: fixes y :: "'b llist" obtains "y = LNil"   | x21 :: "'b" and x22 :: "'b llist" where "y = LCons x21 x22"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool" shows "pred_cstate P Void"
oneDr: fixes x :: "'a" shows "D (1::'a) x"
sigma_finite_measure_axioms: shows "sigma_finite_measure (Pi\<^sub>M I M)"
ae2: shows "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
coprime_2_n: fixes A :: "'a set"   and B :: "'a set" assumes "ideal R A"   and "ideal R B" shows "R /\<^sub>r A \<Oplus>\<^sub>r R /\<^sub>r B = r\<Pi>\<^bsub>{j. j \<le> Suc 0}\<^esub> prodB1 (R /\<^sub>r A) (R /\<^sub>r B)"
dvd_mult_unit_iff': fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b dvd (1::'b)" shows "(a dvd b * c) = (a dvd c)"
resize_vec_index: fixes i :: "nat"   and n :: "nat"   and v :: "'c vec" assumes "i < n" shows "resize_vec n v $ i = v $ i"
empty_correct_1: shows "BinomialHeapStruc.invar []"
finite_borel_measure_M: fixes M :: "real measure" assumes "real_distribution M" shows "finite_borel_measure M"
strategyproofness_2: shows "pmf (sds R36) b + pmf (sds R36) a \<le> pmf (sds R10) b + pmf (sds R10) a"
OclIncludes_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "x->includes\<^sub>S\<^sub>e\<^sub>t(y) = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then \<lfloor>\<lfloor>y \<tau> \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil>\<rfloor>\<rfloor> else \<bottom>)"
flat_def: fixes xqq :: "'a lazy_sequence lazy_sequence" shows "flat xqq = lazy_sequence_of_list (concat (list.map list_of_lazy_sequence (list_of_lazy_sequence xqq)))"
zmult_div_aux1: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "(a - a mod b) div b = a div b"
Right_Cosets_cardinality: fixes H :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c" assumes "subgroup_of_group H G composition unit"   and "x \<in> G"   and "y \<in> G" shows "card (coset_notation.Right_Coset composition H x) = card (coset_notation.Right_Coset composition H y)"
one_j: shows "\<forall>j. j < length l' \<and> l' ! j = m \<longrightarrow> j = length l' - 1"
phi_7: shows "\<forall>x\<in>{0..1}. finite (\<phi> -` {x})"
jj: shows "jj \<in> UNlift01 cl dl P F II"
sim_def: fixes c :: "'a list option"   and q :: "'b" shows "sim c q \<equiv> Pause q Done \<bind> (\<lambda>lo. case lo of None \<Rightarrow> Done (None, c) | Some n \<Rightarrow> if c = None then lift_spmf (spmf_of_set (nlists UNIV n)) \<bind> (\<lambda>x. Done (Some x, Some x)) else Done (c, c))"
Ng_def: shows "Ng = (\<lambda>y. norm (g y - g x - g' (y - x)) / norm (y - x))"
vsubset_iff: fixes A :: "V"   and B :: "V" shows "(A \<subseteq>\<^sub>\<circ> B) = (\<forall>t. t \<in>\<^sub>\<circ> A \<longrightarrow> t \<in>\<^sub>\<circ> B)"
strict_implies_not_eq: fixes a :: "'b list"   and b :: "'b list" assumes "strict_prefix a b" shows "a \<noteq> b"
retraction_maps_section_image2: fixes X :: "'c topology"   and Y :: "'d topology"   and r :: "'c \<Rightarrow> 'd"   and s :: "'d \<Rightarrow> 'c" assumes "retraction_maps X Y r s" shows "subtopology X (s ` topspace Y) homeomorphic_space Y"
subspace: shows "subspace class_ring (local.span (set (cols (A + B)))) (vs (subspace_sum W1 W2))"
spanU: shows "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
prime_elem_int_abs_iff: fixes p :: "int" shows "prime_elem \<bar>p\<bar> = prime_elem p"
Ipol_closed: fixes l :: "'a list"   and p :: "pol" assumes "in_carrier l" shows "Ipol l p \<in> carrier R"
rep_nat_pp_inj: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'"   and "rep_nat_pp x = rep_nat_pp y" shows "x = y"
iTILL_inext: fixes t :: "nat"   and n :: "nat" assumes "t < n" shows "inext t [\<dots>n] = Suc t"
r_one: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
NotNotNNF: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F, \<Gamma> \<Rightarrow>\<^sub>n" shows "\<^bold>\<not> (\<^bold>\<not> F), \<Gamma> \<Rightarrow>\<^sub>n"
CSP_is_CSP2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P" shows "P is CSP2 healthy"
matching_symmetric: fixes sigma :: "Runs.var \<Rightarrow> msg option"   and sigma' :: "Runs.var \<Rightarrow> msg option" assumes "matching sigma sigma'" shows "matching sigma' sigma"
1: shows "kauff_mat (basic [] \<circ> ws) = rat_poly.matrix_mult (kauff_mat (basic [])) (kauff_mat ws)"
h2c: shows "(Av\<^sup>2 = (- Bv * sqrt Cv)\<^sup>2) = (Av\<^sup>2 = Bv\<^sup>2 * (sqrt Cv)\<^sup>2)"
assms: shows "\<exists>k. k \<noteq> 0 \<and> k \<le> n \<and> c k \<noteq> (0::'a)"
backward_lockstep_simulation: fixes s2 :: "('fenv_ubx, 'henv, ('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame) state"   and s2' :: "('fenv_ubx, 'henv, ('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame) state"   and s1 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state" assumes "Subx.step s2 s2'"   and "s1 \<sim> s2" shows "\<exists>s1'. Sinca.step s1 s1' \<and> s1' \<sim> s2'"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>a b. \<parallel>a *\<^sub>v b\<parallel> \<le> \<parallel>a\<parallel> * \<parallel>b\<parallel> * K"
IH1_1: shows "factor_node_info (product_factor_tree p l) = Mp (\<Prod>\<^sub># (factors_of_factor_tree (product_factor_tree p l)))"
orthogonal_complement_zero_intersection: fixes M :: "'b set" assumes "(0::'b) \<in> M" shows "M \<inter> orthogonal_complement M = {0::'b}"
Inf_apply: fixes A :: "('c \<Rightarrow> 'b) set"   and x :: "'c" shows "(\<Sqinter> A) x = (\<Sqinter>f\<in>A. f x)"
meet_idem: fixes x :: "'c" shows "x \<sqinter> x = x"
gsi: fixes i :: "nat" assumes "i \<le> k" shows "gs ! i \<in> carrier_vec n"
TryRedsFail: fixes e\<^sub>1 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and 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 D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<rightarrow>* \<langle>Throw a,(h, l)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Throw a,(h, l)\<rangle>"
disj_commute: fixes x :: "'a"   and y :: "'a" shows "x \<^bold>\<squnion> y = y \<^bold>\<squnion> x"
hyps_4: fixes b :: "(name \<times> lam) list" assumes "b closes (a, \<sigma>) # \<Gamma>" shows "b<t> \<in> RED \<tau>"
abs_fun_eq3: fixes a :: "'c"   and b :: "'c"   and x :: "'b"   and y :: "'b" assumes "pt TYPE('b) TYPE('c)"   and "at TYPE('c)"   and "a \<noteq> b"   and "x = [(a, b)] \<bullet> y"   and "a \<sharp> y" shows "[a].x = [b].y"
trans_def: fixes r :: "('b \<times> 'b) set" shows "trans r = (\<forall>x y z. x \<^bsub>r\<^esub>\<preceq> y \<longrightarrow> y \<^bsub>r\<^esub>\<preceq> z \<longrightarrow> x \<^bsub>r\<^esub>\<preceq> z)"
simps_5: fixes x :: "'b trm"   and y :: "'b trm"   and "next" :: "indices list"   and S :: "'b trm" shows "subterm (x \<cdot> y) (indices.Right # next) S = subterm y next S"