hcju/isabelleps huggingface dataset:
3: assumes "\<not> \<delta> \<le> real_of_int (snd p\<^sub>0 - snd p)" shows "t_find_closest p (min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (p\<^sub>2 # ps) \<le> 1 + length (filter (\<lambda>q. real_of_int (snd q - snd p) \<le> min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (p\<^sub>2 # ps))"
pseudo_rationalizable_on_path_independent_on: fixes A :: "'b set"   and Rs :: "('b \<times> 'b) set set"   and f :: "'b set \<Rightarrow> 'b set" assumes "pseudo_rationalizable_on A Rs f" shows "path_independent_on A f"
assms: shows "filter ((\<noteq>) a) l \<noteq> takeWhile ((\<noteq>) a) l"
coboundedI2: fixes b :: "'e multiset"   and c :: "'e multiset"   and a :: "'e multiset" assumes "b \<subseteq># c" shows "a \<inter># b \<subseteq># c"
cinner_basis_enum_of_vec: fixes x :: "complex vec"   and y :: "complex vec" assumes "dim_vec x = length canonical_basis"   and "dim_vec y = length canonical_basis" shows "basis_enum_of_vec x \<bullet>\<^sub>C basis_enum_of_vec y = y \<bullet>c x"
assms_4: shows "moebius_pt M1 z1 = w1"
pfun_override_dist_comp: fixes f :: "('c, 'b) ffun"   and g :: "('c, 'b) ffun"   and h :: "('a, 'c) ffun" shows "f + g \<circ>\<^sub>f h = (f \<circ>\<^sub>f h) + (g \<circ>\<^sub>f h)"
q: shows "2 * (x div 2 div y) = x div y - x div y mod 2"
fmrel_filter: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap"   and Q :: "'e \<Rightarrow> bool" assumes "fmrel P m n" shows "fmrel P (fmfilter Q m) (fmfilter Q n)"
plus_top: fixes x :: "'a" shows "x\<^sup>+ ; 1 = x ; 1"
adds_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a" assumes "class.ninv_comm_monoid_add plus zero" shows "plus.adds plus s zero = (s = zero)"
even_sum_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "finite A" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (comm_monoid_add.sum plus zero f A) = even (card {a \<in> A. \<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (f a)})"
ebounds_def: fixes as :: "atom list" shows "ebounds as = concat (map (\<lambda>x. case x of Less nat1 nat2 \<Rightarrow> [] | Eq 0 nat2 \<Rightarrow> [] | Eq (Suc i) 0 \<Rightarrow> [i] | Eq (Suc i) (Suc nata) \<Rightarrow> []) as) @ concat (map (\<lambda>x. case x of Less nat1 nat2 \<Rightarrow> [] | Eq 0 0 \<Rightarrow> [] | Eq 0 (Suc i) \<Rightarrow> [i] | Eq (Suc nat) nat2 \<Rightarrow> []) as)"
MakeCatId: fixes X :: "'d"   and C :: "('d, 'e, 'f) Category_scheme" assumes "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X = id\<^bsub>MakeCat C\<^esub> X"
subst_drop: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "name"   and t :: "'a"   and env :: "(name, 'a) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "x |\<notin>| frees t" shows "subst t (fmdrop x env) = subst t env"
or_bet_out: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "Bet A B C \<or> B Out A C \<or> \<not> Col A B C"
advantage_def: fixes \<A> :: "('a \<Rightarrow> (('a \<times> 'a) \<times> 'c) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "local.ind_cpa.advantage \<A> = \<bar>spmf (local.ind_cpa.ind_cpa \<A>) True - 1 / 2\<bar>"
rInput: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and K :: "'a"   and Tvec :: "'a list" assumes "cP = M\<lparr>\<lambda>*xvec N\<rparr>.P"   and "cRs = K\<lparr>N[xvec::=Tvec]\<rparr> \<prec> P[xvec::=Tvec]"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "xvec \<sharp>* Tvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop"
invertible: shows "\<exists>F. inverse_functors (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G"
Union_Int_subset: fixes A :: "'c set set"   and B :: "'c set set" shows "\<Union> (A \<inter> B) \<subseteq> \<Union> A \<inter> \<Union> B"
assms_1: shows "(red F)\<^sup>*\<^sup>* p q"
assms: shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
mon_pl_unconc: fixes a :: "('d set \<times> 'd set) list"   and b :: "('d set \<times> 'd set) list" shows "mon_pl (a @ b) = mon_pl a \<union> mon_pl b"
i_Exec_Stream_Acc_LocalState_nth_f_nth: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and n :: "nat" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_LocalState k output_fun trans_fun input c n = f_Exec_Comp_Stream_Acc_LocalState k output_fun trans_fun (input \<Down> Suc n) c ! n"
sumUnionDisjoint3: fixes X :: "'c set set"   and XX :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>X. finite x"   and "X partitions XX" shows "sum f XX = sum (sum f) X"
C6: fixes x :: "'a"   and y :: "'a" shows "-r x l\<rightarrow> y = -l y r\<rightarrow> x"
in_outgoing: fixes s1 :: "'a"   and s2 :: "'b"   and t :: "'c"   and e :: "(('a \<times> 'b) \<times> 'c) fset"   and s :: "'a" shows "(((s1, s2), t) |\<in>| outgoing_transitions e s) = (((s1, s2), t) |\<in>| e \<and> s1 = s)"
that: fixes a :: "'x \<Rightarrow> 'a" assumes "ideal F = ideal ((\<lambda>x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0) ` X)" shows "thesis"
bin_sc_sc_same: fixes n :: "nat"   and c :: "bool"   and b :: "bool"   and w :: "int" shows "bin_sc n c (bin_sc n b w) = bin_sc n c w"
rm_vars_apply: fixes v :: "'a"   and V :: "'a set"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<in> subst_domain (rm_vars V s)" shows "rm_vars V s v = s v"
RECT_gfp_def: fixes B :: "('f \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> 'e"   and x :: "'f" shows "REC\<^sub>T B x = (if trimono B then gfp B x else top)"
strict_part_mono_singleton: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "strict_part_mono {x} f"
boundary_of_region_is_subset_of_partition_boundaries_1: shows "\<gamma> \<subseteq> two_chain_boundary two_chain_typeI"
center_on_geodesic_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "center x y z \<in> {x--z}"
vsubstEnv_substEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var" assumes "goodEnv rho"   and "good Y"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "rho &[Y / y]_ys &[z1 // z]_zs = rho &[z1 // z]_zs &[Y #[z1 // z]_zs / y]_ys"
strip_eq_Seq: fixes C :: "'a acom"   and c1 :: "com"   and c2 :: "com" shows "(strip C = c1;; c2) = (\<exists>C1 C2. C = C1;; C2 \<and> strip C1 = c1 \<and> strip C2 = c2)"
obj_trg: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.obj (S.trg \<mu>)"
lesspoll_imp_Card_less: fixes a :: "V"   and b :: "V" assumes "elts a \<prec> elts b" shows "vcard a < vcard b"
max2_def: fixes a :: "'b"   and b :: "'b" shows "s.max2 a b \<equiv> if (a, b) \<in> s then b else a"
assms_4: shows "pmdl (set gs) = pmdl (set (init_syzygy_list bs))"
appr_rel_br: shows "appr_rel = br (\<lambda>xs. eucl_of_list ` set_of_appr xs) (\<lambda>xs. length xs = DIM('c))"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "poly_mapping.lookup (x + xa) = (\<lambda>k. poly_mapping.lookup x k + poly_mapping.lookup xa k)"
D: shows "\<not> (\<exists>v\<in>sinks I D u ys. (v, u') \<in> I)"
content_ball_pos: fixes r :: "real"   and c :: "'a" assumes "0 < r" shows "0 < content (ball c r)"
csum_czero1: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r +c czero =o r"
mk_BaseDim_def: fixes d :: "'c" shows "mk_BaseDim d = dim_lambda (\<lambda>i. if i = d then 1 else 0)"
range_inter: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a..b} \<inter> {c..d} = {}) = (\<forall>x. \<not> (a \<le> x \<and> x \<le> b \<and> c \<le> x \<and> x \<le> d))"
enforce_option_K_True: fixes x :: "'c option" shows "enforce_option (\<lambda>_. True) x = x"
openin_INT2: fixes I :: "'a set"   and T :: "'b topology"   and U :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)" shows "openin T (\<Inter> (U ` I))"
square_free_normalize: fixes p :: "'a poly" assumes "square_free p" shows "square_free (normalize p)"
skip: fixes P :: "'b set" shows "\<turnstile> P SKIP P"
multiplicity_lessI: fixes x :: "'a"   and p :: "'a"   and n :: "nat" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p"   and "\<not> p ^ n dvd x" shows "multiplicity p x < n"
cases: fixes a1 :: "('a \<times> 'b) list"   and a2 :: "'b \<Rightarrow> 'a set" assumes "crdt_ops a1 a2" obtains deps :: "'b \<Rightarrow> 'a set" where "a1 = []" and "a2 = deps__"   | xs :: "('a \<times> 'b) list" and deps :: "'b \<Rightarrow> 'a set" and oid :: "'a" and oper :: "'b" where "a1 = xs @ [(oid, oper)]" and "a2 = deps" and "crdt_ops xs deps" and "oid \<notin> set (map fst xs)" and "\<forall>ref\<in>deps oper. ref \<in> set (map fst xs) \<and> ref < oid"
Inr_inject: fixes x :: "'e"   and y :: "'e" assumes "Inr x = Inr y" shows "x = y"
add_commute: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> b = b \<sqinter> a"
lookup_def: fixes r :: "'a ref" shows "!r = tap (\<lambda>h. Ref.get h r)"
T: shows "T \<in> N \<rightarrow>\<^sub>M P"
assms: shows "I \<subseteq> \<Union> (f ` A)"
update_convs_2: fixes p_oiface' :: "char list \<Rightarrow> char list"   and p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_oiface_update p_oiface' \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = \<lparr>p_iiface = p_iiface, p_oiface = p_oiface' p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr>"
pred_mem: fixes x :: "hf" assumes "Ord x"   and "x \<noteq> 0" shows "pred x \<^bold>\<in> x"
set_inorder: fixes t :: "('b \<times> 'c) tree" shows "set (Tree2.inorder t) = Tree2.set_tree t"
add_mset_not_empty: fixes a :: "'a"   and A :: "'a multiset" shows "add_mset a A \<noteq> {#}"
fps_one_nth_Nil: shows "1 $ [] = (1::'c)"
simps: fixes G :: "prog"   and a1 :: "qtname \<times> memberdecl"   and a2 :: "qtname" shows "G \<turnstile> a1 member_of a2 = ((\<exists>m C. a1 = m \<and> a2 = C \<and> G\<turnstile> mbr m declared_in C \<and> declclass m = C) \<or> (\<exists>m C S. a1 = m \<and> a2 = C \<and> G \<turnstile> m inheritable_in pid C \<and> G\<turnstile> memberid m undeclared_in C \<and> G\<turnstile>C\<prec>\<^sub>C1S \<and> G \<turnstile> Class S accessible_in pid C \<and> G \<turnstile> m member_of S))"
w2_def: shows "w2 = Abs_freelist (rev (pair_relator_halflist t s) @ pair_relator_halflist t s)"
long_division_theorem: fixes K :: "'a set"   and p :: "'a list"   and b :: "'a list" assumes "subring K R"   and "polynomial K p"   and "polynomial K b"   and "b \<noteq> []"   and "lead_coeff b \<in> Units (R\<lparr>carrier := K\<rparr>)" shows "\<exists>q r. polynomial K q \<and> polynomial K r \<and> p = b \<otimes>\<^bsub>K [X]\<^esub> q \<oplus>\<^bsub>K [X]\<^esub> r \<and> (r = [] \<or> degree r < degree b)"
M10: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ F \<and> Unchanged v \<longrightarrow> \<circle>F" shows "|~ F \<and> \<circle>\<box>F \<longrightarrow> \<box>F"
take_Cons': fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs I) = (rev_I a \<noteq> a)"
summable_Cauchy: fixes f :: "nat \<Rightarrow> 'b" shows "summable f = (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e)"
pole_mk_moebius: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and z :: "complex_homo" assumes "is_pole (mk_moebius a b c d) z"   and "c \<noteq> 0"   and "a * d - b * c \<noteq> 0" shows "z = of_complex (- d / c)"
l10_8: fixes P :: "'p"   and A :: "'p"   and B :: "'p" assumes "P P Reflect A B" shows "Col P A B"
ZassenhausTr5: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and H1 :: "'e set"   and K :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<natural>\<^bsub>G\<^esub>K \<triangleright> K1" shows "(\<natural>\<^bsub>G\<^esub>H \<inter> K / (H1 \<inter> K \<diamondop>\<^bsub>G\<^esub> H \<inter> K1)) \<cong> (\<natural>\<^bsub>G\<^esub>H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K / (H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K1))"
Quotient_rel_abs: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and r :: "'j"   and s :: "'j" assumes "Quotient R Abs Rep T"   and "R r s" shows "Abs r = Abs s"
word_arith_wis_2: fixes a :: "'b word"   and b :: "'b word" shows "a - b = word_of_int (uint a - uint b)"
closure_of_mono: fixes S :: "'c set"   and T :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> T" shows "X closure_of S \<subseteq> X closure_of T"
of_int_eq_numeral_iff: fixes z :: "int"   and n :: "num" shows "(of_int z = numeral n) = (z = numeral n)"
alpha_def: shows "alpha = PZ.prob {x. a0 < \<bar>x\<bar>} / 4"
positive_rot_to_x_axis: fixes z :: "complex" assumes "z \<noteq> 0" shows "0 < Re (cis (- arg z) * z)"
snd_splitAt_upt: fixes j :: "nat"   and i :: "nat"   and k :: "nat" assumes "j \<le> i"   and "i < k" shows "snd (splitAt i [j..<k]) = [i + 1..<k]"
seq_tail1_3: fixes s :: "'a"   and x :: "'b \<times> 'a"   and xs :: "('b \<times> 'a) list"   and lars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, x # xs) \<in> seq lars" shows "lst (s, x # xs) = lst (snd x, xs)"
mono_on_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "Basis.mono_on A f = (\<forall>x\<in>A. \<forall>y\<in>A. x \<le> y \<longrightarrow> f x \<le> f y)"
graph_restrict_simps_4: fixes G :: "('b, 'a) graph_rec_scheme"   and R :: "'b set" shows "graph_rec.more (graph_restrict G R) = graph_rec.more G"
ht_height: fixes t :: "('b \<times> nat) tree" assumes "avl t" shows "ht t = height t"
assms: shows "((s, c), e, s', c') \<in> ntrs fg"
UnionP_graph_aux_def: shows "UnionP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>UnionP_aux i x y z. x1 = (x, y, z) \<and> x2 = SyntaxN.All i (Var i IN z IFF Var i IN x OR Var i IN y) \<and> atom i \<sharp> (x, y, z))"
divisor_count_upper_bound': fixes n :: "nat" shows "real (divisor_count n) \<le> 2 * sqrt (real n)"
lin_dep_iff_in_span: fixes S :: "'a vec set"   and v :: "'a vec" assumes "S \<subseteq> carrier_vec n"   and "lin_indpt S"   and "v \<in> carrier_vec n"   and "v \<notin> S" shows "(v \<in> local.span S) = lin_dep (S \<union> {v})"
comm_monoid_mult_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_monoid_mult_isom hom \<equiv> monoid_mult_isom hom"
coeff_mpoly_times: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (p * q) mon = prod_fun (MPoly_Type.coeff p) (MPoly_Type.coeff q) mon"
UN_simps_10: fixes B :: "'e \<Rightarrow> 'd set"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "\<Union> (B ` f ` A) = (\<Union>a\<in>A. B (f a))"
arbitrary_intersection_of_Int: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "(arbitrary intersection_of P) S"   and "(arbitrary intersection_of P) T" shows "(arbitrary intersection_of P) (S \<inter> T)"
po_antisym: fixes x :: "'a"   and y :: "'a" assumes "x << y"   and "y << x" shows "x = y"
\<psi>_def: shows "F'.\<psi> \<equiv> THE \<psi>. \<guillemotleft>\<psi> : D'.unity \<rightarrow>\<^sub>D F C'.unity\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = F'.\<iota>\<^sub>1 \<cdot>\<^sub>D D'.T (\<psi>, \<psi>)"
pmf_set_integral_code_alt_finite: fixes A :: "'e set"   and p :: "'e pmf"   and f :: "'e \<Rightarrow> real" assumes "finite A" shows "pmf_set_integral p f A = (\<Sum>x\<in>A. pmf p x * f x)"
comp_arr_dom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a" assumes "VoV.arr f"   and "VoV.dom f = a" shows "VoV.comp f a = f"
cardSuc_greater: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "r <o cardSuc r"
path_components_of_subset_singleton: fixes X :: "'b topology"   and S :: "'b set" shows "(path_components_of X \<subseteq> {S}) = (path_connected_space X \<and> (topspace X = {} \<or> topspace X = S))"
subset_emptyI: fixes A :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> False" shows "A \<subseteq> {}"
delta: fixes S :: "'d set"   and a :: "'d"   and b :: "'d \<Rightarrow> 'e" assumes "finite S" shows "(\<Sum>k\<in>S. if k = a then b k else (0::'e)) = (if a \<in> S then b a else (0::'e))"
sum_zero_representation: fixes v :: "'a" assumes "\<And>b. b \<in> B \<Longrightarrow> representation B v b = 0"   and "v \<in> S" shows "v = (0::'a)"
value_subst_value: fixes e :: "Syntax.term"   and e' :: "Syntax.term"   and x :: "Syntax.var" assumes "value e"   and "value e'" shows "value (e[e' / x])"
Re_rcis: fixes r :: "real"   and a :: "real" shows "Re (rcis r a) = r * cos a"
field_iff_prime: fixes a :: "'a list" assumes "a \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}" shows "field (K [X] Quot PIdl\<^bsub>K [X]\<^esub> a) = pprime K a"
simps_2: fixes x :: "'e"   and xs :: "'e list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
ncoplanar_perm_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B D C"
unbind_lambda_lambda1: fixes x :: "name" shows "Lam [x].Lam [x].Var x \<mapsto> [x, x],Var x"
fd_def: fixes x :: "'a"   and y :: "'a" shows "domain_semiringl.fd (\<lambda>x y. y \<cdot> x) range_op x y = range_op (y \<cdot> x)"
Hash_Tag: fixes X :: "msg"   and X' :: "tag" shows "Hash X \<noteq> Tag X'"
pt_ex_eqvt: fixes pi :: "('b \<times> 'b) list"   and P :: "'a \<Rightarrow> bool" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> (\<exists>x. P x) = (\<exists>x. pi \<bullet> P (rev pi \<bullet> x))"
UN_singleton: fixes A :: "'a set" shows "(\<Union>x\<in>A. {x}) = A"
bis_raw_coind: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and theta :: "('a \<times> 'a) set" assumes "Bisim.mono Retr"   and "sym theta"   and "theta \<subseteq> Retr theta" shows "theta \<subseteq> bis Retr"
lsc_hull_of_convex_agrees: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "\<forall>x. f x = lsc_hull f x \<or> x \<in> rel_frontier (domain f)"
sat_def: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Bexp.sat e = (\<exists>\<sigma>. e \<sigma>)"
rep_eq: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and x :: "'b"   and xa :: "('e, 'c, 'd) formula" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid satisfies transition x xa = nominal_ts.valid_Tree\<^sub>\<alpha> satisfies transition x (Rep_formula xa)"
disc_51: fixes x61 :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and x62 :: "'a expr list" shows "is_Operation (Operation x61 x62)"
simps_2: fixes a :: "'c \<times> ('d, 'e) fmap"   and as :: "('c \<times> ('d, 'e) fmap) list" shows "no_effectless_act (a # as) = (fmdom' (snd a) \<noteq> {} \<and> no_effectless_act as)"
zero_divisor_def: fixes a :: "'b" shows "zero_divisor a \<equiv> \<exists>b. b \<noteq> (0::'b) \<and> a * b = (0::'b)"
that: fixes a :: "'a"   and b :: "'a" assumes "q = Fraction_Field.Fract a b"   and "b \<noteq> (0::'a)"   and "coprime a b" shows "thesis"
supp_sum_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.supp_sum plus zero f S = comm_monoid_add.sum plus zero f (monoid_add.support_on zero S f)"
chain_lemma: fixes S :: "nat \<Rightarrow> 'c"   and i :: "nat"   and k :: "nat" assumes "chain S" shows "S i \<sqsubseteq> S k \<or> S k \<sqsubseteq> S i"
IH: assumes "x \<in> set ys" shows "\<exists>pre suf. ys = pre @ x # suf \<and> (\<forall>y\<in>set pre. x \<noteq> y)"
poly_ge_refl: fixes p :: "('c monom \<times> 'd) list" shows "p \<ge>p p"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
equals_dom_if_value_is_ide: assumes "C.ide a" shows "A.map = A.F.map"
bv_to_nat_zero_prepend: fixes a :: "bit list" shows "bv_to_nat a = bv_to_nat (\<zero> # a)"
a_fresh_def: fixes antecedent :: "('b, 'a) antecedent" shows "a_fresh antecedent = (case antecedent of Antecedent x1 x2 x3 \<Rightarrow> x3)"
l_cancellation: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice" assumes "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "(x::'b::refinement_lattice) \<sqsubseteq> ((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> lower_galois_connections.l_adjoint F) x"
ABind_def: fixes v :: "Vars.var"   and e :: "exp" shows "ABind v e = (\<Lambda> ae. \<A>\<^sup>\<bottom>\<^bsub>ae v\<^esub> e)"
p: shows "Ipoly xs (map_poly real_of_float p) = Ipoly xs (map_poly real_of_float l) + Ipoly xs (map_poly real_of_float r)"
assms_5: shows "Lfilled i lholed (vs @ [$Br i]) LI"
notx: shows "\<forall>w. P w (\<midarrow> x) = (\<not> O w x)"
mem_Collect_eq: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "s2p (Collect P) a = P a"
nonzero_append: fixes xs :: "'a list"   and ys :: "'a list" shows "nonzero (xs @ ys) = (nonzero xs \<or> nonzero ys)"
lsum_simps_2: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list"   and a :: "'a" shows "lsum f (al @ [a]) = lsum f al + f a"
spoly_same: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.spoly p p = 0"