hcju/isabelleps huggingface dataset:
coprime_exp_mod: fixes p :: "int"   and lu :: "int"   and n :: "nat" assumes "poly_mod_prime p"   and "comm_monoid_mult_class.coprime lu p"   and "n \<noteq> 0" shows "lu mod p ^ n \<noteq> 0"
cgenideal_minimal: fixes J :: "int set"   and a :: "int" assumes "ideal J \<Z>"   and "a \<in> J" shows "PIdl\<^bsub>\<Z>\<^esub> a \<subseteq> J"
lt_monomial_plus: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "c \<noteq> (0::'c)"   and "lt p \<prec>\<^sub>t v" shows "lt (monomial c v + p) = v"
apsnd_apfst: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'a"   and x :: "'d \<times> 'c" shows "apsnd f (apfst g x) = (g (fst x), f (snd x))"
functor_ToCT: shows "functor CCC.comp (\<cdot>) T.ToCT"
assms_2: shows "\<forall>ops\<in>set \<psi>. \<exists>op. ops = [op]"
comp_arr_dom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a" assumes "B.VV.arr f"   and "B.VV.dom f = a" shows "B.VV.comp f a = f"
continuous_map_minus_eq: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X euclidean (\<lambda>x. - f x) = continuous_map X euclidean f"
abs_eq: fixes y :: "'d" shows "const_bcontfun y = Bcontfun (\<lambda>_. y)"
st_eq_approx: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite"   and "st x = st y" shows "x \<approx> y"
abc_append_frist_steps_eq_pre: fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat"   and B :: "abc_inst list" assumes "abc_notfinal (abc_steps_l (0, lm) A n) A"   and "A \<noteq> []" shows "abc_steps_l (0, lm) (A @ B) n = abc_steps_l (0, lm) A n"
that: fixes A :: "((real, 'n) vec, 'n) vec" assumes "orthogonal_matrix A"   and "A *v axis k 1 = a" shows "thesis"
length_filter_snd_zip: fixes ys :: "'b list"   and xs :: "'c list"   and p :: "'c \<Rightarrow> bool" assumes "length ys = length xs" shows "length (filter (p \<circ> snd) (zip ys xs)) = length (filter p xs)"
Snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> items H" shows "Y \<in> items H"
app_Some: fixes i :: "'c instr"   and P :: "'d prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and \<tau> :: "ty list \<times> ty err list" shows "app i P mxs T\<^sub>r pc mpc xt \<lfloor>\<tau>\<rfloor> = (app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', s')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc))"
index_to_residue_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat"   and m :: "nat" shows "index_to_residue s k m = s m k"
perm_simps_3: fixes p :: "perm"   and assn :: "assn"   and exp :: "exp" shows "p \<bullet> LetA assn exp = LetA (p \<bullet> assn) (p \<bullet> exp)"
lcf_deg_0: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "deg R p = 0"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "p \<otimes>\<^bsub>UP R\<^esub> q = p (deg R p) \<odot>\<^bsub>UP R\<^esub> q"
walk_edges_vertices_2: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "Digraph G"   and "(v, w) \<in> Digraph.edges_of_walk xs" shows "w \<in> set xs"
structure_is_natural_isomorphism: shows "natural_isomorphism \<F>C.CC.comp (\<cdot>\<^sub>S) D.\<phi> D.FoT\<^sub>C.map D.\<phi>"
Derives1_nonterminal_head: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and N :: "symbol"   and v :: "symbol list" assumes "Derives1 u i r (N # v)"   and "is_nonterminal N" shows "\<exists>u' M. u = M # u' \<and> is_nonterminal M"
rep_eq: shows "Rep_moebius_mat id_moebius_mmat = id_moebius_cmat"
graph_homomorphismI: fixes s :: "('e, 'd) labeled_graph"   and h :: "('d \<times> 'f) set"   and t :: "('e, 'f) labeled_graph" assumes "vertices s = Domain h"   and "h `` vertices s \<subseteq> vertices t"   and "univalent h"   and "edge_preserving h (edges s) (edges t)"   and "graph s"   and "graph t" shows "graph_homomorphism s t h"
null_def: fixes xs :: "'g list" shows "List.null xs = (xs = [])"
hom_mult: fixes x :: "'c poly poly"   and y :: "'c poly poly"   and xa :: "'c"   and ya :: "'c" shows "poly2 (x * y) xa ya = poly2 x xa ya * poly2 y xa ya"
inj_def: fixes f :: "'d \<Rightarrow> 'e" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
Rep_cfun_strict1: fixes x :: "'d" shows "\<bottom>\<cdot>x = \<bottom>"
arc_to_block_def: fixes e :: "'b" shows "arc_to_block e \<equiv> {tail G e, head G e}"
of_nat_sum: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> nat"   and A :: "'c set" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (sum f A) = comm_monoid_add.sum plus zero (\<lambda>x. semiring_1.of_nat one plus zero (f x)) A"
\<pi>'_init_eq: shows "\<pi>' initQ init\<pi> r = (\<lambda>u. if (u, r) \<in> edges rg then Some r else None)"
sumE: fixes s :: "'a + 'b" obtains x :: "'a" where "s = Inl x"   | y :: "'b" where "s = Inr y"
exhaust: fixes y :: "'a\<cdot>'b error" obtains "y = \<bottom>"   | e :: "'b" where "y = Err\<cdot>e"   | a :: "'a" where "y = Ok\<cdot>a"
subst_all_mono: fixes M :: "('a, 'b) Term.term set"   and M' :: "('a, 'b) Term.term set"   and s :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "M \<subseteq> M'" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t s \<subseteq> M' \<cdot>\<^sub>s\<^sub>e\<^sub>t s"
IH: assumes "f \<in> set fs"   and "set fs \<subseteq> carrier G" shows "f divides foldr (\<otimes>) fs \<one>"
d_const: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" assumes "const X" shows "const (d\<^sub>y X)"
f6: shows "\<forall>p ts. adjacent p ts = (Leaf \<notin> set ts \<and> list_all2 (\<lambda>t ta. l t = Suc (r ta)) (butlast ts) (tl ts) \<and> (ts = [] \<or> l (last ts) = fst p \<and> r (hd ts) = snd p))"
tendsto_Zfun_iff: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'e"   and F :: "'d filter" shows "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
msg_zhops_def: fixes m :: "msg" shows "msg_zhops m \<equiv> case m of Rreq hopsc x dipc xa xb oipc xc sipc xd \<Rightarrow> hopsc = 0 \<longrightarrow> oipc = sipc | Rrep hopsc dipc x xa sipc \<Rightarrow> hopsc = 0 \<longrightarrow> dipc = sipc | _ \<Rightarrow> True"
VectorSpaceEnd_inner_dirsum_el_decomp_nth: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and Ns :: "'b set list"   and n :: "nat" assumes "VectorSpace smult V"   and "\<forall>N\<in>set Ns. RModule UNIV smult N \<and> N \<subseteq> V"   and "add_independentS Ns"   and "n < length Ns" shows "VectorSpaceEnd smult (\<Oplus>N\<leftarrow>Ns. N) (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>Ns. G) then \<Oplus>Ns\<leftarrow> a ! n else (0::'b))"
override_on_belowI: fixes S :: "'d set"   and y :: "'d \<Rightarrow> 'e"   and z :: "'d \<Rightarrow> 'e"   and x :: "'d \<Rightarrow> 'e" assumes "\<And>a. a \<in> S \<Longrightarrow> y a \<sqsubseteq> z a"   and "\<And>a. a \<notin> S \<Longrightarrow> x a \<sqsubseteq> z a" shows "x ++\<^bsub>S\<^esub> y \<sqsubseteq> z"
space_def: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real"   and ts :: "traffic"   and v :: "view"   and c :: "cars" assumes "sensors sensors" shows "sensors.space sensors ts v c \<equiv> Abs_real_int (pos ts c, pos ts c + sensors (own v) ts c)"
norm_of_nat: fixes n :: "nat" shows "norm (of_nat n) = real n"
A5_NotSourceSet_level1: shows "\<forall>C\<in>AbstrLevel level1. sA5 \<notin> Sources level1 C"
is_universal: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> D.cones_map f \<chi> = \<chi>'"
space_stream_space_M_ne: fixes x :: "'a" assumes "x \<in> space M" shows "space (stream_space M) \<noteq> {}"
ds_p: shows "ds = a # b \<and> (a \<longrightarrow> branch b T\<^sub>1) \<and> (\<not> a \<longrightarrow> branch b T\<^sub>2)"
rel_funD: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i"   and x :: "'f"   and y :: "'g" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
image_restrict_eq: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "restrict f A ` A = f ` A"
mset_eq: shows "image_mset f (mset_set A) = image_mset f' (mset_set A')"
birkhoff_sum_meas: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> borel_measurable M" shows "birkhoff_sum f n \<in> borel_measurable M"
init_def: fixes \<G> :: "'a cyclic_group"   and x :: "nat"   and h :: "'a \<times> 'a"   and w :: "nat" assumes "chaum_ped_\<Sigma>_base \<G>" shows "chaum_ped_\<Sigma>_base.init \<G> x h w = (let (h, h') = h in sample_uniform (order \<G>) \<bind> (\<lambda>r. return_spmf (r, \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> r, chaum_ped_\<Sigma>_base.g' \<G> x [^]\<^bsub>\<G>\<^esub> r)))"
fixp_above_induct: fixes P :: "'x set \<Rightarrow> bool"   and a :: "'x set"   and ds :: "'d set" assumes "ccpo.admissible \<Union> (\<lambda>x y. (x, y) \<in> {(x, y). x \<subseteq> y}) P"   and "P a"   and "\<And>x. P x \<Longrightarrow> P (cop_F ds x)" shows "P (COP.fixp_above ds a)"
reduction_signs_set_helper_lemma: fixes C :: "rat vec"   and B :: "rat list list"   and A :: "rat list set" assumes "dim_vec C = length B"   and "A \<subseteq> set B"   and "\<forall>n<dim_vec C. C $ n = 0 \<longrightarrow> B ! n \<notin> A" shows "A \<subseteq> set (take_indices B (find_nonzeros_from_input_vec C))"
subst_preserves_beta': fixes r :: "dB"   and s :: "dB"   and t :: "dB"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s" shows "r[t/i] \<rightarrow>\<^sub>\<beta>\<^sup>* s[t/i]"
buffon_set_def: fixes l :: "real"   and d :: "real" shows "buffon_set l d = {(x, \<phi>). (x, \<phi>) \<in> {- d / 2..d / 2} \<times> {- pi..pi} \<and> d / 2 - \<bar>sin \<phi>\<bar> * l / 2 \<le> \<bar>x\<bar>}"
phis'_fst: fixes g :: "'g"   and n :: "'node"   and v :: "'var \<times> 'node \<times> Def"   and vs :: "('var \<times> 'node \<times> Def) list"   and v' :: "'var \<times> 'node \<times> Def" assumes "phis' g (n, v) = Some vs"   and "v' \<in> set vs" shows "var g v' = var g v"
subgroupE_2: fixes H :: "'c set" assumes "subgroup H H" shows "H \<noteq> {}"
ce_rels_def1: shows "ce_rels = {ce_set_to_rel A |A. A \<in> ce_sets}"
coprime_commute: fixes b :: "'a"   and a :: "'a" shows "coprime b a = coprime a b"
hyps: fixes ds :: "'a list"   and cs :: "'a list" assumes "j < length ds"   and "length cs = length ds" shows "(take j ds @ drop j cs)[j := cs ! j] = take j ds @ drop j cs"
assms: fixes u :: "nat \<Rightarrow> 'a" assumes "u \<longlonglongrightarrow> a" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
del_vert_simps_2: fixes u :: "'a" shows "arcs (del_vert u) = {a \<in> arcs G. tail G a \<noteq> u \<and> head G a \<noteq> u}"
vconverse_vempty: shows "0\<inverse>\<^sub>\<circ> = 0"
finite_gpv_Fail: fixes \<I> :: "('d, 'e) \<I>" shows "finite_gpv \<I> Fail"
exhaust: fixes y :: "cospan_shape.Arr" obtains "y = cospan_shape.Null"   | "y = cospan_shape.AA"   | "y = cospan_shape.BB"   | "y = cospan_shape.TT"   | "y = cospan_shape.AT"   | "y = cospan_shape.BT"
split_sfoldl_op: fixes pat :: "[:'r:]" assumes "pat \<noteq> \<bottom>" shows "sfoldl\<cdot>(op\<cdot>pat)\<cdot>([::], pat) = split\<cdot>pat"
bsok_simps_13: fixes e :: "'a expr1"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'a expr1"   and n :: "nat" shows "bsok (e\<bullet>F{D} := e') n = (bsok e n \<and> bsok e' n)"
distinct_9: fixes x21 :: "tm"   and x22 :: "tm"   and x31 :: "fm"   and x32 :: "fm" shows "x21 EQ x22 \<noteq> x31 OR x32"
truncate_down_shift_int: fixes p :: "nat"   and x :: "real"   and k :: "int" shows "truncate_down p (x * 2 powr real_of_int k) = truncate_down p x * 2 powr real_of_int k"
step: fixes p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and l :: "'d"   and \<xi> :: "'b"   and a :: "'c seq_action"   and \<xi>' :: "'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "p \<in> ctermsl (\<Gamma> pn)"   and "not_call p"   and "l \<in> labels \<Gamma> p"   and "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   and "((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<xi>, pp) \<in> reachable A I"   and "p \<in> sterms \<Gamma> pp"   and "(\<xi>', q) \<in> reachable A I"   and "I a" shows "P ((\<xi>, l), a, \<xi>', l')"
ostep_invariant_def': fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool" shows "(A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. \<forall>a s'. (s, a, s') \<in> automaton.trans A \<and> S (fst s) (fst s') a \<longrightarrow> P (s, a, s'))"
capacity_const: shows "\<forall>e. (0::'capacity) \<le> f e \<and> f e \<le> c e"
cindex_polyE_inverse_add: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" assumes "coprime p q" shows "cindex_polyE a b q p + cindex_polyE a b p q = cindex_polyE a b 1 (q * p)"
rep_eq: fixes x :: "'h pmf" shows "pmf x = (\<lambda>xa. measure_pmf.prob x {xa})"
rbd_surj: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "\<exists>R. bd\<^sub>\<R> R = \<phi>"
fps_const_add: fixes c :: "'a"   and d :: "'a" shows "fps_const c + fps_const d = fps_const (c + d)"
division: fixes d :: "'b set set"   and a :: "'b"   and b :: "'b" assumes "d division_of cbox a b" shows "F ((integrable_on) f) d = (f integrable_on cbox a b)"
AND_lower: fixes x :: "int"   and y :: "int" assumes "0 \<le> x" shows "0 \<le> x AND y"
euler_theorem: fixes a :: "nat"   and m :: "nat" assumes "coprime a m" shows "[a ^ totient m = 1] (mod m)"
abs_eq: shows "0 = abs_fract (0::'b, 1::'b)"
drop_butlast: fixes n :: "nat"   and xs :: "'g list" shows "drop n (butlast xs) = butlast (drop n xs)"
cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b stream" assumes "class.order less_eq less"   and "order.sdescending less_eq a" obtains a :: "'b" and b :: "'b" and w :: "'b stream" where "aa__ = a ## b ## w" and "less_eq b a" and "order.sdescending less_eq (b ## w)"
row_echelon_form_one_dim0: shows "row_echelon_form (1\<^sub>m 0)"
is_rbt_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" shows "ord.is_rbt less t = (inv1 t \<and> inv2 t \<and> color_of t = color.B \<and> ord.rbt_sorted less t)"
extremum_uniqueI: fixes a :: "'a \<times> 'b" assumes "less_eq_prod top_prod a" shows "a = top_prod"
mult_strict_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times c a) (times c b)"
tiny_dg_Arr_in_Vset: shows "\<CC>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
expands_to_const: fixes basis :: "(real \<Rightarrow> real) list"   and c :: "real" assumes "basis_wf basis"   and "length basis = expansion_level TYPE('a)" shows "((\<lambda>_. c) expands_to const_expansion c) basis"
three_in_set3: fixes X :: "'b set" assumes "3 \<le> card X" obtains x :: "'b" and y :: "'b" and z :: "'b" where "x \<in> X" and "y \<in> X" and "z \<in> X" and "x \<noteq> y" and "x \<noteq> z" and "y \<noteq> z"
map_add_dom_app_simps_1: fixes m :: "'b"   and l2 :: "'b \<Rightarrow> 'c option"   and l1 :: "'b \<Rightarrow> 'c option" assumes "m \<in> dom l2" shows "(l1 ++ l2) m = l2 m"
bit_unset_bit_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and a :: "'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (ring_bit_operations.unset_bit one push_bit and not m a) n = (bit a n \<and> m \<noteq> n)"
LID\<^sub>LE: fixes l :: "'a"   and s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" assumes "l \<in> LID\<^sub>L s" obtains "\<And>\<sigma> \<tau> e. \<lbrakk>s = (\<sigma>, \<tau>, e); l \<in> LID\<^sub>S \<sigma> \<Longrightarrow> thesis; l \<in> LID\<^sub>S \<tau> \<Longrightarrow> thesis; l \<in> LID\<^sub>E e \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
ring_list: fixes i :: "nat" assumes "i < Suc n" shows "ring (map (\<lambda>i. R Quot I i) [0..<Suc n] ! i)"
vertex_bipartite_web_1: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a" assumes "web \<Gamma>" shows "vertex (bipartite_web_of \<Gamma>) (Inl x) = (vertex \<Gamma> x \<and> x \<notin> B \<Gamma>)"
add_zmset_add_single: fixes a :: "'a"   and A :: "'a zmultiset" shows "add_zmset a A = A + {#a#}\<^sub>z"
im_sheaf_morphisms_def: fixes U :: "'e set"   and V :: "'e set" shows "im0.im_sheaf_morphisms U V \<equiv> identity {0::'d}"
simple_path_join_loop_eq: fixes g2 :: "real \<Rightarrow> 'b"   and g1 :: "real \<Rightarrow> 'b" assumes "pathfinish g2 = pathstart g1"   and "pathfinish g1 = pathstart g2" shows "simple_path (g1 +++ g2) = (arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2})"
diff_intersect_right_idem: fixes M :: "'c multiset"   and N :: "'c multiset" shows "M - N \<inter># M = M - N"
at_swap_simps_2: fixes a :: "'a"   and b :: "'a" assumes "at TYPE('a)" shows "[(a, b)] \<bullet> b = a"
compat_NS_S_point: fixes x :: "('f, 'v) Term.term"   and y :: "('f, 'v) Term.term"   and z :: "('f, 'v) Term.term" assumes "(x, y) \<in> NS"   and "(y, z) \<in> S" shows "(x, z) \<in> S"
special_points_on_x_axis'_1: shows "on_circline x_axis 0\<^sub>h"
f: fixes z :: "complex" assumes "z \<in> S"   and "z \<in> \<real>" shows "f z \<in> \<real>"
red_single_self: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "red_single p 0 p (0::'a)"
add_mset_commute: fixes x :: "'a"   and y :: "'a"   and M :: "'a multiset" shows "x, y, M = y, x, M"
inj_on_id: fixes A :: "'d set" shows "inj_on id A"
hyps_4: fixes i :: "nat"   and \<delta> :: "('c, 'b) ta_rule set"   and q :: "'c" assumes "i < length qs"   and "\<delta>' = remap_rule f ` \<delta>"   and "qs ! i = f q"   and "inj_on f Q"   and "q \<in> Q"   and "\<delta>_states \<delta> \<subseteq> Q" shows "accs \<delta> (ts ! i) q"
interior_rel_interior_gen: fixes S :: "'c set" shows "interior S = (if aff_dim S = int DIM('c) then rel_interior S else {})"
vdomain_atE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> I" obtains b :: "V" where "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f" and "f\<lparr>a\<rparr> = b"
roofed_mono': fixes S :: "'a set"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S' :: "'a set" assumes "S \<subseteq> roofed_gen G B S'" shows "roofed_gen G B S \<subseteq> roofed_gen G B S'"
find_lasso_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "find_lasso_code node_eq_impl node_hash_impl node_def_hash_size G_impl \<equiv> find_ce_code node_eq_impl node_hash_impl node_def_hash_size G_impl \<bind> case_option (RETURN None) (\<lambda>(x, xa). reconstruct_lasso_code node_eq_impl node_hash_impl node_def_hash_size G_impl x xa \<bind> (\<lambda>xb. RETURN (Some xb)))"
IH_2: assumes "receive\<langle>t\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t'. t = t' \<cdot> \<sigma> \<and> receive\<langle>t'\<rangle> \<in> set S"
comm_monoid_add_hom_0_axioms: shows "comm_monoid_add_hom_0 of_real"
integrable_norm_cancel: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M (\<lambda>x. norm (f x))"   and "f \<in> borel_measurable M" shows "integrable M f"
bezout_coefficients_def: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b = fst (euclidean_ring_gcd.euclid_ext_aux divide minus zero times one modulo unit_factor normalize one zero zero one a b)"
step_op_inl_hit: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and opinl :: "'opinl"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and x :: "'dyn"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = IOpInl opinl"   and "\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl) = ar"   and "ar \<le> length \<Sigma>"   and "\<II>\<ss>\<II>\<nn>\<ll> opinl (take ar \<Sigma>)"   and "\<II>\<nn>\<ll>\<OO>\<pp> opinl (take ar \<Sigma>) = x" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
cod_char: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "Cop_S.cod f = (if Cop_S.arr f then Cop_S.MkIde (Cop_S.Cod f) else Cop_S.null)"
lift_lift: fixes i :: "nat"   and k :: "nat"   and t :: "db" assumes "i < k + 1" shows "lift (lift t i) (Suc k) = lift (lift t k) i"
Inf_Set_fold: fixes t :: "('b, unit) RBT.rbt" shows "Inf (Set t) = (if RBT.is_empty t then top else r_min_opt t)"
list_rev_it_correct: fixes s :: "'s" assumes "invar s" shows "set_iterator_rev_linord (list_rev_it s) (\<alpha> s)"
cos_abs: fixes x :: "real" shows "cos \<bar>x\<bar> = cos x"
dom_dom: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.dom (V.dom f) = V.dom f"
cycle_free_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "cycle_free m n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (\<forall>j\<le>n. len m i j (rem_cycles i j xs) \<le> len m i j xs) \<and> \<one> \<le> len m i i xs"
filter_mset_is_empty_iff: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a multiset" shows "(filter_mset P A = {#}) = (\<forall>x. x \<in># A \<longrightarrow> \<not> P x)"