hcju/isabelleps huggingface dataset:
merges_property2: fixes qs :: "(nat \<times> nat list) list"   and ss :: "nat list list"   and q :: "nat" assumes "\<forall>(q, \<tau>)\<in>set qs. q < length ss \<and> \<tau> \<in> A"   and "set ss \<subseteq> A"   and "length ss = n"   and "distinct (map fst qs)"   and "q \<notin> set (map fst qs)"   and "q < length ss" shows "merges f qs ss ! q = ss ! q"
sort_key_on_def: fixes r :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "sort_key_on r f xs = foldr (insort_key_on r f) xs []"
compatible_refl: fixes cmds :: "'Com list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "list_all com_sifum_secure cmds"   and "mem\<^sub>1 =\<^sup>l mem\<^sub>2" shows "makes_compatible (add_initial_modes cmds, mem\<^sub>1) (add_initial_modes cmds, mem\<^sub>2) (replicate (length cmds) (mem\<^sub>1, mem\<^sub>2))"
simps_10: fixes v :: "'b formula"   and va :: "'b formula" shows "is_lit_plus (v \<^bold>\<or> va) = False"
span_some_basis: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "vector_space scale"   and "module.subspace scale V" shows "module.span scale (vector_space.some_basis scale V) = V"
tensor_relations1_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and MN :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "TR1 R M N MN = {x. \<exists>m1\<in>carrier M. \<exists>m2\<in>carrier M. \<exists>n\<in>carrier N. x = (m1 \<plusminus>\<^bsub>M\<^esub> m2, n) \<plusminus>\<^bsub>MN\<^esub> -\<^sub>a\<^bsub>MN\<^esub> ((m1, n) \<plusminus>\<^bsub>MN\<^esub> (m2, n))}"
eql\<^sub>2: shows "override_on (l\<^sub>2 ++ l\<^sub>3) l\<^sub>2 ({this} \<union> set pns) = l\<^sub>2"
zero_Integer_def: shows "0 = MkI\<cdot>0"
ring_distribs_1: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a * (b + c) = a * b + a * c"
vimage2p_rel_fun: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (BNF_Def.vimage2p f g R) R f g"
gbinomial_0_2: fixes k :: "nat" shows "(0::'a) gchoose Suc k = (0::'a)"
bij_conv: fixes x :: "'a" shows "is_bij x = is_bij (x\<^sup>\<smile>)"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(a + b = (zero_class.zero::'a)) = (a = (zero_class.zero::'a) \<and> b = (zero_class.zero::'a))"
hconf_new: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" assumes "P \<turnstile> h \<surd>"   and "h a = None"   and "P,h \<turnstile> obj \<surd>" shows "P \<turnstile> h(a \<mapsto> obj) \<surd>"
simps: fixes recf :: "recf" shows "tm_of_rec recf = (let (ap, k, fp) = rec_ci recf; tp = tm_of (ap [+] dummy_abc k) in tp @ shift (mopup k) (length tp div 2))"
negligible_interval_1: fixes a :: "'c"   and b :: "'c" shows "negligible (cbox a b) = (box a b = {})"
assms: shows "small {[a, b]\<^sub>\<circ> |a b. P a b}"
match_rews_1: fixes k :: "'a match" shows "match_snil\<cdot>\<bottom>\<cdot>k = \<bottom>"
v11_vconverse: fixes r :: "V" assumes "v11 r" shows "v11 (r\<inverse>\<^sub>\<circ>)"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and \<epsilon> :: "ennreal" assumes "hindered_by \<Gamma> \<epsilon>" obtains f :: "'a \<times> 'a \<Rightarrow> ennreal" where "hindrance_by \<Gamma> f \<epsilon>" and "current \<Gamma> f" and "wave \<Gamma> f"
x15: shows "Max (p ` fst ` f) \<in> p ` fst ` f"
def_homo: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<delta> x->includes\<^sub>S\<^sub>e\<^sub>t(y) = (\<delta> x and \<upsilon> y)"
scale_minus_left: fixes a :: "'a"   and x :: "('a, 'b) vec" shows "- a *s x = - (a *s x)"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> Pdds'.CD_on ds X) = (x \<in> Pdds'.Cd (Xd x) X \<and> Xd x \<in> ds)"
substT_subtype: fixes \<Delta> :: "binding list"   and X :: "tyvrs"   and Q :: "ty"   and \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty"   and P :: "ty" assumes "(\<Delta> @ TVarB X Q # \<Gamma>)\<turnstile>S<:T"   and "\<Gamma>\<turnstile>P<:Q" shows "(\<Delta>[X \<mapsto> P]\<^sub>e @ \<Gamma>)\<turnstile>S[X \<mapsto> P]\<^sub>\<tau><:T[X \<mapsto> P]\<^sub>\<tau>"
cone_of_cols: fixes A :: "'a mat"   and nr :: "nat"   and b :: "'a vec" assumes "A \<in> carrier_mat n nr"   and "b \<in> carrier_vec n" shows "(b \<in> cone (set (cols A))) = (\<exists>x\<ge>0\<^sub>v nr. A *\<^sub>v x = b)"
tree_rank_estimate_upper: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) SkewBinomialTree list" assumes "tree_invar (Node e a r ts)" shows "size (tree_to_multiset (Node e a r ts)) \<le> 2 ^ Suc r - 1"
support_inv: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l. P) = {}"
dickson_grading_def: fixes d :: "'b \<Rightarrow> nat" shows "dickson_grading d = ((\<forall>s t. d (s + t) = max (d s) (d t)) \<and> (\<forall>n. almost_full_on (adds) {x. d x \<le> n}))"
strict_trans2: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "strict_prefix a b"   and "prefix b c" shows "strict_prefix a c"
subPA: shows "submatrix (P * A) I UNIV \<in> carrier_mat k n"
fixpt_wf: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "wf_tr M T" shows "wf_tr M (fixpt M T)"
axioms_3: fixes S1 :: "'c set"   and S2 :: "'d set"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "module_hom_on S1 S2 s1 s2 f" shows "module_hom_on_axioms S1 s1 s2 f"
fmdom'_fmsubset_restrict_set: fixes X1 :: "'b set"   and X2 :: "'b set"   and s :: "('b, bool) fmap" assumes "X1 \<subseteq> X2"   and "fmdom' s = X2" shows "fmdom' (fmrestrict_set X1 s) = X1"
approximating_bigstep_fun_seq_wf: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and s :: "state" assumes "wf_ruleset \<gamma> p rs\<^sub>1" shows "approximating_bigstep_fun \<gamma> p (rs\<^sub>1 @ rs\<^sub>2) s = approximating_bigstep_fun \<gamma> p rs\<^sub>2 (approximating_bigstep_fun \<gamma> p rs\<^sub>1 s)"
Pow_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "Pow A \<subseteq> Pow B"
min_max_xyz: shows "maximum (i \<squnion> j) + 1 = minimum k"
lookup_single: fixes k :: "'c"   and v :: "'b"   and k' :: "'c" shows "lookup (Poly_Mapping.single k v) k' = (v when k = k')"
nonnegative: fixes x :: "'a" shows "0 \<le> \<parallel>f *\<^sub>v x\<parallel>"
in_sub_out21: shows "inside (path_image c2 \<union> path_image c) \<subseteq> outside (path_image c1 \<union> path_image c)"
collapse_2: fixes "term" :: "('a, 'b) Term.term" assumes "is_Fun term" shows "Fun (un_Fun1 term) (args term) = term"
has_field_fun: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and b' :: "staticb"   and T' :: "ty" assumes "P \<turnstile> C has F,b:T in D"   and "P \<turnstile> C has F,b':T' in D" shows "b = b' \<and> T' = T"
vdash_properties_9: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]" shows "[\<psi> \<^bold>\<rightarrow> \<phi> in v]"
Q'''Trans: shows "Q''' \<longmapsto>\<alpha> \<prec> Q''"
det_state_cover_initial: fixes M :: "('e, 'f, 'g) FSM"   and V :: "'e list set" assumes "is_det_state_cover M V" shows "[] \<in> V"
assms_2: fixes u :: "'b"   and v :: "'b" assumes "box u v = {}" shows "d (cbox u v) = \<^bold>1"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "lookup (swapPoly\<^sub>0 x xa xb) = (\<lambda>mon. lookup xb (swap0 x xa mon))"
b_e_check_single_top_sound: fixes x1 :: "ct list"   and t_in :: "t list"   and t_out :: "t list"   and x2 :: "ct list"   and tm :: "t list"   and \<C> :: "t_context"   and e :: "b_e" assumes "type_update (TopType x1) (to_ct_list t_in) (Type t_out) = TopType x2"   and "c_types_agree (TopType x2) tm"   and "\<C> \<turnstile> [e] : t_in _> t_out" shows "\<exists>tn. c_types_agree (TopType x1) tn \<and> \<C> \<turnstile> [e] : tn _> tm"
exists_exp_of_Bk: fixes list :: "cell list"   and mr :: "nat"   and rn :: "nat" assumes "Bk # list = Oc \<up> mr @ Bk \<up> rn" shows "\<exists>rn. list = Bk \<up> rn"
maintain_invar2_first_step_2: fixes r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and d :: "nat"   and g :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "Prim2.prim_invar2_init r Q \<pi>"   and "Q u = enat d" shows "Prim2.T_measure2 g r (Prim2.Q' w g r Q \<pi> u) (Prim2.\<pi>' w g r Q \<pi> u) < Prim2.T_measure2 g r Q \<pi>"
unit_in_hom_2: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<i>\<^sub>C[a] : a \<star>\<^sub>C a \<Rightarrow>\<^sub>C a\<guillemotright>"
prv_eqvI: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (imp \<phi> \<chi>)"   and "prv (imp \<chi> \<phi>)" shows "prv (Syntax_with_Connectives.eqv cnj imp \<phi> \<chi>)"
chainD: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Complete_Partial_Order.chain ord S"   and "x \<in> S"   and "y \<in> S" shows "ord x y \<or> ord y x"
irreducibleI: fixes a :: "'c"   and G :: "('c, 'd) monoid_scheme" assumes "a \<notin> Units G"   and "\<And>b. \<lbrakk>b \<in> carrier G; properfactor G b a\<rbrakk> \<Longrightarrow> b \<in> Units G" shows "irreducible G a"
map_proj_thd_update_until: fixes args :: "Monitor.args"   and nt :: "nat"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set" shows "map proj_thd (takeWhile (check_before (args_ivl args) nt) auxlist) = map proj_thd (takeWhile (check_before (args_ivl args) nt) (update_until args rel1 rel2 nt auxlist))"
inject_1: fixes x1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and y1 :: "('a, 'b, 'c, 'd) Tree set['a]" shows "(tConj x1 = tConj y1) = (x1 = y1)"
fp: shows "\<forall>x\<in>F. f x \<le> p x"
sorted_wrt_list_of_set_linorder_of_list: fixes xs :: "'b list" assumes "distinct xs" shows "sorted_wrt_list_of_set (linorder_of_list xs) (set xs) = xs"
splus_sminus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.sminus pair_of_term term_of_pair (term_powerprod.splus pair_of_term term_of_pair t v) t = v"
facets: fixes w :: "'w"   and s :: "'w" assumes "w \<in> W"   and "s \<in> S" shows "dual_order.poset_simplex_map \<P> (w +o \<langle>{s}\<rangle>) \<lhd> dual_order.poset_simplex_map \<P> {w}"
add_neg_neg: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<subset># {#}"   and "b \<subset># {#}" shows "a + b \<subset># {#}"
distinct_35: fixes x6 :: "Arraytype" shows "ShortT \<noteq> ArrT x6"
preserves_arr: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "B.arr (uncurry G f)"
insert_iff: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
out_diff2: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Out B C" shows "C \<noteq> A"
cases: fixes x :: "int \<times> (real \<Rightarrow> real \<times> real)" obtains k :: "int" and \<gamma> :: "real \<Rightarrow> real \<times> real" where "x = (k, \<gamma>)"
bounded_linear_def: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_linear f \<equiv> linear f \<and> bounded_linear_axioms f"
remove_all_distinct: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "distinct (x # remove_all x xs)"
n_in_Nsetn: fixes n :: "nat" shows "n \<in> {i. i \<le> n}"
stirling_trapezium: fixes x :: "real" assumes "0 < x" shows "D x \<in> {0..1 / (12 * x\<^sup>2) - 1 / (12 * (x + 1)\<^sup>2)}"
sum: fixes g :: "'a \<Rightarrow> 'n"   and S :: "'a set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
vmc_path_ldropn: fixes n :: "nat" assumes "enat n < llength P" shows "vmc_path G (ldropn n P) (P $ n) p \<sigma>"
chain': shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>g e. f e - g e) ` Y)"
covering_space_lift_homotopic_path: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and f :: "real \<Rightarrow> 'd"   and f' :: "real \<Rightarrow> 'd"   and g :: "real \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "covering_space C p S"   and "homotopic_paths S f f'"   and "path g"   and "path_image g \<subseteq> C"   and "pathstart g = a"   and "pathfinish g = b"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (g t) = f t" obtains g' :: "real \<Rightarrow> 'c" where "path g'" and "path_image g' \<subseteq> C" and "pathstart g' = a" and "pathfinish g' = b" and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (g' t) = f' t"
essentially_surjective: fixes b :: "'b" assumes "F.B.ide b" shows "\<exists>a. F.A.ide a \<and> F.B.isomorphic (F a) b"
exp_i_ne_1: fixes x :: "real" assumes "0 < x"   and "x < 2 * pi" shows "exp (\<i> * complex_of_real x) \<noteq> 1"
LIMSEQ_incseq_SUP: fixes X :: "nat \<Rightarrow> 'a" assumes "bdd_above (range X)"   and "incseq X" shows "X \<longlonglongrightarrow> Sup (range X)"
comp_converter_parallel2: fixes conv1l :: "('y, 'aa, 'ag, 'ai) converter"   and conv1r :: "('z, 'ab, 'ah, 'aj) converter"   and conv2l :: "('ag, 'ai, 'ac, 'ae) converter"   and conv2r :: "('ah, 'aj, 'ad, 'af) converter" shows "(conv1l |\<^sub>= conv1r) \<odot> (conv2l |\<^sub>= conv2r) = conv1l \<odot> conv2l |\<^sub>= conv1r \<odot> conv2r"
redT_preserves_ts_inv_ok: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "'t \<Rightarrow> 'b option"   and P :: "'b \<Rightarrow> 't \<Rightarrow> 'x \<Rightarrow> 'm \<Rightarrow> bool" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "ts_inv_ok (thr s) I" shows "ts_inv_ok (thr s') (upd_invs I P \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>)"
complex_eq_if_Re_eq: fixes z1 :: "complex"   and z2 :: "complex" assumes "is_real z1"   and "is_real z2" shows "(z1 = z2) = (Re z1 = Re z2)"
belowI: fixes d :: "'a defl"   and f :: "'a \<rightarrow> 'a" assumes "\<And>x. cast\<cdot>d\<cdot>x = x \<Longrightarrow> f\<cdot>x = x" shows "cast\<cdot>d \<sqsubseteq> f"
ACI_norm_derivs_alt: fixes w :: "'a list"   and r :: "'a rexp" shows "\<guillemotleft>derivs w r\<guillemotright> = fold (\<lambda>a r. \<guillemotleft>deriv a r\<guillemotright>) w \<guillemotleft>r\<guillemotright>"
lossless_spmf_conv_spmf_of_pmf: fixes p :: "'a spmf" shows "lossless_spmf p = (\<exists>p'. p = spmf_of_pmf p')"
IH: fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "distinct vs"   and "set vs \<inter> V' = {}"   and "nonneg_cexpr (set vs \<union> V') \<Gamma> e"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> set vs \<union> V'" shows "0 \<le> extract_real (cexpr_sem \<rho> (integrate_vars \<Gamma> vs e))"
U_cases: fixes s :: "'s" shows "U s s = 1 \<or> U s s < 1"
flip_altsublist_chain_map_Cons_grow: fixes tss :: "'w list list"   and t :: "'w" assumes "flip_altsublist_chain tss" shows "flip_altsublist_chain (map ((#) t) tss)"
vcard_vemptyD: fixes A :: "V" assumes "vcard A = []\<^sub>\<circ>" shows "A = []\<^sub>\<circ>"
rev_subspaceD: fixes x :: "'c"   and U :: "'c set"   and V :: "'c set" assumes "x \<in> U"   and "Subspace.subspace U V" shows "x \<in> V"
\<delta>_def: fixes u :: "'v"   and v :: "'v" shows "\<delta> u v \<equiv> LEAST w. w \<in> sum_list ` paths u v"
starfun_n_def: fixes F :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" shows "*fn* F = Ifun (star_n F)"
grey_def: fixes r :: "'k" shows "grey r = (\<lambda>s. \<exists>x. r \<in> WL x s)"
isLub_least: fixes cl :: "'b potype"   and S :: "'b set"   and L :: "'b"   and z :: "'b" assumes "CL cl"   and "isLub S cl L"   and "z \<in> pset cl"   and "\<forall>y\<in>S. (y, z) \<in> order cl" shows "(L, z) \<in> order cl"
gmn_correct: fixes nd :: "('c, 'd) Node" assumes "is_measured_node nd" shows "gmn nd = sum_list (map snd (nodeToList nd))"
path_component_mono: fixes S :: "'b set"   and T :: "'b set"   and x :: "'b" assumes "S \<subseteq> T" shows "path_component_set S x \<subseteq> path_component_set T x"
eq: shows "Polynomial.smult (lead_coeff (of_int_poly f)) (\<Prod>a\<leftarrow>complex_roots_complex (of_int_poly f). [:- a, 1:]) = of_int_poly f"
ord_eq_less_trans: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a = b"   and "b \<subset># c" shows "a \<subset># c"
contradicting_in_cp: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k :: "nat"   and k' :: "nat"   and v :: "'var" assumes "\<sigma> =\<^sub>L \<sigma>'"   and "cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"   and "v \<in> local.reads (path \<sigma> k)"   and "(\<sigma>\<^bsup>k\<^esup>) v \<noteq> (\<sigma>'\<^bsup>k'\<^esup>) v" shows "((\<sigma>, k), \<sigma>', k') \<in> cp"
simps_5: fixes C :: "ty"   and D :: "ty"   and X :: "ntrm set" shows "NEGc (C IMP D) X = AXIOMSc (C IMP D) \<union> BINDINGc (C IMP D) X \<union> IMPRIGHT (C IMP D) (\<parallel>(C)\<parallel>) (\<parallel><D>\<parallel>) (\<parallel>(D)\<parallel>) (\<parallel><C>\<parallel>)"
restrict_spmf_trivial: fixes p :: "'h spmf"   and A :: "'h set" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> x \<in> A" shows "p \<upharpoonleft> A = p"
dense: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a" assumes "class.dlo less_eq less"   and "less x z" shows "\<exists>y. less x y \<and> less y z"
zero_in_cone: fixes n :: "nat"   and Vs :: "'b vec set" shows "0\<^sub>v n \<in> gram_schmidt.cone n Vs"
simps_1: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist" shows "lrev_body3\<cdot>r\<cdot>lnil = ID"
elem_in_uniq_set: fixes x :: "'b"   and A :: "'b set"   and P :: "'b set set" assumes "x \<in> A"   and "P partitions A" shows "\<exists>!X. X \<in> P \<and> x \<in> X"
zmset_of_less: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(zmset_of M < zmset_of N) = (M < N)"
list_comb_cases': 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 f :: "'a"   and xs :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" obtains "pre_term.is_app unapp (term.list_comb app f xs)"   | "term.list_comb app f xs = f" and "xs = []"
L_in_iff: fixes l :: "'lbl" shows "(l \<in> set (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A)))) = (\<exists>a\<in>set A. is_LabelN l a)"
span_insert: fixes a :: "'b"   and S :: "'b set" shows "span (insert a S) = {x. \<exists>k. x - k *s a \<in> span S}"
ccontr: shows "valid_unMultigraph.adjacent (del_unEdge v w u G) v u"
fps_mult_nth: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "fps_nth (f * g) n = (\<Sum>i = 0..n. fps_nth f i * fps_nth g (n - i))"
enc_Inr: fixes \<Sigma> :: "'b list"   and x :: "('b \<times> bool list) list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and M :: "nat"   and P :: "nat set"   and p :: "nat" assumes "formula \<Sigma>"   and "x \<in> formula.enc \<Sigma> (w, I)"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "M < length I"   and "I ! M = Inr P" shows "(p \<in> P) = (p < length x \<and> snd (x ! p) ! M)"
coeffs_monom: fixes a :: "'b"   and n :: "nat" shows "coeffs (monom a n) = (if a = (0::'b) then [] else replicate n (0::'b) @ [a])"
bound_compare_defs_11: fixes c :: "'a"   and b :: "'a option" shows "c \<le>\<^sub>u\<^sub>b b = \<unlhd>\<^sub>u\<^sub>b (<) c b"
abs_zero: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
mat_add_limit: fixes B :: "complex mat"   and m :: "nat"   and X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" assumes "B \<in> carrier_mat m m"   and "limit_mat X A m" shows "limit_mat (mat_add_seq B X) (B + A) m"
bounded_linear_inner_left_comp: fixes g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "bounded_linear g" shows "bounded_linear (\<lambda>x. g x \<bullet> b)"
pdo_denote_while_n_iter: fixes \<rho> :: "complex mat"   and m :: "nat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and M0 :: "complex mat"   and n :: "nat" assumes "\<rho> \<in> carrier_mat m m"   and "partial_density_operator \<rho>"   and "M1 \<in> carrier_mat m m"   and "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m m"   and "\<And>\<rho>. \<rho> \<in> carrier_mat m m \<and> partial_density_operator \<rho> \<Longrightarrow> partial_density_operator (DS \<rho>)"   and "\<And>\<rho>. \<rho> \<in> carrier_mat m m \<and> partial_density_operator \<rho> \<Longrightarrow> DS \<rho> \<in> carrier_mat m m" shows "partial_density_operator (denote_while_n_iter M0 M1 DS n \<rho>)"
assms_2: shows "X \<longlonglongrightarrow> x"
IH: fixes P :: "pi"   and Q :: "pi" assumes "depth P + depth Q \<le> n"   and "valid P"   and "valid Q"   and "uhnf P"   and "uhnf Q"   and "P \<sim> Q" shows "P \<equiv>\<^sub>e Q"
elims_3: fixes x :: "('b \<times> bal) tree" assumes "\<not> is_bal x" obtains l :: "('b \<times> bal) tree" and a :: "'b" and b :: "bal" and r :: "('b \<times> bal) tree" where "x = \<langle>l, (a, b), r\<rangle>" and "b \<noteq> Bal"   | "x = \<langle>\<rangle>" and "\<not> undefined"
sp_same_src_imp_same_tgt: fixes g :: "('d, 'e) rgraph_scheme"   and v :: "'d"   and es :: "'d edge list"   and v1 :: "'d"   and v2 :: "'d" assumes "Graph.subpath g v es v1"   and "Graph.subpath g v es v2" shows "v1 = v2"
dg_prod_Dom_vsv: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "vsv (dg_prod I \<AA>\<lparr>Dom\<rparr>)"
f_ok: shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (f n s)))"