hcju/isabelleps huggingface dataset:
bisim1CallThrowParams: fixes P :: "'b prog"   and h :: "'heap"   and ps :: "'addr expr1 list"   and vs :: "'addr val list"   and a :: "'addr"   and ps' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and obj :: "'addr expr1"   and M :: "String.literal"   and v :: "'addr val" assumes "P,ps,h \<turnstile> (map Val vs @ Throw a # ps', xs) [\<leftrightarrow>] (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "P,obj\<bullet>M(ps),h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 obj) + pc, \<lfloor>a\<rfloor>)"
mult_scalar_distrib_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p + q) \<odot> r = p \<odot> r + q \<odot> r"
A: fixes nr :: "nat"   and nc :: "nat"   and A :: "'b mat" assumes "Matrix_Kernel.kernel nr nc A" shows "A \<in> carrier_mat nr nc"
OclOr_false1: fixes Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(false or Y) = Y"
hoare_r_def: fixes p :: "'c upred"   and Q :: "'c hrel"   and r :: "'c upred" shows "\<lbrace>p\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u = ((\<lceil>p\<rceil>\<^sub>< \<Rightarrow> \<lceil>r\<rceil>\<^sub>>) \<sqsubseteq> Q)"
WT_converterI: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "\<And>q. q \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g run_converter conv q \<surd>"   and "\<And>q r conv'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, conv') \<in> results_gpv \<I>' (run_converter conv q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"
l3_dy_isLtKey_refines_skip: fixes ltk :: "ltkey" shows "{R23s} Id, l3_dy (LtK ltk) {> R23s}"
ih: fixes u' :: "lam" assumes " u \<longrightarrow>\<^sub>\<beta> u'"   and "u' \<in> RED \<tau>" shows "App t u' \<in> RED \<sigma>"
MC: fixes n :: "nat"   and \<omega>' :: "nat \<Rightarrow> 's"   and t :: "'s" assumes "prob {\<omega> \<in> space (T' I). \<forall>i\<le>n. \<omega> !! i = \<omega>' i} \<noteq> 0" shows "cond_prob (T' I) (\<lambda>\<omega>. stl \<omega> !! n = t) (\<lambda>\<omega>. \<forall>i\<le>n. \<omega> !! i = \<omega>' i) = pmf (K (\<omega>' n)) t"
winding_number_around_inside: fixes \<gamma> :: "real \<Rightarrow> complex"   and S :: "complex set"   and z :: "complex"   and w :: "complex" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "closed S"   and "connected S"   and "S \<inter> path_image \<gamma> = {}"   and "z \<in> S"   and "winding_number \<gamma> z \<noteq> 0"   and "w \<in> S \<union> inside S" shows "winding_number \<gamma> w = winding_number \<gamma> z"
cet_snd: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e : PRODUCT t t'" shows "\<Gamma> \<turnstile>\<^sub>c snd\<^sub>c e : t'"
is_unitE: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 a :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" obtains b :: "'c" where "a \<noteq> zero" and "b \<noteq> zero" and "dvd.dvd times b one" and "divide one a = b" and "divide one b = a" and "times a b = one" and "divide c a = times c b"
adm_wf_def: fixes R :: "('d \<times> 'd) set"   and F :: "('d \<Rightarrow> 'e) \<Rightarrow> 'd \<Rightarrow> 'e" shows "adm_wf R F = (\<forall>f g x. (\<forall>z. (z, x) \<in> R \<longrightarrow> f z = g z) \<longrightarrow> F f x = F g x)"
numer_cf2_pos: fixes x :: "real" shows "0 < x\<^sup>2 + 6 * x + 12"
commutative: fixes x :: "'b"   and y :: "'b" assumes "x \<in> A"   and "y \<in> A" shows "addA x y = addA y x"
extra_wt_subseteq: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t" shows "extra_wt \<rho> s \<le> extra_wt \<rho> t"
IH: fixes \<Gamma> :: "'a formula multiset" assumes "image_mset disj_of_clause (mset F) + \<Gamma> \<Rightarrow>\<^sub>n"   and "image_mset disj_of_clause (mset G) + \<Gamma> \<Rightarrow>\<^sub>n" shows "mset (concat (map (\<lambda>c. map (\<lambda>d. disj_of_clause c \<^bold>\<or> disj_of_clause d) G) F)) + \<Gamma> \<Rightarrow>\<^sub>n"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C I\<^sub>C.map I\<^sub>C.cmp"
lformula_conjuncts: fixes \<phi> :: "('a, 'k) aformula" shows "Ball (conjuncts \<phi>) lformula = lformula \<phi>"
alphaEnv_imp_asEnv_equal: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qrho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGoodEnv qrho"   and "qrho &= qrho'" shows "asEnv qrho = asEnv qrho'"
num_val_iff: fixes e :: "'b infty" assumes "e \<noteq> Infty" shows "Num (val e) = e"
pp_np_rv_2: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "f \<in> rv M" shows "np f \<in> rv M"
insert_ops_def: fixes list :: "('b \<times> 'b option) list" shows "insert_ops list \<equiv> spec_ops list set_option"
simplex_dim_ge: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "- 1 \<le> n"
less_length_cut_same_Inr: fixes I :: "(nat + nat set) list"   and r :: "nat"   and P :: "nat set"   and w :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "r < length I"   and "I ! r = Inr P" shows "\<forall>p\<in>P. p < length (cut_same (any, replicate (length I) False) (stream_enc (w, I)))"
ntrp_mon_loc_e_no_ctx: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), LOC e, s', c') \<in> ntrp fg" shows "mon_w fg e \<inter> mon_c fg c = {}"
assms_1: shows "\<forall>i<length as. \<forall>j<length as. i \<noteq> j \<longrightarrow> as ! i \<inter> as ! j = {}"
cons_cluster_has_cons_cluster_slice: fixes C :: "'node set"   and p :: "'node" assumes "is_cons_cluster C"   and "p \<in> C" obtains Sl :: "'node set" where "Sl \<in> slices p" and "Sl \<subseteq> C"
rel_frontier_not_sing: fixes S :: "'a set"   and a :: "'a" assumes "bounded S" shows "rel_frontier S \<noteq> {a}"
subgroup_indicator_le: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and n :: "nat"   and H :: "'c set" assumes "Group.group G"   and "a [^]\<^bsub>G\<^esub> n \<in> H"   and "0 < n"   and "a \<in> carrier G" shows "subgroup_indicator G H a \<le> n"
length_preorder: fixes t :: "'a tree" shows "length (preorder t) = size t"
subset_Un_eq: fixes A :: "'g set"   and B :: "'g set" shows "(A \<subseteq> B) = (A \<union> B = B)"
rel_fun_eq_pcr_ennreal: fixes f :: "'a \<Rightarrow> ereal"   and g :: "'a \<Rightarrow> ennreal" shows "rel_fun (=) pcr_ennreal f g = (f = enn2ereal \<circ> g)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) \<chi>'.A.map D \<chi>'"
paste_outside_restrict: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "P = Outside' X P +< (P || X)"
all_prems_active_after: assumes "0 < m" shows "\<forall>c\<in>set (prems_of \<iota>). \<forall>k\<ge>n. enat k < llength Ns \<longrightarrow> c \<in> active_subset (snd (lnth Ns k))"
rtrancl_into_rtrancl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
measure_pmf_posI: fixes x :: "'b"   and p :: "'b pmf"   and A :: "'b set" assumes "x \<in> set_pmf p"   and "x \<in> A" shows "0 < measure_pmf.prob p A"
reduced_run_project: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "project visible w\<^sub>1 = project visible w"
solve_lhr_aux: fixes f :: "nat \<Rightarrow> 'a"   and cs :: "'a list"   and fs :: "'a list"   and fctrs :: "'a \<times> ('a \<times> nat) list" assumes "linear_homogenous_recurrence f cs fs"   and "is_factorization_of fctrs (lr_fps_denominator' cs)" shows "f = interp_ratfps_solution (solve_factored_ratfps' (lhr_fps_numerator (length fs + 1 - length cs) cs ((!) fs)) fctrs)"
ComplD: fixes c :: "'b"   and A :: "'b set" assumes "c \<in> - A" shows "c \<notin> A"
fls_divide_X_intpow_conv_times': fixes f :: "'a fls"   and i :: "int" assumes "inverse (1::'a) = (1::'a)" shows "f / fls_X_intpow i = f * fls_X_intpow (- i)"
1: shows "countable (range (\<lambda>s. trees_cyl (map_tree (\<lambda>_. space M) s)))"
p_eq_sum_p_u: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a"   and n :: "nat" shows "MC_syntax.p K x y (Suc n) = (\<Sum>i\<le>n. MC_syntax.p K y y (n - i) * MC_syntax.u K x y i)"
elims: fixes x :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and y :: "'a literal multiset set" assumes "N_of_state x = y" obtains N :: "'a literal multiset set" and P :: "'a literal multiset set" and Q :: "'a literal multiset set" where "x = (N, P, Q)" and "y = N"
simps_2: fixes v :: "int \<times> int" shows "degreen (C v) = (\<lambda>m. 0)"
circlepath_def: fixes z :: "complex"   and r :: "real" shows "circlepath z r \<equiv> part_circlepath z r 0 (2 * pi)"
calc_ideal_point1_cvec_def: fixes A :: "complex"   and B :: "complex" shows "calc_ideal_point1_cvec A B = (case let discr = Re (cor ((cmod B)\<^sup>2 - (Re A)\<^sup>2)) in (B * (- A - \<i> * cor (sqrt discr)), (cmod B)\<^sup>2) of (x, y) \<Rightarrow> (x, cor y))"
accepting_pair\<^sub>R_def: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and P :: "('c \<times> 'd \<times> 'c) set \<times> ('c \<times> 'd \<times> 'c) set"   and w :: "nat \<Rightarrow> 'd" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 P w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> fst P = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> snd P \<noteq> {}"
ennreal_mult_cancel_left: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a * b = a * c) = (a = \<top> \<and> b \<noteq> 0 \<and> c \<noteq> 0 \<or> a = 0 \<or> b = c)"
chine_composite'_1: shows "\<mu>\<nu>\<pi>.chine = \<langle>\<mu>.chine \<cdot> doms.Prj\<^sub>1 \<lbrakk>\<mu>.cod.leg0, \<nu>.cod.leg1 \<cdot> \<p>\<^sub>1[\<nu>.cod.leg0, \<pi>.cod.leg1]\<rbrakk> \<langle>\<nu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>0 \<lbrakk>\<nu>.cod.leg0, \<pi>.cod.leg1\<rbrakk> \<pi>.chine \<cdot> doms.Prj\<^sub>0\<^sub>0\<rangle>\<rangle>"
of_nat_dvd_of_nat_gauss_int_iff: fixes m :: "nat"   and n :: "nat" shows "(of_nat m dvd of_nat n) = (m dvd n)"
hyp2_drop_perp_not_same_line: fixes a :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2" shows "drop_perp a l \<noteq> l"
num_sylow_dvd_remainder: shows "card (subgroups_of_size (p ^ a)) dvd m"
F_def: shows "F \<equiv> inf at_infinity (principal (complex_cone' \<alpha>))"
that: fixes v :: "nat"   and u :: "nat" assumes "0 < v"   and "v < u"   and "coprime u v"   and "\<not> (odd u \<and> odd v)"   and "z = u\<^sup>2 + v\<^sup>2"   and "x = 2 * u * v \<and> y = u\<^sup>2 - v\<^sup>2 \<or> x = u\<^sup>2 - v\<^sup>2 \<and> y = 2 * u * v" shows "thesis"
fmul_def: fixes m :: "roundmode"   and a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "fmul m a b = (if is_nan a \<or> is_nan b \<or> is_zero a \<and> is_infinity b \<or> is_infinity a \<and> is_zero b then some_nan else if is_infinity a \<or> is_infinity b then if sign a = sign b then \<infinity> else - \<infinity> else zerosign (if sign a = sign b then 0 else 1) (IEEE.round m (valof a * valof b)))"
g: shows "\<forall>i. (g (Suc i), g i) \<in> measure f \<or> f (g (Suc i)) = f (g i) \<and> (g (Suc i), g i) \<in> r"
YMapInvNatTrans: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and X :: "'d"   and x :: "ZF" assumes "LSCategory C"   and "Ftor F : Op C \<longrightarrow> SET"   and "X \<in> obj\<^bsub>C\<^esub>"   and "x |\<in>| (F @@ X)" shows "NatTrans (YMapInv C X F x)"
intro: fixes R :: "('a, 'b) ring_scheme"   and I :: "'a set" assumes "carrier R \<noteq> I"   and "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R" shows "maximalideal_axioms I R"
arr_RelD_7: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
rotater_drop_take: fixes n :: "nat"   and xs :: "'b list" shows "rotater n xs = drop (length xs - n mod length xs) xs @ take (length xs - n mod length xs) xs"
Collect_conv_if2: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" shows "{x. a = x \<and> P x} = (if P a then {a} else {})"
discrete_compact_space_eq_finite: fixes X :: "'b topology" assumes "X derived_set_of topspace X = {}" shows "compact_space X = finite (topspace X)"
lprefix_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lprefix xs (xs @\<^sub>l ys)"
pp_dist_sum: fixes f :: "'d \<Rightarrow> 'c" shows "sup_monoid.sum (\<lambda>k. - - f k) {k. True} = - - sup_monoid.sum f {k. True}"
assms_1: shows "adjoint_functors (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G"
prod_ord_pairD: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'a" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "prod_ord_pair_tc P xs ys"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "P k (lookup_pair_tc xs k) (lookup_pair_tc ys k)"
a1: shows "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> \<not> snd (dispatch_instruction instr s1) \<and> \<not> snd (dispatch_instruction instr s2) \<and> t1 = snd (fst (dispatch_instruction instr s1)) \<and> t2 = snd (fst (dispatch_instruction instr s2))"
prj_simps_13: shows "C.dom Prj\<^sub>1 = \<mu>\<nu>\<pi>.chine"
stimuli_connected_def: fixes \<C> :: "'a set" shows "stimuli_connected \<C> = (\<forall>X\<^sub>1 X\<^sub>2. X\<^sub>1 \<inter> X\<^sub>2 = {} \<and> X\<^sub>1 \<union> X\<^sub>2 = \<C> \<and> X\<^sub>1 \<noteq> {} \<and> X\<^sub>2 \<noteq> {} \<longrightarrow> (\<exists>A B. A \<in> X\<^sub>1 \<and> B \<in> X\<^sub>2 \<and> (A \<rightarrow>\<^sub>\<S>\<^sup>+ B \<or> B \<rightarrow>\<^sub>\<S>\<^sup>+ A)))"
transpose_transpose_id: fixes A :: "'b matrix" shows "transpose_matrix (transpose_matrix A) = A"
a3: shows "Im (infsetsum f A) = (\<Sum>\<^sub>ax\<in>A. Im (f x))"
univ_asAbs_alphaAbsGood: fixes fAbs :: "('c, 'd, 'varSort, 'var, 'e) qAbs \<Rightarrow> 'f"   and A :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "fAbs respectsP alphaAbsGood"   and "qGoodAbs A" shows "Equiv_Relation2.univ fAbs (asAbs A) = fAbs A"
3: fixes x :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and y :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "(x, y) \<in>\<^sub>p snd ` set ps" shows "x \<in> set (gs @ bs @ hs)"
HRB_slice_refl: fixes m :: "'node"   and S :: "'node SDG_node set" assumes "valid_node m"   and "CFG_node m \<in> S" shows "CFG_node m \<in> HRB_slice S"
add_eq_conv_diff: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a"   and N :: "'a multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> M = add_mset b (N \<ominus> a) \<and> N = add_mset a (M \<ominus> b))"
b: shows "real br / 2 ^ r = real b / 2 ^ p"
eq_fold: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "F (insert x A) = Finite_Set.fold (\<^bold>*) x A"
O_1: shows "tree_invar (Node e1 a1 r ts1)"
simps_2: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "height (Node ts t) = Suc (Max (height ` set (subtrees ts @ [t])))"
that: fixes ub :: "real" assumes "\<forall>x. poly p x = 0 \<longrightarrow> x < ub"   and "\<forall>x\<ge>ub. sgn (poly p x) = sgn_pos_inf p" shows "thesis"
f_def: fixes x :: "'b" shows "f x = (THE i. P i x)"
h2ba: shows "\<forall>q. List.member (qs1 @ qs2) q \<longrightarrow> List.member qs1 q \<or> List.member qs2 q"
simps: fixes l :: "int"   and r :: "int" shows "int_index_def.size (Bound l r) = nat (r - l)"
assms_3: shows "f ` {0..1} \<subseteq> cbox (- 1) 1"
count_insert1: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "p partitions n" shows "sum (p(k := p k + 1)) {..n + k} = sum p {..n} + 1"
path_is_empty_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list" assumes "GS_invar s" shows "path_is_empty_impl s = (GS.p_\<alpha> s = [])"
l_cancel_one: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and a :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> a = x) = (a = \<one>\<^bsub>G\<^esub>)"
zer_iff: fixes ls :: "real poly list"   and x :: "real" shows "(poly (sum_list (map power2 ls)) x = 0) = (\<forall>i\<in>set ls. poly i x = 0)"
continuous_on_compose: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "continuous_on (\<P> f s) g" shows "continuous_on s (g \<circ> f)"
that: fixes p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list" assumes "p = p1 @ p2"   and "isPath s p1 u"   and "isPath u p2 t" shows "thesis"
bigo_plus_absorb_lemma1: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(g)" shows "f +o O(g) \<subseteq> O(g)"
card_of_ordIso_finite: fixes A :: "'a set"   and B :: "'b set" assumes "|A| =o |B|" shows "finite A = finite B"
hyps: fixes qs :: "real poly list" assumes "length qs < length qsa__" shows "p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> set (find_consistent_signs_at_roots p qs) = set (characterize_consistent_signs_at_roots_copr p qs)"
wf_times_shiftD: fixes t' :: "real"   and t :: "real"   and \<omega> :: "(real \<times> 'a) stream" assumes "wf_times t' (smap (\<lambda>(t', y). (t' + t, y)) \<omega>)" shows "wf_times (t' - t) \<omega>"
rep_eq: fixes x :: "'b mod_ring"   and xa :: "'b mod_ring" shows "Rep_mod_ring (x + xa) = (Rep_mod_ring x + Rep_mod_ring xa) mod int CARD('b)"
assms_1: shows "H \<turnstile> HaddP x (SUCC y) z"
prime_square_sum_nat_decomp_eqI: fixes p :: "nat"   and x :: "nat"   and y :: "nat" assumes "prime p"   and "x\<^sup>2 + y\<^sup>2 = p"   and "x \<le> y" shows "prime_square_sum_nat_decomp p = (x, y)"
f: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> f) = (\<exists>k l u. meets k l \<and> meets l p \<and> meets p u \<and> meets k q \<and> meets q u)"
frontier_of_Int: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X frontier_of (S \<inter> T) = X closure_of (S \<inter> T) \<inter> (X frontier_of S \<union> X frontier_of T)"
simps_2: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b option" shows "Some x \<bind> f = f x"
UP_ring: fixes R :: "('a, 'b) ring_scheme" assumes "UP_ring R" shows "ring (UP R)"
simps_6: fixes wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" shows "alphabet.wf wf_atom n (Times r s) = (alphabet.wf wf_atom n r \<and> alphabet.wf wf_atom n s)"
proj_Cons_1: fixes n :: "'d"   and a :: "'e"   and A :: "('d strand_label \<times> 'e) list" shows "proj n ((ln n, a) # A) = (ln n, a) # proj n A"
mapmeasure_PiM: fixes M :: "('i \<Rightarrow> 'e) measure"   and n :: "nat"   and N :: "'e measure"   and X :: "('i \<Rightarrow> 'e) set" assumes "space M = space (Pi\<^sub>M (J n) (\<lambda>_. N))"   and "sets M = sets (Pi\<^sub>M (J n) (\<lambda>_. N))"   and "space N = UNIV"   and "X \<in> sets M" shows "emeasure M X = emeasure (mapmeasure n M (\<lambda>_. N)) (fm n ` X)"
face_cycle_succ_permutes: shows "G.face_cycle_succ permutes arcs G"
eigen_value_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and k :: "'b" shows "eigen_value A k = (\<exists>v. eigen_vector A v k)"
has_sort_imp_exehas_sort: fixes sub :: "(String.literal \<times> String.literal) list"   and tcs :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and T :: "typ"   and S :: "String.literal set" assumes "exe_osig_conds (sub, tcs)"   and "has_sort (translate_osig (sub, tcs)) T S" shows "exehas_sort (sub, tcs) T S"
simps_2: fixes \<E> :: "char list \<Rightarrow> nat option"   and v :: "char list" shows "eval \<E> (Var v) = \<E> v"
transp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
set_not_emptyI: fixes x :: "'d"   and S :: "'d set" assumes "x \<in> S" shows "S \<noteq> {}"
minus_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "r.eq_m (f - r.Mp g) (f - g)"
assms_2: shows "P' \<Longrightarrow>\<^sub>\<tau> P''"
det_eq: fixes A :: "('b, 'c) sq_matrix" shows "Square_Matrix.det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. to_fun A i (p i)))"
transform_simps_5: fixes a :: "Arity"   and b :: "bool" shows "\<T>\<^bsub>a\<^esub> (Bool b) = Bool b"
\<theta>\<^sub>g'_def: shows "\<theta>\<^sub>g' = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
nonnegative_absolutely_integrable: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "f integrable_on A"   and "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> 0 \<le> f x \<bullet> b" shows "f absolutely_integrable_on A"
p_odd_int: obtains z :: "int" where "int p = 2 * z + 1" and "0 < z"
dgrad_p_set_le_subset: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat" assumes "F \<subseteq> G" shows "dgrad_p_set_le d F G"
set_cases: fixes e :: "'d"   and a :: "'d list" assumes "e mem a" obtains z2 :: "'d list" where "a = e # z2__"   | z1 :: "'d" and z2 :: "'d list" where "a = z1 # z2" and "e mem z2"