hcju/isabelleps huggingface dataset:
injectivity_4: fixes P :: "nat set set"   and j :: "nat"   and P' :: "nat set set"   and j' :: "nat"   and B' :: "nat set"   and k :: "nat"   and B'' :: "nat set"   and k' :: "nat"   and Q :: "nat set set"   and Q' :: "nat set set"   and f :: "nat \<Rightarrow> nat set"   and g :: "nat \<Rightarrow> nat set"   and P'' :: "nat set set"   and P''' :: "nat set set" assumes "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "P = P'"
finite_imp_card_positive: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G"   and "finite (carrier G)" shows "0 < card H"
suffix_append: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "suffix xs (ys @ zs) = (suffix xs zs \<or> (\<exists>xs'. xs = xs' @ zs \<and> suffix xs' ys))"
r_psi_2: fixes i :: "nat"   and x :: "nat" shows "eval r_psi [i, x] = \<psi> i x"
IH: shows "\<forall>f. (\<forall>i. i < 2 ^ (n + 1) \<and> even i \<longrightarrow> f i = 0) \<longrightarrow> sum f {0..<2 ^ (n + 1)} = (\<Sum>k = 0..<2 ^ n. f (2 * k + 1))"
hh: shows "(- (cor r)\<^sup>2, 0, 0, 4 - (cor r)\<^sup>2) \<in> hermitean_nonzero"
some: fixes b :: "'b"   and \<sigma>' :: "'a" assumes "ioprog a \<sigma> = Some (b, \<sigma>')"   and "\<sigma>' \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e S ioprog) (P (b # s))" shows "Q"
inf_le1: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<inter># y \<subseteq># x"
trimmed_pos_hd_coeff: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "trimmed_pos (MS (MSLCons (C, e) xs) f)" shows "trimmed_pos C"
int_pow_inv: fixes a :: "'a"   and n :: "int" assumes "a \<in> Units G" shows "a [^] - n = inv a [^] n"
r_one: fixes x :: "int" assumes "x \<in> carrier \<Z>" shows "x \<otimes>\<^bsub>\<Z>\<^esub> \<one>\<^bsub>\<Z>\<^esub> = x"
PQ: fixes \<xi> :: "'a"   and \<xi>' :: "'a" assumes "P \<xi> \<xi>'" shows "Q \<xi> \<xi>'"
f_join_Nil: fixes I :: "nat set" shows "[] \<Join>\<^sub> I = []"
finite_list_set_rel: fixes A :: "('a \<times> 'b) set" shows "(\<lambda>_. True, finite) \<in> \<langle>A\<rangle>list_set_rel \<rightarrow> bool_rel"
circ_top_edge_reparam_polar_circ_split: shows "reparam (rec_join [circle_top_edge]) (rec_join [subcube 0 (1 / 4) (1, circle_polar), subcube (1 / 4) (1 / 2) (1, circle_polar)])"
UP_smult_l_distr: fixes a :: "'a"   and b :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "p \<in> carrier P" shows "(a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
ctr: fixes xs :: "'a msstream" shows "msllist_of_msstream xs = MSLCons (case xs of MSSCons x xs \<Rightarrow> x) (case xs of MSSCons x xs \<Rightarrow> msllist_of_msstream xs)"
is_execution_fragment_def: fixes A :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and ex :: "(nat \<Rightarrow> 'd option) \<times> (nat \<Rightarrow> 'e)" shows "is_execution_fragment A ex \<equiv> let act = fst ex; state = snd ex in \<forall>n a. (act n = None \<longrightarrow> state (Suc n) = state n) \<and> (act n = Some a \<longrightarrow> (state n, a, state (Suc n)) \<in> trans_of A)"
simps_4: fixes p :: "real mpoly" shows "aNeg (Neq p) = Eq p"
antisym: fixes a :: "nat"   and b :: "nat" assumes "a dvd b"   and "b dvd a" shows "a = b"
is_nonterminal_def: fixes s :: "symbol" shows "is_nonterminal s = (s \<in> \<NN>)"
cil_assoc2: fixes w :: "'a list"   and w1 :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and wr :: "'a list"   and w2 :: "'a list"   and w3 :: "'a list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> wr"   and "wr \<in> w2 \<otimes>\<^bsub>\<alpha>\<^esub> w3" shows "\<exists>wl. w \<in> wl \<otimes>\<^bsub>\<alpha>\<^esub> w3 \<and> wl \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2"
normal_lconjby_closed: fixes H :: "'g set"   and g :: "'g"   and h :: "'g" assumes "Group H \<and> order.greater_eq G H"   and "normal H"   and "g \<in> G"   and "h \<in> H" shows "lconjby g h \<in> H"
iTL_defs_8: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iTrigger I P Q \<equiv> (\<box> t I. Q t) \<or> (\<diamond> t I. P t \<and> (\<box> t' I \<down>\<ge> t. Q t'))"
r_step_Pr: fixes e :: "nat" shows "eval r_step_Pr [e] \<down>= estep_Pr e"
restrict_in: fixes x :: "'c"   and A :: "'c set"   and m :: "'c \<Rightarrow> 'd option" assumes "x \<in> A" shows "(m |` A) x = m x"
D_def: fixes r :: "('c \<times> 'c) set"   and \<tau> :: "'c list"   and \<sigma> :: "'c list"   and \<sigma>' :: "'c list"   and \<tau>' :: "'c list" shows "D r \<tau> \<sigma> \<sigma>' \<tau>' = ((r|\<sigma> @ \<tau>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r \<and> (r|\<tau> @ \<sigma>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r)"
P: fixes p :: "real poly" assumes "0 < lead_coeff p"   and "poly p 0 < 0" shows "\<exists>x>0. poly p x = 0"
singleton_greek_less: fixes as :: "(accent \<times> 'a) list"   and r :: "('a \<times> 'a) set"   and b :: "'a"   and a :: "accent" assumes "snd ` set as \<subseteq> under r b" shows "(as, [(a, b)]) \<in> greek_less r"
compact_space_def: fixes X :: "'b topology" shows "compact_space X \<equiv> compactin X (topspace X)"
rep_eq: fixes x :: "'c stone_phi_pair"   and xa :: "'c stone_phi_pair" shows "Rep_stone_phi_pair (x \<sqinter> xa) = stone_phi.pairs_inf (Rep_stone_phi_pair x) (Rep_stone_phi_pair xa)"
Suc: shows "snd (foldl Gauss_Jordan_column_k_iarrays_PA (B, 0, A) [0..<k]) = foldl Gauss_Jordan_column_k_iarrays (0, A) [0..<k]"
strategy_attracts_via_SucE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts_via p \<sigma> v A W"   and "v \<notin> W" shows "\<exists>n. enat (Suc n) < llength P \<and> P $ Suc n \<in> W \<and> lset (ltake (enat (Suc n)) P) \<subseteq> A"
abs_eq: fixes x :: "'c \<Longrightarrow> 'd" shows "x = mk\<^sub>e get\<^bsub>x\<^esub>"
positively_invariant_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.positively_invariant f X M = (\<forall>x\<in>M. auto_ll_on_open.trapped_forward f X x M)"
not_neutral_obtains_not_neutral: fixes g :: "'d \<Rightarrow> 'c" assumes "Sum_any g \<noteq> (0::'c)" obtains a :: "'d" where "g a \<noteq> (0::'c)"
cases: fixes a1 :: "'a list"   and a2 :: "'b sequent" assumes "upRulesp a1 a2" obtains c :: "'b sequent" and R :: "'b" and Fs :: "'b form list" and ps :: "'a list" where "a1 = ps__" and "a2 = c__" and "c__ \<equiv>  \<LM> Compound R Fs__  \<RM> \<Rightarrow>* Em" and "ps__ \<noteq> []"   | c :: "'b sequent" and F :: "'b" and Fs :: "'b form list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "c \<equiv>  \<Empt> \<Rightarrow>* Compound F Fs" and "ps \<noteq> []"
None: shows "solve_depressed_cubic_Cardano_real e f = None"
tighten_bounds_3_1: fixes x :: "real_alg_3" shows "real_of_3 (tighten_bounds_3 x) = real_of_3 x"
assms_1: shows "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"
degv': fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "degree (Knuth_ex_4_6_2_22_main.v' q_ty b u v w) < degree v"
part1: shows "irreducible_mat a = (\<forall>i j. (Bij_Nat.to_nat i, Bij_Nat.to_nat j) \<in> (graph_of_mat a)\<^sup>+)"
response_AND_def: fixes R :: "'i \<times> 'c"   and W :: "'h \<times> 'b"   and s :: "'e" shows "response_AND R W s = (let (r0, r1) = R; (w0, w1) = W in response0 r0 w0 s \<bind> (\<lambda>z0. response1 r1 w1 s \<bind> (\<lambda>z1. return_spmf (z0, z1))))"
xfer_lrun_aux2: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and cl :: "nat \<Rightarrow> 'd"   and cs :: "nat \<Rightarrow> 'e" assumes "simulationI as\<^sub>0 alstep cs\<^sub>0 clstep R"   and "df_system.is_lrun cs\<^sub>0 clstep cl cs" obtains as :: "nat \<Rightarrow> 'b" where "df_system.is_lrun as\<^sub>0 alstep cl as" and "\<forall>i. R (as i) (cs i)"
test_de_morgan: fixes x :: "'a"   and y :: "'a" shows "!(\<tt>\<tt> x \<cdot> \<tt>\<tt> y) = !x + !y"
simps: fixes t :: "('a, 'sz) trm"   and i :: "'sz" shows "singleton t i = (if i = vid1 then t else Const 0)"
A: shows "X \<inter> {x. xs @ [x] \<in> T} = {}"
dense_lemma: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "carrier - (closure A - A) dense"
word_sint_1: shows "sint 1 = (if LENGTH('b) = 1 then - 1 else 1)"
last_snoc: fixes xs :: "'f list"   and x :: "'f" shows "last (xs @ [x]) = x"
decomp_rm\<^sub>e\<^sub>s\<^sub>t_single_2: fixes t :: "('fun, 'var) Term.term" shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)] = [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)]"
cong_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "local.cong R \<equiv> equivp R \<and> rel_fun (rel R) R eval eval"
rt_fresher_refl: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> rt"
ex_bij_betw_nat_finite: fixes M :: "'c set" assumes "finite M" shows "\<exists>h. bij_betw h {0..<card M} M"
similar_into_equiv: fixes x :: "'a itrm"   and y :: "'a itrm" assumes "x \<cong> y" shows "x \<simeq> y"
F: shows "s\<^sub>0 \<Turnstile> u\<^sub>1 \<and> u\<^sub>1 \<turnstile> v\<^sub>1 \<and> \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle> \<notin> spied u\<^sub>1 \<and> \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle> \<in> spied v\<^sub>1"
WT_swap_lassocr: fixes \<I>1 :: "('h, 'k) \<I>"   and \<I>2 :: "('i, 'l) \<I>"   and \<I>3 :: "('j, 'm) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3), \<I>2 \<oplus>\<^sub>\<I> (\<I>1 \<oplus>\<^sub>\<I> \<I>3) \<turnstile>\<^sub>C swap_lassocr \<surd>"
equal_pair_count_maximality_aux: fixes i :: "nat"   and ps :: "('b \<times> 'b) list" assumes "i = equal_pair_count ps" shows "length ps = i \<or> \<not> equal_pair (ps ! i)"
inv_two_faces: fixes g :: "graph" assumes "Invariants.inv g" shows "2 \<le> |faces g|"
simps_4: fixes uu :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom" shows "time ({uu/uv/e} CONSEQ C) = e"
resComm: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x><\<nu>y>P \<sim> <\<nu>y><\<nu>x>P"
leading_coeff_neq_0: fixes p :: "'d poly" assumes "p \<noteq> 0" shows "lead_coeff p \<noteq> (0::'d)"
iMODb_div_mod_gr0_not_ex: fixes k :: "nat"   and m :: "nat"   and c :: "nat"   and r :: "nat" assumes "0 < k"   and "k < m"   and "0 < m mod k"   and "k \<le> c" shows "\<nexists>r' m' c'. [ r', mod m', c' ] = [ r, mod m, c ] \<oslash> k"
simps_2: fixes l :: "'a list" shows "merge_list [] [l] = l"
unique_root_cong: fixes l :: "rat"   and r :: "rat"   and p :: "int poly"   and q :: "int poly" assumes "\<And>x. \<lbrakk>real_of_rat l \<le> x; x \<le> real_of_rat r\<rbrakk> \<Longrightarrow> (poly (real_of_int_poly p) x = 0) = (poly (real_of_int_poly q) x = 0)" shows "(\<exists>!x. root_cond (p, l, r) x) = (\<exists>!x. root_cond (q, l, r) x)"
foundation22: fixes \<tau> :: "'b state \<times> 'b state"   and X :: "'b state \<times> 'b state \<Rightarrow> 'c"   and Y :: "'b state \<times> 'b state \<Rightarrow> 'c" shows "(\<tau> \<Turnstile> X \<triangleq> Y) = (X \<tau> = Y \<tau>)"
l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_owner_document get_parent remove_child remove_child_locs get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs \<equiv> adopt_node = l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node get_owner_document get_parent remove_child get_disconnected_nodes set_disconnected_nodes \<and> adopt_node_locs = l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs"
DupElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some dup" shows "\<exists>v ops. os = v # ops \<and> n = Suc 0 \<and> t = (v # os, S, h) \<and> ll = l + 1"
frechet_derivative_eq_vector_derivative: fixes f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "f differentiable at x" shows "frechet_derivative f (at x) = (\<lambda>r. r *\<^sub>R vector_derivative f (at x))"
det_row_add_iterate_upt_n: fixes n :: "nat"   and A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c" assumes "n < nrows A" shows "det (row_add_iterate A n i j) = det A"
merge_Suc_aux: fixes \<rho> :: "nat \<Rightarrow> val"   and t :: "pdf_type" assumes "\<rho> \<in> space (state_measure (Suc ` V') (case_nat t \<Gamma>))" shows "(\<lambda>\<sigma>. \<delta> (merge V V' (\<sigma>, \<rho> \<circ> Suc))) \<in> borel_measurable (state_measure V \<Gamma>)"
l_set_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs \<equiv> ((\<forall>document_ptr disc_nodes h h'. writes (set_disconnected_nodes_locs document_ptr) (set_disconnected_nodes document_ptr disc_nodes) h h') \<and> (\<forall>h document_ptr disc_noded. type_wf h \<longrightarrow> document_ptr |\<in>| document_ptr_kinds h \<longrightarrow> h \<turnstile> ok set_disconnected_nodes document_ptr disc_noded)) \<and> (\<forall>h document_ptr disc_noded. h \<turnstile> ok set_disconnected_nodes document_ptr disc_noded \<longrightarrow> document_ptr |\<in>| document_ptr_kinds h) \<and> (\<forall>w document_ptr h h'. w \<in> set_disconnected_nodes_locs document_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w document_ptr h h'. w \<in> set_disconnected_nodes_locs document_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')"
\<nu>\<tau>: shows "\<guillemotleft>Map \<nu> \<star>\<^sub>B Map \<tau> : EVAL (Dom \<nu> \<^bold>\<star> Dom \<tau>) \<rightarrow>\<^sub>B EVAL (Cod \<nu> \<^bold>\<star> Cod \<tau>)\<guillemotright>"
is_factorization_of_def: fixes fctrs :: "'a \<times> ('a \<times> nat) list"   and p :: "'a poly" shows "is_factorization_of fctrs p = (interp_factorization fctrs = p \<and> distinct (map fst (snd fctrs)))"
N0: fixes n :: "nat" assumes "N0 \<le> n" shows "PS.prob (space M \<inter> (T ^^ n) -` K n) * indicat_real A0 n < alpha"
is_lbD: fixes S :: "'a set"   and l :: "'a"   and x :: "'a" assumes "S >| l"   and "x \<in> S" shows "l \<sqsubseteq> x"
generate_valid_stateful_policy_IFSACS_2_stateful_policy_compliance: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list"   and \<T> :: "'b stateful_policy" assumes "valid_reqs M"   and "wf_graph G"   and "TopoS_Composition_Theory.all_security_requirements_fulfilled M G"   and "set edgesList = edges G"   and "\<T> = TopoS_Stateful_Policy_Algorithm.generate_valid_stateful_policy_IFSACS_2 G M edgesList" shows "stateful_policy_compliance \<T> G M"
fix_bottom_iff: fixes F :: "'a \<rightarrow> 'a" shows "(fix\<cdot>F = \<bottom>) = (F\<cdot>\<bottom> = \<bottom>)"
coF_dtor_o_corec: fixes s :: "'e \<Rightarrow> ('f, 'f coF + 'e) coF0" shows "dtor_coF \<circ> dtor_corec_coF s = map_pre_coF id (case_sum id (dtor_corec_coF s)) \<circ> s"
assms_6: shows "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"
sublist_NIL: fixes l1 :: "'a list" shows "subseq l1 [] = (l1 = [])"
isovarspase_monom: fixes x :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "real" assumes "x \<notin> keys m"   and "a \<noteq> 0" shows "isolate_variable_sparse (MPoly_Type.monom m a) x 0 = MPoly_Type.monom m a"
simps_45: fixes x61 :: "msg"   and x62 :: "msg"   and x5 :: "msg" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Hash x5"
rep_eq: fixes x :: "'a ltln\<^sub>C"   and xa :: "'a ltln\<^sub>C" shows "local.ltln\<^sub>C_eq_test x xa = rep_ltln\<^sub>C x \<sim>\<^sub>C rep_ltln\<^sub>C xa"
rel_compp: fixes R1 :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and S1 :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and S2 :: "'h \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_prod (R1 OO S1) (R2 OO S2) = rel_prod R1 R2 OO rel_prod S1 S2"
FactGroup_iso: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h"   and "h ` carrier G = carrier H" shows "G Mod kernel G H h \<cong> H"
wt_getTvars: fixes \<sigma>l :: "'tp list" shows "list_all wt (getTvars \<sigma>l)"
span_add: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> cspan S"   and "y \<in> cspan S" shows "x + y \<in> cspan S"
fds_abs_converges_remainder_iff: fixes m :: "nat"   and f :: "'b fds"   and s :: "'b" shows "fds_abs_converges (fds_remainder m f) s = fds_abs_converges f s"
LCons: fixes x :: "'b"   and xs :: "'b llist" assumes "x \<notin> llist.lset xs"   and "ldistinct xs" shows "ldistinct (x ## xs)"
assms_2: fixes x :: "'a" assumes "x \<in> B" shows "infdist x A \<le> D"
r_x: shows "poly (sturm_squarefree' p ! (i + 2)) x = - poly (sturm_squarefree' p ! i) x"
fls_lr_inverse_X_power'_2: fixes n :: "nat"   and y :: "'a" shows "fls_right_inverse (fls_X ^ n) y = fls_const y * fls_X_inv ^ n"
bind_rule: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'd nres"   and \<Phi> :: "'d \<Rightarrow> bool" assumes "M \<le> SPEC (\<lambda>x. f x \<le> SPEC \<Phi>)" shows "M \<bind> f \<le> SPEC \<Phi>"
complete_game_return_true: fixes h :: "'pub_input"   and w :: "'witness"   and e :: "'challenge" assumes "(h, w) \<in> Rel"   and "completeness"   and "\<forall>h w. lossless_spmf (init h w)"   and "\<forall>r w e. lossless_spmf (response r w e)"   and "e \<in> challenge_space" shows "completeness_game h w e = return_spmf True"
assms_1: shows "\<pi> =\<^bsub>k\<^esub> \<pi>'"
NSLIMSEQ_minus: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a" shows "(\<lambda>n. - X n) \<longlonglongrightarrow>\<^sub>N\<^sub>S - a"
simps_18: fixes x21 :: "'b"   and x22 :: "('b, 'a) Term.term list" shows "vars_term (Fun x21 x22) = \<Union> (vars_term ` set x22)"
zs_gt_ys: shows "msetext_huet gt zs ys"
bind_return: fixes x :: "'m2" shows "bind2 x return2 = x"
Abs_type': fixes \<Gamma> :: "binding list"   and S :: "POPLmark.type"   and s :: "trm"   and U :: "POPLmark.type"   and U' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> (\<lambda>:S. s) : U \<rightarrow> U'" obtains S' :: "POPLmark.type" where "\<Gamma> \<turnstile> U <: S" and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'" and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'"
Lemma1_2: fixes x :: "ValD"   and x' :: "('o ValK \<rightarrow> 'o) \<rightarrow> 'o" assumes "\<eta>: x \<mapsto> x'" shows "x = Lub KMtoD_i\<cdot>x'"
fin_sum_periodic_arithmetic_set: fixes A :: "'a set"   and h :: "'a \<Rightarrow> nat \<Rightarrow> complex"   and k :: "nat" assumes "\<forall>i\<in>A. periodic_arithmetic (h i) k" shows "periodic_arithmetic (\<lambda>n. \<Sum>i\<in>A. h i n) k"
A: fixes nr :: "nat"   and nc :: "nat"   and A :: "'a mat" assumes "Matrix_Kernel.kernel nr nc A" shows "A \<in> carrier_mat nr nc"
matrix_match_condn_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and A1 :: "'b list list"   and A2 :: "'b list list"   and B1 :: "'b list list"   and B2 :: "'b list list"   and i :: "nat"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "plus_mult.matrix_match A1 A2 B1 B2 \<and> i < mult.row_length A1 * mult.row_length B1 \<and> j < length A2 * length B2" shows "mat_multI zer g f (mult.row_length (mult.Tensor f A1 B1)) (mult.Tensor f A1 B1) (mult.Tensor f A2 B2) ! j ! i = plus_mult.scalar_product f zer g (mult.vec_vec_Tensor f (row A1 (i div mult.row_length B1)) (row B1 (i mod mult.row_length B1))) (mult.vec_vec_Tensor f (col A2 (j div length B2)) (col B2 (j mod length B2)))"
iterateoi_rule: fixes s :: "'s" assumes "invar s" shows "set_iterator_linord (iterateoi s) (\<alpha> s)"
geq: fixes x :: "'b" assumes "x \<in> t" shows "(g \<circ> h \<circ> k) x = g x"
show_law_float10: fixes r :: "float10" shows "show_law showsp_float10 r"
app3: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list" assumes "list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs"   and "sequence_result rs = Rval vs"   and "do_opapp (rev vs) = None" shows "cupcake_evaluate_single env (App Opapp es) (Rerr (Rabort Rtype_error))"
eq_defined: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> \<bottom>"   and "y \<noteq> \<bottom>" shows "eq\<cdot>x\<cdot>y \<noteq> \<bottom>"
sigma_finite_measure_axioms: fixes M :: "'b pmf" shows "sigma_finite_measure (measure_pmf M)"
op_bfs_def: fixes c :: "(nat \<Rightarrow> nat list) \<times> (nat \<times> nat \<Rightarrow> int)"   and s :: "nat"   and t :: "nat" shows "bfs.op_bfs c s t \<equiv> Graph.bfs2 (snd c) (PR_CONST (\<lambda>(x, y). rg_succ2 x y) c) s t"
abstract_primitive_in_doubt_allow_Allow: fixes \<beta> :: "'c common_primitive \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'c common_primitive match_expr"   and p :: "('c, 'd) tagged_packet_scheme"   and disc :: "'c common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "matches (\<beta>, in_doubt_allow) m Accept p" shows "matches (\<beta>, in_doubt_allow) (abstract_primitive disc m) Accept p"