hcju/isabelleps huggingface dataset:
consistentPrefix: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" assumes "isPrefix valuation1 valuation2"   and "consistent valuation2" shows "consistent valuation1"
is_ring_iso_def: fixes R :: "('g, 'h) ring_scheme"   and S :: "('i, 'j) ring_scheme" shows "R \<simeq> S = (ring_iso R S \<noteq> {})"
hyps_2: shows "P,E(V \<mapsto> T) \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2)\<rangle>"
typ_ok_Tv: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and idn :: "variable"   and S :: "String.literal set" shows "typ_ok thy (Tv idn S) = wf_sort (subclass (osig (sig thy))) S"
mibi: shows "m\<inverse> \<inter> b\<inverse> = {}"
is_none_simps_2: fixes x :: "'a" shows "\<not> Option.is_none (Some x)"
trans: fixes r :: "'c"   and s :: "'c"   and t :: "'c" assumes "r = s"   and "s = t" shows "r = t"
indRelLTPO_impl_TRel_weakly_preserves_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelLTPO TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs TRel TWB"
cos_plus_pi: fixes z :: "'a" shows "cos (z + of_real pi) = - cos z"
assms_4: shows "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"
expands_to_root: fixes n :: "nat"   and F :: "'a"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real" assumes "0 < n"   and "trimmed_pos F"   and "basis_wf basis"   and "(f expands_to F) basis" shows "((\<lambda>x. root n (f x)) expands_to powr_expansion False F (inverse (real n))) basis"
vdisjnt_vemptyI: fixes A :: "V"   and B :: "V" assumes "\<And>x. \<lbrakk>x \<in>\<^sub>\<circ> A; x \<in>\<^sub>\<circ> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>"
list_all2_zip: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'd \<Rightarrow> 'c"   and xs :: "('a \<times> 'b \<times> 'c) list"   and ys :: "'a list"   and zs :: "'d list"   and Q :: "'a \<Rightarrow> bool"   and x :: "'a \<times> 'b \<times> 'c" assumes "list_all2 (\<lambda>x y. triple_eq_pair x y f g) xs (zip ys zs)"   and "\<And>y. y \<in> set ys \<Longrightarrow> Q y"   and "x \<in> set xs" shows "Q (fst x)"
maintained_def: fixes R :: "('a, 'b) labeled_graph \<times> ('a, 'b) labeled_graph"   and G :: "('a, 'c) labeled_graph" shows "maintained R G \<equiv> \<forall>f. graph_homomorphism (fst R) G f \<longrightarrow> extensible R G f"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs H" shows "rev_H a \<in> arcs H"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_6: fixes l :: "'d strand_label"   and a :: "poscheckvariant"   and t :: "('e, 'f) Term.term"   and s :: "('e, 'f) Term.term"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, \<langle>a: t \<in> s\<rangle>) # A) = (l, \<langle>a: t \<in> s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
map_scale_eq_monom_mult: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "c \<cdot> p = monom_mult c (0::'a) p"
matricize_def: fixes rmodes :: "nat set"   and T :: "'a tensor" shows "matricize rmodes T = Matrix.mat (prod_list (nths (dims T) rmodes)) (prod_list (nths (dims T) (- rmodes))) (\<lambda>(r, c). Tensor.lookup T (weave rmodes (digit_encode (nths (dims T) rmodes) r) (digit_encode (nths (dims T) (- rmodes)) c)))"
polys_4: fixes qa :: "int poly"   and ria :: "root_info" assumes "(qa, ria) \<in> set polys" shows "irreducible qa"
Suc_diff_Suc: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "Suc (m - Suc n) = m - n"
assms_4: shows "proots_count (map_poly complex_of_real P) (upper_circle_01 \<union> lower_circle_01) = 1"
nth_list_update_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i \<noteq> j" shows "xs[i := x] ! j = xs ! j"
indicator_UN_disjoint: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and x :: "'b" assumes "finite A"   and "disjoint_family_on f A" shows "indicator (\<Union> (f ` A)) x = (\<Sum>y\<in>A. indicator (f y) x)"
llength_ltake': fixes n :: "nat"   and xs :: "'b llist" assumes "enat n < llength xs" shows "llength (ltake (enat n) xs) = enat n"
rel_spmf_mono: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c spmf"   and g :: "'d spmf"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
length_suffixes: fixes xs :: "'c list" shows "length (suffixes xs) = Suc (length xs)"
assms_2: shows "g \<in> carrier (function_ring (carrier R) R)"
bt_map_append: fixes f :: "'d \<Rightarrow> 'c"   and t :: "'d bt"   and u :: "'d bt" shows "bt_map f (Binary_Tree.append t u) = Binary_Tree.append (bt_map f t) (bt_map f u)"
rlex_irrefl: fixes x :: "nat list" shows "\<not> x <\<^sub>r\<^sub>l\<^sub>e\<^sub>x x"
simple_distinct_ident: fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex" assumes "r \<in> simple_rotations"   and "distinct [a, b, c, d]"   and "\<forall>e\<in>{a, b, c}. r e \<noteq> e" shows "r d = d"
Span_smult_closed: fixes Us :: "'a list"   and k :: "'a"   and v :: "'a" assumes "set Us \<subseteq> carrier R"   and "k \<in> K"   and "v \<in> Span K Us" shows "k \<otimes> v \<in> Span K Us"
is_arg_min_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "ord.is_arg_min less f P x = (P x \<and> (\<nexists>y. P y \<and> less (f y) (f x)))"
sign_simps_35: fixes am :: "'b"   and bm :: "'b"   and ck :: "'b" shows "(am + bm) * ck = am * ck + bm * ck"
implies_dist_sup: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "- (x \<squnion> y) \<squnion> z = (- x \<squnion> z) \<sqinter> (- y \<squnion> z)"
fps_integral0_one: shows "fps_integral0 1 = fps_X"
exec_step_ind_IfFalse_nFalse: fixes stk :: "val list"   and i :: "int"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "hd stk \<noteq> Bool False" shows "exec_step_ind (StepI (IfFalse i)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
\<Sigma>fpub_unfold: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool" assumes "intruder_model arity Ana" shows "((\<Union>n. {f. arity f = n}) - {f. arity f = 0}) \<inter> (\<Union>n. {f. public f} \<inter> {f. arity f = n}) = {f. 0 < arity f \<and> public f}"
base_le: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s" shows "base j \<le> a j"
subset_diff: fixes B :: "'a set"   and A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "B \<subseteq> A"   and "finite A" shows "sum g A = sum g (A - B) + sum g B"
silent_moves_Append: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as' :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"   and "S,f \<turnstile> (ms'',s'') =as'\<Rightarrow>\<^sub>\<tau> (ms',s')" shows "S,f \<turnstile> (ms,s) =as @ as'\<Rightarrow>\<^sub>\<tau> (ms',s')"
\<pi>_upper_bound': fixes x :: "real" assumes "2 \<le> x" shows "\<pi> x < 443 / 139 * (x / ln x)"
strict_order_iff: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "(a \<prec> b) = (a = linordered_monoid.min a b \<and> a \<noteq> b)"
nonzero_coeffs_def: fixes a :: "('a \<times> 'b) multiset" shows "nonzero_coeffs a = ((0::'b) \<notin># snd `# a)"
eq2: shows "a ^ n * Measure.measure M {w. a \<le> f w} = a ^ n * \<integral> \<chi> {w. a \<le> f w} \<partial>M"
CESem_simps_6: fixes as :: "(Vars.var \<times> exp) list"   and body :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Terms.Let as body \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> (C\<cdot>r). (\<N>\<lbrakk> body \<rbrakk>\<^bsub>\<N>\<lbrace>as\<rbrace>\<rho>\<^esub>)\<cdot>r)"
assms_8: shows "\<forall>m\<le>k. ack (Suc (Suc (t + m))) = [connection_ok]"
canonical_successive_distinct: fixes n :: "nat"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and xs :: "nat list"   and i :: "nat"   and j :: "nat"   and B :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "canonical A n"   and "set xs \<subseteq> {0..n}"   and "i \<le> n"   and "j \<le> n"   and "distinct xs"   and "i \<notin> set xs"   and "j \<notin> set xs" shows "\<exists>ys. len (\<lambda>i j. min (A i j) (B i j)) i j ys \<le> len (\<lambda>i j. min (A i j) (B i j)) i j xs \<and> set ys \<subseteq> set xs \<and> successive (\<lambda>(a, b). min (A a b) (B a b) = A a b) (arcs i j ys) \<and> distinct ys \<and> i \<notin> set ys \<and> j \<notin> set ys"
f: shows "f \<in> \<aA> d \<beta> ve"
subdg_Obj_vsubset: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
smc_1_Comp_app: fixes \<aa> :: "V"   and \<ff> :: "V" shows "\<ff> \<circ>\<^sub>A\<^bsub>smc_1 \<aa> \<ff>\<^esub> \<ff> = \<ff>"
gcd_eq_gcd_factorial: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm" shows "gcd a b = factorial_semiring.gcd_factorial zero times one normalize a b"
cases: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'c option"   and a2 :: "'d option" assumes "ord_option ord a1 a2" obtains x :: "'d option" where "a1 = None" and "a2 = x__"   | x :: "'c" and y :: "'d" where "a1 = Some x" and "a2 = Some y" and "ord x y"
path_join_eq: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" assumes "path g1"   and "path g2" shows "path (g1 +++ g2) = (pathfinish g1 = pathstart g2)"
trans_gt: fixes zs :: "'a list"   and ys :: "'a list"   and xs :: "'a list" assumes "ext gt zs ys"   and "ext gt ys xs" shows "ext gt zs xs"
ndec_seqn: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "ndec_seq A a (Suc n) < ndec_seq A a n"
simps_1: fixes x :: "'a"   and ys :: "'a list" shows "replace x ys [] = []"
monom_zero: shows "MPoly_Type.monom 0 (0::'b) = 0"
pwd_spy_sep_agr_def: fixes n :: "nat" shows "pwd_spy_sep_agr n \<equiv> case pwd_spy_sep_map n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, PubKey C) s, S, A, B, C, D)"
sumElim: fixes \<Psi> :: "'b"   and \<phi> :: "'c"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual" assumes "\<Psi> \<rhd> P \<oplus>\<^sub>\<phi> Q \<longmapsto> Rs"   and "\<not> \<Psi> \<turnstile> \<phi>" shows "False"
spoly_def: fixes cmp_term :: "('a, 'b) pp nat_term_order"   and p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c"   and q :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "spoly_punit cmp_term p q = (let v1 = lt_punit cmp_term p; v2 = lt_punit cmp_term q in if () = () then let t1 = v1; t2 = v2; l = lcs t1 t2 in monom_mult_punit ((1::'c) / lookup p v1) (l - t1) p - monom_mult_punit ((1::'c) / lookup q v2) (l - t2) q else 0)"
0: shows "bprv (all yy (imp (NN \<langle>\<phi>\<rangle> (Var yy)) (eql \<langle>neg \<phi>\<rangle> (Var yy))))"
peek_res_subst_res: fixes P :: "vals \<Rightarrow> 'b assn"   and w :: "vals" shows "peek_res P\<leftarrow>w = P w\<leftarrow>w"
Lt_lt_conv: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Lt) = lt_of_comp comp x y"
enumerator_not_empty: fixes ls :: "nat list"   and m :: "nat"   and n :: "nat" assumes "ls \<in> set (enumerator m n)" shows "ls \<noteq> []"
cfun_eq_iff: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c" shows "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
rep_eq: fixes x :: "perm"   and xa :: "'a fset" shows "fset (x \<bullet> xa) = x \<bullet> fset xa"
usubstappf_and_usubstappp_det_1: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<phi> :: "fml"   and V :: "variable set" assumes "usubstappf \<sigma> U \<phi> \<noteq> undeff"   and "usubstappf \<sigma> V \<phi> \<noteq> undeff" shows "usubstappf \<sigma> U \<phi> = usubstappf \<sigma> V \<phi>"
bounded_linear_right: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'b" assumes "bounded_bilinear prod" shows "bounded_linear (prod a)"
vfsequence_vcons_vrange: fixes xs :: "V"   and x :: "V" assumes "vfsequence xs" shows "\<R>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = vinsert x (\<R>\<^sub>\<circ> xs)"
inj_on_add': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and A :: "'c set" assumes "class.cancel_semigroup_add plus" shows "inj_on (\<lambda>b. plus b a) A"
Sem_no_zero_height_derivs: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c :: "OBJ"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" assumes " s , c \<rightarrow>\<^sub>0  t " shows "False"
index_smult_vec_2: fixes a :: "'a"   and v :: "'a vec" shows "dim_vec (a \<cdot>\<^sub>v v) = dim_vec v"
bounded: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_bilinear prod" shows "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
H_2: fixes i :: "'a" shows "openin euclidean (X i)"
SS: fixes r :: "'d rexp"   and r' :: "'d rexp" assumes "r ~~ r'" shows "Star r ~~ Star r'"
ide_in_hom: fixes a :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" shows "S.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>S a\<guillemotright>"
matrix_to_iarray_fst_Gauss_Jordan_in_ij_det_P': fixes i :: "'c"   and j :: "'d"   and A :: "(('b, 'd) vec, 'c) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "fst (Gauss_Jordan_in_ij_det_P A i j) = fst (Gauss_Jordan_in_ij_det_P_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j))"
assms_2: shows "(v, b, v') \<in> \<delta> s r s'"
regular_homogenous_system: fixes a11 :: "'a"   and a12 :: "'a"   and a21 :: "'a"   and a22 :: "'a"   and x1 :: "'a"   and x2 :: "'a" assumes "det2 a11 a12 a21 a22 \<noteq> (0::'a)"   and "a11 * x1 + a12 * x2 = (0::'a)"   and "a21 * x1 + a22 * x2 = (0::'a)" shows "x1 = (0::'a) \<and> x2 = (0::'a)"
isSym_keys: fixes K :: "nat" shows "K \<in> symKeys"
elts_Sup: fixes X :: "V set" assumes "small X" shows "elts (\<Squnion> X) = \<Union> (elts ` X)"
is_valid_operator_sas_plus_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" shows "is_valid_operator_sas_plus \<Psi> op \<equiv> let pre = sas_plus_operator.precondition_of op; eff = effect_of op; vs = \<Psi>\<^sub>\<V>\<^sub>+; D = range_of \<Psi> in list_all (\<lambda>(v, a). ListMem v vs) pre \<and> list_all (\<lambda>(v, a). D v \<noteq> None \<and> ListMem a (the (D v))) pre \<and> list_all (\<lambda>(v, a). ListMem v vs) eff \<and> list_all (\<lambda>(v, a). D v \<noteq> None \<and> ListMem a (the (D v))) eff \<and> list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') pre) pre \<and> list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') eff) eff"
derivable_well_related: assumes "antisymmetric A (\<sqsubseteq>)"   and "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"   and "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x" shows "well_related_set (Collect derivable) (\<sqsubseteq>)"
second_vertex_def: shows "G.second_vertex \<equiv> \<lambda>xs. hd (tl xs)"
rep_eq: fixes x :: "('b, 'a) uexpr"   and xa :: "('b, 'a) uexpr" shows "\<lbrakk>x \<squnion> xa\<rbrakk>\<^sub>e = (\<lambda>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<squnion> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
source_typedef_lens: shows "\<S>\<^bsub>typedef\<^sub>L\<^esub> = A"
multiset_diff_union_assoc: fixes C :: "'c multiset"   and B :: "'c multiset"   and A :: "'c multiset" assumes "C \<subseteq># B" shows "A + B - C = A + (B - C)"
assms_2: shows "0 < emeasure M (\<Union> (range A))"
mset_factors_imp_dvd: fixes F :: "'a multiset"   and x :: "'a"   and f :: "'a" assumes "mset_factors F x"   and "f \<in># F" shows "f dvd x"
intros: 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 tl :: "'t \<times> ('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" assumes "redT s tl s'"   and "m\<tau>move s tl s'" shows "\<tau>mredT s s'"
foldl_break_append: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s :: "'a \<Rightarrow> bool"   and a :: "'a"   and xs :: "'b list"   and ys :: "'b list" shows "foldl_break f s a (xs @ ys) = (if s (foldl_break f s a xs) then foldl_break f s a xs else foldl_break f s (foldl_break f s a xs) ys)"
never_winning_is_losing_even: fixes A :: "'b llist set"   and N :: "enat"   and p :: "'b list"   and \<sigma> :: "'b list \<Rightarrow> 'b" assumes "GaleStewartDeterminedGames.closed_GSgame A N"   and "GSgame.position N p"   and "\<forall>n. \<not> (GSgame.position N ((augment_list \<sigma> ^^ n) p) \<and> (\<exists>\<sigma>'. GSgame.strategy_winning_by_Even A N \<sigma>' ((augment_list \<sigma> ^^ n) p)))" shows "GSgame.induced_play N \<sigma> p \<notin> A"
lookup_add_monom_1: fixes lp :: "(nat, rat) fmap"   and v :: "nat"   and c :: "rat" assumes "get_var_coeff lp v + c \<noteq> 0" shows "fmlookup (add_monom c v lp) v = Some (get_var_coeff lp v + c)"
y: fixes i :: "'a" assumes "i \<in> I" shows "y i \<in> S i"
assms: fixes u :: "'a"   and v :: "'a" assumes "u \<in> nodes g"   and "v \<in> nodes g" shows "(u, v) \<in> (edges g)\<^sup>*"
select_convs_1: fixes Functions :: "'a \<Rightarrow> (real, 'b) vec \<Rightarrow> real"   and Predicates :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> bool"   and Contexts :: "'c \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set \<Rightarrow> ((real, 'b) vec \<times> (real, 'b) vec) set"   and Programs :: "'b \<Rightarrow> (((real, 'b) vec \<times> (real, 'b) vec) \<times> (real, 'b) vec \<times> (real, 'b) vec) set"   and ODEs :: "'b \<Rightarrow> (real, 'b) vec \<Rightarrow> (real, 'b) vec"   and ODEBV :: "'b \<Rightarrow> 'b set"   and more :: "'d" shows "Functions \<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> = Functions"
real_root_gt_1_iff: fixes n :: "nat"   and y :: "real" assumes "0 < n" shows "(1 < root n y) = (1 < y)"
ln: shows "l = Bk \<up> ln @ Bk # Bk # ires"
VectorSpace_VectorSpaceEndSet: shows "VectorSpace (\<cdot>\<cdot>) VectorSpaceEndSet"
rel_gpv''_map_gpv_1: fixes A :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'b \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and gpv :: "('g, 'h, 'f) gpv" shows "rel_gpv'' A C R (map_gpv f g gpv) \<equiv> rel_gpv'' (\<lambda>a. A (f a)) (\<lambda>c. C (g c)) R gpv"
strict_implies_order: fixes b :: "'k"   and a :: "'k" assumes "b < a" shows "b \<le> a"
simps_1: fixes P :: "(name, value) fmap \<Rightarrow> (Term_Class.term \<times> sterm) list \<Rightarrow> bool"   and Q :: "name \<Rightarrow> bool"   and R :: "name fset \<Rightarrow> bool"   and name :: "name"   and vs :: "value list" shows "value_pred.pred P Q R (Vconstr name vs) = (Q name \<and> list_all id (map (value_pred.pred P Q R) vs))"
le_supI: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" assumes "a \<preceq> x"   and "b \<preceq> x" shows "a \<oplus> b \<preceq> x"
hyps_4: assumes "list_all is_Insert (unlabel A)" shows "list_all is_Insert (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
distinct_1: fixes a' :: "'b" shows "\<bottom> \<noteq> Def a'"
scalar_product_Nil2: fixes xs :: "nat list" shows "scalar_product xs [] = 0"
nonneg_linorder_cases: 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" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less_eq zero a"   and "less_eq zero b" obtains "less a b"   | "a = b"   | "less b a"
simps_1: shows "FingerTreeStruc.viewRn Empty = None"
denote_trace: fixes S :: "com"   and \<rho> :: "complex mat" assumes "well_com S"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "trace (denote S \<rho>) \<le> trace \<rho>"
npowf_inv1: fixes x :: "'a"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>"   and "\<not> 0 \<le> j" shows "x\<^bsub>K\<^esub>\<^bsup>j\<^esup> = x\<^bsup>\<hyphen> K\<^esup>\<^bsub>K\<^esub>\<^bsup>- j\<^esup>"
inner_simps_2: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<bullet> (y + z) = x \<bullet> y + x \<bullet> z"
map_simp_ide: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<nu>\<tau>.map a = C (\<nu> a) (\<tau> a)"
cnj_sin: fixes z :: "complex" shows "cnj (sin z) = sin (cnj z)"
node_arriveTE': fixes i :: "nat"   and s :: "'c"   and R :: "nat set"   and H :: "nat set"   and K :: "nat set"   and m :: "'d"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i s R, H\<not>K:arrive(m), s') \<in> node_sos S" obtains s' :: "'c" where "s'a__ = NodeS i s' R" and "(s, receive m, s') \<in> S" and "H = {i}" and "K = {}"   | "s'a__ = NodeS i s R" and "H = {}" and "K = {i}"
Pair_inject: fixes a :: "'d"   and b :: "'e"   and a' :: "'d"   and b' :: "'e" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
fold_Cons_rev: fixes xs :: "'b list" shows "fold (#) xs = (@) (rev xs)"
simps_3: fixes var :: "nat"   and p :: "real mpoly"   and as :: "atom list" shows "find_eq var (Eq p # as) = (let (A, B) = find_eq var as in if MPoly_Type.degree p var < 3 \<and> MPoly_Type.degree p var \<noteq> 0 then (p # A, B) else (A, Eq p # B))"
le_infE: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a \<cdot> b" obtains "x \<le> a" and "x \<le> b"
simple_alt_def_all: shows "\<forall>bl\<in>#\<B>. multiplicity bl = 1"
X_def: shows "(X::'a::refinement_lattice set) = {x::'a::refinement_lattice}"