hcju/isabelleps huggingface dataset:
dur_monoD: fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat"   and j :: "nat" assumes "dur xs i < dur xs j" shows "i < j"
root_info_cond_def: fixes ri :: "root_info"   and p :: "int poly" shows "root_info_cond ri p \<equiv> (\<forall>a b. a \<le> b \<longrightarrow> root_info.l_r ri a b = card {x. root_cond (p, a, b) x}) \<and> (\<forall>a. root_info.number_root ri a = card {x. x \<le> real_of_rat a \<and> poly (real_of_int_poly p) x = 0})"
jipsen2r: fixes y :: "'a"   and x :: "'a" shows "y \<leftarrow> ((y \<leftarrow> x) \<rightarrow> y) = y \<leftarrow> x"
null: shows "g ` {x \<in> S. det (matrix (g' x)) \<noteq> 0} - g ` S = {}"
bij_betw_id: fixes A :: "'a set" shows "bij_betw id A A"
abs_eq: fixes x :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "Square_Matrix.det (of_fun x) = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. x i (p i)))"
insertions_last_del: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and ref :: "'c" shows "insertions (xs @ [(oid, Delete ref)]) = insertions xs"
subst_sends_fv_to_img: fixes t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "fv (t \<cdot> s) \<subseteq> fv t \<union> range_vars s"
minus_carrier_mat': fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A - B \<in> carrier_mat nr nc"
simps: fixes a :: "16 word"   and b :: "16 word"   and c :: "16 word"   and d :: "16 word"   and e :: "16 word"   and f :: "16 word"   and g :: "16 word"   and h :: "16 word" shows "ipv6preferred_to_int (IPv6AddrPreferred a b c d e f g h) = (UCAST(16 \<rightarrow> 128) a << 16 * 7) || (UCAST(16 \<rightarrow> 128) b << 16 * 6) || (UCAST(16 \<rightarrow> 128) c << 16 * 5) || (UCAST(16 \<rightarrow> 128) d << 16 * 4) || (UCAST(16 \<rightarrow> 128) e << 16 * 3) || (UCAST(16 \<rightarrow> 128) f << 16 * 2) || (UCAST(16 \<rightarrow> 128) g << 16 * 1) || (UCAST(16 \<rightarrow> 128) h << 16 * 0)"
admissability_conditions_2: shows "\<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1)) = \<b> * \<k>"
comp_of_ord_eq_comp_of_ords: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "antisymp le" shows "comp_of_ord le = comp_of_ords le (strict le)"
Rep_int'_inverse: fixes x :: "int'" shows "Abs_int' (Rep_int' x) = x"
affine_hull_finite_step: fixes S :: "'a set"   and a :: "'a"   and w :: "real"   and y :: "'a" assumes "finite S" shows "(\<exists>u. sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y) = (\<exists>v u. sum u S = w - v \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a)"
assms_5: shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"
subset_cls_val_process: fixes A :: "'c set"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "finite A"   and "support_set p \<subseteq> A" shows "\<forall>n w. cls_val_process Mkt p (Suc n) w = (\<Sum>x\<in>A. prices Mkt x (Suc n) w * p x (Suc n) w)"
pos_bounded: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
indRelTEQ_impl_TRel_is_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "\<forall>P Q P'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q')" shows "strong_reduction_simulation (TRel\<^sup>*) Target"
D: shows "bn_count ns \<le> foldl (+) 0 ns"
intersects_x_axis_intersects_y_axis: fixes H :: "circline" shows "intersects_x_axis (moebius_circline (moebius_rotation (pi / 2)) H) = intersects_y_axis H"
indep_vars_def2: fixes M' :: "'e \<Rightarrow> 'f measure"   and X :: "'e \<Rightarrow> 'a \<Rightarrow> 'f"   and I :: "'e set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
elims_2: fixes x :: "ipt_tcp_flags"   and xa :: "tcp_flag set" assumes "match_tcp_flags x xa" obtains fmask :: "tcp_flag set" and c :: "tcp_flag set" and flags :: "tcp_flag set" where "x = TCP_Flags fmask c" and "xa = flags" and "flags \<inter> fmask = c"
intro: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h shadow_root_ptr h' r ptr'. \<lbrakk>h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'; r \<in> get_disconnected_nodes_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_delete_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"
simps_4: fixes e\<^sub>1 :: "'a allegorical_term"   and e\<^sub>2 :: "'a allegorical_term" shows "translation (A_Int e\<^sub>1 e\<^sub>2) = (let G\<^sub>1 = translation e\<^sub>1; G\<^sub>2 = translation e\<^sub>2 in graph_union G\<^sub>1 (map_graph_fn G\<^sub>2 (\<lambda>x. if x < 2 then x else x + card (vertices G\<^sub>1) - 2)))"
Conc_eq_map_rexp_iff_2: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e rexp"   and r :: "'d rexp"   and s :: "'d rexp" shows "(map_rexp f x = Conc r s) = (\<exists>r' s'. x = Conc r' s' \<and> map_rexp f r' = r \<and> map_rexp f s' = s)"
matrix_to_iarray_snd_QR_decomposition: fixes A :: "((real, 'c) vec, 'd) vec" shows "matrix_to_iarray (snd (QR_decomposition A)) = snd (QR_decomposition_iarrays (matrix_to_iarray A))"
simps: fixes h :: "'b \<Rightarrow> int"   and t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" shows "rm h (T t1 x t2) = (if t2 = Tip then x else rm h t2)"
compat_act: fixes g :: "Vertex \<Rightarrow> Vertex"   and h :: "Vertex \<Rightarrow> Vertex"   and x :: "Vertex" shows "g \<in> carrier tetrahedral_group \<and> h \<in> carrier tetrahedral_group \<longrightarrow> apply_rotation g (apply_rotation h x) = apply_rotation (g \<otimes>\<^bsub>tetrahedral_group\<^esub> h) x"
size_mset_set_zip_leq: fixes ss :: "'c list"   and ts :: "'d list" shows "size_mset (mset (zip ss ts)) \<le> size_list size ss + size_list size ts"
M'_def: shows "M' \<equiv> \<lambda>cfg. distr (MDP.MC.T (repc' cfg)) (stream_space (count_space UNIV)) (smap absc)"
cast_shadow_root_ptr_not_node_ptr_2: fixes node_ptr :: "(_) node_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" shows "cast node_ptr \<noteq> cast shadow_root_ptr"
A1: fixes x :: "i"   and y :: "i" shows "(E (dom x) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (cod y)) \<^bold>\<or> dom x \<noteq> cod y) \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E (x \<cdot> y) \<^bold>\<leftarrow> E (dom x) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (cod y)) \<^bold>\<or> dom x \<noteq> cod y))"
simplicial_simplex_def: fixes p :: "nat"   and S :: "('a \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "simplicial_simplex p S f \<equiv> singular_simplex p (subtopology (powertop_real UNIV) S) f \<and> (\<exists>l. f = oriented_simplex p l)"
psubst_fresh_coname: fixes a :: "coname"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and M :: "trm" assumes "a \<sharp> \<theta>_n"   and "a \<sharp> \<theta>_c"   and "a \<sharp> M" shows "a \<sharp> \<theta>_n,\<theta>_c<M>"
intersection_of_def: fixes P :: "'c set set \<Rightarrow> bool"   and Q :: "'c set \<Rightarrow> bool" shows "P intersection_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Inter> \<U> = S"
assms_2: shows "is_set_to_list Rk Rs tsl"
norm_equiv: fixes x :: "'a itrm" shows "norm x \<simeq> x"
lt_def: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lpp p = (if p = 0 then punit.min_term else ordered_powerprod_lin.Max (keys p))"
simps_2: fixes \<mu> :: "'c" shows "Ide \<^bold>\<langle>\<mu>\<^bold>\<rangle> = C.ide \<mu>"
measurable_op_sem: fixes oper :: "pdf_operator"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "op_type oper t = Some t'" shows "op_sem oper \<in> stock_measure t \<rightarrow>\<^sub>M stock_measure t'"
list_def: fixes f :: "'a \<Rightarrow> xml_tree list"   and xs :: "'a list" shows "xml.list f xs = map (xml.node \<circ> f) xs"
neval_S_n_3: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and cmn' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "n < length cms"   and "lcn'' = (cms ! n, mem)"   and "lcn' = (cmn', mem')" shows "(cms, mem) \<rightarrow>\<^bsub>replicate k n\<^esub> (cms[n := cmn'], mem')"
correctCompositionDiffLevelsA41: shows "correctCompositionDiffLevels sA41"
subspace_0: fixes S :: "('c \<times> 'e \<Rightarrow>\<^sub>0 'd) set" assumes "pprod.pmdl.subspace S" shows "0 \<in> S"
cprops: shows "\<forall>w\<in>space (sigma (space M) {{}, space M}). disc_rfr_proc r n w = c"
in_rel: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('a, 'c) tllist"   and b :: "('b, 'd) tllist" shows "tllist_all2 R1 R2 a b = (\<exists>z. z \<in> {x. tset x \<subseteq> {(x, y). R1 x y} \<and> set2_tllist x \<subseteq> {(x, y). R2 x y}} \<and> tmap fst fst z = a \<and> tmap snd snd z = b)"
llist_all2_lfiniteD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist" assumes "llist_all2 P xs ys" shows "lfinite xs = lfinite ys"
getGMUserCom: fixes st :: "'c"   and u :: "'a"   and c :: "'b" shows "getGMUserCom (State st u c) = (u, c)"
f_join_cut_less_eq: fixes xs :: "'b list"   and t :: "nat"   and I :: "nat set" assumes "length xs \<le> t" shows "xs \<Join>\<^sub> (I \<down>< t) = xs \<Join>\<^sub> I"
Try: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"
p_def: fixes a :: "'a" shows "p a = (if a \<in> A then p' (THE X. a \<in> X \<and> X \<in> P) a else a)"
dvd_gcd_mult: fixes k :: "'a"   and p :: "'a"   and q :: "'a"   and r :: "'a" assumes "k dvd p * q"   and "k dvd p * r" shows "k dvd p * gcd q r"
class_Object: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "\<exists>C fs ms. class P Object = \<lfloor>(C, fs, ms)\<rfloor>"
pdevs_val_eqI: fixes y :: "'b pdevs"   and x :: "'b pdevs"   and e :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "\<And>i. \<lbrakk>i \<in> pdevs_domain y; i \<in> pdevs_domain x\<rbrakk> \<Longrightarrow> e i *\<^sub>R pdevs_apply x i = f i *\<^sub>R pdevs_apply y i"   and "\<And>i. \<lbrakk>i \<in> pdevs_domain y; i \<notin> pdevs_domain x\<rbrakk> \<Longrightarrow> f i *\<^sub>R pdevs_apply y i = (0::'b)"   and "\<And>i. \<lbrakk>i \<in> pdevs_domain x; i \<notin> pdevs_domain y\<rbrakk> \<Longrightarrow> e i *\<^sub>R pdevs_apply x i = (0::'b)" shows "pdevs_val e x = pdevs_val f y"
elementary_symmetric_monoidal_category_axioms: shows "elementary_symmetric_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit runit assoc sym"
conf_list_widen: fixes G :: "prog"   and s :: "st"   and vs :: "val list"   and Ts :: "ty list"   and Ts' :: "ty list" assumes "ws_prog G"   and "list_all2 (conf G s) vs Ts"   and "G\<turnstile>Ts[\<preceq>]Ts'" shows "list_all2 (conf G s) vs Ts'"
subterms_refl: fixes p :: "('d, 'e, 'f, 'g) seqp" shows "p \<in> subterms p"
mono_aval'': fixes S1 :: "'a st option"   and S2 :: "'a st option"   and e :: "aexp" assumes "S1 \<le> S2" shows "aval'' e S1 \<le> aval'' e S2"
map_simp_2: fixes f :: "'a" assumes "A.arr f" shows "local.map f = \<tau> f \<cdot>\<^sub>B \<sigma> (A.dom f)"
closure_of_closedin: fixes X :: "'c topology"   and S :: "'c set" assumes "closedin X S" shows "X closure_of S = S"
induct: fixes P :: "'d seq_action \<Rightarrow> bool"   and seq_action :: "'d seq_action" assumes "\<And>x. P (broadcast x)"   and "\<And>x1a x2. P (groupcast x1a x2)"   and "\<And>x1a x2. P (unicast x1a x2)"   and "\<And>x. P (\<not>unicast x)"   and "\<And>x. P (send x)"   and "\<And>x. P (deliver x)"   and "\<And>x. P (receive x)"   and "P \<tau>\<^sub>s" shows "P seq_action"
iT_Div_iSince_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "0 < k" shows "(P t1. t1 \<S> t2 I \<oslash> k. Q t2) = (P (t1 div k). t1 \<S> t2 I. Q (t2 div k))"
Suc_5: fixes k :: "nat" assumes "k < Suc n" shows "f k = g k + h k"
is_sublist_at5: fixes as :: "'a list"   and bs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct (as @ bs)"   and "is_sublist [x, y] (as @ bs)" shows "is_sublist [x, y] as \<or> is_sublist [x, y] bs \<or> x = last as \<and> y = hd bs"
cong_symmetry: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong C D A B"
idx_inj: fixes bound :: "'a bound"   and x :: "'a"   and y :: "'a" assumes "in_bound index_class.less_eq index_class.less bound x"   and "in_bound index_class.less_eq index_class.less bound y"   and "idx bound x = idx bound y" shows "x = y"
Comp: fixes G :: "'a triple set"   and P :: "'a assn"   and c :: "com"   and Q :: "'a assn"   and d :: "com"   and R :: "'a assn" assumes "G|-{P}. c .{Q}"   and "G|-{Q}. d .{R}" shows "G|-{P}. (c;; d) .{R}"
eq_impD: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and s :: "'e"   and s' :: "'e" assumes "eq_imp f g"   and "\<forall>x. f x s = f x s'" shows "g s = g s'"
smult_closed: fixes x :: "'a vec"   and a :: "'a" assumes "True"   and "x \<in> carrier_vec n" shows "a \<cdot>\<^sub>v x \<in> carrier_vec n"
that: fixes B :: "real" assumes "0 \<le> B"   and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> B" shows "thesis"
rep_eq: fixes x :: "'b interval"   and xa :: "'b interval" shows "bounds_of_interval (x + xa) = (case bounds_of_interval x of (a, b) \<Rightarrow> \<lambda>(c, d). (a + c, b + d)) (bounds_of_interval xa)"
\<mu>_p: shows "C\<mu> = C \<cdot> \<mu> \<and> is_ground_subst \<mu>"
inv_is_inverse: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.inverse_arrows f (C.VV.inv f)"
map_fun_apply: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'e"   and h :: "'g \<Rightarrow> 'h"   and x :: "'f" shows "map_fun f g h x = g (h (f x))"
af_G_ltl_continuation: fixes \<phi> :: "'a ltl"   and w' :: "nat \<Rightarrow> 'a set"   and w :: "'a set list" assumes "\<forall>\<psi>\<in>\<^bold>G \<phi>. w' \<Turnstile> \<psi> = (w \<frown> w') \<Turnstile> \<psi>" shows "(w \<frown> w') \<Turnstile> \<phi> = w' \<Turnstile> af\<^sub>G \<phi> w"
derived_set_consistent: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H G" shows "derived_set (G\<lparr>carrier := H\<rparr>) K = derived_set G K"
hyps_2: fixes t' :: "'a tree" assumes "delete x r = Some t'" shows "set_of t' \<subseteq> set_of r"
exp_strong_induct: fixes P :: "'a \<Rightarrow> exp \<Rightarrow> bool"   and c :: "'a"   and exp :: "exp" assumes "\<And>var c. P c (Var var)"   and "\<And>exp var c. (\<And>c. P c exp) \<Longrightarrow> P c (App exp var)"   and "\<And>\<Gamma> exp c. \<lbrakk>atom ` domA \<Gamma> \<sharp>* c; \<And>c x. x \<in> domA \<Gamma> \<Longrightarrow> P c (the (map_of \<Gamma> x)); \<And>c. P c exp\<rbrakk> \<Longrightarrow> P c (Let \<Gamma> exp)"   and "\<And>var exp c. \<lbrakk>{atom var} \<sharp>* c; \<And>c. P c exp\<rbrakk> \<Longrightarrow> P c (Lam [var]. exp)"   and "\<And>b c. P c (Bool b)"   and "\<And>scrut e1 e2 c. \<lbrakk>\<And>c. P c scrut; \<And>c. P c e1; \<And>c. P c e2\<rbrakk> \<Longrightarrow> P c (scrut ? e1 : e2)" shows "P c exp"
MinRVarsEq_def: fixes min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat" shows "MinRVarsEq min_rvar_incdec_eq \<equiv> ((\<forall>dir s eqa is. min_rvar_incdec_eq dir s eqa = Inl is \<longrightarrow> (\<forall>x\<in>rvars_eq eqa. \<not> (0 < coeff (rhs eqa) x \<and> \<lhd>\<^sub>u\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.UB dir s x) \<or> coeff (rhs eqa) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.LB dir s x))) \<and> set is = {Direction.LI dir s (lhs eqa)} \<union> {Direction.LI dir s x |x. x \<in> rvars_eq eqa \<and> coeff (rhs eqa) x < 0} \<union> {Direction.UI dir s x |x. x \<in> rvars_eq eqa \<and> 0 < coeff (rhs eqa) x} \<and> (dir = Positive \<or> dir = Negative \<longrightarrow> Direction.LI dir s (lhs eqa) \<in> indices_state s \<longrightarrow> set is \<subseteq> indices_state s)) \<and> (\<forall>dir s eqa x\<^sub>j. min_rvar_incdec_eq dir s eqa = Inr x\<^sub>j \<longrightarrow> x\<^sub>j \<in> rvars_eq eqa)) \<and> (\<forall>dir s eqa x\<^sub>j. min_rvar_incdec_eq dir s eqa = Inr x\<^sub>j \<longrightarrow> 0 < coeff (rhs eqa) x\<^sub>j \<and> \<lhd>\<^sub>u\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x\<^sub>j) (Direction.UB dir s x\<^sub>j) \<or> coeff (rhs eqa) x\<^sub>j < 0 \<and> \<rhd>\<^sub>l\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x\<^sub>j) (Direction.LB dir s x\<^sub>j)) \<and> (\<forall>dir s eqa x\<^sub>j. min_rvar_incdec_eq dir s eqa = Inr x\<^sub>j \<longrightarrow> (\<forall>x\<in>rvars_eq eqa. x < x\<^sub>j \<longrightarrow> \<not> (0 < coeff (rhs eqa) x \<and> \<lhd>\<^sub>u\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.UB dir s x) \<or> coeff (rhs eqa) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.LB dir s x))))"
coherent_Assoc'_Ide: fixes a :: "'c local.term"   and b :: "'c local.term"   and c :: "'c local.term" assumes "Ide a"   and "Ide b"   and "Ide c"   and "Src a = Trg b"   and "Src b = Trg c" shows "coherent \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[a, b, c\<^bold>]"
l_child_def: fixes l :: "int"   and p :: "int" shows "l_child l p = 2 * p - l + 1"
that: fixes s :: "'a" assumes "s \<le> t \<and> s \<in>\<^sub>A frontier M" shows "thesis"
ecard_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "ecard (f ` A) = ecard A"
prod_mset_dict: shows "\<Prod>\<^sub># \<equiv> comm_monoid_mult.prod_mset (*) (1::'a)"
ord_option_simps_4: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a" shows "ord_option ord (Some x) None = False"
valid_empty_edges_iff_exists_offending_flows: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph G"   and "\<not> sinvar G nP" shows "sinvar \<lparr>nodes = nodes G, edges = {}\<rparr> nP = (SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<noteq> {})"
rmge2_init_total: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "total1 f" shows "rmge2 (f \<triangleright> n) = (if \<forall>i<Suc n. the (f i) < 2 then Some 0 else f (GREATEST i. i < Suc n \<and> 2 \<le> the (f i)))"
permute_multiset_def: fixes p :: "perm"   and M :: "'b multiset" shows "p \<bullet> M = image_mset (permute p) M"
negative_cycle_dest_diag: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "\<not> cycle_free_up_to m k n"   and "k \<le> n" shows "\<exists>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<and> len m i i xs < (0::'a)"
path_from_to_first: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto> w" shows "v \<notin> set (tl xs)"
finite_measure_finite_Union: fixes S :: "'a set"   and A :: "'a \<Rightarrow> 'key set" assumes "finite S"   and "A ` S \<subseteq> K.events"   and "disjoint_family_on A S" shows "K.prob (\<Union> (A ` S)) = (\<Sum>i\<in>S. K.prob (A i))"
2: fixes H' :: "ocircline" assumes "A \<in> ocircline_set H' \<and> B \<in> ocircline_set H' \<and> C \<in> ocircline_set H'" shows "H' = moebius_ocircline (moebius_inv M) o_x_axis \<or> H' = opposite_ocircline (moebius_ocircline (moebius_inv M) o_x_axis)"
almost_all_commutative: fixes S :: "'b set"   and P :: "'b \<Rightarrow> nat \<Rightarrow> bool" assumes "finite S" shows "(\<forall>x\<in>S. \<forall>\<^sub>\<infinity>i. P x i) = (\<forall>\<^sub>\<infinity>i. \<forall>x\<in>S. P x i)"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "'state" assumes "a1 \<rightarrow>t a2" obtains atm :: "'atom" and s :: "'state" where "a1 = (Atm atm, sa__)" and "a2 = aval atm sa__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, s)" and "a2 = s" and "\<not> tval tst s"
rel_ar_def: fixes R :: "('a \<times> 'a) set" shows "rel_ar R = {(y, y) |y. \<nexists>x. (x, y) \<in> R}"
f_Exec_Stream_Acc_Output__State_Idle_nth: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "nat"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and xs :: "'f message_af list"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. n \<le> i \<Longrightarrow> State_Idle localState output_fun trans_fun (f_Exec_Comp_Stream_Acc_LocalState l localState trans_fun xs c ! n)"   and "i < length xs" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c ! i = f_Exec_Comp_Stream_Acc_Output l output_fun trans_fun xs c ! i"
mustbeint: shows "rat_of_int (coeff (pCons b2 c2) (degree q)) / rat_of_int (lead_coeff q) = coeff (pCons (rat_of_int b2) (of_int_poly c2)) (degree (of_int_poly q)) / lead_coeff (of_int_poly q)"
ordinal_rec0_mono_lemma: fixes s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and w :: "ordinal"   and z :: "'a" assumes "\<forall>p x. x << s p x" shows "\<forall>y\<le>w. \<forall>x\<le>y. ordinal_rec0 z s lub x << ordinal_rec0 z s lub y"
lemma_9_4: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and g1 :: "'t \<Rightarrow>\<^sub>0 'b"   and h :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_rewrite_ord rword"   and "is_RB_upt d rword G u"   and "inj_on lt G"   and "\<not> is_syz_sig d u"   and "is_canon_rewriter rword G u g1"   and "h \<in> G"   and "is_sig_red (\<prec>\<^sub>t) (=) {h} (monom_mult (1::'b) (pp_of_term u - lp g1) g1)"   and "d (pp_of_term u) \<le> dgrad_max d" shows "lt (spair g1 h) = u"
simps: fixes ff_ops :: "'a arith_ops_record"   and modulus :: "'a list \<Rightarrow> 'a list"   and a :: "'a list"   and n :: "nat" shows "power_poly_f_mod_i ff_ops modulus a n = (if n = 0 then modulus (one_poly_i ff_ops) else let (d, r) = divmod_nat n 2; rec = power_poly_f_mod_i ff_ops modulus (modulus (times_poly_i ff_ops a a)) d in if r = 0 then rec else modulus (times_poly_i ff_ops rec a))"
Lyndon_std: fixes w :: "'a list" assumes "Lyndon w"   and "1 < \<^bold>|w\<^bold>|" obtains l :: "'a list" and m :: "'a list" where "w = l \<cdot> m" and "Lyndon l" and "Lyndon m" and "l <lex m"
Standard_hIm: fixes x :: "complex star" assumes "x \<in> SComplex" shows "hIm x \<in> Standard"