hcju/isabelleps huggingface dataset:
dgrad_sig_set_closed_sig_red_rtrancl: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_red :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "F \<subseteq> dgrad_sig_set' (length fs) d"   and "p \<in> dgrad_sig_set' (length fs) d"   and "(sig_red sing_red top_tail F)\<^sup>*\<^sup>* p q" shows "q \<in> dgrad_sig_set' (length fs) d"
simps_4: fixes v :: "nat formula" shows "cnf_to_dimacs (\<^bold>\<not> v) = [disj_to_dimacs (\<^bold>\<not> v)]"
antilinear_def: fixes f :: "'d \<Rightarrow> 'e" shows "antilinear f \<equiv> additive f \<and> antilinear_axioms f"
add_diff_add: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" shows "a + c - (b + d) = a - b + (c - d)"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<alpha> \<prec> P'"   and "guarded P"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) \<alpha> P' A\<^sub>P (\<Psi>\<^sub>P \<otimes> \<one>)"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (!P) \<alpha> P' [] (\<one>)"
fps_unit_dvd_left: fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "\<exists>g. 1 = f * g"
leadsETo_def: fixes A :: "'d set"   and CC :: "'d set set"   and B :: "'d set" shows "A leadsTo[CC] B = {F. (A, B) \<in> elt CC F}"
f2: assumes "x \<noteq> y"   and "atom y \<sharp> T"   and "atom x \<sharp> (y \<leftrightarrow> x) \<bullet> T" shows "atom y \<sharp> f x T"
Rep_point_ext_inverse: fixes x :: "'a point_scheme" shows "Abs_point_ext (Rep_point_ext x) = x"
closed_cdi_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "closed_cdi \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> (\<forall>s\<in>M. \<Omega> - s \<in> M) \<and> (\<forall>A. range A \<subseteq> M \<and> A 0 = {} \<and> (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow> \<Union> (range A) \<in> M) \<and> (\<forall>A. range A \<subseteq> M \<and> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M))"
eij_def: fixes i :: "nat"   and j :: "nat" shows "eij i j = lcm (a ! i) (b ! j) div b ! j"
intros_3: fixes v :: "Vertices"   and w :: "Vertices"   and vs :: "Vertices list" assumes "v \<rightarrow>\<^bsub>T3\<^esub> w"   and "walk (w # vs)" shows "walk (v # w # vs)"
rev_it_def: fixes xs :: "'b list" shows "rev_it xs = rev_it' [] xs"
the_\<nu>_props_2: shows "local.iso the_\<nu>"
l_delete_shadow_root_get_shadow_root_def: fixes get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_shadow_root get_shadow_root_locs \<equiv> \<forall>h shadow_root_ptr h' r ptr'. 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' \<longrightarrow> r \<in> get_shadow_root_locs ptr' \<longrightarrow> r h h'"
hom_closed: fixes x :: "(bool \<times> 'd) list" assumes "x \<in> carrier \<F>\<^bsub>I\<^esub>" shows "G.lift g x \<in> carrier G"
sinks_aux_member_ipurge_tr: fixes u :: "'b"   and I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and v :: "'b"   and U :: "'b set"   and xs :: "'c list"   and w :: "'b" assumes "u \<in> sinks_aux I D (insert v U) xs"   and "(u, w) \<in> I"   and "\<not> ((v, w) \<in> I \<or> (\<exists>v'\<in>sinks I D v xs. (v', w) \<in> I))" shows "u \<in> sinks_aux I D U (ipurge_tr I D v xs)"
PreCoxeterSystemWithDeletion_axioms: shows "PreCoxeterSystemWithDeletion S"
Vector_Spaces_linear_vec: shows "Vector_Spaces.linear (*) (*s) vec"
simps_1: fixes E :: "'b uprod set"   and u :: "'b"   and v :: "'b" shows "epath E u [] v = (u = v)"
cases: fixes a1 :: "'Var \<Rightarrow> Sec option"   and a2 :: "'AExp"   and a3 :: "Sec" assumes "a1 \<turnstile>\<^sub>a a2 \<in> a3" obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" and e :: "'AExp" where "a1 = \<Gamma>" and "a2 = e" and "a3 = max_dom (to_total \<Gamma> ` aexp_vars e)"
adjacent_int_facet1: fixes x :: "'b set"   and y :: "'b set" assumes "x \<sim> y"   and "x \<noteq> y" shows "x \<inter> y \<lhd> x"
small_steps_determ: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "small_steps \<pi> cs None" shows "\<not> small_steps \<pi> cs (Some (SKIP, t))"
snil_append_iff: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "(xs :@ ys = [::]) = (xs = [::] \<and> ys = [::])"
distinct_139: shows "GreaterThan \<noteq> ShiftRightSigned"
prems: shows "H \<in> set (wall_crossings (C # D # Ds))"
big_step_t3_post_dom_conv: fixes c :: "com"   and ps :: "char list \<Rightarrow> int option"   and t :: "nat"   and ps' :: "char list \<Rightarrow> int option" assumes "(c, ps) \<Rightarrow>\<^sub>A t \<Down> ps'" shows "dom ps' = dom ps"
InDomConc: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators" assumes "p \<noteq> []"   and "x \<in> dom (C (list2FWpolicy p))" shows "x \<in> dom (C (list2FWpolicy (a # p)))"
Prec_trans: fixes D :: "'g"   and A :: "'f"   and B :: "'f"   and C :: "'f" assumes "Prec_F_g D A B"   and "Prec_F_g D B C" shows "Prec_F_g D A C"
ereal_Sup: fixes A :: "real set" assumes "\<bar>\<Squnion> (ereal ` A)\<bar> \<noteq> \<infinity>" shows "ereal (\<Squnion> A) = \<Squnion> (ereal ` A)"
simps_1: fixes uu :: "'a extB" shows "(extB.Bot \<le> uu) = True"
birkhoff_theorem_AE_nonergodic: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> real" assumes "integrable M' f" shows "AE x in M'. (\<lambda>n. T.birkhoff_sum f n x / real n) \<longlonglongrightarrow> real_cond_exp M' T.Invariants f x"
IH: fixes t :: "'a trm"   and s :: "'a trm" assumes "(v \<lhd> \<sigma>, u \<lhd> \<sigma>) \<in> trm_ord"   and "subterm t q u"   and "replace_subterm t q v s" shows "(s \<lhd> \<sigma>, t \<lhd> \<sigma>) \<in> trm_ord"
exists_indexed_eval_aux_monomial: fixes P :: "'d multiset \<Rightarrow> 'a"   and Qs :: "('d multiset \<Rightarrow> 'a) list"   and n :: "'d multiset"   and i :: "'d"   and k :: "nat" assumes "carrier_coeff P"   and "list_all carrier_coeff Qs"   and "count n i = k"   and "P n \<noteq> \<zero>"   and "list_all (\<lambda>Q. index_free Q i) Qs" obtains m :: "'d multiset" where "count m i = length Qs + k" and "indexed_eval_aux (Qs @ [P]) i m \<noteq> \<zero>"
add_strict_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.strict_ordered_ab_semigroup_add plus less_eq less"   and "less a b"   and "less c d" shows "less (plus a c) (plus b d)"
subtrees_split: fixes l :: "('b \<times> 'c) list"   and a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) list" shows "set (subtrees (l @ (a, b) # r)) = set (subtrees l) \<union> set (subtrees r) \<union> {a}"
aux: fixes xs :: "bit list"   and n :: "nat" shows "length (delblockhelp xs n) \<le> length xs"
simps_6: fixes I :: "\<I>"   and \<phi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Prev I \<phi>) = MFOTL.future_reach \<phi> - enat (left I)"
add_ac_3: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "sup_prod b (sup_prod a c) = sup_prod a (sup_prod b c)"
T_Assign: fixes p :: "local_expr"   and \<tau> :: "thread_state"   and l' :: "V + bool \<Rightarrow> nat"   and v :: "V"   and e :: "local_expr"   and a :: "abs_level" assumes "eval_bool p \<tau>"   and "l' = (l \<tau>)(Inl v := eval_word e \<tau>)" shows "step_t a (\<tau>, Assign (Var v) e, p) (\<tau>\<lparr>l := l'\<rparr>)"
const_term_eq_last: fixes p :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R" shows "const_term (p @ [a]) = a"
algebraic_imp_represents_unique: fixes x :: "'a" assumes "algebraic x" shows "\<exists>!p. p represents x \<and> irreducible p \<and> 0 < lead_coeff p"
zero_upt_k_i: shows "is_zero_row_upt_k i k A"
simps: fixes x :: "Concrete.var"   and e :: "exp"   and s :: "Concrete.var \<Rightarrow> nat" shows "aval (Assign x e) s = s(x := eval e s)"
Hyp: shows "\<forall>ys xs. (ys, xs) \<in> {(ys, xs). (\<forall>y\<in>set ys. SN_on {(s, t). fst (g s t)} {y}) \<and> fst (mul_ext g ys xs)} \<longrightarrow> SN_on {(ys, xs). (\<forall>y\<in>set ys. SN_on {(s, t). fst (g s t)} {y}) \<and> fst (mul_ext g ys xs)} {ys}"
intro: fixes x :: "'c \<Longrightarrow> 'd" assumes "weak_lens x"   and "pbij_lens_axioms x" shows "pbij_lens x"
possible_substitute: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and x :: "'b" shows "possible (substitute f T t) x = possible t x"
induct: fixes x1 :: "'a multiset"   and x2 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and P :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" assumes "(x1, x2) \<in> multpw ns"   and "P {#} {#}"   and "\<And>x y X Y. \<lbrakk>(x, y) \<in> ns; (X, Y) \<in> multpw ns; P X Y\<rbrakk> \<Longrightarrow> P (add_mset x X) (add_mset y Y)" shows "P x1 x2"
h'_norm_pres: fixes h' :: "'c \<Rightarrow> real"   and x0 :: "'c"   and H :: "'c set"   and h :: "'c \<Rightarrow> real"   and xi :: "real"   and H' :: "'c set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real" assumes "\<And>x. h' x = (let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"   and "H' = H + lin x0"   and "x0 \<notin> H"   and "x0 \<in> E"   and "x0 \<noteq> (0::'c)"   and "vectorspace E"   and "Subspace.subspace H E"   and "seminorm E p"   and "linearform H h"   and "\<forall>y\<in>H. h y \<le> p y"   and "\<forall>y\<in>H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y" shows "\<forall>x\<in>H'. h' x \<le> p x"
mod_0_imp_dvd: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_modulo plus minus zero times one divide modulo"   and "modulo a b = zero" shows "dvd.dvd times b a"
cfcomp2: fixes f :: "'c \<rightarrow> 'a"   and g :: "'b \<rightarrow> 'c"   and x :: "'b" shows "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
ext_inject: fixes NTDom :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NTCod :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NatTransMap :: "'d \<Rightarrow> 'g"   and more :: "'j"   and NTDom' :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NTCod' :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NatTransMap' :: "'d \<Rightarrow> 'g"   and more' :: "'j" shows "(\<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = \<lparr>NTDom = NTDom', NTCod = NTCod', NatTransMap = NatTransMap', \<dots> = more'\<rparr>) = (NTDom = NTDom' \<and> NTCod = NTCod' \<and> NatTransMap = NatTransMap' \<and> more = more')"
ring_hom_cringI: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h"   and "cring R"   and "cring S" shows "ring_hom_cring R S h"
csk: shows "cs\<^bsup>\<pi>\<^esup> k = xs @ [\<pi> k]"
converse_rtranclE: fixes x :: "'c"   and z :: "'c"   and r :: "('c \<times> 'c) set" assumes "(x, z) \<in> r\<^sup>*" obtains "x = z"   | y :: "'c" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>*"
cases: fixes a1 :: "'opSym"   and a2 :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsInp a1 a2" obtains delta :: "'opSym" and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" where "a1 = delta" and "a2 = inp" and "wlsOpS delta" and "\<And>i. arOf delta i = None \<and> inp i = None \<or> (\<exists>s X. arOf delta i = Some s \<and> inp i = Some X \<and> wls s X)"
complement_index: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>" shows "\<B>\<^sup>C index ps = size {#b \<in># \<B>. \<forall>x\<in>ps. x \<notin> b#}"
lattice_ow_axioms: shows "lattice_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1"
f2: shows "y' \<in> (r' |\<^bsub>rel\<^esub> s')"
SMP_I: fixes s :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "s \<in> M"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "t \<sqsubseteq> s \<cdot> \<delta>"   and "\<And>v. wf\<^sub>t\<^sub>r\<^sub>m (\<delta> v)" shows "t \<in> SMP M"
ir: shows "root n (real_of_1 x) \<le> real_of_rat ir"
prv_imp_com: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> (imp \<chi> \<psi>))" shows "prv (imp \<chi> (imp \<phi> \<psi>))"
scf_alts: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "scf R \<subseteq> alts"
wellformed_defP: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp" shows "wellformed \<Gamma> = wfP (\<lambda>q p. p \<leadsto>\<^bsub>\<Gamma>\<^esub> q)"
distrib_right: fixes plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m"   and c :: "'m" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
transpose_index: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat"   and j :: "nat" assumes "mat nr nc m"   and "i < nr"   and "j < nc" shows "Matrix_Legacy.transpose nr m ! i ! j = m ! j ! i"
dg_Cod_vrange: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
Abs_quot_inject: fixes x :: "'d set"   and y :: "'d set" assumes "x \<in> quot"   and "y \<in> quot" shows "(Abs_quot x = Abs_quot y) = (x = y)"
set_less_eq_aux''_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq>''' B"   and "B \<sqsubseteq>''' A"   and "A \<in> infinite_complement_partition"   and "B \<in> infinite_complement_partition" shows "A = B"
in_set_insert: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> ran\<^sub>l xs" shows "List.insert x xs = xs"
diff_add: fixes M :: "'a multiset"   and N :: "'a multiset"   and Q :: "'a multiset" shows "M - (N + Q) = M - N - Q"
C: shows "bn_inv p q (u, Suc (Suc m) # ns, xs)"
Tree_wf_eqvt_aux: fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree"   and p :: "perm" assumes "(t1, t2) \<in> Tree_wf" shows "(p \<bullet> t1, p \<bullet> t2) \<in> Tree_wf"
analz_synth_subset: fixes X :: "msg set"   and X' :: "msg set"   and Y :: "msg set"   and Y' :: "msg set" assumes "analz X \<subseteq> synth (analz X')"   and "analz Y \<subseteq> synth (analz Y')" shows "analz (X \<union> Y) \<subseteq> synth (analz (X' \<union> Y'))"
simps_2: fixes pf :: "'form"   and i :: "nat"   and "is" :: "nat list" shows "nodeOf (pf, i # is) = iNodeOf (tree_at (it' pf) is)"
simps: fixes v :: "(real, 'a) vec \<times> (real, 'a) vec"   and x :: "'a"   and r :: "real" shows "repd v x r = (fst v, \<chi>y. if x = y then r else snd v $ y)"
multiset_eqI: fixes A :: "'c multiset"   and B :: "'c multiset" assumes "\<And>x. count A x = count B x" shows "A = B"
add_lattice: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.aggregation_lattice inf less_eq less sup bot top plus" shows "plus x y = plus (sup x y) (inf x y)"
l2_complete: fixes f :: "nat \<Rightarrow> real \<Rightarrow> real"   and S :: "real set" assumes "\<And>i. f i square_integrable S"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. l2norm S (\<lambda>x. f m x - f n x) < e" obtains g :: "real \<Rightarrow> real" where "g square_integrable S" and "(\<lambda>n. l2norm S (\<lambda>x. f n x - g x)) \<longlonglongrightarrow> 0"
145: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- (x \<sqinter> y) \<sqinter> x \<squnion> z) \<sqinter> y = z \<sqinter> y"
rt_fresher_update: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and ip :: "nat" assumes "dip \<in> kD rt"   and "1 \<le> the (dhops rt dip)"   and "update_arg_wf r" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> update rt ip r"
leg1_simps_4: shows "trg (tab\<^sub>1 g) = trg g"
assms_2: shows "\<exists>n w. b n w \<noteq> 0"
adj_mat_2: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "A * adj_mat A = det A \<cdot>\<^sub>m 1\<^sub>m n"
alphaRes: fixes y :: "name"   and P :: "('d, 'e, 'f) psi"   and x :: "name" assumes "y \<sharp> P" shows "\<lparr>\<nu>x\<rparr>P = \<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)"
weakenP: fixes \<sigma> :: "'a"   and m :: "'a" assumes "P \<sigma> m" shows "P' \<sigma> m"
strict_iff_order: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry" shows "(b \<prec> a) = (b \<preceq> a \<and> a \<noteq> b)"
delete_eq: fixes k :: "'c" shows "AList.delete k = filter (\<lambda>(k', uu_). k \<noteq> k')"
assms: shows "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
set_of_mul_contains_zero: fixes A :: "'a interval"   and B :: "'a interval" assumes "(0::'a) \<in>\<^sub>i A \<or> (0::'a) \<in>\<^sub>i B" shows "(0::'a) \<in>\<^sub>i A * B"
zipWithL_strict_1: fixes f :: "'d \<rightarrow> 'c \<rightarrow> 'b"   and ys :: "'c LList" shows "zipWithL\<cdot>f\<cdot>\<bottom>\<cdot>ys = \<bottom>"
ord_isom_restrictedS: fixes D :: "'c Order"   and E :: "'d Order"   and f :: "'c \<Rightarrow> 'd"   and D1 :: "'c set" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "D1 \<subseteq> carrier D" shows "ord_isom (SIod D D1) (SIod E (f ` D1)) (restrict f D1)"
diff_add_cancel: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a -\<^sub>o\<^sub>w b +\<^sub>o\<^sub>w b = a"
lookup_not_eq_zero_eq_in_keys: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "(lookup f k \<noteq> (0::'b)) = (k \<in> keys f)"
init: shows "nfa.init M \<subseteq> nfa.states M"
bounded_linear: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_antilinear f" shows "bounded_linear f"
square_free_separable_main: fixes f :: "'c poly" assumes "square_free f"   and "\<not> separable f" shows "\<exists>g k. f = g * k \<and> degree g \<noteq> 0 \<and> pderiv g = 0"
dependent_single: fixes x :: "'b" shows "dependent {x} = (x = (0::'b))"
apply_swap_simp_1: fixes a :: "'a"   and b :: "'a" shows "\<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> a = b"
assms_6: shows "(_Entry_) -as'\<rightarrow>* (_Exit_)"
scc_ofI_reachable: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v" shows "u \<in> H.scc_of v"
assms: shows "fdisjnt (frees t) (fmdom env\<^sub>1)"
analz_Un_implInsecSet: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({asym.implInsec A B M |A B M. M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
card_degen_lotteries_equals_outcomes: fixes out :: "'a set" shows "card {x \<in> \<P> out. card (set_pmf x) = 1} = card out"
c_def: shows "c \<equiv> \<lambda>x. midpoint (a x) (b x)"
simp_thms_22: fixes P :: "bool" shows "(True \<and> P) = P"
in_KeysI: fixes s :: "'e"   and f :: "'e \<Rightarrow>\<^sub>0 'f"   and F :: "('e \<Rightarrow>\<^sub>0 'f) set" assumes "s \<in> keys f"   and "f \<in> F" shows "s \<in> Keys F"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (domain_semigroup.ds_ord d (\<cdot>) b a \<and> domain_semigroup.ds_ord d (\<cdot>) a b)"
monitorable_formula_code: fixes \<phi> :: "'a MFOTL.formula" shows "mmonitorable \<phi> = mmonitorable_exec \<phi>"
bounded: fixes y :: "'t \<Rightarrow>\<^sub>0 'b" assumes "y \<in> Q"   and "y \<noteq> 0" shows "y \<in> dgrad_p_set d m \<and> dickson_less_v d m (lt y) u"
insertion_monom: fixes i :: "nat \<Rightarrow> 'a"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "insertion i (monom m a) = a * (\<Prod>k\<in>keys m. i k ^ lookup m k)"
conc_respects_priv: fixes x\<^sub>C :: "'Var\<^sub>C"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val"   and cms\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and n :: "nat"   and cm\<^sub>Cn' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)"   and cms\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and \<R>n :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" assumes "x\<^sub>C \<notin> range var\<^sub>C_of"   and "mem\<^sub>C x\<^sub>C \<noteq> mem\<^sub>C' x\<^sub>C \<or> dma\<^sub>C mem\<^sub>C x\<^sub>C \<noteq> dma\<^sub>C mem\<^sub>C' x\<^sub>C"   and "(cms\<^sub>C ! n, mem\<^sub>C) \<leadsto>\<^sub>C (cm\<^sub>Cn', mem\<^sub>C')"   and "((cms\<^sub>A ! n, mem\<^sub>A), cms\<^sub>C ! n, mem\<^sub>C) \<in> \<R>n"   and "preserves_local_guarantee_compliance \<R>n"   and "abs.sound_mode_use (cms\<^sub>A, mem\<^sub>A)"   and "n < length cms"   and "length cms\<^sub>A = length cms"   and "length cms\<^sub>C = length cms" shows "x\<^sub>C \<notin> snd (cms\<^sub>C ! n) GuarNoWrite \<and> x\<^sub>C \<notin> snd (cms\<^sub>C ! n) GuarNoReadOrWrite"
flow_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and t0 :: "real"   and x0 :: "'a" assumes "ll_on_open T f X" shows "ll_on_open.flow T f X t0 x0 = (\<lambda>t. if t \<in> ll_on_open.existence_ivl T f X t0 x0 then ll_on_open.csol T f X t0 x0 t t else (0::'a))"
distinct_7: fixes x5 :: "char list" shows "Void \<noteq> Class x5"
simps: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" shows "Diagonal_to_Smith_row_i_PQ_JNF i bezout (P, A, Q) = diagonal_to_Smith_i_PQ_JNF [i + 1..<min (dim_row A) (dim_col A)] i bezout (P, A, Q)"
in_\<R>: fixes i :: "nat"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "i < length cms" shows "((cms\<^sub>A ! i, mem\<^sub>A_of mem\<^sub>C), cms\<^sub>C ! i, mem\<^sub>C) \<in> \<R>_rel (cms ! i)"
Union_eq: fixes A :: "'c set set" shows "\<Union> A = {x. \<exists>B\<in>A. x \<in> B}"
rem_condless_valid_1: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "exec_plan s as = exec_plan s (rem_condless_act s [] as)"
Smith_1xn_aux_S'_AQ': fixes S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "(S', Q') = Smith_1xn_aux i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "S \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n"   and "S = A * Q"   and "i < n" shows "S' = A * Q'"