hcju/isabelleps huggingface dataset:
of_nat_diff: fixes plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
extract_matrix_cong: fixes m :: "nat"   and n :: "nat"   and a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "\<And>i. i < m * n \<Longrightarrow> a i = b i" shows "extract_matrix a m n = extract_matrix b m n"
conjugate_def: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "conjugate f g \<equiv> \<forall>x y. (f x \<sqinter> y = bot) = (x \<sqinter> g y = bot)"
adds_triv_right: fixes a :: "'a"   and b :: "'a" shows "a adds b + a"
eq_upto_seq_inconsist_def: fixes P :: "'f prog" shows "eq_upto_seq_inconsist P = foldl_list_all2 (\<lambda>ob ob' vs. mrw_value P vs ob) (\<lambda>ob ob' vs. case ob of NormalAction (ReadMem ad al v) \<Rightarrow> \<exists>b. vs (ad, al) = \<lfloor>(v, b)\<rfloor> | NormalAction _ \<Rightarrow> True | _ \<Rightarrow> True) (\<lambda>ob ob' vs. if case ob of NormalAction (ReadMem ad al v) \<Rightarrow> \<exists>b. vs (ad, al) = \<lfloor>(v, b)\<rfloor> | NormalAction _ \<Rightarrow> True | _ \<Rightarrow> True then ob = ob' else ob \<approx> ob')"
deriv_linear: fixes c :: "'b" shows "deriv ((*) c) = (\<lambda>z. c)"
summable'_2: fixes p :: "nat" assumes "prime p" shows "norm (f p) < 1"
order_iff: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
simps_51: fixes I :: "\<I>"   and r :: "formula Regex.regex" shows "safe_formula (MatchF I r) = safe_regex Futu Strict r"
pos_mod_bound: fixes b :: "'g"   and a :: "'g" assumes "(0::'g) < b" shows "a mod b < b"
Subobjs_isClass: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs P C Cs" shows "is_class P C"
ord_of_nonzero_1: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "0 \<le> ord_Zp x" shows "x \<noteq> \<zero>"
Eq1: shows "\<bar>te p i - te q i\<bar> \<le> \<beta>"
bis_coind2: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and theta :: "('a \<times> 'a) set" assumes "Bisim.mono Retr"   and "theta \<subseteq> Retr (theta \<union> bis Retr)"   and "theta\<inverse> \<subseteq> Retr (theta\<inverse> \<union> bis Retr)" shows "theta \<subseteq> bis Retr"
that: fixes r :: "'a list" assumes "vpath r G"   and "hd r = u"   and "last r = w" shows "thesis"
extend_measure_def: fixes \<Omega> :: "'c set"   and I :: "'d set"   and G :: "'d \<Rightarrow> 'c set"   and \<mu> :: "'d \<Rightarrow> ennreal" shows "extend_measure \<Omega> I G \<mu> = (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0) then measure_of \<Omega> (G ` I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>') else sigma \<Omega> (G ` I))"
chanEqSupp: fixes M :: "'a"   and N :: "'a" shows "supp (M \<leftrightarrow> N) \<subseteq> supp M \<union> supp N"
fls_of_nat_nth: fixes n :: "nat"   and k :: "int" shows "fls_nth (of_nat n) k = (if k = 0 then of_nat n else (0::'a))"
microstep_choiceI1: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" shows "(p1 \<oplus> p2) \<leadsto>\<^bsub>\<Gamma>\<^esub> p1"
right_lesseq: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a l\<rightarrow> b = (1::'b))"
set_sel_1: fixes a :: "('a, 'b) hd" assumes "is_Sym a" shows "hd.sym a \<in> syms_hd a"
l1_inv4_def: shows "l1_inv4 \<equiv> {s. \<forall>Rb A B gnx. 0 < signalsResp s (Commit A B (Exp gnx (NonceF (Rb $ ny)))) \<longrightarrow> guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Rb xgnx = Some gnx}"
dfs_invariant: fixes S :: "('a \<times> 'b \<times> 'a) list"   and xs :: "('a \<times> 'b \<times> 'a) list" assumes "set S \<subseteq> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"   and "list_all (\<lambda>t. t \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0) xs" shows "set (dfs S xs) \<subseteq> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"
assms_3: shows "pf risk_free_asset (Suc n) w \<noteq> 0"
assms_1: shows "\<forall>\<^sub>F x in F. f x = c"
ev_introR: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<phi> \<pi>l" shows "Shallow.ev \<phi> \<pi>l"
simps_2: fixes x2 :: "'m"   and y2 :: "'m" shows "(Ok x2 = Ok y2) = (x2 = y2)"
Infinity: shows "Elem Empty HOLZF.Inf \<and> (\<forall>x. Elem x HOLZF.Inf \<longrightarrow> Elem (SucNat x) HOLZF.Inf)"
zero_less_divide_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide one a) = less zero a"
islimpt_greaterThanLessThan2: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "b islimpt {a<..<b}"
bv_prepend: fixes x :: "nat"   and b :: "'a"   and bv :: "'a list" shows "bv_prepend x b bv = replicate x b @ bv"
functor_axioms_def: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" shows "functor_axioms AA BB F \<equiv> (F \<^bsub>\<a>\<^esub> \<in> Ar\<^bsub>AA\<^esub> \<rightarrow> Ar\<^bsub>BB\<^esub> \<and> F \<^bsub>\<o>\<^esub> \<in> Ob\<^bsub>AA\<^esub> \<rightarrow> Ob\<^bsub>BB\<^esub> \<and> (\<forall>f\<in>Ar\<^bsub>AA\<^esub>. F \<^bsub>\<o>\<^esub> (Dom\<^bsub>AA\<^esub> f) = Dom\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f)) \<and> (\<forall>f\<in>Ar\<^bsub>AA\<^esub>. F \<^bsub>\<o>\<^esub> (Cod\<^bsub>AA\<^esub> f) = Cod\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f) \<and> (\<forall>A\<in>Ob\<^bsub>AA\<^esub>. F \<^bsub>\<a>\<^esub> (Id\<^bsub>AA\<^esub> A) = Id\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A) \<and> (\<forall>f\<in>Ar\<^bsub>AA\<^esub>. \<forall>g\<in>Ar\<^bsub>AA\<^esub>. Cod\<^bsub>AA\<^esub> f = Dom\<^bsub>AA\<^esub> g \<longrightarrow> F \<^bsub>\<a>\<^esub> (g \<bullet>\<^bsub>AA\<^esub> f) = F \<^bsub>\<a>\<^esub> g \<bullet>\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f)"
external_from_internal_invariant: fixes S :: "('a, 'b) spec"   and P :: "'a set"   and Q :: "'b set" assumes "reach S \<subseteq> P"   and "obs S ` P \<subseteq> Q" shows "oreach S \<subseteq> Q"
removel_correct_2: fixes s :: "'s" assumes "invar s"   and "\<alpha> s \<noteq> []" shows "\<alpha> (snd (removel s)) = tl (\<alpha> s)"
path_component_subset_connected_component: fixes S :: "'b set"   and x :: "'b" shows "path_component_set S x \<subseteq> connected_component_set S x"
interchange: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a"   and \<sigma> :: "'a" assumes "B.seq \<nu> \<mu>"   and "B.seq \<tau> \<sigma>" shows "\<nu> \<cdot>\<^sub>B \<mu> \<star>\<^sub>B \<tau> \<cdot>\<^sub>B \<sigma> = (\<nu> \<star>\<^sub>B \<tau>) \<cdot>\<^sub>B (\<mu> \<star>\<^sub>B \<sigma>)"
\<A>\<^sub>d_2: shows "(\<S>\<^sub>0, []) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>, \<A>\<^sub>d)"
dist_ne_ninf: fixes v :: "'a" assumes "v \<in> verts G" shows "dist v \<noteq> - \<infinity>"
char_distr_add: fixes X1 :: "'a \<Rightarrow> real"   and X2 :: "'a \<Rightarrow> real"   and t :: "real" assumes "indep_var borel X1 borel X2" shows "char (distr M borel (\<lambda>\<omega>. X1 \<omega> + X2 \<omega>)) t = char (distr M borel X1) t * char (distr M borel X2) t"
cases: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b" assumes "r\<^sup>+\<^sup>+ a1 a2" obtains a :: "'b" and b :: "'b" where "a1 = a__" and "a2 = b__" and "r a__ b__"   | a :: "'b" and b :: "'b" and c :: "'b" where "a1 = a" and "a2 = c" and "r\<^sup>+\<^sup>+ a b" and "r b c"
bigo_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "O[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
right_neutral: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a :: "'c" assumes "Groups.monoid f z" shows "f a z = a"
finite_UNIV_llength_eq: fixes n :: "nat" assumes "finite UNIV" shows "finite {w. llength w = enat n}"
exhaust: fixes y :: "(_) object_ptr" obtains x :: "'e" where "y = object_ptr.Ext x"
bind_def: fixes e :: "'d \<Rightarrow> 'g"   and c :: "'g \<Rightarrow> ('d, 'e, 'f) com" shows "bind e c = DynCom (\<lambda>s. c (e s))"
simps_7: fixes X :: "'d list"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and G :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>) \<or>\<notin>: G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>\<rangle>"
nn_integral_snd: fixes f :: "'c \<times> 'd \<times> 'e \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>S \<partial>T \<Otimes>\<^sub>M P = integral\<^sup>N (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) f"
char_distr_sum: fixes X :: "'b \<Rightarrow> real \<Rightarrow> real"   and A :: "'b set"   and t :: "real" assumes "\<mu>.indep_vars (\<lambda>i. borel) X A" shows "char (distr \<mu> borel (\<lambda>\<omega>. \<Sum>i\<in>A. X i \<omega>)) t = (\<Prod>i\<in>A. char (distr \<mu> borel (X i)) t)"
VA: shows "VSagree (sol t) (\<chi>i. if i \<in> FVO ODE then sol t $ i else ab $ i) (FVO ODE)"
Quotient_to_transfer: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'a"   and c' :: "'b" assumes "Quotient R Abs Rep T"   and "R c c"   and "c' \<equiv> Abs c" shows "T c c'"
simps: fixes i :: "int" shows "around_zero i = (if i < 0 then [] else if i = 0 then [0] else around_zero (i - 1) @ [i, - i])"
case_prod_Pair: shows "(\<lambda>(x, y). (x, y)) = id"
P_mod_multTr: fixes I :: "'a set"   and J :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "ideal S I"   and "ideal S J"   and "pol_coeff S (n, f)"   and "P_mod R S X I (polyn_expr R X n (n, f))" shows "\<forall>g. pol_coeff S (m, g) \<and> P_mod R S X J (polyn_expr R X m (m, g)) \<longrightarrow> P_mod R S X (I \<diamondsuit>\<^sub>r\<^bsub>S\<^esub> J) (polyn_expr R X n (n, f) \<cdot>\<^sub>r polyn_expr R X m (m, g))"
pderiv_PLUS: fixes a :: "'a"   and x :: "'a rexp"   and xs :: "'a rexp list" shows "pderiv a (PLUS (x # xs)) = pderiv a x \<union> pderiv a (PLUS xs)"
induct: fixes P :: "'c list \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'b"   and a2 :: "'b" assumes "\<And>x y. P [] x y"   and "\<And>i x y. P [i] x y"   and "\<And>i v va x z. (\<And>xa. P (v # va) xa z) \<Longrightarrow> P (i # v # va) x z" shows "P a0 a1 a2"
af_letter_alt_def_1: fixes \<phi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_letter \<phi> \<nu> = step (Unf \<phi>) \<nu>"
cases: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and final :: "'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a behaviour" assumes "semantics step final"   and "semantics.sem_behaves step final a1 a2" obtains s1 :: "'a" and s2 :: "'a" where "a1 = s1b__" and "a2 = Terminates s2a__" and "step\<^sup>*\<^sup>* s1b__ s2a__" and "finished step s2a__" and "final s2a__"   | s1 :: "'a" where "a1 = s1c__" and "a2 = Diverges" and "Inf.inf step s1c__"   | s1 :: "'a" and s2 :: "'a" where "a1 = s1" and "a2 = Goes_wrong s2" and "step\<^sup>*\<^sup>* s1 s2" and "finished step s2" and "\<not> final s2"
is_pfa_def: fixes c :: "nat"   and l :: "'a list" shows "is_pfa c l \<equiv> \<lambda>(a, n). \<exists>\<^sub>Al'. a \<mapsto>\<^sub>a l' * \<up> (c = length l' \<and> n \<le> c \<and> l = take n l')"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "B.dom (F f) = F (A.dom f)"
map_2: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "map_tree f \<langle>x21, x22, x23\<rangle> = \<langle>map_tree f x21, f x22, map_tree f x23\<rangle>"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<tau>s :: "'a list" shows "merges f [] \<tau>s = \<tau>s"
lessThan_iff: fixes i :: "'c word"   and k :: "'c word" shows "(i \<in> signed.lessThan k) = (i <s k)"
f_def: shows "f = Float (normal_mantissa x) (normal_exponent x)"
sig_red_single_top_tailI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "sig_red_single sing_reg (\<preceq>) p q f t"
upper_asymptotic_density_zero_union: fixes A :: "nat set"   and B :: "nat set" assumes "upper_asymptotic_density A = 0"   and "upper_asymptotic_density B = 0" shows "upper_asymptotic_density (A \<union> B) = 0"
unitary_proj_psi_minus_1: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat" assumes "grover_state_sig n f dims R K" shows "unitary (2 \<cdot>\<^sub>m grover_state_sig.proj_psi n - 1\<^sub>m (grover_state.N n))"
finvimage_eq_fdomain_frestriction: fixes r :: "V"   and A :: "V" shows "r -`\<^sub>\<bullet> A = \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A)"
boolean_axioms_imp_Pf: fixes \<alpha> :: "fm"   and e :: "(name, hf) finfun" assumes "\<alpha> \<in> boolean_axioms" shows "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e"
map_scale_sum_distrib_left: fixes k :: "'d"   and f :: "'e \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'd"   and A :: "'e set" shows "k \<cdot> sum f A = (\<Sum>a\<in>A. k \<cdot> f a)"
distinct_89: shows "LessThan \<noteq> ShiftRightSigned"
assm: shows "0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
delta_price_def: fixes N :: "bool stream measure"   and cash_flow :: "bool stream \<Rightarrow> real"   and T :: "nat" shows "delta_price N cash_flow T = (\<lambda>n w. if Suc n \<le> T then (rn_price N cash_flow T (Suc n) (pseudo_proj_True n w) - rn_price N cash_flow T (Suc n) (pseudo_proj_False n w)) / (geom_proc (Suc n) (spick w n True) - geom_proc (Suc n) (spick w n False)) else 0)"
is_processT9: fixes P :: "'b process" shows "\<forall>s. s @ [tick] \<in> D P \<longrightarrow> s \<in> D P"
tag_name_eq_h: fixes ptr' :: "(_) element_ptr"   and disc_nodes :: "char list" assumes "ptr' \<noteq> new_element_ptr" shows "h \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes = h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
subst_ground_ident: fixes t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t = {}" shows "t \<cdot> s = t"
leftderives1_implies_leftderives: fixes a :: "symbol list"   and b :: "symbol list" assumes "leftderives1 a b" shows "leftderives a b"
Lang_def: fixes R :: "'b rexp set" shows "Lang R \<equiv> \<Union> (lang ` R)"
endofunctor_L: shows "endofunctor (\<cdot>\<^sub>B) B.L"
distinct_65: fixes x61 :: "MIR.num"   and x62 :: "MIR.num"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "Sub x61 x62 \<noteq> CF x91 x92 x93"
simps_5: fixes a :: "Ferrack.num" shows "bound0 (Gt a) = numbound0 a"
wf_eq_append''': fixes V :: "'d set"   and S :: "('e, 'd) strand_step list"   and t' :: "('e, 'd) Term.term"   and a :: "poscheckvariant"   and t :: "('e, 'd) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "fv t' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
hcomp_assoc: fixes P :: "interface"   and Q :: "interface"   and R :: "interface" shows "(P \<otimes> (Q \<otimes> R)) = ((P \<otimes> Q) \<otimes> R)"
p1: shows "s = rev r \<cdot> concat (drop (\<^bold>|ws\<^bold>| - j) ws)"
mat_gt_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sd :: "nat"   and A :: "'b mat"   and B :: "'b mat" shows "ord.mat_gt less_eq gt sd A B = (ord.mat_ge less_eq A B \<and> (\<exists>i<sd. \<exists>j<sd. gt (A $$ (i, j)) (B $$ (i, j))))"
prems: shows " Lam [x].P \<longrightarrow>\<^sub>1 M2"
Times_Plus_TimesLR_One: fixes r :: "'a language" shows "Times_Plus (TimesLR r One) = r"
group_axioms: shows "Group_Theory.group G' (\<cdot>') \<one>'"
in_listsD: fixes xs :: "'d list"   and A :: "'d set" assumes "xs \<in> lists A" shows "\<forall>x\<in>set xs. x \<in> A"
abs_fun_supp: fixes x :: "'a"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)" shows "supp ([a].x) = supp x - {a}"
\<psi>_G_char: fixes f :: "'a"   and x :: "'a"   and x' :: "'a" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>A x'\<guillemotright>" shows "F (G f) = f \<cdot>\<^sub>A \<epsilon>o x"
S00_dvd_all_A: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and Q :: "'a mat"   and S :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "invertible_mat P"   and "invertible_mat Q"   and "S = P * A * Q"   and "Smith_normal_form_mat S"   and "i < m"   and "j < n" shows "S $$ (0, 0) dvd A $$ (i, j)"
inject: fixes x2 :: "enat"   and y2 :: "enat" shows "(eSuc x2 = eSuc y2) = (x2 = y2)"
simps: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "(('b, 'a, 'c) Stmt \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'd)"   and a2 :: "(('b, 'a, 'c) Stmt \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'd)"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "((a1, a2) \<in> sifum_types.\<R> aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>) = ((\<exists>lc lc'. a1 = lc \<and> a2 = lc' \<and> (case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). (((x, xa), xaa), (xaaa, xaaaa), xaaaaa) \<in> sifum_types.\<R>\<^sub>1 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>) xa) lc') \<or> (\<exists>lc lc'. a1 = lc \<and> a2 = lc' \<and> (case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). (((x, xa), xaa), (xaaa, xaaaa), xaaaaa) \<in> sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>) xa) lc') \<or> (\<exists>lc lc'. a1 = lc \<and> a2 = lc' \<and> sifum_types.\<R>\<^sub>3_aux aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma> lc lc'))"
lunit_simps_1: fixes f :: "'a" assumes "S.ide f" shows "S.arr (S.lunit f)"
induced_automorphism_surj_on_vertexset: shows "\
 ` \<Union> X = \<Union> X"
rel_of_succ_def: fixes s :: "'g \<Rightarrow> 'h set" shows "rel_of_succ s \<equiv> {(a, b). b \<in> s a}"
DBM_and_sound2: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat" assumes "u \<turnstile>\<^bsub>v,n\<^esub> And M1 M2" shows "u \<turnstile>\<^bsub>v,n\<^esub> M2"
proper_imp_closed_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "proper_map X Y f" shows "closed_map X Y f"
assms_7: shows "b \<otimes> snd h [^] e = g' [^] z"
update_unsat_id: fixes s :: "('i, 'a) state"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<U> (update x c s) = \<U> s"
\<phi>: shows "natural_isomorphism (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<phi>"
assms: shows "lambda C -quasi_isometry_between UNIV UNIV f"
gc_W_empty_mut_inv_def: shows "gc_W_empty_mut_inv = (\<lambda>s. sys_W s = {} \<and> sys_ghost_hs_in_sync m s \<and> WL (mutator m) s \<noteq> {} \<longrightarrow> (\<exists>x. \<not> sys_ghost_hs_in_sync x s \<and> WL (mutator x) s \<noteq> {}))"
rel_mono_strong: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e set['g]"   and y :: "'f set['g]"   and Ra :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_bset R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_bset x; yb \<in> set_bset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_bset Ra x y"
omega_sum_ge: fixes m :: "nat"   and ms :: "nat list" assumes "m \<noteq> 0" shows "\<omega> \<up> ord_of_nat (length ms) \<le> omega_sum (m # ms)"
VAR_neqv_PLUS: fixes m :: "nat"   and X :: "freeExp"   and Y :: "freeExp" assumes "VAR m \<sim> PLUS X Y" shows "False"
timesE: fixes c :: "hf"   and A :: "hf"   and B :: "hf" assumes "c \<^bold>\<in> A * B" obtains x :: "hf" and y :: "hf" where "x \<^bold>\<in> A" and "y \<^bold>\<in> B" and "c = \<langle>x, y\<rangle>"
openin_subset: fixes U :: "'c topology"   and S :: "'c set" assumes "openin U S" shows "S \<subseteq> topspace U"
square_diff_square_factored: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.comm_ring plus zero minus uminus times" shows "minus (times x x) (times y y) = times (plus x y) (minus x y)"
set_disconnected_nodes_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_disconnected_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
Rats_of_rat: fixes r :: "rat" shows "of_rat r \<in> \<rat>"
G'_fcs: shows "G'.face_cycle_succ = perm_swap a b face_cycle_succ"
append_self_conv2: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs @ ys = ys) = (xs = [])"
Lcm_1_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> '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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "(Lcm A = one) = (\<forall>a\<in>A. dvd.dvd times a one)"