hcju/isabelleps huggingface dataset:
sequence_to_failure_alt_def: fixes PM :: "('e, 'f, 'd \<times> 'd) FSM"   and io :: "('e \<times> 'f) list"   and FAIL :: "'d \<times> 'd"   and M1 :: "('e, 'f, 'd) FSM"   and M2 :: "('e, 'f, 'd) FSM" assumes "io_targets PM (initial PM) io = {FAIL}"   and "OFSM M1"   and "OFSM M2"   and "productF M2 M1 FAIL PM" shows "sequence_to_failure M1 M2 io"
record_call_other: fixes x' :: "'a"   and x :: "'a"   and f :: "'a \<Rightarrow> one\<^sub>\<bottom>" assumes "x' \<noteq> x" shows "(record_call x\<cdot>f) x' = f x'"
valid2_hd_Rcons: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and tr :: "('a, 'b, 'c) trans list"   and tran :: "('a, 'b, 'c) trans" assumes "IO_Automaton.valid2 step tr" shows "hd (tr ## tran) = hd tr"
times_octo_def: fixes a :: "octo"   and b :: "octo" shows "a * b = (let t0 = Ree a * Ree b - Im1 a * Im1 b - Im2 a * Im2 b - Im3 a * Im3 b - Im4 a * Im4 b - Im5 a * Im5 b - Im6 a * Im6 b - Im7 a * Im7 b; t1 = Ree a * Im1 b + Im1 a * Ree b + Im2 a * Im4 b + Im3 a * Im7 b - Im4 a * Im2 b + Im5 a * Im6 b - Im6 a * Im5 b - Im7 a * Im3 b; t2 = Ree a * Im2 b - Im1 a * Im4 b + Im2 a * Ree b + Im3 a * Im5 b + Im4 a * Im1 b - Im5 a * Im3 b + Im6 a * Im7 b - Im7 a * Im6 b; t3 = Ree a * Im3 b - Im1 a * Im7 b - Im2 a * Im5 b + Im3 a * Ree b + Im4 a * Im6 b + Im5 a * Im2 b - Im6 a * Im4 b + Im7 a * Im1 b; t4 = Ree a * Im4 b + Im1 a * Im2 b - Im2 a * Im1 b - Im3 a * Im6 b + Im4 a * Ree b + Im5 a * Im7 b + Im6 a * Im3 b - Im7 a * Im5 b; t5 = Ree a * Im5 b - Im1 a * Im6 b + Im2 a * Im3 b - Im3 a * Im2 b - Im4 a * Im7 b + Im5 a * Ree b + Im6 a * Im1 b + Im7 a * Im4 b; t6 = Ree a * Im6 b + Im1 a * Im5 b - Im2 a * Im7 b + Im3 a * Im4 b - Im4 a * Im3 b - Im5 a * Im1 b + Im6 a * Ree b + Im7 a * Im2 b in Let (Ree a * Im7 b + Im1 a * Im3 b + Im2 a * Im6 b - Im3 a * Im1 b + Im4 a * Im5 b - Im5 a * Im4 b - Im6 a * Im2 b + Im7 a * Ree b) (Octo t0 t1 t2 t3 t4 t5 t6))"
singleton_insert_inj_eq': fixes a :: "'k"   and A :: "'k set"   and b :: "'k" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
mend: shows "mat n n (\<lambda>(i, j). if (i, j) = (i_end, k) then if Suc (i_end - i_begin) = Suc (i_end - i_begin) then A $$ (i_end, k) else (0::'a) else if i_begin + Suc (i_end - i_begin) \<le> i \<and> i \<le> i_end \<and> k < j then A $$ (i, j) - quot * A $$ (l + i - i_end, j) else if (i, j) = (i_end - 0, Suc l - 0) \<and> 0 \<notin> {0, Suc (i_end - i_begin)} then quot else A $$ (i, j)) = A"
conj_disj_distribs_1: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x AND (y OR z) = x AND y OR x AND z"
has_real_derivative_neg_dec_right: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and S :: "real set" assumes "(f has_real_derivative l) (at x within S)"   and "l < 0" shows "\<exists>d>0. \<forall>h>0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x + h) < f x"
A92_DAcc_level1: shows "DAcc level1 sA92 = {}"
T_delete2_def: fixes x :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "T_delete2 x t = time (delete2_tm x t)"
simps_9: fixes f :: "'c \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c list" shows "map f (x21 # x22) = f x21 # map f x22"
le_iff_join: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq> y) = (x = x \<sqinter> y)"
free_tv_subset_gen_le: fixes B :: "type_scheme list"   and A :: "type_scheme list"   and t :: "typ" assumes "free_tv B \<subseteq> free_tv A" shows "gen A t \<le> gen B t"
le_less: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.order (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type) = ((less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) x y \<or> x = y)"
oracle_decrypt_def: fixes k :: "'prf_key \<times> 'upf_key"   and L :: "(bool list \<times> bool list \<times> 'hash) set"   and c :: "bool list \<times> bool list \<times> 'hash" shows "ind_cca'.oracle_decrypt k L c = return_spmf (if c \<in> L then None else None, L)"
atLeast_Suc_lessThan: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m < n" shows "sum g {m..<n} = g m + sum g {Suc m..<n}"
simps_2: fixes x21 :: "e list"   and x22 :: "nat"   and x23 :: "e list"   and x24 :: "Lholed"   and x25 :: "e list"   and y21 :: "e list"   and y22 :: "nat"   and y23 :: "e list"   and y24 :: "Lholed"   and y25 :: "e list" shows "(LRec x21 x22 x23 x24 x25 = LRec y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
minus_Nat_def: fixes a :: "'a"   and b :: "'a" shows "a - b = AbsNat (RepNat a - RepNat b)"
neg_nnf: fixes \<phi> :: "atom fm" shows "\<forall>\<Gamma>. (\<not> eval (nnf (Neg \<phi>)) \<Gamma>) = eval (nnf \<phi>) \<Gamma>"
remove: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'b option" assumes "finite A"   and "x \<in> A" shows "F g A = lift_option (+) (g x) (F g (A - {x}))"
point_in_S_polar_is_tangent: fixes p :: "proj2"   and q :: "proj2" assumes "p \<in> S"   and "q \<in> S"   and "proj2_incident q (polar p)" shows "q = p"
inverse_nonzero_iff_nonzero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = zero) = (a = zero)"
rev_predicate2D: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "P x y"   and "P \<le> Q" shows "Q x y"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs, s\<rangle> \<Rightarrow> t"
the_\<nu>_props_1: shows "\<guillemotleft>w.the_\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright>"
nfoldli_by_idx_gen: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli (drop 0 l) c f s = nfoldli [0..<length l] c (\<lambda>i s. ASSERT (i < length l) \<bind> (\<lambda>_. let x = l ! i in f x s)) s"
B'_def: shows "B' = (*\<^sub>R) 2 -` B"
trans: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
sin_times_cos: fixes w :: "'a"   and z :: "'a" shows "sin w * cos z = (sin (w + z) + sin (w - z)) / (2::'a)"
if_SE_def: fixes c :: "'b \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<times> 'b) option"   and F :: "'b \<Rightarrow> ('c \<times> 'b) option" shows "if\<^sub>S\<^sub>E c E F = (\<lambda>\<sigma>. if c \<sigma> then E \<sigma> else F \<sigma>)"
is_module: fixes W :: "'e set" assumes "subspace K W W" shows "LinearCombinations.submodule K W W"
E0_is: shows "E0 \<cdot> \<eta>2 = E"
cases: fixes a1 :: "Lambda_Z.term"   and a2 :: "Lambda_Z.term" assumes "a1 \<rightarrow>\<^sub>\<beta> a2" obtains x :: "name" and t :: "Lambda_Z.term" and s :: "Lambda_Z.term" where "a1 = App (Abs x__ sb__) tb__" and "a2 = subst x__ tb__ sb__" and "atom x__ \<sharp> tb__"   | s :: "Lambda_Z.term" and t :: "Lambda_Z.term" and u :: "Lambda_Z.term" where "a1 = App sc__ ua__" and "a2 = App tc__ ua__" and "sc__ \<rightarrow>\<^sub>\<beta> tc__"   | s :: "Lambda_Z.term" and t :: "Lambda_Z.term" and u :: "Lambda_Z.term" where "a1 = App u sd__" and "a2 = App u td__" and "sd__ \<rightarrow>\<^sub>\<beta> td__"   | s :: "Lambda_Z.term" and t :: "Lambda_Z.term" and x :: "name" where "a1 = Abs x s" and "a2 = Abs x t" and "s \<rightarrow>\<^sub>\<beta> t"
hq_mod_f: shows "[h ^ CARD('a) = h] (mod f)"
j_of_min_s: fixes k :: "nat"   and j :: "nat" assumes "k < j"   and "j < j_of k" shows "(f (Suc j), f j) \<in> S"
13: shows "1 / 3 + 1 / 3 + 1 / 3 = 1"
Max_wrt_among_nonempty: fixes B :: "'alt set" assumes "B \<inter> alts \<noteq> {}"   and "finite (B \<inter> alts)" shows "Max_wrt_among (R j) B \<noteq> {}"
symmetric: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "(a, b) \<in> E" shows "(b, a) \<in> E"
assms_2: shows "\<forall>n. \<exists>m>n. \<psi> (the (s (f \<triangleright> m))) \<noteq> f"
A_ncong_p: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "GAUSS p a"   and "x \<in> GAUSS.A p" shows "[x \<noteq> 0] (mod int p)"
cbox_Complex_eq: fixes a :: "real"   and c :: "real"   and b :: "real"   and d :: "real" shows "cbox (Complex a c) (Complex b d) = (\<lambda>(x, y). Complex x y) ` (cbox a b \<times> cbox c d)"
dghm_up_def: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" shows "dghm_up I \<AA> \<CC> \<phi> = [\<lambda>a\<in>\<^sub>\<circ>\<CC>\<lparr>Obj\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<phi> i\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>, \<lambda>f\<in>\<^sub>\<circ>\<CC>\<lparr>Arr\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<phi> i\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>, \<CC>, dg_prod I \<AA>]\<^sub>\<circ>"
czeroI: fixes r :: "('c \<times> 'c) set" assumes "Card_order r"   and "Field r = {}" shows "r =o czero"
lunit_in_hom_2: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>B.lunit f : f \<star> src f \<rightarrow>\<^sub>B f\<guillemotright>"
minus_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y \<oplus> x = \<zero>"   and "x \<oplus> y' = \<zero>"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "y' \<in> carrier R" shows "y = y'"
assms: shows "t \<in> ik\<^sub>s\<^sub>t (map Receive L)"
induct: fixes P :: "'d\<cdot>'e writer \<Rightarrow> bool"   and x :: "'d\<cdot>'e writer" assumes "P \<bottom>"   and "\<And>w a. P (Writer\<cdot>w\<cdot>a)" shows "P x"
cross7_simps_3: fixes f :: "7 \<Rightarrow> 'b" shows "sum f UNIV = f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7"
reciprocal_id_iff: fixes z :: "complex_homo" shows "(reciprocal z = z) = (z = of_complex 1 \<or> z = of_complex (- 1))"
inverse_fpxs_def: fixes f :: "'b fpxs" shows "inverse f = fpxs_compose_power (fpxs_of_fls (inverse (fls_of_fpxs f))) (1 / rat_of_nat (fpxs_root_order f))"
lsum_take: fixes f :: "'a \<Rightarrow> nat"   and n :: "nat"   and al :: "'a list" shows "lsum f (take n al) \<le> lsum f al"
setToListFresh_3: fixes x :: "name"   and yvec :: "name list" shows "x \<sharp> set yvec = x \<sharp> yvec"
eventually_at_split: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eventually P (at x) = (eventually P (at_left x) \<and> eventually P (at_right x))"
S_def: shows "S = {k. \<not> suffix k w \<Turnstile> \<phi>}"
init_sim: fixes a0 :: "'a" assumes "a0 \<in> s1.a.V0" shows "\<exists>b0. b0 \<in> s1.b.V0 \<and> (a0, b0) \<in> R1"
BSIA_def: fixes \<rho> :: "'a V_rec \<Rightarrow> 'a set"   and \<V> :: "'a V_rec"   and Tr :: "'a list set" shows "BSIA \<rho> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> Adm \<V> \<rho> Tr \<beta> c \<longrightarrow> (\<exists>\<alpha>'. \<beta> @ [c] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
weak_reduction_coupled_simulation_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_coupled_simulation Rel Cal" shows "weak_reduction_coupled_simulation (Rel\<^sup>+) Cal"
numeral_power_eq_of_nat_cancel_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "num"   and n :: "nat"   and y :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(power.power one times (numeral.numeral one plus x) n = semiring_1.of_nat one plus zero y) = (numeral x ^ n = y)"
inf_llist_def: fixes f :: "nat \<Rightarrow> 'c" shows "inf_llist f = lmap f (iterates Suc 0)"
consistent_def: fixes A :: "'c set"   and S :: "(('d, 'c) fm list \<times> 'c) set" shows "consistent A S \<equiv> \<nexists>S'. set S' \<subseteq> S \<and> A \<turnstile> S'"
Max_ge: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set"   and x :: "'k" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
"2_1": shows "rb_aux_dom ((bs, ss, p # ps), z)"
assms_2: shows "\<forall>x. has_least_fixpoint (f x)"
c_def: shows "c = (\<Prod>ia = 0..<m - 1. real m - 1 - real ia)"
id_1: shows "charpoly (HMA_Connect.map_matrix complex_of_real A) = prod_root_unity ks * map_poly complex_of_real F"
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "F g (insert x A) = g x \<^bold>* F g (A - {x})"
oPrefix_def: fixes c :: "'d \<Rightarrow> 'c"   and P :: "('c, 'd) action" shows "oPrefix c P \<equiv> action_of (R (true \<turnstile> do c \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A))) `;` P"
sort_oalist_ko'_def: fixes ko :: "'a key_order" shows "sort_oalist_ko' ko \<equiv> comparator.sort_oalist (key_compare ko)"
less_eq_option_None: fixes x :: "'a option" shows "None \<le> x"
NotR_inv': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta> \<down> n" shows "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
linear: fixes x :: "char list"   and y :: "char list" shows "ord.lexordp_eq (\<lambda>c d. of_char c < of_char d) x y \<or> ord.lexordp_eq (\<lambda>c d. of_char c < of_char d) y x"
cases: fixes a1 :: "'a"   and a2 :: "'b"   and r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" assumes "(a1, a2) \<in> r O s" obtains a :: "'a" and b :: "'c" and c :: "'b" where "a1 = a" and "a2 = c" and "(a, b) \<in> r" and "(b, c) \<in> s"
assms_1: shows "x \<in> carrier Q\<^sub>p"
insert3_def2: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat" shows "insert3 x (t, n) = (let t' = ins3 n 0 x t in case t' of Same2 \<Rightarrow> (t, n) | Bal2 t' \<Rightarrow> (t', n + 1))"
sub: shows "snd ` set (map (\<lambda>a. (f (atom_var a), a)) C) \<subseteq> snd ` as"
onlyone: fixes n :: "nat"   and tid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'\<ge>n. \<parallel>tid\<parallel>\<^bsub>t n'\<^esub>"   and "\<exists>n'. latestAct_cond tid t n n'" shows "\<exists>!i. \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> i \<and> latestAct_cond tid t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub> i"
hmset_of_enat_of_nat: fixes n :: "nat" shows "hmset_of_enat (of_nat n) = of_nat n"
zero_least2: fixes x :: "'a" shows "o \<cdot> n x = o"
saturated_def: fixes N :: "'f set" shows "saturated N = (Inf_from N \<subseteq> Red_I_\<G> N)"
mult_scaleR_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "real"   and x :: "'c"   and y :: "'c" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times (scaleR a x) y = scaleR a (times x y)"
mult_1_pos: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "int poly \<times> rat \<times> rat" assumes "invariant_1_2 x"   and "invariant_1_2 y"   and "0 < real_of_1 x"   and "0 < real_of_1 y" shows "invariant_2 (mult_1_pos x y) \<and> real_of_2 (mult_1_pos x y) = real_of_1 x * real_of_1 y"
paths_Cons_nxt_iff: fixes t :: "'b ttree"   and x :: "'b"   and xs :: "'b list" assumes "possible t x" shows "(xs \<in> paths (nxt t x)) = (x # xs \<in> paths t)"
veval'_closed: fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and v :: "value" assumes "\<Gamma> \<turnstile>\<^sub>v t \<down> v"   and "closed_except t (fmdom \<Gamma>)"   and "closed_venv \<Gamma>"   and "pre_strong_term_class.wellformed t"   and "wellformed_venv \<Gamma>" shows "vclosed v"
h: fixes x :: "'a" shows "h x = (if stop x then 0 else eSuc (h (next x)))"
prerun_reachable_state: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and \<sigma> :: "nat \<Rightarrow> ('g, 'h, 'i, 'j, 'k) system_state"   and i :: "nat" assumes "prerun sys \<sigma>" shows "reachable_state sys (\<sigma> i)"
tfr\<^sub>s\<^sub>t_def: fixes S :: "('fun, 'var) strand_step list" shows "tfr\<^sub>s\<^sub>t S \<equiv> tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S) \<and> list_all tfr\<^sub>s\<^sub>t\<^sub>p S"
beukers_integral3_integrand_bound: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x \<in> {0<..<1}"   and "y \<in> {0<..<1}"   and "z \<in> {0<..<1}" shows "x * (1 - x) * y * (1 - y) * z * (1 - z) / (1 - (1 - x * y) * z) \<le> 1 / 27"
cauchy_filter_def: fixes "open" :: "'c set \<Rightarrow> bool"   and uniformity :: "('c \<times> 'c) filter"   and F :: "'c filter" assumes "class.uniform_space open uniformity" shows "uniform_space.cauchy_filter uniformity F = (F \<times>\<^sub>F F \<le> uniformity)"
kcomp_def: fixes F :: "'d \<Rightarrow> 'f set"   and G :: "'f \<Rightarrow> 'e set" shows "F ; G = \<mu> \<circ> \<P> G \<circ> F"
add_cancel_right_right: fixes a :: "'c"   and b :: "'c" shows "(a = a + b) = (b = (0::'c))"
aperiodic_def: fixes C :: "'s set" shows "aperiodic C = (C \<in> UNIV // communicating \<and> period C = 1)"
strict_implies_order: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "ord_strict_p a b" shows "ord_p a b"
abs_split: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" shows "P \<bar>a\<bar> = (((0::'c) \<le> a \<longrightarrow> P a) \<and> (a < (0::'c) \<longrightarrow> P (- a)))"
test_absorb1: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "p + p \<cdot> q = p"
simps_2: fixes s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list"   and opss :: "('a, 'b) sas_plus_operator list list" shows "trace_parallel_plan_sas_plus s (ops # opss) = s # (if are_all_operators_applicable_in s ops \<and> are_all_operator_effects_consistent ops then trace_parallel_plan_sas_plus (execute_parallel_operator_sas_plus s ops) opss else [])"
sel_3: fixes x1 :: "'a"   and x2 :: "'a tree"   and x3 :: "'a tree" shows "right (Node x1 x2 x3) = x3"
fin: shows "AE x in M. f x \<noteq> \<top>"
subdg_dom_simp: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = \<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr>"
write0_def: fixes a :: "'d"   and P :: "'d process" shows "a \<rightarrow> P \<equiv> Mprefix {a} (\<lambda>x. P)"
choiceT1: fixes \<xi> :: "'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, p), a, \<xi>', p') \<in> seqp_sos \<Gamma>" shows "((\<xi>, p \<oplus> q), a, \<xi>', p') \<in> seqp_sos \<Gamma>"
check_cert_def: fixes P :: "jvm_prog"   and mxs :: "nat"   and mxl :: "nat"   and n :: "nat"   and cert :: "(ty list \<times> ty err list) option err list" shows "check_cert P mxs mxl n cert \<equiv> check_types P mxs mxl cert \<and> length cert = n + 1 \<and> (\<forall>i<n. cert ! i \<noteq> Err) \<and> cert ! n = OK None"
prv_sdsj_imp: fixes \<chi> :: "'fmla"   and F :: "'fmla set" assumes "\<chi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F"   and "\<And>\<phi>. \<phi> \<in> F \<Longrightarrow> bprv (imp \<phi> \<chi>)" shows "bprv (imp (sdsj F) \<chi>)"
foldli_mono_dres: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'g dres"   and f' :: "'f \<Rightarrow> 'g \<Rightarrow> 'g dres"   and l :: "'f list"   and c :: "'g \<Rightarrow> bool"   and \<sigma> :: "'g dres" assumes "\<And>a x. f a x \<le> f' a x" shows "foldli l (case_dres False False c) (\<lambda>x s. s \<bind> f x) \<sigma> \<le> foldli l (case_dres False False c) (\<lambda>x s. s \<bind> f' x) \<sigma>"
closed_interval_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less" shows "order.closed_interval less_eq a b = {c. less_eq a c \<and> less_eq c b}"
rep_inverse: fixes x :: "('a, 'b) mapping" shows "mapping.Mapping (mapping.rep x) = x"
less_linear: fixes x :: "'b word"   and y :: "'b word" shows "x <s y \<or> x = y \<or> y <s x"
m'_stl: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M" shows "distr (m' N M y) (stream_space (count_space UNIV)) stl = m N M y"
relcompp_apply: fixes R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and S :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and a :: "'k"   and c :: "'m" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
cmp_simps'_2: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "src\<^sub>D (\<Phi> (\<mu>, \<nu>)) = map\<^sub>0 (src\<^sub>C \<nu>)"
minus_divide_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "uminus (divide a b) = divide (uminus a) b"
power_increasing: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat"   and N :: "nat"   and a :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq one a" shows "less_eq (power.power one times a n) (power.power one times a N)"
3: fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "s \<in> SMP (pair ` setops\<^sub>s\<^sub>s\<^sub>t S) - range Var"   and "t \<in> SMP (pair ` setops\<^sub>s\<^sub>s\<^sub>t S) - range Var"   and "\<exists>\<delta>. Unifier \<delta> s t" shows "\<Gamma> s = \<Gamma> t"
i_take_Suc: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> Suc n = f 0 # f \<Up> Suc 0 \<Down> n"
sup_bot_left: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and bot :: "'e"   and x :: "'e" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
simps: fixes x :: "'e"   and xs :: "'e list" shows "last (x # xs) = (if xs = [] then x else last xs)"
admissible\<^sub>3_def: fixes w :: "face \<Rightarrow> nat"   and g :: "graph" shows "admissible\<^sub>3 w g \<equiv> \<forall>v\<in>\<V> g. vertextype g v = (5, 0, 1) \<longrightarrow> \<a> \<le> ListSum (filter triangle (facesAt g v)) w"
d: fixes x :: "'a" assumes "x \<in> A" shows "infsetsum (f x) (B' x) = infsetsum (f x) (B x)"
bounded_translation: fixes S :: "'a set"   and a :: "'a" assumes "bounded S" shows "bounded ((+) a ` S)"
uminus_in_intervalI: fixes x :: "'b"   and X :: "'b interval" assumes "x \<in>\<^sub>i X" shows "- x \<in>\<^sub>i - X"
is_Hcomp_HcompNml: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u"   and "\<not> is_Prim\<^sub>0 t"   and "\<not> is_Prim\<^sub>0 u" shows "is_Hcomp (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
id_upt_k_Gauss_Jordan: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" assumes "invertible A" shows "id_upt_k (Gauss_Jordan A) k"