hcju/isabelleps huggingface dataset:
parallel_def: fixes A :: "point"   and B :: "point"   and C :: "point"   and D :: "point" shows "parallel A B C D = ((abscissa A - abscissa B) * (ordinate C - ordinate D) = (ordinate A - ordinate B) * (abscissa C - abscissa D))"
left_adjoint_extends_to_adjoint_pair: fixes f :: "'a" assumes "C.is_left_adjoint f" shows "C.adjoint_pair f f\<^sup>*\<^sup>C"
euler_char_def: shows "H.euler_char \<equiv> int (card (verts (del_vert u))) - int (card (arcs (del_vert u)) div 2) + int (card H.face_cycle_sets)"
degree_diff_le_max: fixes p :: "'a poly"   and q :: "'a poly" shows "degree (p - q) \<le> max (degree p) (degree q)"
inject_1: fixes x1 :: "('a, 'b) atm"   and y1 :: "('a, 'b) atm" shows "(Pos x1 = Pos y1) = (x1 = y1)"
Rf_def: fixes N :: "'a literal multiset set" shows "Rf N = {C. \<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD \<longrightarrow> I \<Turnstile> C) \<and> (\<forall>D. D \<in># DD \<longrightarrow> D < C)}"
shiftr1_0_or_1: fixes x :: "'b word" assumes "x >> 1 = 0" shows "x = 0 \<or> x = 1"
Im_interval_times: fixes A :: "complex_interval"   and B :: "complex_interval" shows "Im_interval (A * B) = Re_interval A * Im_interval B + Im_interval A * Re_interval B"
lunit'_closed: fixes f :: "'a" assumes "Arr f"   and "B.ide f" shows "Arr (B.inv (B.\<ll> f))"
nextElem_notin_butlast: fixes y :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "y \<notin> set (butlast xs)" shows "nextElem xs x y = x"
reductio_aa_1: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<not>\<phi> in v] \<Longrightarrow> [\<^bold>\<not>\<psi> in v]"   and "[\<^bold>\<not>\<phi> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<phi> in v]"
mono_UNIV: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "order.mono_set less_eq UNIV"
zero_seq_unk_hops_one': fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg (\<lambda>m. rreq_rrep_sn m \<and> msg_zhops m) \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, uu_). \<forall>dip\<in>kD (rt \<xi>). (sqn (rt \<xi>) dip = 0 \<longrightarrow> sqnf (rt \<xi>) dip = unk) \<and> (sqnf (rt \<xi>) dip = unk \<longrightarrow> the (dhops (rt \<xi>) dip) = 1) \<and> (the (dhops (rt \<xi>) dip) = 1 \<longrightarrow> the (nhop (rt \<xi>) dip) = dip))"
is_real_arg1: fixes z :: "complex" assumes "arg z = 0 \<or> arg z = pi" shows "is_real z"
trans_of_def: shows "trans_of \<equiv> snd \<circ> snd"
qSkel_qSwap: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "qSkel (X #[[x \<and> y]]_zs) = qSkel X"
assms: shows "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A))"
assms_3: shows "local \<C> = map typeof (vi @ [v] @ vs)"
kbo'_eq_kbo: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" shows "kbo.kbo w w0 scf least kbo'.pr_strict' kbo'.pr_weak' s t = kbo'.kbo s t"
E_def: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "GAUSS.E p a = GAUSS.C p a \<inter> {(int p - 1) div 2<..}"
sporadic_sub: fixes sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and \<tau> :: "'b tag_const"   and c' :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> c sporadic \<tau> on c' \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> c sporadic \<tau> on c' \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
os_reverse_def: fixes p :: "'b node ref option" shows "os_reverse p = os_reverse_aux None p"
function_uminus_eval': fixes f :: "'c \<Rightarrow> 'a"   and a :: "'c" assumes "f \<in> carrier F"   and "a \<in> S" shows "(\<ominus>\<^bsub>F\<^esub> f) a = function_uminus S R f a"
thetaSeqTT_RetrT: shows "thetaSeqTT \<subseteq> RetrT (thetaSeqTT \<union> BisT)"
\<pi>_root: shows "\<pi>' Q \<pi> u r = None"
DERIV_chain2: fixes f :: "'a \<Rightarrow> 'a"   and Da :: "'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and Db :: "'a"   and s :: "'a set" assumes "(f has_field_derivative Da) (at (g x))"   and "(g has_field_derivative Db) (at x within s)" shows "((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
T_count: fixes l :: "nat" shows "card (T l) = (k * k - k + 1) * k ^ l"
LeftDerivationIntro_bounds_ij: fixes \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<beta> :: "symbol list" assumes "LeftDerivationIntro \<alpha> i r ix D j \<beta>" shows "i < length \<alpha> \<and> j < length \<beta>"
onl_def: fixes \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "'e \<times> 'i \<Rightarrow> bool" shows "onl \<Gamma> P \<equiv> \<lambda>(\<xi>, p). \<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"
take_map_def: fixes n :: "nat"   and l :: "'b list" shows "take_map n l = (if length l \<le> n then l else map ((!) l) [0..<n])"
l11_44_1_a: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "A \<noteq> C"   and "Cong B A B C" shows "B A C CongA B C A"
eta: fixes s :: "dB"   and dummy :: "dB" assumes "\<not> free s 0" shows "Abs (s \<degree> Var 0) \<rightarrow>\<^sub>\<eta> s[dummy/0]"
ran: shows "range f = {j. \<forall>k>j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j}"
hom': shows "ring_hom_ring (R Quot a_kernel R (S Quot P) ((+>\<^bsub>S\<^esub>) P \<circ> h)) (S Quot P) (\<lambda>X. the_elem (((+>\<^bsub>S\<^esub>) P \<circ> h) ` X))"
isVal_graph_aux_def: shows "isVal_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>isVal_aux x. x1 = Var x \<and> x2 = False) \<or> (\<exists>isVal_aux x e. x1 = Lam [x]. e \<and> x2 = True) \<or> (\<exists>isVal_aux e x. x1 = App e x \<and> x2 = False) \<or> (\<exists>isVal_aux as e. x1 = Let as e \<and> x2 = False) \<or> (\<exists>isVal_aux b. x1 = Bool b \<and> x2 = True) \<or> (\<exists>isVal_aux scrut e1 e2. x1 = (scrut ? e1 : e2) \<and> x2 = False))"
measure_Un2: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M (B - A)"
Cd_def: fixes d :: "'d" shows "Cd d \<equiv> set_option \<circ> MaxR.MaxR_opt (Pd d)"
binop_axioms: shows "binop \<real>\<^sub>\<circ> vreal_plus"
box_subset_cbox: shows "box a b \<subseteq> cbox a b"
Un_Int_distrib2: fixes B :: "'b set"   and C :: "'b set"   and A :: "'b set" shows "B \<inter> C \<union> A = (B \<union> A) \<inter> (C \<union> A)"
le: shows "\<forall>n. of_rat (X n) \<le> y"
awalk_empty_ends: fixes u :: "'a"   and v :: "'a" assumes "awalk u [] v" shows "u = v"
t': shows "t = (if p \<in> t then insert p t' else t')"
continuous_minus: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'e" assumes "continuous F f" shows "continuous F (\<lambda>x. - f x)"
subst_bv2_eq: fixes k :: "nat"   and u :: "Core.term" shows "subst_bv2 (Bv k) k u = u"
simps_40: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "mmonitorable_exec (formula.And \<phi> \<psi>) = (mmonitorable_exec \<phi> \<and> (safe_assignment (fv \<phi>) \<psi> \<or> mmonitorable_exec \<psi> \<or> fv \<psi> \<subseteq> fv \<phi> \<and> (is_constraint \<psi> \<or> (case \<psi> of formula.Neg x \<Rightarrow> mmonitorable_exec x | _ \<Rightarrow> False))))"
l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node get_owner_document \<equiv> known_ptr = DocumentClass.known_ptr \<and> get_owner_document = l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node get_disconnected_nodes"
entry_lemma: fixes str :: "nat"   and i :: "nat" shows "rec_exec rec_entry [str, i] = Entry str i"
subdigraph_def: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC> \<equiv> digraph \<alpha> \<BB> \<and> digraph \<alpha> \<CC> \<and> subdigraph_axioms \<BB> \<CC>"
fBall_fempty: fixes P :: "'a \<Rightarrow> bool" shows "fBall {||} P = True"
nat_floor_neg: fixes x :: "real" assumes "x \<le> 0" shows "nat \<lfloor>x\<rfloor> = 0"
atleast_free_SucD_lemma: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and n :: "nat"   and b :: "'d"   and d :: "'c" assumes "m a = None"   and "\<forall>c. atleast_free (m(a \<mapsto> c)) n"   and "a \<noteq> b" shows "atleast_free (m(b \<mapsto> d)) n"
enat_ord_simps_4: fixes q :: "enat" shows "(q < \<infinity>) = (q \<noteq> \<infinity>)"
n_intersect_num_subset_def: fixes b1 :: "'a set"   and n :: "nat"   and b2 :: "'a set" shows "(b1 |\<inter>|\<^sub>n b2) = int (card {x. x \<subseteq> b1 \<inter> b2 \<and> card x = n})"
distinct_13: fixes x21 :: "POPLmark.type"   and x22 :: "trm"   and x51 :: "trm"   and x52 :: "POPLmark.type" shows "(\<lambda>:x21. x22) \<noteq> x51 \<bullet>\<^sub>\<tau> x52"
HEndPhase1_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2c s"   and "HEndPhase1 s s' p"   and "Inv2a s"   and "HInv1 s" shows "Inv2c s'"
rep_eq: shows "pdevs_apply zero_pdevs = (\<lambda>_. 0::'d)"
ex_bad: shows "\<exists>a. inf_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) a"
distinct_concat_iff: fixes xs :: "'a list list" shows "distinct (concat xs) = (distinct (removeAll [] xs) \<and> (\<forall>ys. ys \<in> set xs \<longrightarrow> distinct ys) \<and> (\<forall>ys zs. ys \<in> set xs \<and> zs \<in> set xs \<and> ys \<noteq> zs \<longrightarrow> set ys \<inter> set zs = {}))"
cases: fixes a1 :: "Core.term"   and a2 :: "Core.term" assumes "a1 \<rightarrow>\<^sub>\<beta> a2" obtains T :: "typ" and s :: "Core.term" and t :: "Core.term" where "a1 = Abs T__ s__ $ ta__" and "a2 = subst_bv2 s__ 0 ta__"   | s :: "Core.term" and t :: "Core.term" and u :: "Core.term" where "a1 = sa__ $ u__" and "a2 = tb__ $ u__" and "sa__ \<rightarrow>\<^sub>\<beta> tb__"   | s :: "Core.term" and t :: "Core.term" and u :: "Core.term" where "a1 = u $ sb__" and "a2 = u $ tc__" and "sb__ \<rightarrow>\<^sub>\<beta> tc__"   | s :: "Core.term" and t :: "Core.term" and T :: "typ" where "a1 = Abs T s" and "a2 = Abs T t" and "s \<rightarrow>\<^sub>\<beta> t"
inverse_transformations_axioms: shows "inverse_transformations (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) (\<Phi>'.Y a') \<Phi>\<Psi>'.map \<Phi>'\<Psi>.map"
lasso_cycle_def: fixes L :: "('b, 'c) lasso_scheme" shows "lasso_cycle L = lasso_va L # lasso_cysfx L"
S4RPCUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RPCNext crCh rmCh rst p]_(r p) \<and> $S4 rmhist p \<longrightarrow> unchanged r p"
sqr_of_sqrt_2: fixes z :: "complex" shows "z * 2 / (complex_of_real (sqrt 2) * complex_of_real (sqrt 2)) = z"
final_no_red: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x' :: "'x1"   and m' :: "'m1" assumes "r1.r_syntax t x m ta x' m'"   and "final1 x" shows "False"
step: shows "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
dyn_ref_eq_dyn_ref_red: shows "dynamically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F = dynamically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
NSLIMSEQ_def: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" shows "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L = (\<forall>N\<in>HNatInfinite. (*f* X) N \<approx> star_of L)"
sorted_insort: fixes cmp :: "'b comparator"   and xs :: "'b list"   and x :: "'b" assumes "Sorting_Algorithms.sorted cmp xs" shows "Sorting_Algorithms.sorted cmp (Sorting_Algorithms.insort cmp x xs)"
w2Range: shows "- (2 ^ (size (SCAST(32 \<rightarrow> 64) w2) - 1)) \<le> sint (SCAST(32 \<rightarrow> 64) w2) + sint (- 0x7FFFFFFF) \<and> sint (SCAST(32 \<rightarrow> 64) w2) + sint (- 0x7FFFFFFF) \<le> 2 ^ (size (SCAST(32 \<rightarrow> 64) w2) - 1) - 1"
transfer: shows "pcr_complex_homo 0\<^sub>h\<^sub>c 0\<^sub>h"
typing_subst: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and z :: "'a"   and \<tau> :: "PreSimplyTyped.type"   and M :: "'a trm"   and \<sigma> :: "PreSimplyTyped.type"   and N :: "'a trm" assumes "\<Gamma>(z \<mapsto> \<tau>) \<turnstile> M : \<sigma>"   and "\<Gamma> \<turnstile> N : \<tau>" shows "\<Gamma> \<turnstile> M[z ::= N] : \<sigma>"
unfold_tree_tree_iterate: fixes out :: "'a \<Rightarrow> 'b"   and l :: "'a \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'a" shows "unfold_tree out l r = map_tree out \<circ> tree_iterate l r"
rep_eq: fixes x :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "'c set_dlist" shows "DList_Set.fold x xa = fold x (list_of_dlist xa)"
has_derivative_power: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and S :: "'a set"   and n :: "nat" assumes "(f has_derivative f') (at x within S)" shows "((\<lambda>x. f x ^ n) has_derivative (\<lambda>y. of_nat n * f' y * f x ^ (n - 1))) (at x within S)"
nofail_WHILE_eq_rwof: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres" assumes "nofail (m0 \<bind> WHILE cond step)" shows "m0 \<bind> WHILE cond step = SPEC (\<lambda>s. rwof m0 cond step s \<and> \<not> cond s)"
sorted_lower1: shows "sorted_sublist_map R h xs'' lo (p - Suc 0)"
ident: fixes x :: "'a" assumes "x \<in> D" shows "x \<cdot> e = x"
\<tau>move2_stk_append: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option"   and vs :: "'addr val list" assumes "\<tau>move2 P h stk e pc xcp" shows "\<tau>move2 P h (stk @ vs) e pc xcp"
pong: shows "\<forall>i\<in>I. act (inv\<^bsub>G\<^esub> g i) ` (X - Xin i) \<subseteq> Xout i"
domConc: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C (list2FWpolicy b))"   and "b \<noteq> []" shows "x \<in> dom (C (list2FWpolicy (a @ b)))"
hensel_def: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" shows "hensel p f a \<equiv> padic_integers p \<and> hensel_axioms p f a"
compatible_r_def: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set" shows "compatible_r R1 R2 \<equiv> R2 O R1 \<subseteq> R2"
find_sub_True_proj_list_5_same: fixes xs :: "'b set list"   and ys :: "'b set list"   and w :: "'b set"   and ws :: "'b set list"   and z :: "'b set"   and zs :: "'b set list"   and xs' :: "'c list"   and ys' :: "'c list"   and w' :: "'c"   and ws' :: "'c list"   and z' :: "'c"   and zs' :: "'c list" assumes "find_sub_True xs = Some (ys, w, ws, z, zs)"   and "length xs = length xs'"   and "proj_list_5 xs' (ys, w, ws, z, zs) = (ys', w', ws', z', zs')" shows "xs' = ys' @ w' # ws' @ z' # zs'"
mono_dual_fun: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "mono (dual_fun f)"
separation_t1: fixes x :: "'a"   and y :: "'a" shows "(x \<noteq> y) = (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
Scan_regular: fixes T :: "(symbol \<times> character list) set"   and k :: "nat" shows "regular (Scan T k)"
RSpan_Cons: fixes m :: "'m"   and ms :: "'m list" shows "RSpan (m # ms) = RSpan [m] + RSpan ms"
iso_Inf_subdistl: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "prod.mono f" shows "f \<circ> Inf_prod \<le> Inf \<circ> (`) f"
intros_2: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
intro: fixes exec :: "('a, 'b, 'c) configuration list"   and trace :: "('a, 'b) message list"   and start :: "'a \<Rightarrow> 'c"   and trans :: "'a \<Rightarrow> 'c \<Rightarrow> 'b messageValue \<Rightarrow> 'c"   and sends :: "'a \<Rightarrow> 'c \<Rightarrow> 'b messageValue \<Rightarrow> ('a, 'b) message \<Rightarrow> nat" assumes "1 \<le> length exec"   and "length exec - 1 = length trace"   and "asynchronousSystem.initial start (hd exec)"   and "\<And>i cfg1 cfg2. \<lbrakk>i < length exec - 1; cfg1 = exec ! i; cfg2 = exec ! (i + 1)\<rbrakk> \<Longrightarrow> asynchronousSystem.steps trans sends cfg1 (trace ! i) cfg2" shows "execution trans sends start exec trace"
sorted_add_spmat_helper1: fixes a :: "nat"   and b :: "(nat \<times> 'a) list"   and arr :: "(nat \<times> (nat \<times> 'a) list) list"   and brr :: "(nat \<times> (nat \<times> 'a) list) list"   and ab :: "nat"   and bb :: "(nat \<times> 'a) list"   and list :: "(nat \<times> (nat \<times> 'a) list) list" assumes "add_spmat ((a, b) # arr) brr = (ab, bb) # list" shows "ab = a \<or> brr \<noteq> [] \<and> ab = fst (hd brr)"
F'_apply: fixes XD :: "'a set"   and XH :: "'a set dual" shows "F' (XD, XH) = (- Rf CH (undual XH), dual (- Rf (CD_on (ds - {d'})) XD))"
take_drop_eq_sublist_list: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "m + n \<le> length xs" shows "xs \<up> m \<down> n = sublist_list xs [m..<m + n]"
binop_dfa_reachable: fixes bss :: "bool list list"   and f :: "bool \<Rightarrow> bool \<Rightarrow> bool"   and s\<^sub>1 :: "nat"   and s\<^sub>2 :: "nat" assumes "list_all (is_alph n) bss" shows "(\<exists>m. dfa_reach (binop_dfa f A B) 0 bss m \<and> fst (prod_dfs A B (0, 0)) ! s\<^sub>1 ! s\<^sub>2 = Some m \<and> dfa_is_node A s\<^sub>1 \<and> dfa_is_node B s\<^sub>2) = (dfa_reach A 0 bss s\<^sub>1 \<and> dfa_reach B 0 bss s\<^sub>2)"
solve_for_lhs_def: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "nat list list"   and matr :: "rat mat" shows "solve_for_lhs p qs subsets matr = matr_option (dim_row matr) (mat_inverse_var matr) *\<^sub>v v_vec p qs subsets"
obj_simps_5: fixes a :: "'a" assumes "H.obj a" shows "V.cod a = a"
erf_remainder_integral_aux_nonneg: fixes x :: "real"   and m :: "nat"   and n :: "nat" assumes "0 < x" shows "0 \<le> integral {x..x + real m} (\<lambda>t. exp (- t\<^sup>2) / t ^ (2 * n))"
s'_def: shows "s' = (locks s, ((thr s)(t := None), shr s), wset s, interrupts s)"
holomorphic_transform: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on s"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "g holomorphic_on s"
fold_rev: fixes xs :: "'g list"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "fold f (rev xs) = fold f xs"
itrm_refl: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm" shows "itrm_cong R x x"
cyclicI: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "ccw p q r" shows "ccw q r p"
pickFT_length: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'e) cmd list" assumes "PL cval"   and "WtFT cl = 1" shows "pickFT cl < length cl"
assms: shows "\<rho> f|` fv e = \<rho>' f|` fv e"
assms_2: fixes K :: "'a set" assumes "K \<in> s" shows "K \<subseteq> i"
assms: shows "f x \<le> \<Down> R (f' x')"
mask_sub: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "mask m - mask n = mask m AND NOT (mask n)"
ex_bij_betw_strict_mono_card: fixes M :: "'a set" assumes "finite M" obtains h :: "nat \<Rightarrow> 'a" where "bij_betw h {..<card M} M" and "strict_mono_on h {..<card M}"
Lazy_llist_inject: fixes xs :: "unit \<Rightarrow> ('a \<times> 'a llist) option"   and ys :: "unit \<Rightarrow> ('a \<times> 'a llist) option" shows "(Lazy_llist xs = Lazy_llist ys) = (xs = ys)"
Cs_def: shows "Cs = map (\<lambda>w. w `\<rightarrow> C0) (sums ss)"
interp_subseteq_INTERP: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.interp S N C \<subseteq> ground_resolution_with_selection.INTERP S N"
valid_decomp_shift_list: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and x :: "'x" assumes "valid_decomp X ps"   and "(h, U) \<in> set ps"   and "x \<in> U" shows "valid_decomp X (shift_list (h, U) x ps)"
red_monomial_monomial_setD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p"   and "is_monomial_set B"   and "red B p q" shows "q = 0"
assms_2: shows "CONSTRAINT (IS_PURE IS_LEFT_UNIQUE) K"
f_range_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set" assumes "f_range_on A f"   and "B \<subseteq> A" shows "f B \<subseteq> B"
funcset_mem: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and x :: "'c" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
oseq_id_constant: fixes i :: "nat"   and ANY :: "state \<Rightarrow> state \<Rightarrow> bool" shows "optoy i \<Turnstile> (act TT, other ANY {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (seql i (\<lambda>(\<xi>, uu_). state.id \<xi> = i))"
2: shows "\<Phi> (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x)) (inD \<circ> \<phi> y \<circ> HomC.\<psi> (F y, x))"
dom_map_of_conv_domA: fixes \<Gamma> :: "('c \<times> 'd) list" shows "dom (map_of \<Gamma>) = domA \<Gamma>"
ground_fun: fixes s :: "('b, 'a) tm" assumes "ground s" shows "ground (fun s)"
to_hma\<^sub>v_def: fixes v :: "'d Matrix.vec" shows "Mod_Type_Connect.to_hma\<^sub>v v = (\<chi>i. v $v mod_type_class.to_nat i)"
d_eq_dl: fixes dl :: "nat \<Rightarrow> nat set"   and a :: "nat set"   and b :: "nat \<times> nat \<Rightarrow> nat set"   and P :: "nat set"   and l :: "nat"   and k :: "nat" assumes "(dl, a, b, P) = DF l"   and "k < l" shows "d k = dl k"
simps_9: fixes wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" shows "alphabet.wf wf_atom n (rexp.Inter r s) = (alphabet.wf wf_atom n r \<and> alphabet.wf wf_atom n s)"
cs_path_swap_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and n :: "nat"   and n' :: "nat"   and k :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>'\<^esup> n'"   and "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" obtains k' :: "nat" where "n' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k'" and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"
Inf_lower: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "x \<le> Supi A"
arg1_le_c_pair: fixes x :: "nat"   and y :: "nat" shows "x \<le> c_pair x y"
neq_Var: fixes x :: "nat"   and n :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set" assumes "x < n" shows "wf_mformula \<sigma> j P V n R (MRel empty_table) (formula.Neg (formula.Eq (trm.Var x) (trm.Var x)))"
simps_4: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and v :: "Vars.var"   and S :: "stack_elem list" assumes "CardinalityArityTransformation Aexp Aheap" shows "CardinalityArityTransformation.transform_alts Aheap as (Arg v # S) = Arg v # CardinalityArityTransformation.transform_alts Aheap as S"