hcju/isabelleps huggingface dataset:
3: shows "regular w \<and> regular f \<and> regular (selected_edge h j g)"
rev_nth_Suc: fixes xs :: "'b list"   and x :: "'b"   and v :: "nat" shows "(xs @ [x]) !. Suc v = xs !. v"
mod_natE: fixes a :: "nat"   and n :: "nat"   and b :: "nat" assumes "a mod n = b" shows "\<exists>l. a = n * l + b"
synSingleHeadAdd: fixes a :: "'a event list"   and t :: "'a event list"   and A :: "'a event set"   and u :: "'a event list"   and h :: "'a event" assumes "a setinterleaves ((t, u), A) \<and> h \<notin> A" shows "h # a setinterleaves ((h # t, u), A)"
Min_eq_iff: fixes A :: "'a word set"   and m :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(signed.Min A = m) = (m \<in> A \<and> (\<forall>a\<in>A. m \<le>s a))"
sum_basis_eq: fixes X :: "(('a, 'b) vec, 'b) vec"   and f :: "('a, 'b) vec \<Rightarrow> 'a" assumes "is_basis (set_of_vector X)" shows "(\<Sum>x\<in>set_of_vector X. f x *s x) = (\<Sum>i\<in>UNIV. f (X $ i) *s X $ i)"
fields_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list" shows "fields P C \<equiv> THE FDTs. P \<turnstile> C has_fields FDTs"
new_keys_not_used: fixes evs :: "event list"   and K :: "nat" assumes "evs \<in> set_pur"   and "Key K \<notin> used evs"   and "K \<in> symKeys" shows "K \<notin> keysFor (parts (knows Spy evs))"
G'_3: shows "E \<subseteq> E\<^bsub>G'\<^esub>"
adjunction_data_in_bicategory_axioms: shows "adjunction_data_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f f\<^sup>* \<eta> \<epsilon>"
coboundedI2: fixes c :: "'b"   and b :: "'b"   and a :: "'b" assumes "c \<le> b" shows "c \<le> a \<squnion> b"
skip: fixes L :: "lenv"   and accC :: "qtname"   and s :: "st" shows "P L accC (Norm s) (In1r Skip) \<diamondsuit> (Norm s)"
hyp_3: shows "x \<in> X\<^bsup>Suc n\<^esup>"
cases: fixes a1 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a2 :: "'a expr"   and a3 :: "'a OCL_Types.type" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" and expr :: "'a expr" and expr\<^sub>N :: "'a expr" and \<tau> :: "'a OCL_Types.type" where "a1 = \<Gamma>" and "a2 = expr" and "a3 = \<tau>" and "\<Gamma> \<turnstile> expr \<Rrightarrow> expr\<^sub>N" and "\<Gamma> \<turnstile>\<^sub>E expr\<^sub>N : \<tau>"
Diff_Int: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A - B \<inter> C = A - B \<union> (A - C)"
closure_id: fixes Z :: "('c \<Rightarrow> real) set"   and R :: "('c \<Rightarrow> real) set" assumes "Closure\<^sub>\<alpha> Z \<noteq> {}"   and "Z \<subseteq> R"   and "R \<in> \<R>" shows "Closure\<^sub>\<alpha> Z = R"
UP_ring_def: fixes R :: "('e, 'f) ring_scheme" shows "UP_ring R \<equiv> ring R"
harm_pos: fixes n :: "nat" assumes "0 < n" shows "(0::'a) < harm n"
power_int_of_nat: fixes x :: "'a"   and n :: "nat" shows "x powi int n = x ^ n"
cube_in_cball: fixes r :: "real"   and x :: "'b"   and y :: "'b" assumes "0 < r"   and "\<And>i. i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> r / sqrt (real DIM('b))" shows "y \<in> cball x r"
that: fixes c :: "nat \<Rightarrow> nat option" assumes "c \<in> \<R>"   and "\<forall>i. \<chi> i = \<psi> (the (c i))" shows "thesis"
perron_frobenius_non_neg: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "non_neg_mat A" shows "\<exists>sr ks f. 0 \<le> sr \<and> 0 \<notin> set ks \<and> ks \<noteq> [] \<and> charpoly A = (\<Prod>k\<leftarrow>ks. monom 1 k - [:sr ^ k:]) * f \<and> (\<forall>x. poly (map_poly complex_of_real f) x = 0 \<longrightarrow> cmod x < sr)"
gseq_def: shows "gseq = {(f, g). f \<in> SEQ A \<and> g \<in> SEQ A \<and> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j))}"
index_matrix_prod: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and B :: "'a mat" assumes "i < dim_row A"   and "j < dim_col B"   and "dim_col A = dim_row B" shows "(A * B) $$ (i, j) = (\<Sum>k<dim_row B. A $$ (i, k) * B $$ (k, j))"
inverse_ennreal_def: shows "inverse \<equiv> map_fun enn2ereal e2ennreal inverse"
path_entry_hd_edge: fixes pa :: "nat list"   and p :: "nat" assumes "path_entry (g_E G) pa p"   and "pa \<noteq> []" shows "(hd pa, p) \<in> g_E G"
assms_2: fixes v :: "'a vec"   and a :: "'a" assumes "P v" shows "P (vCons a v)"
FinalAllow_approximating_in_doubt_deny: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "(\<beta>, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
TR3_sub_carr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR3 R M N P \<subseteq> carrier P"
\<Delta>\<^sub>a_def: shows "\<Delta>\<^sub>a = (\<Sum>i\<in>I. \<Sum>j | j \<in> I \<and> i \<noteq> j. prob (A i \<inter> A j))"
antisym_conv1: 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)"   and "\<not> (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type) = (x = y)"
infinite_atLeast_int: fixes n :: "int" shows "infinite {n..}"
invalidate_other: fixes dip :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<notin> dom dests" shows "invalidate rt dests dip = rt dip"
Det_commute: fixes P :: "'c process"   and Q :: "'c process" shows "(P [+] Q) = (Q [+] P)"
init_fin_\<tau>move_simps_3: fixes \<tau>move :: "('c, 'd, 'a, 'b, 'e, 'f) \<tau>moves"   and x :: "'a"   and m :: "'b"   and ta :: "('c, 'd, status \<times> 'a, 'b, 'e, 'f action) thread_action"   and s :: "status"   and x' :: "'a"   and m' :: "'b" shows "\<tau>multithreaded.init_fin_\<tau>move \<tau>move ((Running, x), m) ta ((s, x'), m') = (\<exists>ta'. ta = convert_TA_initial (convert_obs_initial ta') \<and> s = Running \<and> \<tau>move (x, m) ta' (x', m'))"
transpose_diag: fixes c :: "'c" shows "Square_Matrix.transpose (diag c) = diag c"
completeI: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "class.metric_space dist uniformity open"   and "\<And>f. \<lbrakk>\<forall>n. f n \<in> s; uniform_space.Cauchy uniformity f\<rbrakk> \<Longrightarrow> \<exists>l\<in>s. filterlim f (topological_space.nhds open l) sequentially" shows "uniform_space.complete open uniformity s"
simps: fixes a :: "('fun, 'var) extstrand_step list" shows "well_analyzed a = (a = [] \<or> (\<exists>A x. a = A @ [Step x] \<and> well_analyzed A) \<or> (\<exists>A t. a = A @ [Decomp t] \<and> well_analyzed A \<and> t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) - range Var))"
simps_5: fixes v :: "'a" shows "(\<infinity> \<le> Fin v) = False"
ord_defs_1: fixes compare :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.compare_order compare less_eq less" shows "le_of_comp compare = less_eq"
dynlookup_def: fixes G :: "prog"   and statT :: "ref_ty"   and dynC :: "qtname" shows "dynlookup G statT dynC = (case statT of NullT \<Rightarrow> Map.empty | IfaceT I \<Rightarrow> dynimethd G I dynC | ClassT statC \<Rightarrow> dynmethd G statC dynC | ArrayT ty \<Rightarrow> dynmethd G Object dynC)"
le_iff_plus_unchanged: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = (x \<squnion>\<^bsub>f\<^esub> y = y)"
GrpI: fixes f :: "'o \<Rightarrow> 'n"   and x :: "'o"   and y :: "'n"   and A :: "'o set" assumes "f x = y"   and "x \<in> A" shows "Grp A f x y"
SN_trans: fixes p :: "trm"   and p' :: "trm" assumes " p \<mapsto>\<^sup>* p' "   and "SN p" shows "SN p'"
seq_char: fixes g :: "cospan_shape.Arr"   and f :: "cospan_shape.Arr" shows "partial_magma.arr cospan_shape.comp (cospan_shape.comp g f) = (f = cospan_shape.AA \<and> (g = cospan_shape.AA \<or> g = cospan_shape.AT) \<or> f = cospan_shape.BB \<and> (g = cospan_shape.BB \<or> g = cospan_shape.BT) \<or> f = cospan_shape.AT \<and> g = cospan_shape.TT \<or> f = cospan_shape.BT \<and> g = cospan_shape.TT \<or> f = cospan_shape.TT \<and> g = cospan_shape.TT)"
Mapping_filter_cong: fixes m :: "('a, 'b) mapping"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>k v. k \<in> Mapping.keys m \<Longrightarrow> f k v = f' k v" shows "Mapping.filter f m = Mapping.filter f' m"
c_f_list_to_f: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" shows "f = (\<lambda>y x. c_hd (c_f_list f y x))"
eSubst_simp5: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var"   and eX :: "'c withERR" shows "eSubst MOD ys ERR y eX = ERR"
fst_set_def: fixes s :: "('a \<times> 'b) set" shows "fst_set s = {a. \<exists>b. (a, b) \<in> s}"
multiplicity_prod_prime_powers: fixes S :: "'b set"   and p :: "'b"   and f :: "'b \<Rightarrow> nat" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> prime x"   and "prime p" shows "multiplicity p (\<Prod>p\<in>S. p ^ f p) = (if p \<in> S then f p else 0)"
function_of_enum: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and xs :: "'c list"   and i :: "nat" assumes "bij_betw enum {0..<card A} A"   and "length xs = card A"   and "i < card A" shows "function_of A enum xs (enum i) = xs ! i"
i_neq_j_imp_events_neq_inf: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat" assumes "[f[f 0 ..]X]"   and "i \<noteq> j" shows "f i \<noteq> f j"
pd_def: shows "pd = perp_foot (Rep_hyp2 c) l"
assms_4: shows "recfn (length xs) (Pr n f g)"
ran_map_upd: fixes m :: "'e \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and a :: "'e"   and b :: "'d" assumes "m a = None" shows "ran (m(a \<mapsto> b)) = insert b (ran m)"
expand: fixes option :: "'a option"   and option' :: "'a option" assumes "(option = None) = (option' = None)"   and "\<lbrakk>option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" shows "option = option'"
mod_div_injTr: fixes r :: "nat"   and x :: "nat"   and y :: "nat" assumes "0 < r"   and "x mod r = y mod r"   and "x div r = y div r" shows "x = y"
rep_number: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<B>' rep x = \<r>'"
lfinite_Liminf_llist: fixes Xs :: "'b set llist" assumes "lfinite Xs" shows "Liminf_llist Xs = (if lnull Xs then {} else llast Xs)"
simps_2: fixes s :: "'b set"   and ss :: "'b set list"   and ts :: "'b set set" shows "partitioning_nontail (s # ss) ts = addSubsetSet s (partitioning_nontail ss ts)"
components_are_iso: fixes a :: "'c \<times> 'c" assumes "C.CC.ide a" shows "C.CC.iso (C.CC.map a)"
content_dvd_contentI: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "content p dvd content q"
monomial_sum: fixes a :: "'b"   and f :: "'d \<Rightarrow> 'c"   and C :: "'d set" shows "monomial (sum f C) a = (\<Sum>c\<in>C. monomial (f c) a)"
map_component_def: fixes f :: "nat \<Rightarrow> nat"   and v :: "'t" shows "map_component f v = term_of_pair (pp_of_term v, f (component_of_term v))"
rel_refl_strong: fixes x :: "'b exp"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> vars x \<Longrightarrow> Ra z z" shows "rel_exp Ra x x"
ranked: fixes q :: "'c"   and f :: "'b"   and qs :: "'c list" assumes "q \<rightarrow> f qs \<in> ta2.\<delta>" shows "A f = Some (length qs)"
emp_trap_set_def: fixes s :: "'b sparc_state" shows "emp_trap_set s \<equiv> s\<lparr>traps := {}\<rparr>"
comp_arr_inv': fixes f :: "'a" assumes "B.iso f" shows "f \<cdot>\<^sub>B B.inv f = B.cod f"
principalidealI: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R"   and "\<exists>i\<in>carrier R. I = Idl\<^bsub>R\<^esub> {i}" shows "principalideal I R"
resumption_ord_ABORT: fixes r :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord ABORT r"
False: shows "\<not> Sup (g ` X) \<le> Sup (f ` X)"
cet_and: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : BOOL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : BOOL" shows "\<Gamma> \<turnstile>\<^sub>c e1 \<and>\<^sub>c e2 : BOOL"
circle_right_edge_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" assumes "circle i j d" shows "circle.circle_right_edge d = (1, \<lambda>y. (d / 2, 0::'b))"
fiter_induct_nil: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and x :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(x::'b::refinement_lattice) \<sqsubseteq> (nil::'b::refinement_lattice) \<sqinter> (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) x" shows "(x::'b::refinement_lattice) \<sqsubseteq> finite_iteration.fiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice)"
Y_mem: fixes y :: "'a" assumes "y \<in> Y" shows "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. s x = 0} y). s x = 0 \<longrightarrow> x \<in> S"
simps_3: fixes i :: "nat"   and uu :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubst (Bv i) uu = Bv i"
disjunctive_def: shows "Apply.disjunctive = {x. \<forall>y z. x (y \<squnion> z) = x y \<squnion> x z}"
multiset_cases: fixes M :: "'b multiset" obtains "M = {#}"   | x :: "'b" and N :: "'b multiset" where "M = x, N"
add_mono: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset"   and d :: "'d multiset" assumes "a \<subseteq># b"   and "c \<subseteq># d" shows "a + c \<subseteq># b + d"
nrows_iarray_def: fixes A :: "'d iarray iarray" shows "nrows_iarray A = IArray.length A"
tinfinite_def: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and V :: "Preliminaries.var set" assumes "Signature TYPE('a) wtFsym wtPsym" shows "Signature.tinfinite TYPE('a) V \<equiv> \<forall>\<sigma>. infinite (V \<inter> Signature.tpOfV -` {\<sigma>})"
induct: fixes P :: "(name \<times> lam) list \<Rightarrow> name \<Rightarrow> bool"   and a0 :: "(name \<times> lam) list"   and a1 :: "name" assumes "\<And>x. P [] x"   and "\<And>y e \<theta> x. (x \<noteq> y \<Longrightarrow> P \<theta> x) \<Longrightarrow> P ((y, e) # \<theta>) x" shows "P a0 a1"
span_is_subset2: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and S :: "'g set" assumes "module R M"   and "S \<subseteq> carrier M" shows "module.span R M S \<subseteq> carrier M"
composites_are_arrows_2: shows "hseq (f \<star> src f) (src f)"
types': shows "P,E,h \<turnstile> es' [:]\<^bsub>NT\<^esub> Ts''"
take_bit_eq_mask: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "take_bit n a = and a (mask n)"
assms_2: shows "Lower_Semicontinuous.convex_on UNIV g"
Fax_def: shows "Fax \<equiv> {[Pos (Eq (lOfFax f) (rOfFax f))] |f. wtFsym f}"
normal_self: shows "carrier (Z 1) \<lhd> Z 1"
spmf_of_set_def: fixes A :: "'g set" shows "spmf_of_set A = (if finite' A then spmf_of_pmf (pmf_of_set A) else return_pmf None)"
rule_sub_remark_6: fixes \<psi> :: "\<o>"   and \<chi> :: "\<o>"   and v :: "i"   and \<phi> :: "\<o>" assumes "\<And>v. [\<psi> \<^bold>\<equiv> \<chi> in v]"   and "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<chi>) in v]"
nat_def: shows "length cts = Suc (Suc (Suc nat))"
cUnion_cinsert: fixes x :: "'b cset"   and A :: "'b cset cset" shows "cUnion (cinsert x A) = cUn x (cUnion A)"
f2r_Sup_pres: shows "Sup_pres \<R>"
keys_cmul_iff: fixes i :: "'c"   and c :: "int"   and x :: "'c \<Rightarrow>\<^sub>0 int" shows "(i \<in> Poly_Mapping.keys (frag_cmul c x)) = (i \<in> Poly_Mapping.keys x \<and> c \<noteq> 0)"
semilattice_set: shows "semilattice_set (max_torder r)"
closure_cbox: fixes a :: "'b"   and b :: "'b" shows "closure (cbox a b) = cbox a b"
mult_scalar_punit_def: shows "mult_scalar_punit \<equiv> punit.mult_scalar"
perm_4: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list" shows "pi \<bullet> ff = ff"
ccw'_sortedP_implies_distinct: fixes x :: "real \<times> real"   and qs :: "(real \<times> real) list" assumes "ccw'.sortedP x qs" shows "distinct qs"
less_eq_unit: fixes u :: "unit"   and v :: "unit" shows "u \<le> v"
bigo_mult6: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x. f x \<noteq> (0::'d)" shows "O(f * g) = f *o O(g)"
list_emb_Nil2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "list_emb P xs []" shows "xs = []"
normalize_poly_def: fixes p :: "'b poly" shows "normalize p = p div [:unit_factor (Polynomial.lead_coeff p):]"
kleene_qfp: fixes q :: "'a" assumes "extreme_bound A (\<sqsubseteq>) {f^n \<bottom> |. n} q" shows "f q \<sim> q"
has_derivative_within': fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow> norm (f x' - f x - f' (x' - x)) / norm (x' - x) < e))"
spsafe: fixes f :: "'sz"   and f' :: "('sf + 'sz, 'sc, 'sz) formula" assumes "SPredicates \<sigma> f = Some f'" shows "fsafe f'"
step: fixes bs :: "'b list" shows "sum_list (map2 (*s) qs bs) \<in> local.span (set bs)"
join_compl2_top: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<squnion> \<^bold>- x = \<top>"
dst_ipPart: fixes rs :: "'a simple_rule list"   and A :: "'a word set set"   and B :: "'a word set"   and s1 :: "'a word"   and s2 :: "'a word"   and p :: "('a, 'b) simple_packet_scheme" assumes "ipPartition (set (map wordinterval_to_set (extract_IPSets_generic0 dst rs))) A"   and "B \<in> A"   and "s1 \<in> B"   and "s2 \<in> B" shows "simple_fw rs (p\<lparr>p_dst := s1\<rparr>) = simple_fw rs (p\<lparr>p_dst := s2\<rparr>)"
simps: fixes x0 :: "'g"   and x1 :: "'a"   and a0 :: "'j"   and a1 :: "'d"   and s :: "'e"   and z0 :: "'k"   and z1 :: "'f" shows "check_AND (x0, x1) (a0, a1) s (z0, z1) = (check0 x0 a0 s z0 \<and> check1 x1 a1 s z1)"
app_mono: fixes p :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and \<tau> :: "(ty list \<times> ty err list) option"   and \<tau>' :: "(ty list \<times> ty err list) option"   and i :: "instr"   and m :: "nat"   and rT :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>' \<tau>'"   and "app i P m rT pc mpc xt \<tau>'" shows "app i P m rT pc mpc xt \<tau>"
makespan_correct_1: fixes T :: "nat \<Rightarrow> nat" shows "\<forall>x\<in>{1..m}. T x \<le> makespan T"
saux3: fixes a :: "'b"   and b :: "'b"   and list :: "('b, 'c) Combinators list"   and c :: "'b"   and d :: "'b" assumes "DenyAllFromTo a b \<in> set list"   and "DenyAllFromTo c d \<notin> set list" shows "a \<noteq> c \<or> b \<noteq> d"
prob_grw_axioms: shows "prob_grw p M geom_proc u d init"
HaddP_Mem_cases: fixes H :: "fm set"   and k1 :: "tm"   and k2 :: "tm"   and k :: "tm"   and x :: "tm"   and A :: "fm"   and i :: "name" assumes "H \<turnstile> HaddP k1 k2 k"   and "H \<turnstile> OrdP k1"   and "insert (x IN k1) H \<turnstile> A"   and "insert (Var i IN k2) (insert (HaddP k1 (Var i) x) H) \<turnstile> A"   and "atom i \<sharp> (k1, k2, k, x, A)"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (x IN k) H \<turnstile> A"
ff': shows "f\<cdot>\<bottom> = f'\<cdot>\<bottom>"
sep_conj_sep_true_right: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and h :: "'d" assumes "(P \<and>* Q) h" shows "(P \<and>* (\<lambda>s. True)) h"
inject: fixes x1 :: "real"   and x2 :: "real"   and y1 :: "real"   and y2 :: "real" shows "(Point x1 x2 = Point y1 y2) = (x1 = y1 \<and> x2 = y2)"
assms_2: assumes "b"   and "b'" shows "S1 \<le> \<Down>\<^sub>E E R S1'"
cond_ass_if: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  Q) = (b \<and> P \<triangleleft> b \<triangleright>  Q)"
d1: shows "(\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> 0 < poly (prod_list (retrieve_polys qs (snd I))) x \<longrightarrow> z_R I sign = 1"
sublist_append_leftI: fixes xs :: "'b list"   and ps :: "'b list" shows "sublist xs (ps @ xs)"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b AND a AND c = a AND b AND c"
Min_le_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Min less_eq A) x = (\<exists>a\<in>A. less_eq a x)"
hd_butlast_last_id: fixes xs :: "'c list" assumes "1 < length xs" shows "hd xs # tl (butlast xs) @ [last xs] = xs"
that: shows "\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (select_object\<^sub>S\<^sub>e\<^sub>t f s_set \<tau>)\<rceil>\<rceil> = z"
units_eq_var: fixes e1 :: "'a"   and e2 :: "'a" assumes "e1 \<in> E"   and "e2 \<in> E"   and "e1 \<noteq> e2" shows "\<not> D e1 e2"
taylor_deg: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "deg R (T\<^bsub>a\<^esub> p) = deg R p"
wf_theory_translate_iff_exe_wf_theory: fixes a :: "exetheory" shows "wf_theory (translate_theory a) = exe_wf_theory a"
pol_nonzero: fixes p :: "'a" assumes "p \<in> carrier R" shows "(0 \<le> deg R S X p) = (p \<noteq> \<zero>)"
unilateral_substitutes_on_def2: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" shows "unilateral_substitutes_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. \<forall>b\<in>A. Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<longrightarrow> b \<notin> f (B \<union> {a, b}))"